Moment Resisting Frames Coupled with Rocking Walls Subjected to Earthquakes
Authors: Aghagholizadeh Mehrdad , Makris Nicos 
Published: June 2022

The high occupancy levels in urban multistory buildings, in association with current safety considerations inevitably leads to a reconsideration of performance objectives. In view of the appreciable seismic damage and several weak-story failures (some at mid-height) of multistory buildings that have been documented after major earthquakes, there has been a growing effort to develop an alternative hybrid structural system by coupling the response of moment resisting frames with rigid/stiff walls which are allowed to uplift and rock during ground shaking; therefore, enforcing a uniform drift distribution (Meek 1978; Ajrab et al. 2004; Lu 2005; Toranzo et al. 2009; Wada et al. 2011; Hu and Zhang 2012; Qu et al. 2012; Aghagholizadeh and Makris 2018a, 2018b). Part of the reason for this semi-articulated seismic design alternative is that on several occasions, the further strengthening of the building with fixed-based shear walls leads to the attraction of larger seismic forces and the entire approach reaches an impasse given that the resulting forces that develop cannot be accommodated by cost-effective foundations. Another major issue that is a concern in multistory buildings in which their earthquake performance relies on ductile behavior is that after severe shaking the multi-story building may end up with appreciable permanent displacements and there is a need for re-centering which in most cases leads to demolition (an outcome against the emerging trends of functional recovery) as happened after the 2011 Christchurch, New Zealand earthquake (Elwood 2013). In this report we first investigate the inelastic response of a yielding single-degree-of-freedom oscillator coupled with a rocking wall. Configurations of both a stepping rocking wall and a pinned rocking wall that have been reported in the literature are examined. The full nonlinear equations of motions are derived, and the report shows that a stepping wall suppresses peak and permanent displacements, with the heavier wall being most effective. In contrast, when the yielding oscillator is coupled with a pinned rocking wall, both peak and permanent displacements increase, with the heavier wall being most unfavorable. This unfavorable response is mainly because the moment from the weight of the pinned wall works against stability, and in most cases, it contributes to larger permanent displacements. Subsequently, the report investigates the inelastic response of a yielding structure coupled with a vertically restrained rocking wall. The nonlinear equations of motion are extended for of a yielding oscillator coupled with a vertically restrained rocking wall, and the dependability of the one‐degree of freedom idealization is validated against the nonlinear time‐history response analysis the nine‐story SAC steel frame that is coupled with a stepping vertically restrained rocking wall. The planar response analysis is conducted with the open-source software, OpenSees. While the coupling of weak building frames with rocking walls is an efficient strategy that controls inelastic deformations by enforcing a uniform inter-story‐drift distribution, therefore, avoiding mid‐story failures, our analysis shows that even for medium‐rise buildings the effect of vertical tendons on the inelastic structural response is marginal, with the exception of increasing the vertical reactions at the pivoting points of the rocking wall. Accordingly, our planar response analysis concludes that for medium‐rise to high‐rise buildings, vertical tendons in rocking walls are not beneficial. Given that the coupling of a moment-resisting building with a stiff rocking wall enforces a first-mode dominating response, our study proceeds by investigating the dynamic response of a yielding single-degree-of-freedom oscillator coupled to a stepping rocking wall with supplemental damping (either hysteretic or linear viscous) along its sides. The full nonlinear equations of motion are derived, and the study presents an earthquake response analysis in terms of inelastic spectra. The study shows that for structures with pre-yielding period T1 < .0 s, the effect of supplemental damping along the sides of the rocking wall is marginal even when large values of damping are used. The study uncovers that occasionally, the damped response matches or exceeds the undamped response; however, when this happens, the exceedance is marginal. The report concludes that for yielding structures with strength less than 10% of their weight, the use of supplemental damping along the sides of a rocking wall coupled to a yielding structure is not recommended. Our study concludes that supplemental damping along the sides of the rocking wall may have some limited beneficial effects for structures with longer pre-yielding periods (say T1 > 1.0 s). Nevertheless, no notable further response reduction is observed when larger values of hysteretic or viscous damping are used.