Introduction

Ancient pagodas, as exemplary representatives of China's historical and cultural heritage, occupy a significant position in the fields of architectural art and religious culture, and they provide invaluable materials for the exploration of historical geography. These ancient structures not only reflect the social characteristics and technological achievements of history but also offer crucial physical evidence for the study of ancient urban planning, transportation development, military strategy, and economic conditions. Historically, these cultural heritages and other buildings have suffered damage due to different natural disasters, such as the landslides triggered by the Wenchuan earthquake in 2008, which led to the collapse of a large part of the ancient complex of Erwangmiao. In 2020, thunderstorms swept through northern India, destroying the gates of the Taj Mahal complex. In July 2021, heavy rains hit Luoyang City in China's Henan Province, threatening the Longmen Grottoes with natural disasters such as flooding and rockfalls caused by torrential rains. In October 2021, China's Shanxi Province was hit by rare and sustained rainfall, which triggered landslides and caused damage to several ancient buildings in Shanxi. In February 2023, the Gaziantep Castle in Turkey was hit by two earthquakes in February 2023, part of the castle collapsed during the earthquakes and the retaining wall next to the castle collapsed in many places. However, ancient pagodas in mountainous areas of China are threatened by natural disasters such as landslides, debris flows, and rockfalls. In particular, rockfalls have become an urgent issue due to the significant increase in the number of accidents they have caused in China over recent decades [1]. In economically developed areas, where ancient pagodas are close to steep slopes, the rockfall process is often rapid and high-energy, accompanied by a huge impact force, which not only increases the risk of casualties and loss of life but also causes severe damage to infrastructure [2], as shown in Fig. 1. In 2011, multiple landslides occurred around Huayan Temple in Xi'an City, Shaanxi Province, and the potential threat of rockfall from the landslides impacted the ancient pagoda. In 2017, a natural sudden rockfall geologic disaster occurred at Pagoda Mountain in Guilin, Guangxi. There was a large area of huge collapsed rock under the east side of Pagoda Mountain. Fortunately, the collapse had no direct impact on the Shoufo Pagoda on the mountain for the time being, but it did highlight the potential threat of falling rocks to the ancient pagoda. These natural disasters pose a serious threat to the safety of ancient pagodas, especially in mountainous or rugged areas prone to rockfalls. Ancient pagodas must confront not only the problem of unstable foundations but also the multiple threats of material degradation and natural factors such as earthquakes. Therefore, to effectively protect these precious cultural heritages, it is essential to conduct in-depth research on the historical and geographical background of ancient pagodas and to assess their structural safety and stability [3].

Fig. 1
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Natural disaster site

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In the field of rockfall mechanics research, numerous scholars have made significant contributions. For instance, Zhang et al. [4] proposed a logistic curve-based systematic calculation theory for determining key parameters during the rockfall impact process, such as acceleration, impact force, displacement, and velocity. Yu et al. [5] established an impact force model that simplifies multiple parameters affecting the impact force using Buckingham's theorem, analyzing the kinetic energy of the rockfall, the impact angle between the falling rock and the struck object, and the elastic moduli of the rock and object, and validated the model through physical experiments. Zhang et al. [6] based on rigid body motion theory, developed models for rockfalls in various motion forms, taking into account rotational effects and plastic deformation to estimate the maximum impact force on obstacles. Chen et al. [7], based on viscoelastic contact theory, established a simplified calculation model for rockfall impact on the ground and proposed formulas for calculating impact characteristic parameters, validating the viscoelastic results through the Logistic method and laboratory tests.

In the field of rockfall trajectory research, Asteriou et al. [8] conducted numerous experiments with cubic and spherical artificial material blocks, discovering the impact of rock shape, slope angle, and post-impact trajectory deviation. Additionally, the study proposed an empirical model for estimating the deviation of rock trajectories after impact. He [9] simulated the impact area of rockfalls based on the secondary development of the Discrete Element Method (DEM). Despite extensive research and comprehensive evaluation methods for the impact and damage of rockfalls on structures such as bridges [10], He [11] developed a novel, flexible, energy-dissipative anti-collision device to protect bridge piers, simulating the process of rock impact on piers. Sun [12] studied the dynamic response and damage patterns of bridge piers under rock impact through model testing and theoretical analysis. Zhang [13] investigated the impact force, bridge response, and track structure deformation during rock impact on piers through numerical simulation.

Regarding the protection of ancient buildings, Hemeda S [14] pointed out that most of the archaeological sites, which have not been evaluated for seismicity, have not applied any reinforcement measures. Numerous studies have been conducted on archaeological sites, such as the Tomb of the Sons of Ramses II (KV5) in the Valley of the Kings in Luxor, Egypt [15], the Ben Ezra Synagogue in the Old City of Cairo, Egypt [16], and the Yashbak Palace in Egypt [17].

