Dynamics of rocking podium structures

A rocking podium structure is a class of structures consisting of a superstructure placed on top of a rigid slab supported by free‐standing columns. The free‐standing columns respond to sufficiently strong ground motion excitation by uplifting and rocking. Uplift works as a mechanical fuse that limits the forces transmitted to the superstructure, while rocking enables large lateral displacements. Such ‘soft‐story’ system runs counter to the modern seismic design philosophy but has been used to construct several hundred buildings in countries of the former USSR following Polyakov's rule‐of‐thumb guidelines: (i) that the superstructure behave as a rigid body and (ii) that the maximum lateral displacement of the rocking podium frame be estimated using elastic earthquake displacement response spectra. The objectives of this paper are to present a dynamic model for analysis of the in‐plane seismic response of rocking podium structures and to investigate if Polyakov's rule‐of‐thumb guidelines are adequate for the design of such structures. Examination of the rocking podium structure response to analytical pulse and recorded ground motion excitations shows that the rocking podium structures are stable and that Polyakov's rule‐of‐thumb guidelines produce generally conservative designs. Copyright © 2017 John Wiley & Sons, Ltd.     

A finite element model for seismic response analysis of deformable rocking frames

A new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero‐length fiber cross‐section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti‐symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

Rocking bodies with arbitrary interface defects: Analytical development and experimental verification

The rocking response of a rigid, freestanding block in two dimensions typically assumes perfect contact at the base of the block with instantaneous impacts at two distinct, symmetric rocking points. This paper extends the classical two‐dimensional rocking model to account for an arbitrary number of rocking points at the base representing geometric interface defects. The equations of motion of this modified rocking system are derived and presented in general terms. Energy dissipation is modeled assuming instantaneous point impacts, yielding a discrete angular velocity adjustment. Whereas this factor is always less than unity in the classical model, it is possible for this factor to exceed unity in the presented model, yielding a finite increase in the angular velocity at impact and a markedly different rotational response than the classical model predicts. The derived model and the classical model are numerically integrated and compared to the results of recent shake table tests. These comparisons show that the new model significantly enhances agreement in both peak angular displacement and motion decay. The equations of motion and the energy dissipation of the presented model are further investigated parametrically considering the size of the defect, the number of rocking points, and the aspect ratio and size of the block.     

Rolling and rocking of rigid uplifting structures

Allowing structures to uplift modifies their seismic response; uplifting works as a mechanical fuse and limits the forces transmitted to the superstructure. However, engineers are generally reluctant to construct an unanchored structure because the system could overturn due to lacking redundancy. Using a safety factor for the design of a flat rocking foundation, ie, designing it wider, goes against the main idea of this seismic modification method as the force demand for the structure increases. We propose to extend the flat base of a rocking block with curved extensions to better protect the block from overturning, yet not prevent its uplifting. After investigating the seismic response of such rocking blocks, we extend the study to investigate the seismic response of rolling and rocking frames comprising columns with curved base extensions. The equations of motion are derived, time history analyses are performed, and rocking spectra are constructed. We draw two important conclusions: (a) the response of a class of rocking oscillators with curved base extensions is equivalent to the response of a flat-base rocking oscillators of the same slenderness, yet larger size; (b) the rotation demand on two negative stiffness rocking and rolling oscillators with the same uplifting acceleration and the same size is roughly the same as long as the rocking oscillators are not close to overturning. The above findings can serve as a basis for the rational seismic design of structures supported on rocking columns with curved bases, a system that has been used since the 1960s.     

Seismic response of rocking frames with top support eccentricity

The seismic response of rocking frames that consist of a rigid beam freely supported on rigid freestanding rectangular piers has received recent attention in the literature. Past studies have investigated the special case where, upon planar rocking motion, the beam maintains contact with the piers at their extreme edges. However, in many real scenarios, the beam‐to‐pier contact lies closer to the center of the pier, affecting the overall stability of the system. This paper investigates the seismic response of rocking frames under the more general case which allows the contact edge to reside anywhere in‐between the center of the pier and its extreme edge. The study introduces a rocking block model that is dynamically equivalent to a rocking frame with vertically symmetric piers of any geometry. The impact of top eccentricity (ie, the distance of the contact edge from the pier's vertical axis of symmetry) on the seismic response of rocking frames is investigated under pulse excitations and earthquake records. It is concluded that the stability of a top‐heavy rocking frame is highly influenced by the top eccentricity. For instance, a rocking frame with contacts at the extreme edges of the piers can be more seismically stable than a solitary block that is identical to one of the frame's piers, while a rocking frame with contacts closer to the centers of the piers can be less stable. The concept of critical eccentricity is introduced, beyond which the coefficient of restitution contributes to a greater reduction in the response of a frame than of a solitary pier.     

