The basal cavity of a rock block formed due to differential weathering is an important predisposing factor for rockfall in hard–soft interbedded rocks, which induces an eccentricity situation at the base of the rock block. Rock block falling due to the non-uniform distribution with the failure modes of toppling or sliding is defined as biased rockfall in this study. Taking into account the non-uniform stress distribution due to the eccentricity effect, a new analytical method is proposed for three-dimensional stability analysis of biased rockfall. The development of non-uniform stress distribution stress calculated by this analytical method was verified by numerical simulation. The biased rockfall progresses from partial damage of the soft underlying layer, caused by non-uniform distributed stress, to toppling and sliding of overhanging hard rock block due to overall unbalanced force. Therefore, a set of factors of safety (Fos's) against partial damage (compressive and tensile damage of the soft underlying layer) and overall failure (toppling and sliding of the hard rock block) are used to determine the rockfall susceptibility level. The analytical method is applied and validated using biased rockfalls on the northeastern edge of the Sichuan Basin in southwest China, where a significant number of rockfalls consisting of overhanging thick sandstone and underlying mudstone occur. The evolution process of biased rockfalls is divided into four stages, initial state, basal cavity formation, partially unstable and failure. The proposed method is validated by calculating Fos's of the typical unstable rock blocks in the study area. As the cavity continues to grow, the continuous retreat of mudstone causes stress redistribution between the hard and soft rock layers. This results in damage to the underlying soft rock layer due to the development of the non-uniform distribution, ultimately leading to the failure of the hard rock block. The critical retreat ratio is determined to be 0.33, which is used to classify the low and moderate rockfall susceptibility in the eastern Sichuan Basin. The proposed analytical method provides insights into the evolution of biased rockfall and a means for early identification and susceptibility assessment of rockfall.

Rockfall is defined as the detachment of a rock block from a steep slope along a surface, on which little or no shear displacement takes place (Cruden and Varnes, 1996). Rockfalls frequently occur in mountainous ranges, on cut slopes and on coastal cliffs, and they may cause significant facility damage and casualties in residential areas and transport corridors (Chau et al., 2003; Volkwein et al., 2011; Corominas et al., 2018). Stability analysis of rock blocks is crucial for risk management and early warning of rockfall (Kromer et al., 2017).

Potentially unstable blocks and basal cavities caused by differential weathering.

Rockfalls are prone to occur in soft–hard rock formations, and the non-uniform stress distribution caused by differential weathering of rock formations is the main reason for the failure of rockfall. In the eastern Sichuan Basin, southwest China, rockfall is widespread and poses a high risk (Chen et al., 2008; Chen and Tang, 2010; Zhang et al., 2016; Zhou et al., 2017, 2018). The rockfall in this area is attributed to the tectonic setting of Jura-type folds and the stratum sequence, which is characterized by the interbedding of hard and soft layers. An alternation of thick sandstone and thin mudstone layers is formed in the wide and gentle-angle synclines (Zhang et al., 2015; Wu et al., 2018). Weathering is known to be one of the main predisposing factors in rockfall (Jaboyedoff et al., 2021; Zhan et al., 2022). The cliff comprised of hard sandstone is the source of rockfall, and the underlying mudstone is more susceptible to weathering. Along with the retreat of basal cavities in the mudstone layer, the gravity centre of the overhanging sandstone block moves outward relative to the mudstone. In this case, the stress distribution in the contact surface of sandstone and mudstone is non-uniform. The mudstone on the outer side bears higher compressive stress than that on the inner side. This phenomenon can be defined as an eccentricity effect, which leads to mudstone damage and failure of the overhanging sandstone by toppling or sliding. This type of rockfall is defined as biased rockfall in this study (Fig. 1). Similar rockfall patterns have been widely reported in other regions, such as Joss Bay in England (Hutchinson, 1972), Okinawa Island in Japan (Kogure et al., 2006) and the Colorado Plateau of the southwestern United States (Ward et al., 2011). Retreat of the basal cavity is a main cause for the failure of the overhanging block. Therefore, it is necessary to establish an analytical method, considering the development of the basal cavity, to analyse the stress distribution and stability of rock blocks, which is fundamental to the susceptibility assessment and risk control of biased rockfall.