However, research on the impact of rockfalls on ancient pagodas is relatively scarce. Currently, research on the protection of ancient pagodas mainly focuses on seismic performance [18], structural reinforcement [19], and foundation correction and pagoda body alignment [20], with specific research and protective measures against rockfall disasters still lacking.

Due to the complex structural system of ancient pagodas, there are generally two modeling approaches: the holistic modeling approach [21,22,23] and the layered modeling approach. Nonlinear dynamic behavior is a primary concern in the study of ancient masonry structures. Li et al. [24] defined the nonlinear mechanical parameters of contact surfaces between rigid bodies. Ferrante et al. [25] recreated the complex mechanical behavior between blocks and predicted the vulnerability assessment of historical masonry buildings. Bui et al. [26] used the Discrete Element Method (DEM) to predict the ultimate load and failure mode analysis of dry-jointed masonry walls under in-plane and out-of-plane loading. Wu et al. [27] compared the two modeling methods and found that the layered modeling approach offers higher computational accuracy and more realistic simulation effects, especially when dealing with complex structures and variations in material properties, while the holistic modeling approach has advantages in computational efficiency and simplified assumptions.

This study employs the ABAQUS finite element analysis software to simulate and analyze the response of ancient pagodas when subjected to rockfall impact loads. Detailed geometric modeling and material property assignment of the pagoda structure were conducted to establish the corresponding finite element model. Subsequently, the impact of rockfalls with varying masses, velocities, and impact heights on the pagoda was simulated to assess its structural safety and stability. The research aims to provide a theoretical basis for the protection against rockfall disasters of ancient pagodas, effectively safeguarding these precious cultural heritages.

Finite element model

Ancient pagoda overview

The pagoda studied in this paper is located in Hebei Province, China. It is an octagonal, seven-story, pavilion-style brick pagoda, nearly 40 m tall, as shown in Fig. 2. The pagoda was originally constructed during the Kaihuang period of the Sui Dynasty, approximately from 581 to 600 AD. It has been renovated in various dynasties, and the existing structure primarily exhibits characteristics of Song Dynasty pagodas. It is listed as a national key cultural relics protection unit. Each level of the pagoda's south side is equipped with arched windows, while the east and west sides only have arched openings on the third and fourth levels. The pagoda's spire was added in modern times to seal the top of the pagoda, and its base is relatively well-preserved, featuring a five-tiered lotus petal design. The dimensions of the pagoda are shown in Table 1. The pagoda is closely connected to the mountain environment; however, it is this natural geographical ___location that exposes the ancient pagoda to the threat of natural disasters such as rockfalls. The body of the pagoda, especially the lower parts close to the mountain, has shown signs of weathering and damage, which not only affects the integrity of the pagoda but also poses a potential risk to its structural safety. The impact of rockfalls may cause further damage to the brick and stone structure, and in some cases, it could even lead to the local collapse of the pagoda body.

Fig. 2
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Ancient pagoda overview

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Table 1 Pagoda detail dimensions
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Establishment of the numerical model

By measuring the data of the pagoda, a holistic modeling approach was utilized with the modeling software SolidWorks, as shown in Fig. 3. ABAQUS is recognized by a wide range of industries based on its powerful solver, which has obvious advantages in dealing with problems related to collision direction. Its rich material modeling and powerful nonlinear solving capability enable us to accurately simulate the mechanical response of the ancient tower under the impact of rock avalanches, which is crucial for an in-depth study of the stability and safety of the ancient tower. During the modeling process, simplifications were made to the arched openings and the eaves of the pagoda. To study the dynamic response of the ancient pagoda under the impact of rockfalls, the model was imported into the ABAQUS finite element software for nonlinear analysis. The Concrete Damaged Plasticity (CDP) plastic damage model in ABAQUS can effectively simulate the damage behavior of masonry. By integrating the stress–strain relationship from the concrete specification into the CDP constitutive model in ABAQUS, based on the energy equivalence principle of Sidoroff, key indicators such as the plastic damage factor suitable for the CDP model were calculated [28].

Fig. 3
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The overall appearance of the ancient pagoda

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According to the measurement of the mechanical properties of the masonry around the ancient pagoda, as shown in Fig. 4, the relevant parameters of the masonry are obtained, and the specific mechanical parameters of the ancient tower are shown in Table 2. The specific mechanical parameters of the ancient pagoda are shown in Table 2. The specific value curves of the compression and tension damage factors are shown in Fig. 5. The compression damage factor can reach 0.90029 (corresponding to a compressive strength of 4.28 MPa), and the tension damage factor can reach 0.901 (corresponding to a tensile strength of 0.49 MPa). To simulate the damage of concrete under impact loading, the complete failure criteria for masonry are defined by setting the keyword "Concrete Failure" and specifying the type as "Strain." Additionally, element deletion is used to simulate structural damage caused by impact.