Dynamic response analysis of solitary flexible rocking bodies: modeling and behavior under pulse‐like ground excitation

Results obtained for rigid structures suggest that rocking can be used as seismic response modification strategy. However, actual structures are not rigid: structural elements where rocking is expected to occur are often slender and flexible. Modeling of the rocking motion and impact of flexible bodies is a challenging task. A non‐linear elastic viscously damped zero‐length spring rocking model, directly usable in conventional finite element software, is presented in this paper. The flexible rocking body is modeled using a conventional beam‐column element with distributed masses. This model is verified by comparing its pulse excitation response to the corresponding analytical solution and validated by overturning analysis of rocking blocks subjected to a recorded ground motion excitation. The rigid rocking block model provides a good approximation of the seismic response of solitary flexible columns designed to uplift when excited by pulse‐like ground motions. Guidance for development of rocking column models in ordinary finite element software is provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Is rocking motion predictable?

An argument of engineers and researchers against the use of rocking as a seismic response modification technique is that the rocking motion of a structure is chaotic and the existing models are incapable of predicting it well. This argument is supported by the documented inability of rocking models to predict the motion of a specimen excited by a single ground motion. A statistical comparison of the experimental and the numerical responses of a rigid rocking oscillator not to a specific ground motion, but to ensembles of ground motions that have the same statistical properties, is presented. It is shown that the simple analytical model proposed by Housner in 1963 is capable of predicting the statistics of seismic response of a rigid rocking oscillator.     

Dynamics of a sliding-rocking block considering impact with an adjacent wall

A freestanding rigid block subjected to base excitation can exhibit complicated motion described by five response modes: rest, pure rocking, pure sliding, combined sliding-rocking, and free flight. Previous studies on the dynamics of a rocking block have assumed that the block does not interact with neighboring objects. However, there are many applications in which the block may start or come in contact with an adjacent boundary during its motion, for example, a bookcase or cabinet colliding with a partition wall in an earthquake. This paper investigates the dynamics of a sliding-rocking block considering impact with an adjacent wall. A model is developed in which the base and wall are assumed rigid, and impact is treated using the classical impulse and momentum principle. The model is verified by comparing its predictions in numerical simulations against those of an existing general-purpose rigid-body model in which impact is treated using a viscoelastic impact model. The developed model is used to investigate the effects of different parameters on the stability of a block subjected to analytical pulse excitations. It is found that wall placement (left or right) has a dominant effect on the shape of the overturning acceleration spectra for pulse excitations. In general, decreasing the gap distance, base friction coefficient, and wall coefficient of restitution enhance the stability of the block. Similar observations are made when evaluating the overturning probability of a block using earthquake floor motions.     

Response analysis of rigid structures rocking on viscoelastic foundation

In this paper the rocking response of slender/rigid structures stepping on a viscoelastic foundation is revisited. The study examines in depth the motion of the system with a non‐linear analysis that complements the linear analysis presented in the past by other investigators. The non‐linear formulation combines the fully non‐linear equations of motion together with the impulse‐momentum equations during impacts. The study shows that the response of the rocking block depends on the size, shape and slenderness of the block, the stiffness and damping of the foundation and the energy loss during impact. The effect of the stiffness and damping of the foundation system along with the influence of the coefficient of restitution during impact is presented in rocking spectra in which the peak values of the response are compared with those of the rigid block rocking on a monolithic base. Various trends of the response are identified. For instance, less slender and smaller blocks have a tendency to separate easier, whereas the smaller the angle of slenderness, the less sensitive the response to the flexibility, damping and coefficient of restitution of the foundation. Copyright © 2008 John Wiley & Sons, Ltd.     

Seismic control of rocking structures via external resonators

Tall rigid blocks are prevalent in ancient historical constructions. Such structures are prone to rocking behaviour under strong ground motion, which is recognizably challenging to predict and mitigate. Our study is motivated by the need to provide innovative nonintrusive solutions to attenuate the rocking response of historical buildings and monuments. In this paper, we examine a novel scheme that employs external resonators buried next to the rocking structure as a means to control its seismic response. The strategy capitalizes on the vibration absorbing potential of the structure-soil-resonator interaction. Furthermore, the benefits of combining the resonators with inerters in order to reduce their gravitational mass without hampering their motion-control capabilities are also explored. Advanced numerical analyses of discrete models under coherent acceleration pulses with rocking bodies of different slenderness ratios under various ground motion intensities highlight the significant vibration absorbing qualities of the external resonating system. The influence of key system parameters such as the mass, stiffness, and damping of the resonator and those of the soil-structure-resonator arrangement are studied. Finally, a case study on the evaluation of the response of rocking structures with external resonators under real pulse-like ground-motion records confirms the important reductions in peak seismic rotational demands obtained with the proposed arrangement.     

