Modeling of rocking frames under seismic loading

Abstract

A novel modeling approach for the seismic response assessment of rocking frames is presented. Rocking frames are systems with columns that are allowed to fully, or partially, uplift. Despite the apparent lack of a mechanism to resist lateral forces, they have a remarkable capacity against earthquake loading. Rocking frames are found in old structures, for example, ancient monuments, but it is also a promising design concept for modern structures such as bridges or buildings. The proposed modeling can be implemented in a general‐purpose structural analysis software, avoiding the difficulties that come with the need of formulating and solving specifically tailored differential equations, or the use of detailed computational models. Different configurations of a rocking portal frame problem are examined. The model is based on rigid, or flexible, beam elements that describe the members of the frame. Negative‐stiffness rotational springs are smartly positioned at the rocking interfaces in order to simulate the rocking restoring moment, while the mass and the rotational moment of inertia are considered either lumped or distributed. Both the cases of rigid and flexible piers/columns are discussed, while it is shown that frames with restrained columns can be considered in a straightforward manner. A simple alternative based on an equivalent oscillator that follows the generalized rocking equation of motion is also investigated. The efficiency and the accuracy of the proposed modeling is demonstrated with the aid of carefully chosen case studies.

Seismic pounding damage to adjacent reinforced concrete frame–shear wall buildings and freestanding contents

“…The “lollipop” model is identical to the spring model “SM2” proposed by Diamantopoulos and Fragiadakis 40 . The advantage of using the model is its ability to be extended to flexible rocking structures 40,41 . For the rigid rocking block problem under the current investigation, an even simpler model was adopted, which is also sketched in Figure 4(B), consisting of a single rotational mass $$I$$ mounted on the same rocking spring as was defined for the “lollipop” model.…”

“…Using the conservation of angular momentum about the impact corner, the coefficient of restitution can be determined as $$e_{r}=1-1.5\sin ^{2}{(\alpha )}$$. An “event‐based” scheme has been proposed to implement $$e_{r}$$ in OpenSees, 40,41 which required specially designed code (e.g., using Matlab) to detect impact (i.e., the change of sign of angular displacement). To simplify analyses, the impact damping was accounted for in the present study using a linear dashpot placed in parallel with the rocking spring, and the damping coefficient formula proposed by Vassiliou et al 42 .…”

“…40 The advantage of using the model is its ability to be extended to flexible rocking structures. 40,41 For the rigid rocking block problem under the current investigation, an even simpler model was adopted, which is also sketched in Figure 4(B), consisting of a single rotational mass 𝐼 mounted on the same rocking spring as was defined for the "lollipop" model. The equation of motion of the rotational oscillator is simply:…”

“…based" scheme has been proposed to implement 𝑒 𝑟 in OpenSees, 40,41 which required specially designed code (e.g., using Matlab) to detect impact (i.e., the change of sign of angular displacement). To simplify analyses, the impact damping was accounted for in the present study using a linear dashpot placed in parallel with the rocking spring, and the damping coefficient formula proposed by Vassiliou et al 42 was adopted:…”

Analysis of seismically‐isolated two‐block systems using a multi–rocking‐body dynamic model

“…(2018), and Diamantopoulos et al. (2022). The authors draw the conclusion that the formulation of this extremely nonlinear analytical model is challenging, due in part to highly nonlinear equations of motion, impacts, and continually changing contacts (Bao & Konstantinidis, 2020).…”