2.1 Grottoes overview
The Yungang Grottoes in Datong city, Shanxi Province represent the outstanding achievement of Buddhist art in China in the 5th and 6th century, with 252 caves and 51,000 statues. The Yungang Grottoes, Mogao Grottoes in Dunhuang and Longmen Grottoes in Luoyang are known as China's three largest ancient grottoes; they are famous as world cultural heritages. Under the action of long-term natural forces, the surrounding rock of these grottoes has suffered numerous serious geological diseases (Guo, 2009; Li, 2016; Liu, 2012).
The total protection area of Yungang Grottoes is approximately 3.6 km2, with Shili River in the south and low hills in the north. The geographical terrain of the grottoes is slightly undulating, with a maximum height difference of approximately 55 m. The geomorphologic types of the grottoes can be generally typed into two forms: a high platform structure with denudation in the low hills at the top of Yungang Grottoes and valley erosion accumulation on the terrace of the Shili River. (Huang, 2003). The grotto rock mass has moderate softness and hardness, which is suitable for carving but has poor resistance to weathering. The composition of the 19th grotto consists of grayish-white medium sandstone on the top of the grotto with a distribution between 4 and 5 m; medium sand and fine sandstone with mudstone and sandy mudstone in the interlayer with a distribution between 5 and 10 m; and light brown and grayish-white medium sand and fine sandstone 10 m above the grotto (Yan & Fang, 2004; Yang, et al. 2009). The geological section of the grotto cliff body is shown in Fig. 1. To obtain the calculation parameters of the rock mass that occurs in the grottoes, samples were taken from the same stratum of the mountain near the grottoes, and the physical and mechanical parameters of the rock mass were obtained by experiments (Fig. 2). The parameters of the rock mass that occurs in the grottoes are shown in Table 1.
Table 1
Physical and mechanical parameters of the rock mass in the grottoes
Rock type | Density (g∙cm− 3) | Elasticity modulus (GPa) | Poisson's ratio | Cohesion (MPa) | Internal friction angle (˚) | Tensile strength (MPa) |
Gray-white sandstone | 2.16 | 27.90 | 0.198 | 1.0 | 30.45 | 5.39 |
Light brown sandstone | 2.55 | 23.70 | 0.144 | 0.8 | 32.13 | 1.25 |
Argillaceous siltstone | 1.79 | 13.39 | 0.298 | 0.5 | 14.07 | 0.01 |
Mudstone | 1.71 | 8.29 | 0.280 | 0.3 | 13.13 | 0.02 |
The 19th grotto of Yungang Grottoes is one of the most representative grottoes at the research site. The main cave is oval in shape, with a vault roof, a door and clear windows. The height of the seated Buddha in the main cave is 16.8 m, and it is the second highest statue in Yungang Grottoes. One ear grotto is cut approximately 5 m from the ground to the east and west of the grotto, in which a sitting statue with a height of 8 m is carved. The research object of this paper is the west ear grottoes and their cliff bodies, as shown in Fig. 3(a). After an earthquake, the fore-wall of the 20th grotto and the west ear grotto of the 19th grotto collapsed. Part of the cave wall between the ear grotto and the main grotto is relatively thin, and there is a broken hole on the cave wall, as shown in Fig. 3(b). The thinnest part is less than 10 cm; the cause of this damage is unknown.
2.2 Calculation model and boundary conditions
In previous research, the time-history analysis of the seismic dynamics of the grottoes’ cliff bodies is relatively rare. Grottoes are usually reduced to regular caves, while the statues in grottoes are large in size and irregular in shape, which affects the stress distribution of their cliff bodies. The geometries of the grottoes may have greater influence on the dynamic response under dynamic action. Therefore, the shapes of the statues in the grottoes should not be disregarded in the dynamic response analysis of the grottoes’ cliff bodies, and the morphologies of the grottoes should be reflected in the model for better observation or modelling. The three-dimensional model of the grottoes’ cliff bodies is established, and the three-dimensional point cloud of the facade and the inner main grottoes walls of the grottoes’ cliff bodies are obtained. The finite element software ANSYS is employed for auxiliary modeling, and the model is imported into FLAC3D for calculation. The built model is shown in Fig. 4(a). The total height of the model is 29.25 m; the east-west (X-axis direction) width is 12.12 m; the north-south (Y-axis direction) length is 20 m; and the model has 48,029 nodes and 255,463 units. Because the cracks in the grottoes’ cliff bodies have been treated by grouting, the influence of cracks is not considered in this model.
