Analytical solution for seismic response of vertical shaft with primary and secondary linings under SH wave excitation

Abstract

This paper presents a novel mechanical model for seismic response analysis of vertical shafts. The model is based on elastic foundation beam theory and simplifies the shaft linings as parallel Euler Bernoulli beams. Their interaction is simulated using uniformly distributed springs. The governing differential equation for shaft vibration is derived for SH wave excitation. The seismic input motion is applied to the primary lining by converting the free field displacement into harmonic form. The closed-form solutions for seismic response of both primary and secondary linings are obtained via the transfer function method. The accuracy of analytical solution is validated by comparing with finite element analysis results. Additionally, the impact of foundation stiffness, secondary lining thickness, shaft outer diameter, and relative stiffness between linings on seismic response are investigated in the parametric analysis. The analytical solutions of the seismic response analysis of vertical shafts indicates that the relative stiffness between linings significantly affects the secondary lining's peak bending moment. Furthermore, higher stiffness of the elastic connection layer between the two linings can effectively reduce the peak displacement response of the secondary lining.