Rockfall stability analysis methods include statistical analysis (Frattini et al., 2008; Santi et al., 2009), empirical rating systems (Pierson et al., 1990; Ferrari et al., 2016) and mechanical analysis (Lin and Fairhurst, 1988; Jaboyedoff et al., 2004; Derron et al., 2005; Matasci et al., 2018). The statistical analysis and empirical rating systems are suitable for rockfall hazard assessment at a regional scale. The accuracy of statistical analysis depends on the completeness of rockfall inventories (Chau et al., 2003; Guzzetti et al., 2003; D'Amato et al., 2016). However, its application to rockfall hazards is limited due to the lack of complete inventory data (Budetta and Nappi, 2013; Malamud et al., 2004). Empirical and semi-empirical rating systems are used where site-specific rockfall inventories are either unavailable or unreliable. Therefore, rockfall susceptibility can be assessed by heuristic ranking of selected predisposing factors (Frattini et al., 2008; Budetta, 2004). Mechanical analysis based on static equilibrium theory is the main method to analyse the stability of site-specific rockfall using the factor of safety (Fos). Ashby (1971) conducted stability analysis with a parallelepiped block resting on an inclined plane (Fig. 2a), and the solution was subsequently modified by Bray and Goodman (1981) and Sagaseta (1986). Kogure et al. (2006) utilized a cantilever beam model to determine the critical state of limestone cliffs. Frayssines and Hantz (2009) proposed the limit equilibrium method (LEM) to predict block stability against sliding and toppling in steep limestone cliffs (Fig. 2c). Chen and Tang (2010) established a stability analysis method of three types of unstable rocks in the Three Gorges Reservoir area with the LEM. Alejano et al. (2015) studied the influence of rounding of block corners on the block stability. Zhang et al. (2016) defined Fos based on fracture mechanics and studied the progressive failure process by analysing crack propagation. Alejano et al. (2010) and Pérez-Rey et al. (2021) deduced a formula for the Fos of blocks with more complex geometry.

Traditional force analysis diagrams of the rock block. Panels

The supporting force on the contact surface is assumed to be applied at a point in the current LEMs (i.e.

Based on rockfall investigation in the eastern Sichuan Basin, China, the main objective of this study was to propose a new three-dimensional method for the determination of failure modes and the Fos of biased rockfall, considering the non-uniform force distribution on the contact surfaces. Compared with the traditional LEM, this study takes into account the partial damage of the underlying soft rock and the overall instability of the overhanging hard rock blocks and can evaluate the stability of biased rockfall more comprehensively. Fos's of the typical unstable rock blocks in the study area are calculated to validate the proposed method. In addition, the critical mudstone retreat ratio in this area is analysed. This study is an extension of the basic LEM for rockfall, which can promote the accuracy of rockfall stability analysis and facilitate rockfall prevention and risk mitigation.

The study area is located on the northeastern edge of the Sichuan Basin, China (Fig. 3a). Continuous erosion processes generate moderate–low mountain and valley landforms (Yu et al., 2021). The tectonic structure of this area is characterized by a series of ENE anticlines and synclines (Fig. 3b and c). In the anticline area, the rock layers dip relatively steeply, where translational rockslides are the main mode of slope failure. The syncline area is dominated by gently dipping strata and is prone to rockfall (Zhou et al., 2018). The study area is located in the core of the Matouchang syncline, where the rock layers are sub-horizontal (Fig. 3d and e). In this valley, due to the long-standing fluvial incision, the relative relief is approximately 500 m and the valley flanks are extremely steep (Fig. 3e). In addition, the toes of the hillslopes are reshaped because of the construction of the G318 national road, which is the main traffic line and is always threatened by rockfalls from steep rock slopes (shown in Fig. 3d and Table 1).

