Dynamic Analysis of Flexible Stepping Frames for Earthquakes

TL;DR

This paper develops analytical solutions and stability criteria for the nonlinear seismic response of flexible stepping frames, addressing convergence issues in traditional iterative methods and exploring resonance effects during earthquakes.

Contribution

It introduces closed-form solutions and stability criteria for stepping flexible frames under seismic excitation, extending dynamics analysis to include moving resonance effects.

Findings

Closed-form solutions for displacement response

Stability criteria for nonlinear seismic response

Implications for limit-state analysis of stepping frames

Abstract

This paper investigates the nonlinear dynamics of stepping flexible frames under seismic excitation. The conventional iterative method of solution of peak quasi-dynamic displacement of stepping frames is not guaranteed to converge. To address this limitation, we present closed-form solutions and stability criteria for displacement response of stepping flexible frames. Bifurcation of displacements in response of such systems is next studied through the extension of dynamics of stepping rigid bodies. An approximate analytical expression is presented to account for the effects of moving resonance under earthquake ground motions. The closed-form solutions for displacement demand can be readily adjusted to incorporate the influence of moving resonance on the quasi-dynamic response of stepping oscillators. While the quasi-dynamic method of analysis may be useful in the early stages of design, numerical integration of the nonlinear system of differential equations of motion is recommended for the solution of dynamic response in such applications. Implications for formal limit-state analysis of stepping response are discussed, accompanied by several examples demonstrating the procedures.