A simplified model for calculating the seismic responses of the shaft is proposed in this paper. Based on the theory of Winkler elastic foundation beam, the urban shaft is simplified as a vertical beam. The horizontal soil reaction and vertical shear tractions between the shaft circumference and the surrounding soils are considered through horizontal springs and rotating springs on the sidewall of the shaft. The translation and rocking motion of the shaft are considered through horizontal springs and rotating springs at the bottom of the shaft. Then, the dynamic analysis model of the shafts under seismic motion is established, and the control equation of the dynamic response of the shaft in frequency domain is deduced. The analytical solution of the steady state response of the shaft is obtained. Considering the randomness of the earthquake motion, this method can get the shaft kinematic responses under different ground motions efficiently in conceptual design process.

The safety of underground engineering projects is significantly influenced by groundwater. One of the key complexities is identifying the primary seepage paths within underground rock formations, understanding the patterns of seepage, and determining the effects of fracture parameters on the fluid movement inside the rock mass. To address these issues, a probabilistic model is constructed for random fractures using the finite element method, reflecting the random nature of fracture distributions in the real world. This model allows for an in-depth examination of the distribution of pore water pressure and Darcy velocity field, revealing the permeability trends in fractured rock masses. A variety of fracture models were devised to understand the relationship between factors such as fracture density, length, length power law, angle, dispersion coefficient, aperture, and power law, and how they affect the overall permeability of rock masses. The study suggests that, in the context of discrete fractured rock masses, there is a linear increase in permeability with an increase in fracture density and aperture. Moreover, fractures of greater length lead to increased permeability, with fractures aligned with the direction of water pressure having the most impact on seepage velocity. A thorough investigation of the factors that affect each fracture parameter was performed, and the permeability of each model was computed. From these findings, a series of predictive equations were suggested for estimating rock permeability based on fracture geometry parameters.

In numerous subterranean projects, the impact of groundwater on the safety of the engineering undertaking is of paramount significance. Fractures, functioning as the primary channels for seepage within subterranean rock masses, necessitate the complex and challenging task of accurately characterizing seepage patterns and quantitatively investigating the effect of fissure parameters on fluid dynamics within the rock masses. This article presents a stochastic fissure model incorporated within a finite element framework, which captures the probabilistic distribution of fissures found in nature. It provides a comprehensive analysis of the distribution of pore water pressure and Darcy velocity fields. It unveils the permeation patterns of fissured rock masses and establishes a series of fissure models, quantitatively investigating the correlations between matrix permeability, water pressure, fissure density, fissure length, the length power law, fissure angle, the dispersion coefficient, fissure aperture, and the aperture power law, as well as their influence on the equivalent permeability of the rock mass. The findings reveal that in a discrete fissured rock mass, the greater the matrix permeability, the higher the equivalent permeability, and vice versa. Under water pressures of less than 10 MPa, gravity significantly impacts equivalent permeability, and permeability linearly increases with a rise in fissure density. Longer fractures result in higher permeability, and fractures parallel to the direction of water pressure contribute most significantly to the speed of seepage. Moreover, permeability markedly increases with an increase in aperture. This study provides a comprehensive analysis of the impact of matrix permeability and fissure parameters on equivalent permeability and calculates the permeability of each model. We also propose a set of predictive formulas based on fissure geometric parameters to anticipate the permeability of rock masses.