An integrated macroelement formulation for the dynamic response of inelastic deformable rocking bodies

This paper presents a macroelement formulation for the prediction of the planar dynamic response of inelastic deformable rocking bodies. The formulation is based on a previous macroelement developed by the authors able to describe the cyclic response of inelastic rocking bodies, which takes into account the deformability both along the height of the member, as well as near the rocking end. Modifications of this formulation to account for other motion modes of rocking members during their dynamic response, namely, sliding and upthrow, as well as modifications to account for damping in a uniform manner during the whole motion, including impacts, are introduced. The dynamic response predicted by the macroelement for free-standing rigid and deformable rocking bodies is presented and compared with existing theoretical solutions, and the effect of deformability, damping, inelasticity, and friction on the response is discussed.     

Seismic response of a wobbling 3D frame

This paper investigates the 3D response of a slab supported by wobbling columns. The columns are not allowed neither to slide nor to roll out of their initial position. An analytical model is proposed, the equations of motion are derived, and they are solved numerically. The paper concludes that the addition of the slab makes the columns more stable. In fact, the system is almost equivalent to the response of a solitary column with the same aspect ratio yet larger size. However, it is also shown that the system is less stable than its planar counterpart and that planar analysis can only qualitatively describe the behavior of 3D structures. A case study shows that the concept could be used as a seismic isolation technique for bridges. However, more research need to be performed on defining proper intensity measures for uplifting structures, as it is shown that there is large record‐to‐record variability, even when intensity measures developed for rocking structures are used.     

Rocking amplification and strong‐motion duration

This paper characterizes the ability of natural ground motions to induce rocking demands on rigid structures. In particular, focusing on rocking blocks of different size and slenderness subjected to a large number of historic earthquake records, the study unveils the predominant importance of the strong‐motion duration to rocking amplification (ie, peak rocking response without overturning). It proposes original dimensionless intensity measures (IMs), which capture the total duration (or total impulse accordingly) of the time intervals during which the ground motion is capable of triggering rocking motion. The results show that the proposed duration‐based IMs outperform all other examined (intensity, frequency, duration, and/or energy‐based) scalar IMs in terms of both “efficiency” and “sufficiency.” Further, the pertinent probabilistic seismic demand models offer a prediction of the peak rocking demand, which is adequately “universal” and of satisfactory accuracy. Lastly, the analysis shows that an IM that “efficiently” captures rocking amplification is not necessarily an “efficient” IM for predicting rocking overturning, which is dominated by the velocity characteristics (eg, peak velocity) of the ground motion.     

Seismic retrofitting of braced frame buildings by RC rocking walls and viscous dampers

A design procedure for seismic retrofitting of concentrically and eccentrically braced frame buildings is proposed and validated in this paper. Rocking walls are added to the existing system to ensure an almost uniform distribution of the interstorey displacement in elevation. To achieve direct and efficient control over the seismic performance, the design procedure is founded on the displacement‐based approach and makes use of overdamped elastic response spectra. The top displacement capacity of the building is evaluated based on a rigid lateral deformed configuration of the structure and on the ductility capacity of the dissipative members of the braced frames. The equivalent viscous damping ratio of the braced structure with rocking walls is calculated based on semi‐empirical relationships specifically calibrated in this paper for concentrically and eccentrically braced frames. If the equivalent viscous damping ratio of the structure is lower than the required equivalent viscous damping ratio, viscous dampers are added and arranged between the rocking walls and adjacent reaction columns. The design internal forces of the rocking walls are evaluated considering the contributions of more than one mode of vibration. The proposed design procedure is applied to a large set of archetype braced frame buildings and its effectiveness verified by nonlinear dynamic analysis.     

Seismic performance of an existing bridge with scoured caisson foundation

A three-dimensional rigid body on the shape of a parallelepiped is modelled in order to rock on a side or a vertex of the base,in order to evaluate the seismic response of rigid blocks lying on a horizontal support.The center of mass of the body is considered as eccentric with respect to its geometric center.As seismic input,three Italian recorded accelerograms,with different spectral content,are used.The study is mainly conducted to highlight the differences between the seismic response of 2D and 3D models of rigid blocks,with the aim to understand if,in some cases,the use of the 3D model of rigid block is required to obtain safer results.In fact,the outcomes show that in some ranges of the geometrical and mechanical parameters that characterize the excitation and the body,a two-dimensional model,which is not able to consider the 3D rocking on a vertex,can provide unsafe results.In particular,it is found that the overturning process of the three-dimensional block can occur under excitations which are lower than those which overturn a corresponding two-dimensional block.     