Fig. 4
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Compressive test

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Table 2 Pagoda parameter
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Fig. 5
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Evolution law of damage factor

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In this study, Abaqus/Explicit is used to establish the finite element model of the ancient tower under the impact of falling stones. The impact load is applied by defining the initial velocity for the falling stone, and the gravity acceleration is applied to the whole model, and the bottom of the tower is fixed constraint. Granite, which is commonly found in the mountainous region, is selected as the rockfall material, with the following material properties: the elastic modulus E is 60 GPa, the density ρ is 2600 kg/m3, and Poisson's ratio ν is 0.25. During the simulation of the rockfall impact on the ancient pagoda, to simplify the calculations, the rock is assumed to be a rigid body and is analyzed using an elastic model.

The purpose of this study is to investigate the effect of falling stone impact parameters on the dynamic response and damage degree of the ancient pagoda by adjusting the parameters to construct a variety of working conditions, covering the impact of falling stones with different impact speeds (v1 = 5 m/s, v2 = 10 m/s, v3 = 13 m/s, v4 = 16 m/s), radius (R1 = 0.75m, R2 = 0.375m), impact angle (α1 = 30°, α2 = 45°, α3 = 60°), impact height (h1 = 4m, h2 = 11.5m, h3 = 18m) on the impact of the pagoda. The details are shown in Table 3 and Fig. 6 shows the schematic diagram of different cases. In the actual environment, the impact velocity of falling rocks is affected by the slope of the mountain, the height of the initial position and the friction with the slope. Setting speeds of 5 m/s, 10 m/s, 13 m/s, 16 m/s, etc. can simulate the actual situation and formulate protective measures for ancient pagodas in different terrains. The radius of rockfall reflects the size difference, 0.75m and 0.375m etc. cover the common range, the damage of large radius rockfall may be more serious, and the small radius should not be neglected. The impact angle depends on the rolling trajectory and the ___location of the tower, 30°, 45°, 60° covers the main range, and different angles cause different stress distribution.

Table 3 Detailed cases of rockfall impact on ancient pagodas
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Fig. 6
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Schematic of rockfall impact scenarios

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In light of the significant weathering observed on the second and third levels of the ancient pagoda, this study selects heights of 4 m, 11.5 m, and 18 m as reference heights for rock impact, corresponding to the first, second, and third levels of the ancient pagoda, respectively. This setup is designed to more accurately simulate and assess the potential damage effects of rockfall impact on the ancient pagoda.

Dynamic response results analysis of the ancient pagoda

The impact of rockfall velocity on the impulsive action on the ancient pagoda

In this study, the structure of the ancient pagoda was simulated using C3D4 elements, totaling 174,379 elements; for the simulation of the rockfall, C3D8R elements were employed, totaling 7,168 elements. The finite element model is shown in Fig. 7. The figure illustrates the case where the radius R of the rock is 0.5 m, and velocities v1 = 5 m/s, v2 = 10 m/s, v3 = 13 m/s, and v4 = 16 m/s are considered, with the impact angle α set at 0° and the rockfall height h at 4 m. The analysis time step is set to 1s, and the calculated results obtained from the simulation are shown in Figs. 8, 9, 10 and 11.

Fig. 7
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Finite element model

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Figures 8, 9, 10 and 11 present the analysis of impact scenarios on the ancient pagoda under four different rockfall velocities. The instantaneous moments of rock impact on the pagoda are: t1 = 56.016 ms, t2 = 28.018 ms, t3 = 24.010 ms, and t4 = 20.002 ms. In these figures, figure a displays the Mises stress cloud diagram at the moment of rock impact with the pagoda, while Figure b shows the tensile damage cloud diagram of the pagoda at the end of the analysis. Under each velocity scenario, the rock separates from the pagoda after the instantaneous impact and continues to fall, with parts of the pagoda's surface having reached the preset equivalent plastic strain threshold. The finite element software automatically executed element deletion operations accordingly. The stress waves generated by the rock impact caused further damage to the ancient pagoda along the direction. Over time, the stress waves fully diffused within the pagoda, and the cracks in the pagoda body also fully developed.