Historical rockfall events along the G318 national road in the study area.

Characteristics of biased rockfalls in the study area.

The slopes in the study area consist of a sub-horizontally interbedded sandstone and mudstone layer. Therefore, there are multiple layers of potentially unstable rock blocks in the hillslopes (Fig. 4a). The thick sandstone has two sets of sub-vertical joints (Fig. 5), which cut the rock mass into blocks as the potential rockfall source (Fig. 4b). Basal cavities have formed in the underlying mudstone layer (Fig. 4c and d). Joints and bedding planes (BPs) constitute the detachment surfaces between the blocks and steep slope (Fig. 4e). The eccentricity effect produced by the mudstone external erosion plays an important role in the evolution process of rockfall. When the basal mudstone cannot provide adequate supporting force, the blocks detach from the steep slope, and biased rockfall occurs. Sliding and toppling are two possible failure modes of biased rockfall.

Stereonet produced using compass-clinometer survey data, showing the densities and orientations of five clusters. The data were collected in the rockfall-prone area shown in Fig. 3d.

According to the historical rockfall events in this area, precipitation is considered a triggering factor of rock instability. The precipitation mainly infiltrates along the sub-vertical joints or cracks of the sandstone (Fig. 4e). However, the drainage of fissure water is hysteretic due to the obstruction of basal mudstone. Therefore, transient steady flow exists in vertical cracks during heavy rainfall, and the hydrostatic pressure triggers the detachment of rock blocks. Thus, typical scenarios (such as rainfall intensity and earthquake) need to be considered in the stability analysis model.

The unstable blocks were labelled W02, W08, W18, W04 and W21, which are detached by the dominating discontinuities in Fig. 5. Basal cavities can be identified under the bedding planes of sandstone.

The evolution process of rock blocks from a stable state to failure.

A detailed geological investigation of unstable rock blocks was carried out in the study area (Fig. 6). The geological model of the rock block is mainly composed of the overhanging sandstone and the underlying mudstone. The sandstone block is assumed to be a rigid body, which is divided by two sets of orthogonal vertical smooth joints without friction resistance. According to the relatively persistent sub-vertical fractures observed in the field, the vertical joints are assumed to be fully persistent in the geological model. The sandstone block is assumed to be a complete body without persistent discontinuity, and it will not disintegrate before it falls. Due to the basal cavity in mudstone, the contact surface between sandstone and mudstone exhibits an eccentricity situation where non-uniform stresses are distributed at different positions. Mudstone is mainly loaded by compressive stress and tensile stress. When the compressive stress of mudstone exceeds its strength on the outer side, some initial damage appears. The effective contact surface between mudstone and sandstone is reduced, which aggravates the non-uniform distribution of stress. In this way, the ability of mudstone to resist the sliding and toppling of overhanging sandstone is reduced. In the field, compression deformation of mudstone can be observed, which usually manifests as micro-fractures and cleavages (Fig. 4d). The deformation is very slight and slow in the short term. In addition, the LEM is essentially a force and stress approach that does not take into account the deformation. Therefore, in this study, it is assumed that the mudstone is not subjected to deformation. The rock block remains in the state of static equilibrium prior to the final overall failure. Figure 7 displays the four evolutionary stages of biased rockfall. In the initial stage, the base cavity has not yet formed, and the normal force acting on the contact surface is uniform in different positions. The eccentricity effect leads to a non-uniform supporting force as the basal cavity grows, and partial damage gradually develops when the non-uniform stress exceeds the compressive or tensile strength of the mudstone. Under the triggering effects of rainfall or earthquakes, the rock blocks are separated by sliding or toppling.

