Masonry bridges are vulnerable structural systems to the ground motion excitation that their survival in case of such incidents has to be studied in detail. In this work, a simplified model for dynamic analysis of masonry arch bridges based on rocking motion of rigid blocks is proposed. Using this model, nonlinear time integration analyses on these bridges can be done with ease and in a short time. Later, acceptance criteria for three cases of un-cracked, fullyoperational and collapse-prevention pier sections are developed for such bridges. The accuracy of proposed model in representing the behavior of a rocking system has been verified using the results of experimental studies on rocking motion of a masonry-concrete block reported elsewhere. The results show the suitability of the proposed model in representing rocking motion of rigid blocks. In a case study, the proposed model for masonry arch bridges was used in evaluation of seismic performances of a monumental masonry bridge subjected to both horizontal and vertical seismic actions. The study shows the importance of vertical component of ground motion in determination of internal forces and shear-sliding deformation at the bottom of the bridge’ s pier. The proposed model has also shown its ability in defining the effectiveness of a seismic retrofit approach for the same bridge system in a comparative study. According to this investigation, seismic performances of the bridge can be significantly improved in case of adding ductility to its deck assembly. To understand the capacity of bridge system in dealing with earthquake demands, a series of Incremental Dynamic Analyses (IDA) have been carried out on the rocking-pier model of the bridge system using earthquake records. Considering the simplicity of rocking pier model, all the analyses on above-mentioned bridge system have been carried out with ease and in a very short time. According to results, a bridge system subjected to bidirectional seismic actions (vertical and horizontal) has, unexpectedly, more capacity in dealing with seismic demands if it is compared with the same bridge system with unidirectional horizontal seismic excitation. Conversely, the sliding breakdown of the pier in case of bidirectional seismic actions is much higher than that in the case of unidirectional one. Moreover, significant reductions in the level of rotational pitch and shear sliding at rocking joint of the pier is expected in case of adding ductility to the deck of the bridge assembly. As it was expected, ductility in the bridge system also decreases the discrepancy of bridge responses with respect to different earthquake actions, which is attributed to the systems with higher energy dissipation potential.

A growing body of evidence suggests that the possibility of nonlinear rocking oscillations can protect the structure from serious damage. One of the potential concerns about this design approach is the magnitude of residual settlement and maximum rotation. Several improvement techniques have been proposed to ameliorate the nonlinear behavior of rocking foundations based on the vertical factor of safety. Rocking systems with a large factor of safety against vertical load are more prone to toppling collapse under severe ground motions. This research explored the effectiveness of using soft walls next to a rocking foundation for mitigating seismic risk. The vital advantage of this improvement technique is that it is a feasible strategy for both new construction and existing structures. The soil-foundation-structure system has been analyzed using the numerical finite element (FE) method that takes the material and geometric nonlinearities into account. In this case, the nonlinear response mainly involves the interaction between footing uplift and soil failure, which may induce additional (gravitational) aggravating moment (P-Δ effect). In the first step, a three-dimensional (3D) numerical model has been constructed for the rocking foundation-soil system experiment. In order to verify the accuracy of the simulation, the numerical modeling and the experimental test results have been compared. The results of the centrifuge physical testing conducted were used to validate the numerical simulation. All FE analyses were performed in Abaqus software. This software has been utilized by a number of researchers to study complex soil-foundation–structure interaction phenomena. Because the initial conditions play an important role in simulating geotechnical problems, a staged analysis procedure has been adopted. In the dynamic analysis stage, an incremental-iterative procedure was used to integrate the equations of motion. The Hilber-Hughes-Taylor algorithm was used to conduct the transient analysis phase. The modified Newton–Raphson method was employed to decrease the calculation cost needed for the great number of degrees of freedom of the model. Finely refined 3D FE mesh was used to precisely reproduce the mechanism of bearing capacity failure and the rocking behavior. The soil medium and foundation were discretized into eight-node hexahedral continuum elements (C3D8 element type). Two-node linear beam elements were used to model the superstructure (B31 element type). A special surface-to-surface contact formulation between the foundation and soil was used for the realistic simulation of possible uplift and sliding of the foundation. Surface-to-surface contact can calculate contact stresses accurately by reducing the possibility of large-localized penetration of the two surfaces. The properties of the contact element were defined by the interface stiffness in the normal and the tangential directions. Nonlinear soil behavior was modeled using a kinematic hardening model with the Von Mises failure criterion and associated plastic flow rule. This simplified constitutive model is applicable for the prediction of the undrained behavior of clay as normal pressure independent. In contrast to soil, the behavior of the structure-foundation model is assumed to be linear elastic. A parametric study was carried out to explore the sensitivity of the geometric design variables of the diaphragm wall, such as height and thickness on the behavior of the isolated foundation-soil system. It should be mentioned that the values of model parameters were determined based on their practical feasibility. The distance of walls has been determined far from the edge of the foundation to avoid static bearing capacity failure. The results showed that the placement of vertical soft walls next to the foundation could limit the transmitted forces onto the superstructure. This could be lessened the maximum motion of the structure and the following overturning moment of the foundation. Hence, adequate safety margins against toppling collapse could be easily achieved under strong motions.