Fig. 8
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Case 1

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Fig. 9
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Case 2

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Fig. 10
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Case 3

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Fig. 11
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Case 4

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From the figure, it can be observed that the increase in the falling rock velocity leads to an increase in the stresses on the tower, as well as an increase in the degree of damage after impact. When the falling rock velocity is v2 = 10m/s, the tower body starts to crack downward at the three-level coupon. This indicates that at this speed, the impact force of the falling rock is sufficient to cause significant damage to specific parts of the tower. The third floor coupon is a relatively weak point in the structure of the tower, making it the first place to withstand the impact force and crack. When the speed increased to v3 = 10m/s, the cracks in the tower at the third floor coupon not only extended downward, but also extended upward, and the crack width further increased. The extension of the crack indicates that the stress distribution inside the tower body has changed, and the material around the crack is subjected to a further increase in stress, which may lead to a gradual decrease in the stability of the structure. And when the falling rock velocity reached v4 = 16m/s, the tower brake partially collapsed and the crack of the tower body further expanded. The tower brake is usually located at the top of an ancient tower, which is a more vulnerable part. At the higher velocity of falling rocks, the tower brake collapsed because it was subjected to a relatively high impact force and had a relatively small structure. The further expansion of the cracks means that the structural integrity of the tower has been damaged even more, and the stability of the overall structure has been drastically reduced. At this point, the tower may have been on the verge of collapse, and any small external perturbation may lead to the collapse of the overall structure. These results show that the falling stone has a certain mass and speed, and according to the kinetic energy formula \({E}_{k}=\frac{1}{2}m{v}^{2}\), the greater the speed of the falling stone, the greater its kinetic energy. When the falling stone hits the pagoda, the kinetic energy of the falling stone will be converted into the impact force on the pagoda body and the deformation energy of the pagoda body in an instant. The tower in the moment to withstand a huge impact force, will make the tower body internal stress increases rapidly, more than its material bearing limit will occur when the damage. The speed of falling stones is a key factor affecting the structural safety of the pagoda, and the increase in speed significantly aggravates the degree of damage to the pagoda.

In the study, the changes in volume and mass of the ancient pagoda over time for Case 3 and Case 4 are shown in Fig. 12. The initial volume of the pagoda is 1451.23 m3, and the mass is 2612.22 tons. Under the impact, the volume and mass gradually decrease as masonry elements are removed. At the end of the analysis, the damaged volume for Case 3 is 24.12 m3, with a damaged mass of 43.42 tons, resulting in a damage rate of 1.66%. For Case 4, the damaged volume is 41.01 m3, with a damaged mass of 73.82 tons, resulting in a damage rate of 2.83%.

Fig. 12
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Case3 and case4: Mass and volume of the pagoda over time

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Hourglass energy is a non-physical form of energy dissipation that may occur in finite element analysis. In numerical calculations, when the deformation pattern of a cell does not accurately simulate the actual physical phenomenon, an hourglass phenomenon may arise, and thus hourglass energy. If the hourglass energy is too large, the accuracy of the calculation results will be seriously affected. Because the hourglass energy is not real physical energy, its existence will interfere with the calculation of the real energy of the system. The total energy includes kinetic energy, potential energy, strain energy and other forms of energy, which reflects the overall energy state of the system under specific working conditions. In the whole calculation process of working Case 4, the maximum value of hourglass energy is 3654.5 J and the maximum value of total energy is 3950.9 J. It is usually considered that the calculation results are reliable when the proportion of hourglass energy to total energy is less than 10%. This criterion is based on a large number of numerical simulation practices and experience [29]. The hourglass energy is 2.37% < 10% of the total energy of the system, as shown in Fig. 13, and the proportion of the hourglass energy is less than 10% means that the influence of the hourglass phenomenon on the calculation results is relatively small and negligible. In this case, the calculation results are mainly dominated by the real physical energy, which is more reflective of the behavior of the actual system, and the calculation results of this data can be considered reliable.

Fig. 13
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ALLAE/ ALLIE

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The impact of rockfall radius on the impulsive action on the ancient pagoda

While keeping other conditions constant (rock impact velocity of 13 m/s, impact angle of 0°, and impact height of 4 m), the radius of the rock in the simulation is varied from 0.375 m to 0.75 m. By analyzing the calculation results obtained from the impact of rocks of different sizes, the study investigates the influence of rock radius on the damage and dynamic response of the ancient pagoda. The force–time curve of the rock impact at the specified ___location is shown in Fig. 14. As can be seen from Fig. 14, the force–time curve is characterized by nonlinearity, multiple peaks, and a short duration. After the first contact between the rock and the pagoda, the impact force rises nonlinearly and in multiple stages, reaching a peak value when the rock and the pagoda are in full contact.

