Face Stability Analysis Of Shield Tunnel In Sandy Cobble Stratum By Stochastic Multi-scale Numerical Limit Analysis

Shield tunnel method is widely used in subway construction,and as the tunnel is passing through sandy cobble stratum,face collapse can be induced by improper tunneling settings because the surrounding soils are cohesionless,random and susceptible to disturbance.The stability of tunnel face is dependent on the strength of the sandy cobble stratum,which is a composite material whose strength is closely related to its mesostructure such as the dimension,the shape and the dip angle of the cobbles.The diameter of the cobbles is large and the mesostructure is random,making the indoor test of sandy cobble samples very difficult and unrepresentative.As a result,it is very hard to accurately calculate the limit supporting pressure of tunnels in sandy cobble stratum.In the present paper,an efficient and reliable method for stability analysis of sandy cobble stratum considering the mesostructure and its randomness is aimed to be established.Based on the classical bounding theory,we establish an upper bound model which can consider the effect of pore water pressure through incorporating the effective stress of saturated soils.With combination of random generation algorithm of sandy cobble mesostructure,limit analysis model and computational homogenization,an iterative multi-scale numerical limit analysis method is proposed considering strength anisotropy caused by cobble dip angle.Statistical volume elements are sampled by the moving window technique.Stochastic multi-scale numerical limit analysis is proposed based on the Karhunen-Loève(K-L)expansion.Parameter analysis is conducted to study the effect of the mesoscopic property and its randomness on the stability of tunnel face,providing guidance for tunnel construction.The main research and conclusions are as follows:(1)An upper bound finite element method is established using the second order cone programming and the finite element spatial discretization by incorporating the effective stress of saturated soils into the classical bounding theory to reflect the influence of pore water pressure.The established method is reliable and efficient.(2)Particle size distribution function of the cross section of sandy cobble stratum is derived based on the fractal theory.Random generation algorithm of sandy cobble mesostructure is established using ellipse to represent the cross section of the cobble.With the combination of the homogenization theory and the limit analysis,an iterative multi-scale numerical limit analysis method is proposed which can consider the effect of cobble dip angle to study the relationship between the mesoscopic and macroscopic material properties.It is found that,the macroscopic parameters can be effectively identified by the homogenized limit analysis method and the multi-scale limit analysis framework is reliable and efficient.Close agreement can be seen between the results of global mesoscopic analysis and those of multiscale computations.Only a few iterations are needed as the dip angle is considered and any form of strength anisotropy can be analyzed.(3)The moving window technique is employed to sample the statistical volume elements,which are homogenized to obtain the statistical information of strength parameters.Stochastic multi-scale numerical limit analysis method is developed based on the K-L expansion method and multi-scale limit analysis.The effect of mesoscopic parameters and the sampling method is discussed on the statistical results of apparent properties of sandy cobble stratum.It is shown that,the logarithmic incremental apparent strength parameters accord with the normal distribution and the exponential function is able to describe the autocorrelation structure.The K-L expansion method can greatly improve the efficiency of parameter sampling.(4)Based on stochastic multi-scale numerical limit analysis,two-dimensional model of face stability analysis is established.For dry and watery stratra,parameter analysis is conducted to study the effect of the mesoscopic parameters and their randomness on the stability of the tunnel face.It is concluded that,the stability of the tunnel face improves with the decrease of the fractal dimension and the increase of the largest diameter,the interface reduction ratio,the aspect ratio and the soil strength.A large impact is found of the shape,content and diameter of the cobbles and the effect of soil strengths is most obvious.The face stability attains the greatest as the major axis of the cobble is in parallel with the axis of the tunnel.Discreteness of the limit supporting pressure increases with the decrease of the size of the statistical volume element and the results are larger than those of deterministic analysis.The mean value of the limit supporting pressure is less affected by the sampling parameters and is therefore more reliable.