Pasternak Model-Based Tunnel Segment Uplift Model of Subway Shield Tunnel during Construction

Segment uplift is a frequent problem during the construction of shield tunnels. The phenomenon of segment uplift in Xuzhou Metro Line 1 is investigated. The segment uplift magnitude in the shale layer is quite smaller than that in the clay layer. A tunnel segment uplift calculation model based on the Pasternak foundation beam model is proposed, which considers the hardening process of the grouted body with time. The finite difference method is adopted to calculate the magnitude of segment uplift during different tunnel construction stages, and a numerical solution of segment uplift on the longitudinal direction is obtained. The applicability of the numerical solution is studied by comparing with the field test results and parametric analyses are also performed to investigate the effects of different factors on segment uplifting. The results show that the coefficient of subgrade modulus, shear stiffness of the shear layer, and grout pressure all influence the segment uplifting of the tunnel. The influence of the coefficient of subgrade modulus and grout pressure on segment uplift is more significant.

A Simplified Analysis Method for the Deformation Response of an Existing Tunnel to Ground Surcharge Based on the Pasternak Model

Surface surcharge changes the existing equilibrium stress field of the stratum and adversely affects the existing tunnel. This paper presents a simplified analytical solution for calculating the longitudinal displacement of existing tunnels that are subjected to adjacent surcharge loading. Based on the Boussinesq solution, the distribution of the additional load matrix caused by the surface surcharge on the existing tunnel was obtained. A Euler–Bernoulli beam with a Pasternak foundation was used as a simplified model for tunnel stress analysis. Using the corrected reaction coefficient of the foundation bed, the differential equation of tunnel deformation was established, and the solution matrix of the longitudinal displacement of the tunnel was obtained by using the finite difference method. The reliability and applicability of the proposed method were verified by comparing the results with finite element simulation results, field test data, and the calculation results of three simplified elastic analysis methods with different foundation bed coefficients. On this basis, the parameters of the load–tunnel model were analyzed, and the effects of the buried depth, the size of the load, the relative positions of the load and the tunnel, and the relative stiffness of the tunnel soil on the maximum displacement of the existing tunnel were calculated. An empirical formula is proposed for calculating the maximum longitudinal displacement of the existing tunnel subjected to surface surcharge. The findings of this research can provide a basis for the theoretical verification of the deformation response of an existing tunnel subjected to adjacent surface surcharge.

COMPARISON OF SOLUTIONS TO THE BENDING PROBLEMS OF THREE-LAYER PLATES ON THE WINKLER AND PASTERNAK FOUNDATIONS

Solutions of problems on axisymmetric bending of an elastic three-layer circular plate on the Winkler and Pasternak foundations are given. The bearing layers are taken as isotropic, for which Kirchhoff’s hypotheses are fulfilled. In a sufficiently thick lightweight, incompressible in thickness aggregate, the Timoshenko model is valid. The cylindrical coordinate system, in which the statements and solutions of boundary value problems are carried out, is connected with the median plane of the filler. On the plate contour, it is assumed that there is a rigid diaphragm that prevents the relative shear of the layers. The system of differential equations of equilibrium is obtained by the variational method. Three types of boundary conditions are formulated. One- and two-parameter Winkler and Pasternak models are used to describe the reaction of an elastic foundation. The solution to the boundary value problem is reduced to finding three desired functions, plate deflection, shear, and radial displacement in the filler. The general analytical solution to the boundary value problem is written out in the case of the Pasternak model in Bessel functions. At the Winkler foundation, the known solution is given in Kelvin functions. A numerical comparison of the displacements and stresses obtained by both models with a uniformly distributed load and rigid sealing of the plate contour is carried out.

Numerical analysis of methods for calculating the soil base and methods for determining bed coefficients

Despite significant progress in the development of methods for calculating the soil base in a single calculation model with structures and the ability to perform calculations in three dimensions, the most popular in the community of design engineers remains the calculation model of the slab on an elastic basis. This is due to the simplicity of such a model. Probably, the greatest difficulty in applying such a model is to determine the coefficients of flexibility of the base (bed coefficients). In this paper, a study of the reliability of different methods for determining the coefficients of the bed under different variants of soil conditions The paper determines and compares the characteristics of the stress-strain state (sedimentation values, reactive pressure and forces) in three foundation slabs of different geometry and under different engineering and geological conditions of the construction site. A homogeneous base composed of loess soils, an inhomogeneous base composed of alternating layers of both cohesive (loam and sand) and incoherent (fine sand) soils and a close to homogeneous base composed of sand and soils are considered. The research was conducted using the software and computer system LIRA SAPR 2016. Implemented three methods for determining the coefficients of the bed (the flexibility of the base): - Pasternak model with two bed coefficients, which for inhomogeneous soils are determined by the values of the deformation modulus and Poisson's ratio averaged within the depth of the compressible thickness; - Winkler model with one bed ratio; - Pasternak model with two bed coefficients, which are determined by the average values of the deformation modulus and Poisson's ratio when introducing the correction factor to the deformation modulus. The convergence of the absolute values of the controlled parameters obtained using the above methods, depending on the type of soil base and the nature of soil layers within the compressible layer, is analyzed. The dependence of the convergence of the characteristics of the stress-strain state of the slab obtained by different methods on the homogeneity of the base soils is established.

An Energy Solution for Predicting Buried Pipeline Response Induced by Tunneling Based on a Uniform Ground Movement Model

The construction of shield tunnels inevitably causes displacement of the surrounding soil and additional stress and deformation of the buried pipeline. An energy solution for predicting the deformation of buried pipelines caused by tunneling is proposed in this study. First, based on the uniform ground movement model, the interval of the free displacement field of soil around the pipeline induced by tunneling is calculated. Then, we use the Pasternak model to establish the total potential energy equation of the tunnel-soil-pipeline interaction. The final settlement interval of the pipeline is obtained by solving the numerical calculation program with MATLAB. The calculation results of the energy solution are compared with the results of the centrifugal test and the reported theoretical solutions of Winkler and Pasternak, and then the applicability of the solution for predicting the pipeline response under different geotechnical conditions is verified. Combined with an engineering case, the energy method calculation results, numerical simulation results, and measured results are compared to obtain the most unfavorable position of the pipeline caused by tunneling. At the end of this study, the application steps of the proposed method in actual construction are summarized. These steps are used to predict pipeline response in order to take protective measures.

A simplified planar model for geosynthetics reinforced composite foundation subjected to vertical load

A simplified planar model for geosynthetics reinforced composite foundation under large-scale loading was established with a new consolidation analysis. The cushion was modeled by modified Pasternak model and the reaction of pile and subsoil was modeled by Winkler model. The effect of geosynthetics layer was directly considered as an elastic cable and the subsoil was divided into numerous columns with only vertical drainage. The solution was obtained by a finite difference based iterative scheme. The feasibility of the model was demonstrated by a case study. Then a parameter study was executed to analyze the effect of several influential factors. The results showed that there is a critical pile –to-pitch ratio that makes the increase of the stiffness of the geosynthetic material the most conducive to deformation control.