Fig. 14
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Comparison of impact force time history curve

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The first contact of the rock impact results in the first valley in the force–time curve. At this point, the rock and the pagoda are in a separated state. According to the principle of conservation of momentum, let the system momentum of the ancient pagoda before being impacted by the rock be P2 = m2v2, and the system momentum after impact be P2′ = m2v2′. The system momentum of the ancient pagoda before the rock impact is P1 = m1v1, and after the impact, it is P1′ = m1v1′. The momentum reduction m1Δv1 of the ancient pagoda after being impacted by the rock will be converted into the momentum increase m2Δv2 of the ancient pagoda during the impact process (not considering heat energy loss). The reason for the appearance of the first valley is that the velocity v2′ of the ancient pagoda structure after the impact is greater than the velocity v1′ of the rock, leading to separation between the two systems. Due to the fully fixed constraint at the bottom of the ancient pagoda and the elastic deformation recovery and inertial resistance of the lower structure of the ancient pagoda, the velocity of the lower structure of the ancient pagoda decays rapidly. After a brief separation, the two systems undergo a second collision, and so on.

The analysis of the force–time curves and the understanding of the impact dynamics provide insights into how the size of the rock affects the structural response of the ancient pagoda. Larger rocks, with their greater radius, tend to deliver a more substantial impact force, leading to more severe structural damage.

The analysis of the force–time curves and the understanding of the impact dynamics provide insights into how the size of the rock affects the structural response of the ancient pagoda. Larger rocks, with their greater radius, tend to deliver a more substantial impact force, leading to more severe structural damage.

When a rock impacts an ancient pagoda, the larger the radius of the rock, the more complex and greater the impact force tends to be. Due to the varying radii of the rocks, their mass and contact area also differ. According to the collision theory in mechanics, under the same conditions of velocity and impact ___location, the magnitude of the impact force is directly proportional to the mass of the rock, while the relationship with the contact area is more complex. Specifically, larger rocks (i.e., those with a larger radius) will generate a greater impact force due to their larger mass at the same velocity. However, their larger contact area might slightly disperse the impact force, which could result in a slightly different duration of contact. Therefore, the greater the structure's ability to resist deformation at the impact ___location, the stronger its ability to resist the impact force. The structural response to the impact is influenced by several factors, including the material properties of both the rock and the pagoda, the geometry of the rock, and the structural integrity and stiffness of the pagoda. The energy absorbed by the pagoda during the impact event and the subsequent energy dissipation mechanisms also play a crucial role in determining the extent of damage.

As can be seen from Figs. 15, 16, when the radius of the falling stone increases, the contact area between the falling stone and the pagoda will increase accordingly. According to the pressure formula P = F/S, the pressure per unit area will decrease when the contact area increases under the condition of certain force such as gravity of the falling stone. However, since the kinetic energy of the falling stone is related to the mass and velocity, and the falling stone with larger radius usually has a larger mass, the overall impact force may increase although the pressure per unit area may decrease, which leads to the tensile damage of the pagoda to show an increasing trend. When the falling stone impacts the pagoda, a stress wave will be generated to propagate inside the pagoda. The radius of the falling rock increases, the impact energy also increases, and the intensity and propagation range of the stress wave will also increase accordingly. Stress waves in the propagation process will make the internal stress distribution of the pagoda changes, when the stress exceeds the tensile strength of the pagoda material, cracks and other damage will occur. After a falling stone with a larger radius impacts the pagoda, the stress wave may spread farther in all directions from the point of impact, affecting more areas and causing cracks to appear in the surrounding areas that are not adjacent to the impact surface.

Fig. 15
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Case 5: Tensile damage of the pagoda

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Fig. 16
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Case 6: Tensile damage of the pagoda

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When the radius of the falling stone is R2 = 0.375m, the impact effect on the pagoda is only on the surface of the masonry, which did not cause serious damage to the pagoda. This indicates that in this radius, the impact energy of the falling stone is relatively small, not enough to cause serious damage inside the pagoda. When the radius of the falling stone is R1 = 0.75m, the damage area of the pagoda is further enlarged, and the diffusion of the stress wave makes the peripheral area which is not adjacent to the impact surface also have cracks with longer length. This indicates that with the increase of the radius of the falling rock, the impact energy and the influence area of the stress wave both increase significantly. The appearance and expansion of the cracks is an indication that the internal stresses within the pagoda exceed the tensile strength of the material. These cracks will reduce the structural strength and stability of the pagoda and put the pagoda at risk of collapse.

The impact of rockfall impact angle on the ancient pagoda

Keeping all other conditions constant (R = 0.5m, v = 13m/s, h = 4m), the impact angle of the rock in the simulation is altered to 30°, 45°, and 60°. The damage cases obtained under different rock impact angles are illustrated in Figs. 17, 18 and 19.

