Earthquake Response Analysis of Yielding Structures Coupled with Rocking Walls

In an effort to eliminate the appreciable seismic damage in moment-resisting frames that occasionally resulted to a weak-story failure, the concept of a rigid core system gained appreciable ground (Paulay 1969, Fintel 1975, Emori et al. 1978, Bertero 1980, Aktan et al. 1984). When the core walls in tall buildings are fixedbased, the ductility capacity of the base of the core wall may be limited given the significant axial loads; while, the ductility demands are appreciable under long-duration pulse motions (Paulay 1986, Zhang et al. 2000). Furthermore, the base of the core wall may suffer from cyclic degradation under prolonged shaking which usually results to permanent inelastic deformations. Such inelastic response may result to permanent drifts and lead to large repair costs; therefore, the entire design becomes unsustainable. During the last three decades, there has been a growing effort to direct the attention of engineers to the unique advantages associated with allowing major vertical structural elements (piers in bridges or shear wall in buildings) to uplift in an effort to intentionally mobilize a lower "failure" mechanism. In this way failures associated with cyclic degradation are essentially avoided; while, permanent displacements remain small due to the inherent recentering tendency of the rocking mechanism. For instance, as early as the PRESS Program (Priestley 1991(Priestley , 1996)), the jointed shear wall system was allowed to lift-off and rock (Nakaki et al. 1999, Priestley et al. 1999). About the same time Kurama et al. (1999Kurama et al. ( , 2002) ) examined the lateral load behavior of unboded segmented post-tensioned precast walls; while, Mander and Cheng (1997) introduced the damage avoidance design (DAD) in which the free-standing piers of a bridge frame are only vertically restrained through their center line and are allowed to rock atop the pile-cap and bellow the pier-cap beam without inducing any damage. Following these studies, Holden et al. (2003) presented experimental studies on the cyclic loading of a precast, partially prestressed system that incorporated post-tensioned unbonded tendons; while Ajrab et al. (2004) presented a performance-based design methodology for the frame-building-rocking-wall system with various prestressed tendon configurations and energy dissipation devices. In their proposed methodology Ajrab et al. (2004) adopt an "equivalent-static" lateral force procedure, and the study concludes that the proposed performance-based, capacity-demand method predicts larger displacements than those obtained from timehistory analysis.

In the aforementioned studies, central postensioned steel tendons inside the rocking wall or bridge-pier are provided to increase the lateral resistance of the entire structure. The force-deformation curve of the vertically restrained solitary rocking wall reported in these studies has invariably a positive post-uplifting stiffness, indicating that the axial stiffness of the steel tendon is large enough to the extent that the post-uplift stiffness of the rocking wall is positive. By introducing such a stiff tendon that reverses the negative stiffness of the solitary rocking wall, one creates a strong system; nevertheless, at present it is not well understood to what extent a stiff vertical tendon that offers a positive lateral stiffness enhances the seismic stability of the overall structure or it merely contributes to accentuate the crushing of the pivoting points of the rocking wall due to the increased vertical load. Part of the motivation of this study is to build upon the previously referenced work and examine the role of vertical restrainers in the planar seismic response of moment-resisting frames coupled with rocking walls.

With reference to Figure (1), this study examines the dynamic response of a yielding single-degree-of-freedom (SDOF) structure, with mass, ms, pre-yielding stiffness, k1 post yielding stiffness, k2, and strength, 𝑄, that is coupled with a free-standing stepping rocking wall of size, 𝑅 = √𝑏 + ℎ , slenderness, tan 𝛼 = 𝑏/ℎ, mass, mw and moment of inertia about the pivoting (stepping) points O and O’, 𝐼 = 4/3 𝑚 𝑅 , that is vertically restrained with an elastic tendon with axial stiffness EA which can be prestressed with a prestressing force Po.

In the interest of simplicity, it is assumed that the arm with length, L, that couples the motion is articulated at the center of mass of the rocking wall at a height, h, from its foundation as shown in Figure 1.

During rocking motion, the center of mass of the rocking wall uplifts by v; therefore, the initially horizontal coupling arm rotates by an angle ψ. Accordingly, the horizontal translation of the center of mass of the rotating wall, x, is related to the horizontal displacement of the SDOF oscillator, u, via the expression, cos 𝜓 = 1 -(𝑢 -𝑥)/𝐿; whereas, sin 𝜓 = 𝑣/𝐿. In this paper, the coupling arm is assumed to be long enough so that 𝑣 /𝐿 is much smaller that uni

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