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When using the bedded-beam model based on Winkler springs for cavity lining design, it is always difficult for engineers to select an appropriate value for the radial subgrade modulus ( k ) due to the lack of related theoretical research. The solutions of k for a circular tunnel proposed by Wood (1975) and Zhang et al. (2014) have been found to be applicable only when a tunnel is in elastic ground. However, a rock mass will exhibit elastic, brittle and plastic states under different stress conditions. Considering that the plastic state can effectively improve the bearing capacity and reduce the engineering cost, to investigate the radial subgrade modulus in an elastic-brittle-plastic rock mass, modified analytical solutions are derived that are compatible with a generalized nonlinear unified strength criterion. Using the maximum shear stress modulus, these solutions consider the comprehensive influences of the intermediate principal stress, the dilation characteristic and brittle plastic. Then, a comparative verification of the new solutions using a nonlinear Hoek-Brown yield criterion is conducted. The effects of the parameters are discussed in detail. The calculation results show that k is a comprehensive evaluation index for multiple factors, is significantly influenced by the geo-stress and support pressure, and increases with the geological strength index, elastic strain and Young’s modulus in the plastic zone. During cavity contraction, the intermediate principal stress improves the magnitude of k more significantly than dilation, whereas dilation has a greater effect during cavity expansion. Using the perfectly plastic model without considering brittle-plastic behaviour will lead to an overestimation of k , and using different strength criteria directly affects the timing of the plastic deformation of a rock mass. The influence of various factors should be comprehensively considered to select a reasonable radial subgrade modulus to take full advantage of the latent potentialities of a rock mass. The deep-buried diversion tunnel at the Jinping Ⅱ hydropower station is studied as an example to demonstrate the practical application of the proposed model. [ABSTRACT FROM AUTHOR]

• Presented a method to derive subgrade modulus consistent with ground deformation. • Proposed solutions for radial modulus considering the influence of the grout layer. • Ground deformation would lead to a nonuniformly distributed subgrade modulus. Subgrade modulus is a critical parameter when analyzing the segmental lining of shield tunnels. However, prevailing analytical solutions of radial subgrade modulus (k r) are based on predefined ideal deformation modes, either circular or oval, which are not consistent with the actual ground deformation, and the influence of the grout layer is commonly neglected. In view of this, this study presents a solution of k r considering the actual deformation of the ground and the grout layer between linings and surrounding ground. A displacement fitting method is proposed to relate k r to the arbitrary distribution shape of ground deformation. With this method, k r for actual deformation consists of a series of radial subgrade moduli (k r m) for single deformation modes described by sine and cosine functions. Theoretical solutions of k r m considering the influence of the grout layer for single deformation modes are then derived based on the composed ground which is made up of the ground and the grout layer. The influence of the grout layer on k r m is then discussed by a parameter study. Moreover, the applicability of the proposed method is illustrated through a design case, where the calculated internal forces agree well with the data from field measurement. The influence of actual ground deformation and grout layer on k r , internal forces, and displacement of shield tunnel is then discussed. Results show that an accurate prediction of k r requires the consideration of the actual deformation of the lining and the influence of the grout layer. [ABSTRACT FROM AUTHOR]

Abstract: In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (k r ). The widely used solution k r for a circular tunnel in elastic ground proposed by Wood (1975) was found to be applicable only when the tunnel radial deformation is oval-shaped. On the basis of the Wood׳s solution, this note presents a general solution for k r when the radial deformation of the tunnel is described by a Fourier series. This modified Wood׳s solution of k r using compatible stress functions is validated by a numerical example. The modified solution for the example shows good consistency with the original Wood׳s solution when the tunnel becomes an oval shape with deformations. The example indicates that the magnitude of k r is significantly affected by the distribution shape of the tunnel radial deformation. The value of k r is no longer a constant value around the tunnel when the tunnel deforms into a general shape described by a Fourier series. It is quite different from the value of k r for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for k r is illustrated by a design case using a bedded beam model. [Copyright &y& Elsevier]