Fig. 17
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Case 7: Localized damage of the pagoda

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Fig. 18
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Case 8: Localized damage of the pagoda

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Fig. 19
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Case 9: Localized damage of the pagoda

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The analysis results from Figs. 17, 18 and 19 reveal that the ancient pagoda experiences significantly different degrees of damage under the impact of rocks at various angles. In the most severe scenario (α = 45°), the pagoda endures the most extensive damage. This is likely due to the fact that at this angle, the kinetic energy of the rock is most effectively transferred to the structure of the ancient pagoda, leading to exacerbated structural damage. This is manifested by a more extensive distribution of cracks and potential loss of structural integrity. As the impact angle decreases to 30°, the degree of damage to the pagoda is somewhat mitigated, but it remains considerable. At this angle, the rock's impact may not be directly aligned with the most vulnerable parts of the pagoda's structure, but it is still sufficient to cause significant damage, such as local masonry crushing and the formation of cracks. When the impact angle increases further to 60°, the degree of damage to the ancient pagoda is minimal. This indicates that at this angle, the kinetic energy of the rock is more dispersed upon contact, reducing the destructive effect on the pagoda's structure. The pagoda may be subjected to a larger impacted area at this angle, but the dispersion of the impact force results in lighter damage at individual points, and thus the overall degree of damage is the least.

As can be seen from Table 4, the degree of damage to the ancient pagoda shows a significant change with the gradual increase in the rock impact angle. The damage peaks when the impact angle reaches 45°, and then the damage decreases as the impact angle continues to increase. At the smallest impact angle (30°), the maximum displacement of the pagoda's spire is relatively large. At 45°, the pagoda suffers the most severe damage, which is characterized by a notable increase in the displacement of the spire, as well as the damaged volume and area. However, when the impact angle further increases to 60°, the degree of damage to the ancient pagoda is significantly reduced. The maximum horizontal displacement of the pagoda's spire decreases from an initial value of 6.95mm to 5.82mm, a reduction of 16.26%, and the damaged volume decreases by 0.72%.

Table 4 Ancient pagoda damage parameters with varying rockfall impact angles
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The reason for this is that when the falling stone impacts the pagoda at a small angle, the velocity component in the horizontal direction accounts for a larger portion, which means that the falling stone has a higher linear momentum and kinetic energy in the horizontal direction. At the moment of collision between the falling stone and the pagoda, the kinetic energy in the horizontal direction is converted into the impact force on the pagoda, resulting in deformation and stress concentration in the horizontal direction of the pagoda. The pagoda structure usually shows good stability and bearing capacity in the vertical direction, which is due to its large vertical stiffness; and relatively weak in the horizontal direction, mainly due to the horizontal bending stiffness and shear stiffness is relatively small. The impact force in the horizontal direction is easy to make the pagoda's walls, columns and other structures appear lateral displacement and deformation, which in turn destroys the structural integrity of the pagoda and reduces its structural reliability and durability.

The wall of the pagoda mainly bears its own gravity and the pressure of the upper structure in the vertical direction, and lacks sufficient support and constraint in the horizontal direction. When subjected to a large horizontal impact, the wall is prone to cracks, tilting and even collapse due to insufficient tensile strength and shear strength in the horizontal direction. It can be determined that during the impact of falling stones on the pagoda, it is mainly the velocity component in the horizontal direction that causes damage to the pagoda. When the impact angle of the falling stone is small, the horizontal velocity component is larger, and the damage to the pagoda is more serious. The impact of the velocity component in the vertical direction on the pagoda is relatively small, because the impact force in the vertical direction is mainly balanced by the pagoda's own gravity and the vertical load-bearing capacity of the structure. With the increase of the impact angle, the horizontal velocity component gradually decreases, while the vertical velocity component increases, which makes the damage degree of the pagoda gradually decrease. Therefore, the pagoda has relatively good impact resistance and less damage at larger impact angles.

The impact of impact height on the ancient pagoda

While keeping other conditions constant (R = 0.5m, v = 10m/s, α = 0°), the impact height of the rock in the simulation is varied, with the first level at 4m, the second level at 11.5m, and the third level at 18m. The calculation results of the ancient pagoda under different rock impact heights are studied to investigate their influence on the damage and collapse of the ancient pagoda. The damage conditions of the ancient pagoda are shown in Figs. 20, 21 and 22.

Fig. 20
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Case 10: Damage of the pagoda

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Fig. 21
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Case 11: Damage of the pagoda

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Fig. 22
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Case 12: Damage of the pagoda

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It can be observed from the figure that the damage degree of the pagoda shows a significant increasing trend when the height of the falling rock impact increases. This shows that different heights of pagodas have different vulnerabilities when facing the same impact of falling rocks. In the higher level, the pagoda is more vulnerable to natural environmental factors, such as wind, sun, rain, etc., resulting in more serious weathering phenomenon. Weathering makes pagoda material properties gradually deteriorate, such as masonry material strength, loose surface. This makes the pagoda more prone to damage when subjected to the impact of falling rocks.