Figure 8 represents the mechanical model of the force equilibrium analysis of a rock block with two or three free faces. The rock block (the overhanging sandstone) is generalized as a parallelepiped block. The underlying mudstone is impermeable, so rainfall can fill the joints and transmit horizontal hydrostatic pressure. The shear strength of the underlying mudstone is assumed to obey the Mohr–Coulomb criterion. Rainfall and earthquakes decrease Fos by generating hydrostatic pressure

Diagram of the force equilibrium analysis of the rock block model. Panels

A Cartesian coordinate system is established in three-dimensional space for the force analysis. The origin

The following formulas are used to calculate the apparent dip of

As shown in Fig. 8b, with respect to the

Based on bending theory (Adrian, 2010), the eccentricity distance along the

According to the Mohr–Coulomb criterion, the ultimate shear strength

With regard to stability against toppling, along the

Therefore, Fos with the consideration of compressive strength (Fos

Summarizing, four Fos types of unstable rock block are obtained. Fos

Calculation process of the Fos of the unstable rock blocks.

The damage mechanisms at the base of the rock block play an important role in the rockfall evolution process. However, the stress distribution on the contact surface calculated by the proposed analytical methods is difficult to validate with the field data. Therefore, numerical simulation of a biased rockfall was conducted in this study to determine the stress distribution on the contact surface between overhanging sandstone and underlying mudstone. Numerical simulations can take into account material deformation, unlike the analytical methods. FLAC3D, professional software that utilizes the finite-difference method (FDM) for three-dimensional analysis of rocks, soils and other materials, was employed for the 3D numerical simulation. Based on the geological models, a 3D numerical simulation model was conducted with FLAC3D 6.00 to analyse the stress distribution on the contact surface (Fig. 10).

Numerical model built in FLAC3D.

The model is mainly composed of sandstone and mudstone, where “Overhanging sandstone 1” represents an unstable rock block (dimensions

Diagram of stress distribution in the vertical direction on the contact interface through different methods:

When there is no cavity present, represented by

In the context of the limit equilibrium method, the contact area plays a vital role in stability analysis, as shown in Eqs. (21)–(30) in Sect. 3. The numerical simulation process provides an intuitive understanding of the influence of non-uniform stress distribution on the contact surfaces on the stability of rock blocks. Whether subjected to tension or compression, the rock layer has an ultimate strength. In Fig. 11, when

The traditional LEM does not account for distributed forces and fails to consider changes in the contact surface. The method proposed in this study addresses this issue and is applied to the calculation of Fos

A detailed field investigation was carried out in the source area of rockfall (Fig. 3d). The size of the blocks was determined by on-site measurement with tape and a laser rangefinder. The external erosion in mudstone was measured with a steel ruler, and the morphological characteristics of mudstone foundation were mainly described with the average erosion depth of the basal cavity. The attitude of discontinuities was measured by compass. The mechanical parameters are given in Sect. 4. The height of the water level (

Geometric parameters of rock blocks in the study area and Fos results.

Note: when there is no tensile stress in the mudstone foundation, Fos

There are up to 12 results of Fos per potentially unstable block with the consideration of three scenarios and four failure modes (i.e. partial damage and overall failure). Most Fos

Distribution of Fos values in different scenarios. Shapes represent different scenarios, and colours represent different failure modes.

Figure 13 presents the relationship between Fos

Correlation between Fos values and the dip of contact surface and retreat ratio. Here,

To explore the variation in Fos values with the progressive erosion process of the mudstone on the blocks, the mudstone retreat velocities in different directions are assumed to be equal (5 mm yr

Variation in Fos values with

Rockfall susceptibility based on the combination of four Fos types. The susceptibility is defined in three levels, represented by red, yellow and blue. Panel

These results further elucidate the stability analysis model proposed in this study. Fos

This result is consistent with Fig. 12, in which 63.7 % of the yellow and green points (Fos

Based on the meaning of four Fos types, rockfall susceptibility can be divided into three levels. When both Fos

The basal erosion in the mudstone plays an important role in the progressive failure process of biased rockfall. To analyse the effect of the retreat ratio on the stability of rock blocks, all blocks in the study area were selected to calculate their Fos values and susceptibility level with the increasing

Effect of the retreat ratio (

This study involves the development of an analytical model for the three-dimensional stability of biased rockfall, combining the basic LEM and the consideration of the non-uniform distribution. Due to the complexity of rock structure and force analysis, it is necessary to highlight the limitations of this model.