After an earthquake, plastic deformation of structural elements can render repairing uneconomical, thus necessitating the adoption of low-damage systems that mitigate the residual drift of structures. Among these, self-centering rocking-core systems have been extensively explored. However, due to the geometric nonlinearity of such systems, higher modes dominate their seismic responses, which necessitates the incorporation of secondary rocking joints to minimize their effects. Nevertheless, identifying the optimal location of such joints is challenging, given the redistribution of internal forces between the rocking plates. In this context, a bi-rocking steel braced frame is designed using the modified modal superposition method (MMS), accounting for higher mode effects. Subsequently, the secondary joint is moved floor by floor to determine its optimal location that minimizes shear, overturning moment, peak floor acceleration, and drift. To represent damage to non-structural components, peak floor acceleration and drift are chosen as key parameters. Three sets of seven ground motions, namely Far-Field (FF), Near-Field-Pulse (NF-Pulse), and Near-Field-no-Pulse (NF-No Pulse), are considered for frames of 12, 18, and 24 stories, modeled using OpenSees software in 2-dimensional frameworks. A total of 1071 non-linear time-history analyses are carried out, and the results indicate that the conventional practice of placing the secondary joint in the mid-height is inadequate for the 12-story frame under NF-Pulse records, causing a 15.1% deviation from the optimal state of overturning moment. In most cases, placing the joint at 40% height reduces all four demands. To evaluate demand sensitivity, the standard deviation of their percentage difference with the optimal state is computed, with higher values indicating greater unpredictability. Among the sets of records, FF and among demands, overturning moment exhibit the highest sensitivity to the location of the secondary joint, with changes in overturning moment being correlated with shear. Therefore, we suggest selecting overturning moment as an optimization objective function.

The research leading to this paper was prompted by the need to estimate strength and stiffness of Rigid Rocking Cores (RRCs) as essential elements of resilient earthquake resisting structures. While a limited number of such studies have been reported, no general study in terms of physical properties of RRCs, their appendages and adjoining structures have been published. Despite the growing knowledge on RRCs there are no design guidelines on their applications for seismic protection of buildings. The purpose of the present article is to propose effective rigidity limits beyond which it would be unproductive to use stiffer cores and to provide basic guidelines for the preliminary design of RRCs with a view to collapse prevention, re-centering and post-earthquake repairs/replacements. Several examples supported by computer analysis have been provided to demonstrate the applications and the validity of the proposed solutions.

This paper presents a novel rocking system for bridge piers that aims to minimize structural damage and reduce residual drift during seismic events. This system relies on gravity for re-centering and does not utilize post-tensioned tendons. It consists of a pair of tubular steel columns attached to the foundation and cap beam with rocking connections. Each connection utilizes a friction damper that absorbs energy through friction at low lateral displacement and plastic deformation at large lateral displacement. Finite element (FE) simulation was utilized to evaluate the behavior of this system under vertical and lateral loading. The finite element models under two different gravity loads were subjected to lateral cyclic loading to study the re-centering characteristics of the system. The analyses have demonstrated that the system exhibits a flag-shaped hysteresis response with minimal residual drift. A supplementary analysis has also shown that any residual drift in this system can be recovered by loosening bolts in the frictional connection.

In recent years, considering vertical movement in structures, which is known as self-centering motion, is one of the methods used to dissipate the earthquake energy. In this study, 4-, 8-, and 12-Story steel structures with special concentrically braced frame (SCBFs) and eccentrically braced frames (SEBFs) were considered in order to investigate the effect of selfcentering motion under seven near-field earthquakes using Sap 2000 software. The results of this research indicate that considering the self-centering motion, the behavior of structures are changing in SCBFs and SEBFs. In, 4-, 8-, and 12-Story SEBFs structures with considering the self-centering motion, the column and bracing forces were increased which this was different with SCBFs structures. Considering the self-centering motion, more plastic joints are formed in the beams. Therefore, in the self-centering of the columns, the beams outside the bay are more affected. In general, the results indicate that the self-centering motion is better in the SCBFs.