In addition, the pagoda usually adopts the structural form of layer by layer convergence, and the thickness of the wall gradually decreases with the increase of height, which affects the structural rigidity of the pagoda to a certain extent. The decrease in wall thickness means that the pagoda's bearing capacity and deformation resistance are relatively weak at higher locations. The thinner walls are more prone to deformation and damage during the impact of falling stones. Comparing the thicknesses of the walls at the bottom and the top of the pagoda, it is clear that the top wall is thinner, and under the impact of falling stones, the top wall may not be able to withstand the larger impact force, resulting in more serious damage.

Specifically, when the falling stones hit the bottom layer of the pagoda, the damage is relatively small, and only pits and cracks are formed at the point of impact. This is because the bottom layer of the pagoda is usually thicker in structure, and bears the weight of the entire tower body, with relatively high strength and stability. The masonry structure of the bottom layer has formed a certain degree of compactness under long-term pressure, and is able to withstand the impact of falling stones to a certain extent. However, even such relatively minor damage may gradually weaken the structural integrity of the bottom layer of the pagoda, laying a hidden danger for subsequent potential damage.

Figure 23 shows the damage differences between the different models for the same impulse conditions. For Case 10, the greater wall thickness in the first story gives it greater structural stability. A thicker wall means a larger cross-sectional area and moment of inertia, which increases the resistance of the subgrade to impact forces. When subjected to lateral impact forces, thicker walls can better distribute the stresses and reduce localized stress concentrations, so that the impact forces can be spread over a wider area and the risk of localized damage can be reduced. For Case 11, the thinner wall in the middle of the pagoda is an important factor in its susceptibility to collapse under impact. The thinner wall corresponds to a smaller cross-sectional area and moment of inertia, and the structural stiffness and strength are relatively low. When subjected to the impact force, the thin wall is more prone to deformation and damage, and it is difficult to effectively resist the impact force. The stress wave generated by the impact of falling stones causes further damage to the pagoda along the impact direction. When the stress wave spreads sufficiently in the pagoda, it will lead to the generation of cracks in the pagoda. Cracks cause stress concentration when the structure is stressed, which can easily lead to crack expansion and structural damage. The old damage may weaken the material strength and structural load carrying capacity, making the middle storey more susceptible to collapse under the impact force. For Case 12, as the height of the impact increases, the thickness of the wall decreases and the shape of the structure's cross-section changes, which results in a reduction in structural rigidity. The upper part of the pagoda is usually more vulnerable to the natural environment because it is exposed to a larger area in the air and is more strongly affected by natural factors such as wind, sun and rain.

Fig. 23
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Comparison of impact force time history curve

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The layer shear envelope values of the pagoda under the impact of different scenarios are shown in Fig. 24. In Case 10, the impact ___location is the first layer of the pagoda, and the maximum value of the shear force also appears in the first layer. The first layer of the pagoda is usually close to the ground and bears most of the weight of the entire tower, and the structural wall is thicker, which is able to block and absorb the energy to a certain extent, and the connection with the foundation is closer, which generates a larger shear force. In Case 11, the impact ___location is the second floor of the pagoda, the second floor of the pagoda is in the middle and lower intermediate position, both from the upper layer of gravity and pressure, but also to transfer part of their own and the upper layer of the load to the lower layer. Part of the energy generated by the impact will be transferred upward to the third floor, part of the energy will be transferred downward to the first floor, and at the same time part of the energy will be dissipated and redistributed inside the second floor, resulting in its shear change in the direction shown in the figure. In Case 12, the impact ___location is the third layer of the pagoda, which is in the middle of the city and its stability may be more challenged compared to the first and second layers. In the case of rockfall impact, due to the higher height, the impact force on the third floor may produce more shaking and deformation. In order to reduce the overall weight and load of the pagoda, the wall of the third floor may be thinner and the structure is usually relatively lightweight, which makes the third floor relatively weaker in resisting the impact of the falling rock, and the shear force changes will be more complex and sensitive.

Fig. 24
figure 24

Layer shear force envelope of pagodas under different scenario impacts

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When the impact occurs on the upper floors, crack extension is more significant and may lead to severe structural damage, including cracking of the tower body and collapse of the tower brakes. The upper layer of the pagoda is more affected by wind load and other natural factors due to its higher height, and the structure is relatively weak. At the same time, with the increase in height, the tower body cross-section size gradually decreases, and the bearing capacity and impact resistance of the upper layer is relatively low. When the falling stones hit the upper layer, the impact force spreads rapidly in the relatively weak structure, resulting in cracks that are more likely to expand. Cracking of the pagoda body seriously affects the overall stability of the pagoda, and the collapse of the pagoda brake, as an important decorative and structural part, not only destroys the appearance, but also may further aggravate the instability of the pagoda body.

