Analytical Solution for the Evaluation of Kinematic Demands on Underground Linear Structures Subjected to S-Waves

Abstract

Seismic shear waves that propagate along the axis of underground linear structures (e.g., tunnels, pipes, and piles) generally impose deformations that need to be accounted for in structural design or verification. Preliminary analyses are typically carried out following a free-field deformation approach based on conventional Euler–Bernoulli beam theory for bending behavior and a pure-shear beam model for the evaluation of distortions. This simplified approach, widely used in practice, may lead to inconsistent and overly conservative results. Solutions accounting for soil–structure interaction have also been proposed in the literature, considering an Euler–Bernoulli beam model supported on an elastic foundation subjected to simple harmonic shear waves. This approach, however, may not adequately represent cases where shear behavior dominates over bending, particularly in very stiff ground conditions. This paper presents a simplified analytical solution to evaluate kinematic demands in underground linear structures subjected to axially propagating harmonic S-waves by means of a Timoshenko beam model supported on a Winkler foundation, including transverse and rotational springs, which are in turn derived following a modified Novak approach. The proposed analytical solution can be construed as a generalization of the free-field and soil–structure interaction solutions currently available in the literature, which is also able to represent situations where both shear and bending mechanisms are present, overcoming the mechanical inconsistencies of the current solutions commonly used in practice. Based on the proposed model, simple analytical expressions are proposed, considering both a complete and a simplified solution, which allow the evaluation of peak kinematic demands in an expedited manner. The analytical formulation can thus be used to evaluate displacements, rotations, bending, and shearing demands in tunnels, piles, pipes, shafts, among other buried linear structures when subjected to harmonic S-waves. Results obtained by means of the proposed approach are validated against a full 3D finite-element model. The present paper presents a simplified analytical solution to evaluate kinematic demands in underground linear structures subjected to axially propagating S-waves by means of a Timoshenko beam model supported on a Winkler foundation, including transverse and rotational springs, which are in turn derived following a modified Novak approach. Based on the proposed model, simple closed-form solutions can be used to calculate displacements, rotations, bending, and shearing demands in tunnels, piles, pipes, shafts, among other buried linear structures subjected to harmonic S-waves. The analytical expressions proposed in this paper are validated against a full 3D finite-element model (FEM) and allow the evaluation of peak kinematic bending moments and shear forces in an expedited manner.