Stability evaluation model of toppling unstable rock based on natural vibration frequency

The instability of toppling unstable rock is usually characterized by a sudden failure without obvious displacement. Therefore, it is difficult to achieve early-warning by conventional displacement monitoring. In addition, it is difficult to obtain one of the important parameters influencing unstable rock stability, namely the area of bonding surface between unstable rock and parent rock. Due to these problems, the stability evaluation of unstable rock remains challenging. The objective of this study is to develop a stability evaluation method of toppling unstable rock based on a dynamic characteristic model. Assuming that the unstable rock is homogenous and isotropic, the control fissure is a single plane, the damping ratio of the system is <1, and the deformation is elastic deformation within the amplitude range, then the vibration model of unstable rock can be simplified into a pendulum vibration model. The relationship between natural vibration frequency, bonding surface area, elastic modulus, pendulum length, bonding surface width, and mass of toppling unstable rock is established by theoretical derivation. Then a limit equilibrium model is combined to obtain the quantitative relationship between natural frequency and safety factor of toppling unstable rock, thereby realizing the stability evaluation of toppling unstable rock based on natural frequency. Further, a case study is conducted using two toppling unstable rock blocks from the left bank of Baihebao Reservoir in Beijing, China. The test results verify the feasibility of stability evaluation of toppling unstable rock based on natural frequency, which is faster and more accurate than the conventional method.