In higher-level impacts, the structural rigidity is further reduced and the degree of damage is increased, which may even lead to the collapse of the entire tower. With increasing height, the pagoda's structural rigidity continues to weaken, making it even more vulnerable in the face of falling rock impacts. Higher-level impacts may trigger a chain reaction, causing the existing cracks to expand further and the structural deformation to become more severe. In extreme cases, the entire tower may not be able to withstand the cumulative damage and collapse, which would not only be a great loss to the historical and cultural heritage, but could also pose a serious threat to the surrounding environment and the safety of people.

Discussion

In the practice of pagoda conservation, a comprehensive protection strategy is needed to deal with the risk factor of falling rocks. For example, wire rope netting and ring netting systems are installed at the foot of the mountain to intercept falling rocks and disperse the impact of falling rocks, as shown in Fig. 25. At the same time, the mountain protection zone is delineated to limit the activities that destabilize the mountain, and for the pagoda as a tourist attraction, the tourist routes are reasonably planned to control the number of tourists, professionals are arranged to regularly check and repair the related problems, and a perfect emergency plan is formulated.

Fig. 25
figure 25

Rockfall protection net

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Conclusions

Stability problems caused by natural disasters on pagodas have received great attention, among which the impact of falling rocks on pagodas is a typical case. Based on the dimensions of the pagoda model measured in the field and the masonry constitutive relation measured through tests. The process of rockfall impact on the pagoda is analyzed and simulated by finite element software, which is found to be of great significance for understanding this kind of problem and protecting the site. The finite element simulation results show that there is a clear quantitative relationship between the impact velocity, radius, impact angle, and impact height of the falling stones and the extent of damage to the pagoda.

  1. 1.

    When the impact velocity is below 13 m/s, the damage to the pagoda is not obvious, and only pits are formed on the surface of the pagoda, and when the impact velocity is 13 m/s, the damage volume of the pagoda is 43.42t, and the damage rate is 1.66%. When the impact velocity exceeds 16 m/s, the pagoda tower brake causes damage, the cracks at the impact place start to spread, the damage volume is 73.82t, the damage rate is 2.83%, and the possibility of the pagoda collapsing increases dramatically.

  2. 2.

    When the radius of the falling stone is 0.375 m, the impact of the pagoda is only on the surface of the masonry, which did not cause serious damage to the pagoda. This indicates that under this radius, the impact energy of the falling stone is relatively small, which is not enough to cause serious damage to the internal part of the pagoda. When the radius of the falling stone is 0.75 m, the damage area of the pagoda is further enlarged, and the diffusion of the stress wave makes the peripheral area which is not adjacent to the impact surface also have cracks of longer length. This indicates that with the increase of the radius of the falling stone, the impact energy and the influence area of the stress wave both increase significantly. The appearance and expansion of the cracks is an indication that the internal stresses within the pagoda exceed the tensile strength of the material. These cracks will reduce the structural strength and stability of the pagoda and put the pagoda at risk of collapse.

  3. 3.

    For the impact angle, at 30°, the horizontal velocity component is 10.816 m/s, the horizontal displacement caused to the pagoda is 6.11 mm, the damage volume is 16.2 m3, and the damage rate is 1.12%; at the impact angle of 45°, the horizontal velocity component is 9.191 m/s, the horizontal displacement caused to the pagoda is 6.95 mm, the damage volume is 17.32 m3, and the damage rate is 1.19%. The damage rate is 1.19%; when the impact angle is 60°, the horizontal velocity component is 6.699 m/s, the horizontal displacement of the pagoda is 5.82 mm, the damage volume is 6.77 m3, and the damage rate is 0.47%.

  4. 4.

    When the impact occurred at the upper level, the crack extension was more significant, which may lead to serious structural damage, including cracking of the tower body and collapse of the tower brake. As the height of impact increases, the tower body cross-section size gradually decreases, and the upper layer bearing capacity and impact resistance are relatively low. When the falling stone hits the upper layer, the impact force propagates rapidly in the relatively weak structure, resulting in cracks that are more likely to expand, and the cracking of the pagoda body seriously affects the overall stability of the pagoda.

Each layer of the pagoda can be regarded as an independent structural unit with different impact resistance. By implementing the comprehensive measures mentioned above, the risk of impact on pagodas from falling rocks can be reduced, thus strongly guaranteeing the safety of these precious cultural heritages under natural disasters, ensuring their long-term preservation and inheritance, and providing a solid physical basis for the study of history, culture and architectural art for future generations.