Analysis of Tall Stepping Viaduct Piers under Seismic Excitation

This paper presents an approximate solution for the peak quasi-dynamic displacement response of stepping flexible frames subjected to earthquake ground motions. The conventional iterative method available in codes of practice is not guaranteed to converge, necessitating an alternative approach in design. We present stability criteria for the displacement response of stepping frames in this context. Stepping response, first implemented in a tall railway viaduct in New Zealand, is particularly advantageous in seismic regions. It reduces significant design force demands at the base of piers, eliminating the requirement for heavy footings or deep foundation elements. In return, substantial reduction in construction costs and seismic damage is realized, as a consequence of period elongation in the stepping response similar to displacement-dependent period of vibration in an inverted pendulum. Another benefit is that stepping structure aligns with the design philosophy of fast recovery after exceptional loads, termed “resiliency,” because it mitigates permanent offset, the cumulative post-elastic cyclic deformations, that correspond with the increased repair costs and reduced functionality. We further study the bifurcation of displacements in the response of flexible stepping portal frames through the extension of the dynamics of stepping rigid bodies and present an approximate analytical expression to account for the effects of moving resonance under earthquake ground motions. The closed-form solution for displacement demand can be readily adjusted to incorporate the influence of moving resonance on the quasi-dynamic response of stepping oscillators. The quasi-dynamic method of analysis is useful in the early stages of design. We discuss implications for the formal limit-state analysis of stepping response with several examples demonstrating the procedures.