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Dynamic Analysis of Flexible Stepping Frames for Earthquakes

Arzhang Alimoradi, James L. Beck

TL;DR

This work tackles nonlinear dynamics of stepping flexible frames under seismic excitation, addressing the unreliable convergence of traditional peak quasi-dynamic displacement iterations by deriving closed-form displacement solutions and stability criteria. It extends rocking analysis from rigid blocks to flexible portal frames and incorporates moving-resonance effects, formulating a quasi-dynamic framework based on equivalent linearization and nonlinear displacement maps. The authors derive two governing maps for short- and long-period regimes, analyze their fixed points and linear stability, and identify conditions under which iterative design procedures converge versus when direct numerical integration is warranted. The findings have practical implications for limit-state design of stepping structures, offering analytical tools for initial design checks while highlighting the necessity of full nonlinear time integration for accurate dynamic response estimation.

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.

Dynamic Analysis of Flexible Stepping Frames for Earthquakes

TL;DR

This work tackles nonlinear dynamics of stepping flexible frames under seismic excitation, addressing the unreliable convergence of traditional peak quasi-dynamic displacement iterations by deriving closed-form displacement solutions and stability criteria. It extends rocking analysis from rigid blocks to flexible portal frames and incorporates moving-resonance effects, formulating a quasi-dynamic framework based on equivalent linearization and nonlinear displacement maps. The authors derive two governing maps for short- and long-period regimes, analyze their fixed points and linear stability, and identify conditions under which iterative design procedures converge versus when direct numerical integration is warranted. The findings have practical implications for limit-state design of stepping structures, offering analytical tools for initial design checks while highlighting the necessity of full nonlinear time integration for accurate dynamic response estimation.

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.
Paper Structure (8 sections, 21 equations, 5 figures)

This paper contains 8 sections, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Evidence, Relevance, and Justification
  3. Background
  4. Quasi-dynamic Response
  5. Equivalent Linearization of a Single Degree of Freedom Oscillator
  6. Analytical Design Demand Solutions and Their Stability for Linearized Response
  7. The Iterated Maps of Design Displacement
  8. Fixed Points of Displacement and Their Linear Stability

Figures (5)

  • Figure 1.1: Evidence of rocking and sliding motion of a rigid wall inside Tunnel No. 3 during July 21, 1952 Kern County, California, earthquake Caltech-1.
  • Figure 1.2: Example of precarious rocks in Meadowcliff Canyon near the California-Nevada border (Image Courtesy of D. Trugman).
  • Figure 1.3: South Rangitikei Viaduct in New Zealand (image courtesy of Wikimedia Commons/D B W), bridge and pier elevations, and cross section of the deck S_Rangitikei_Const.
  • Figure 2.1: Rocking of a single column Priestley-2.
  • Figure 2.2: A multi-period design spectrum (solid curve) and its simplification (dashed curve).