Closed-form elastic solution for irregular frozen wall of inclined shaft considering the interaction with ground

An analytical solution is derived that provides a closed-form formulation for stresses and displacements of a deep inclined shaft with irregular frozen wall liner in an infinite elastic medium, subjected to a non-uniform stresses. This solution is based on: (i) the conformal mapping of a doubly connected domain of prescribed shape onto a circular ring by an appropriate numerical optimization scheme which is achieved by MATLAB program, (j) the complex variable method. For a particular case, the distributions of hoop stress and radial displacement inside frozen wall are presented with analytical solution. The accuracy of stress functions is verified by comparison of different number of coefficients in stress functions and it show that the number of coefficients k≥15 is advised. The validity of analytical solution is verified by a comparison between its result and that obtained from the finite element program ANSYS. Noting that the shaft excavation can be regarded as relaxation of initial ground stresses around shaft excavation boundary, this paper presents the mechanical problem as two separate problems: (i) initial stresses and displacements model where stresses keep constant and displacements equal zero around frozen wall and ground, (j) stresses relaxation model with relaxed normal and shear stresses acted on excavated boundary. The solution is worked out by the principle of superposition in elastic medium. The Young's modulus ratio (β) of the frozen wall to the ground reflects the influence of ground to frozen wall, and the influence decreases with the growth of β. The proposed solution can provide analytical basis for frozen wall design and be used as a quick-solver for back-analysis of in situ stresses and displacements.