A modified solution of radial subgrade modulus for a circular tunnel in elastic ground

In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (kr). The widely used solution kr 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 kr when the radial deformation of the tunnel is described by a Fourier series. This modified Wood׳s solution of kr 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 kr is significantly affected by the distribution shape of the tunnel radial deformation. The value of kr 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 kr for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for kr is illustrated by a design case using a bedded beam model.