First, this study uses a three-dimensional coordinate system and bending theory. It is difficult to consider diverse shapes of rock blocks, and the rock block was simplified as a prismatic column. The assumption of fully persistent discontinuities may underestimate the stability of rock blocks and ignores the stress transmission in joints or rock bridges. Then, following the basic framework of the general LEM, this study assumed that the rock is not subjected to deformations. The complete stress–strain behaviour, such as the deformation in the mudstone layer, is not considered in this study. The mode of tension failure is very difficult to observe in the field, and it is currently verified by means of numerical simulation. Furthermore, the block stability is strongly influenced by the uncertainty in mechanical parameters. However, because of the difficulties in sampling strongly weathered mudstone, it is difficult to obtain adequate parameter values for uncertainty statistics. These limitations will be important considerations in future studies.

Due to differential weathering in sub-horizontally interbedded hard rock and soft rock, multi-layer biased rockfalls develop on steep slopes. In mountainous ranges, cut slopes and coastal cliffs, rockfall may cause significant facility damage and casualties in residential areas and transport corridors. The aim of this study was to present a new three-dimensional analytical method for the stability of rock blocks with basal cavities. In this method, a non-uniform distributed stress due to the eccentricity effect is applied at the contact surface instead of a point force. The development of non-uniform distributed stress calculated by the proposed analytical methods was validated by numerical simulation, which presents the evolution process of biased rockfall from partial damage of the soft underlying layer, caused by non-uniform distributed stress, to toppling and sliding of overhanging hard rock block due to overall unbalanced force. The method considers four failure modes according to the rockfall evolution process, including partial damage of the soft foundation (Fos

Taking the northeastern edge of the Sichuan Basin in southwest China as the study area, the proposed method is used to calculate the Fos of biased unstable rock blocks. The results show that in the natural scenario, the underlying mudstone of some rock blocks has been partially damaged, and compression failure of the mudstone has been observed in the field. Some rock blocks are expected to fail as a whole in rainfall or earthquake scenarios. The statistical analysis indicates that the retreat ratio is the crucial factor influencing the Fos of biased rockfall. On the basis of different combinations of four Fos types, rockfall susceptibility was classified into three levels. As the retreat rate increases, the rock blocks undergo an evolution process from stability to partial instability and then overall instability. Based on the current mechanical parameters of the eastern Sichuan Basin, the critical retreat ratio from low to moderate rockfall susceptibility is 0.33.

The proposed method improves the three-dimensional mechanical model of a rock block with the basal erosion by considering non-uniform distributed stress at the contact surface, which could promote the accuracy of rockfall stability analysis. Due to the assumptions adopted and the complexity of the failure mechanism of biased rockfall, there are some limitations in this method, mainly including the simplification of boundary conditions and rock deformation. These limitations will be important considerations in future studies.

All raw data can be provided by the corresponding authors upon request.

XS, BC and JD planned the campaign; XS and BC performed the field measurements; XS, BC, WW and BL designed and developed the methodology. XS, BC and JD analysed the data; XS and BC wrote the manuscript draft; JD and WW reviewed and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research is funded by the National Natural Science Foundation of China (nos. 42172318 and 42177159). The first author thanks Chengjie Luo and Yu Wang for data collection in the field. We also appreciate the assistance of the Research Center for Geohazards Monitoring and Warning in Three Gorges Reservoir, China. We also thank the reviewers for their suggestions that improved the quality of this paper.

This research has been supported by the National Natural Science Foundation of China (grant nos. 42172318 and 42177159).

This paper was edited by Daniele Giordan and reviewed by two anonymous referees.