Generalized Block Theory for the Stability Analysis of Blocky Rock Mass Systems Under Seismic Loads

Abstract

The stability analysis of rock blocks on man-made excavation faces (e.g. tunnel, cavern, and slope) subject to seismic loads is an important issue in the field of rock engineering. This paper proposes a generalized block theory (GBT) by combining a pseudo-static method and the traditional block theory to evaluate the stability of blocky rock masses during earthquake activities. In our analysis, the basic safety factors are derived considering time-varying seismic loads to determine the stability of a rock block at each time step. Afterwards, two new parameters, P^u and V^u, are used to evaluate the seismic stability of a rock block, where P^u is the instability probability defined as the ratio of the time for the block becoming unstable to the total seismic loading time, and V^u is the probabilistic instability volume defined as P^u times the block volume. As for a blocky rock mass system, its probabilistic instability volume is the sum of V^u of all seismically unstable blocks and the instability probability is the ratio of its probabilistic instability volume and total volume of seismically unstable blocks. Through the simulation of a generic slope excavation, we observe that seismic loads significantly affect the stability and kinematics of a rock block during an earthquake. For a blocky rock mass, both P^u and V^u decay with the epicentral distance, in general following an inverse power law trend. Furthermore, it is found that the local site effect also has a strong influence on the slope stability under seismic loads.

Appendix A: Comparative Analysis of BLKLAB and 3DEC

1.1 Initial Stability Status Subject to Gravity

For the comparison purpose, we build a model in 3DEC using exactly the same configuration as that in BLKLAB (as described in Sect. 3.1). Since the GBT follows the rigid block assumption of the traditional block theory (Goodman and Shi 1985), the 3DEC model also assumes blocks are rigid. The model in 3DEC utilizes the Coulomb slip model to simulate joints, of which the parameters are listed in Table 1. In addition, the cohesion of joints is set as 0, and the normal and shear stiffnesses have an equal value of 10 GPa/m. The 3DEC model uses roller boundary conditions on the side and bottom boundaries while the upper boundary is a free surface. The geometrical model and the initial displacement field after geo-stress initialization in 3DEC are shown in Fig.

Fig. 18

The 3DEC model of a blocky rock slope: a the geometrical model; b the initial displacement field after geo-stress initialization

18a and b, respectively. The maximum magnitude of the block displacement is about 8.03 × 10^–4 m. In 3DEC, we could also calculate the safety factor of joints in a rigid blocky rock mass system based on the strength reduction method, which gives the minimum safety factor of the system (3DEC Manual 2019). It should be noted that the safety factor calculation should be executed after the geo-stress initialization. The initial safety factor of the 3DEC model is 1.0, i.e. the slope is initially stable, which is smaller than the safety factor (1.24 or 1.59 as shown in Table 3) obtained by BLKLAB. The difference between the two software platforms is attributed to the different calculation algorithms. The GBT in BLKLAB only solves the safety factor of removable blocks and ignores other blocks that may be secondary unstable blocks (Noroozi et al. 2012; Fu and Ma 2014). As a contrary, 3DEC loop all the blocks to fetch a minimum safety factor of the rock system.

1.2 Time-Sequential Safety Factors of the Rock System Under Seismic Loads

Within the scope of pseudo-static method (Zhang 2018), the time-sequential safety factor of the rock system under seismic loads in 3DEC is also calculated by altering the resultant force ${\mathbf{F}}^{a,t}$ derived in Eq. (4). The sequential safety factor of the rock system subject to the Cape Mendocino earthquake (M_w 7.01 as shown in Fig. 5; Giardini et al. 2013) is illustrated in Fig. 

Fig. 19

Time-sequential safety factors of the rock slope system calculated by the 3DEC

19. The minimum safety factor during the earthquake is 0.11 at 7.22 s, which is close to the one given by BLKLAB, i.e. 0.1 at 2.62 s. This consistency indicates that the removable blocks resolved by GBT are indeed the controlling parts of the slope under seismic loads. Similar to the GBT results shown in Figs. 5 and 10, the safety factor derived by 3DEC fluctuates more significantly in the early phase of the earthquake than in the later phase. Among all the 1500 time-steps, there are 717 time-steps with the safety factor less than unity. According to Eq. (21), the instability probability of the rock system in 3DEC model is calculated as 717/1500 = 47.89%, which is larger than the result given by BLKLAB, i.e. 11.18%. The smaller safety factor and higher instability probability in 3DEC imply that the discrete element method gives a more conservative evaluation for the rock system under seismic loads than the GBT. However, the GBT is capable of determining the exact location of unfavorable blocks during the earthquake and their kinematic modes. In addition, the GBT has a higher computational efficiency since it does not require any contact detection and interaction calculation.

1.3 Relationship Between the Instability Probability and Earthquake Magnitude

We then select 11 earthquakes (Yenier et al. 2010; Giardini et al. 2013; Ancheta et al. 2013) with the moment magnitude ranging from 5.20 to 7.51 as a database to investigate the sensitivity of the GBT and 3DEC results to the earthquake magnitude. We choose the time series of signals recorded by several representative stations for each earthquake as listed in Table

Table 5 Earthquakes information

5. All the acceleration signals, resampled to 0.02 s and transformed into the standard directions, are plotted in Fig. 

Fig. 20

Ground accelerations monitored during different earthquakes

20. The instability probabilities of the rock system by 3DEC and BLKLAB are shown in Fig.

Fig. 21

Instability probability of the blocky rock mass under different earthquakes calculated by a the BLKLAB and b the 3DEC models

21a and b, respectively, where they both tend to exhibit a positive linear correlation with the earthquake magnitude. The aforementioned results support the validity and effectiveness of our proposed GBT method for analyzing blocky rock mass stability under seismic loads.