2.3 Anchorage design and calculation parameters
Small bolts are applied to reinforce the upper rock mass of the ear grottoes, as shown in Fig. 4(b). A full-length bonded bolt is adopted for support: the length of the bolt l = 7 m; transverse and vertical spacing D = 2 m; angle = 10 °, bolt diameter r = 12 mm; and design load is 40 kN. From the left boundary of the model, a row of bolts was installed with an interval of 2 m, 5 vertical rows, 4 bolts in each row, and a total of 20 small bolts. For the convenience of analysis, 5 rows of bolts were numbered as groups X1 to X5 from left to right along the x-axis. From low to high along the elevation direction, the bolts were numbered as groups Z1 to Z4. According to the position, the bolts could be numbered as Groups XZ, in which X and Z represent the x direction of the bolts and the z direction of the bolts, respectively.
2.4 Cable Structural Elements
Structural units provided for anchor/cable simulation in FLAC3D include Cable and Pile (Itasca, 2009). The axial stress of the small bolt, which has a small diameter and a weak bending resistance, is considered. Therefore, a cable is chosen to simulate the anchor rod within the research. Each cable structural element is defined by its geometric, material and grout properties. A cableSEL is assumed to be a straight segment of uniform cross-sectional and material properties that is located between two nodal points. An arbitrarily curved structural cable can be modeled as a curvilinear structure that is composed of a collection of cableSELs. The cableSEL behaves as an elastic perfectly plastic material that can yield in tension and compression but cannot resist a bending moment.
Each cableSEL has a unique coordinate system, as shown in Fig. 5, to define the average axial cable direction. The cableSEL coordinate system is defined by the locations of its two nodal points, which are labeled 1 and 2. The cableSEL coordinate system is defined as follows:
(1) the centroidal axis coincides with the x-axis,
(2) the x-axis is directed from node-1 to node-2, and
(3) the y-axis is aligned with the projection of the global y- or x-direction (whichever is not parallel with the local x-axis) onto the cross-sectional plane.
The two active degrees-of-freedom of the cable finite element are shown in Fig. 5. For each axial displacement shown in the figure, there is a corresponding axial force. The stiffness matrix of the cable finite element includes a single degree of freedom at each node, which represents axial action within a cable structure.
Naturally, the shear behavior of the cable-rock interface is cohesive and frictional. Within this model, the system is conceptualized and represented numerically as a spring-slider system that is located at the nodal points along the cable axis. In evaluating the axial forces that develop in the reinforcement, displacements are computed at the nodal points along the reinforcement axis, as shown in Fig. 6. Out-of-balance forces at each node are computed from the axial force in the reinforcement, and the shear forces contributed via shear interaction along the grout annulus. Axial displacements are computed by integrating the nodal accelerations using the out-of-balance axial force and a mass that is lumped at each node.
In the calculation of anchoring, the grotto model without anchoring is analyzed, and the boundary conditions of dynamic-static force, monitoring point setting and input seismic waves remain the same as the original model. Only the bolt element is added to the upper rock mass of the original model. The material parameters of the small bolt in the model are shown in Table 2.
Table 2
Material parameters of the small bolt
Modulus of elasticity (GPa) | Yield load (kN) | Bond stiffness (N/m2) | Bond strength (N/m) | Anchor bar diameter (mm) | Anchor hole diameter (mm) |
200 | 300 | 1.0 × 109 | 2 × 108 | 12 | 30 |
2.5 Boundary conditions and seismic waves
Numerical analysis of the seismic response of surface structures, such as dams, requires the discretization of a region of the material adjacent to the foundation. The seismic input is normally represented by plane waves that propagate upward through the underlying material. The boundary conditions at the sides of the model must be accounted for in the free-field motion that exists in the absence of the structure. These boundaries need to be placed at distances that are sufficient for minimizing the wave reflections and achieve free-field conditions. To apply the free-field boundary in FLAC3D (Fig. 7), the model should be oriented such that the base is horizontal, its normal is in the direction of the z-axis, the sides are vertical and their normals are in the direction of either the x- or y-axis.
The Kobe seismic wave with a maximum positive acceleration of 2 m/s2 is selected as the input seismic wave in the calculation of the seismic force within the grotto cliff body. The duration of the seismic wave is 18 s, and the preeminent frequency range is 1–3 Hz. The input direction is positive in the y-axis. The time-history curve of acceleration is shown in Fig. 8.
2.6 Monitoring stations
To facilitate the subsequent analysis and generate the final calculation results, monitoring points were established before processing the calculation. The distribution of the monitoring points is shown in Fig. 9. A total of 38 monitoring points in three groups—E1-E12, M1-M14 and W1-W12—were set up along the elevation at different x coordinates of the model facade. A group of 12 monitoring points were set up along the elevation of the inner wall of the model, namely, the wall of the main grotto.