Shake table tests of stiff,unattached, asymmetric structures

Stiff, unattached structures are highly vulnerable to damage and failure during an earthquake, as evidenced following numerous past events. This class of structures encompasses a wide range of objects and systems such as electrical transformers, radiation shields, office furniture, and marble statues. The vulnerability of these objects is exacerbated when it is highly asymmetric and unattached. Although a number of studies have focused on rigid blocks, few have concentrated on blocks with asymmetric geometries. In an effort to better understand the implications of asymmetries, an extensive shake table testing campaign including more than 150 tests was conducted. These tests incorporate a systematic variation of the mass eccentricities of stiff, unattached structures. The primary modes of rocking, sliding, and twisting as well as interactive modes were recorded for the duration of numerous earthquake motions. The magnitude and direction of response are experimentally correlated with the geometric variations in the various models. These tests indicate that even for symmetric structures with uniaxial shaking, multiple modes and three‐dimensional responses are probable. Furthermore, certain asymmetric geometries exhibited both increased rocking (and overturning) as well as increased sliding when compared with their symmetric counterparts. A final aspect of this study compared the free rocking response of symmetric and asymmetric structures to classical, two‐dimensional rocking analysis. While the theoretical values for the coefficient of restitution yielded a significant overestimation in the simulation (up to ≈90%), reduced coefficients greatly improved the performance of the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical modelling of multi‐mass flexible rocking structures

This paper presents a new analytical model for describing the large rocking response of an elastic multi‐mass structure resting on ideally rigid ground. Using the experimental results from a rocking steel column, the ability of the proposed analytical model to estimate the rocking and translational acceleration response under free vibration, pulse and earthquake excitations is evaluated. It is observed that the classical treatment of impact may result in an unrealistically large transfer of energy to vibrations. Therefore a new Dirac‐delta type impact model that spreads the effects of impact over time and space is proposed. The use of a Dirac‐delta model and accurate restitution factors play a pivotal role in prediction of rocking and acceleration responses. In order to characterize the nonlinear response better, a modal analysis of the linearized system is proposed. With this approach, the vibration mode frequencies and shapes during rocking action were determined. A comparison of analytical and experimental modal estimations suggests good agreement. The results emphasize that the vibration characteristics of several vibration modes are affected by rocking action, and these modes may be excited at impact. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling of rocking frames under seismic loading

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.     

The interaction of elasticity and rocking in flexible structures allowed to uplift

Numerous structures uplift under the influence of strong ground motion. Although many researchers have investigated the effects of base uplift on very stiff (ideally rigid) structures, the rocking response of flexible structures has received less attention. Related practical analysis methods treat these structures with simplified ‘equivalent’ oscillators without directly addressing the interaction between elasticity and rocking. This paper addresses the fundamental dynamics of flexible rocking structures. The nonlinear equations of motion, derived using a Lagrangian formulation for large rotations, are presented for an idealized structural model. Particular attention is devoted to the transition between successive phases; a physically consistent classical impact framework is utilized alongside an energy approach. The fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures. In particular, parametric analysis is performed to quantify the effect of elasticity on uplift, overturning instability, and harmonic response, from which an uplifted resonance emerges. The contribution of stability and strength to the collapse of flexible rocking structures is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Seismic fragilities of single-column highway bridges with rocking column-footing

Rocking isolation has been increasingly studied as a promising design concept to limit the earthquake damage of civil structures. Despite the difficulties and uncertainties of predicting the rocking response under individual earthquake excitations (due to negative rotational stiffness and complex impact energy loss), in a statistical sense, the seismic performance of rocking structures has been shown to be generally consistent with the experimental outcomes. To this end, this study assesses, in a probabilistic manner, the effectiveness of using rocking isolation as a retrofit strategy for single-column concrete box-girder highway bridges in California. Under earthquake excitation, the rocking bridge could experience multi-class responses (eg, full contacted or uplifting foundation) and multi-mode damage (eg, overturning, uplift impact, and column nonlinearity). A multi-step machine learning framework is developed to estimate the damage probability associated with each damage scenario. The framework consists of the dimensionally consistent generalized linear model for regression of seismic demand, the logistic regression for classification of distinct response classes, and the stepwise regression for feature selection of significant ground motion and structural parameters. Fragility curves are derived to predict the response class probabilities of rocking uplift and overturning, and the conditional damage probabilities such as column vibrational damage and rocking uplift impact damage. The fragility estimates of rocking bridges are compared with those for as-built bridges, indicating that rocking isolation is capable of reducing column damage potential. Additionally, there exists an optimal slenderness angle range that enables the studied bridges to experience much lower overturning tendencies and significantly reduced column damage probabilities at the same time.