Displacement controlled rocking behaviour of rigid objects

The capacity of a gravity structure to counter seismically induced overturning can only be estimated with good accuracy using a dynamic analysis of the rotational (rocking) motion involving large displacement theory. Seismic assessment employing quasi‐static analysis can be overly conservative if the reserve capacity of this type of rocking structure to displace without overturning is not taken into account. It was revealed through dynamic testing on a shaking table that the overturning hazards of ground shaking are best represented by the peak displacement demand (PDD) parameter and that the vulnerability to overturning instability decreases with the increasing size of the object when the aspect ratio is held constant. This finding has important implications on the engineering of structures for countering moderate ground shaking in regions of low and moderate seismicity. Experimental data were validated and supplemented by computer simulations that involved generating artificial accelerograms of designated earthquake scenarios and non‐linear time‐history analyses of the overturning motions. Based on these simulations, fragility curves were constructed for estimating the probability of overturning for given levels of PDD and for different specimen dimensions. An expression was developed for estimating the level of PDD required to overturn rectangular objects of given dimensions for 5% probability of exceedance. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Seismic response of slender rigid structures with foundation uplifting

The rocking of rigid structures uplifting from their support under strong earthquake shaking is investigated. The structure is resting on the surface of either a rigid base or a linearly elastic continuum. A large-displacement approach is adopted to extract the governing equations of motion allowing for a rigorous calculation of the nonlinear response even under near-overturning conditions. Directivity-affected near-fault ground motions, idealized as Ricker wavelets or trigonometric pulses, are used as excitation. The conditions under which uplifting leads to large angles of rotation and eventually to overturning are investigated. A profoundly nonlinear rocking behavior is revealed for both rigid and elastic soil conditions. This geometrically nonlinear response is further amplified by unfavorable sequences of long-duration pulses in the excitation. Moreover, through the overturning response of a toppled tombstone, it is concluded that the practice of estimating ground accelerations from overturning observations is rather misleading and meaningless.     

Freestanding block overturning fragilities: Numerical simulation and experimental validation

The overturning fragilities of symmetric and asymmetric freestanding blocks, ranging in height from 0.54 to 3.6 m and with height‐to‐width ratios ranging from 2.1 to 6.6, are determined numerically. A probabilistic formulation regularizes the overturning responses when exposed to earthquake‐like random‐vibration waveforms. The peak amplitude of the forcing excitation (peak ground acceleration or PGA) is parameterized as a function of the block size, block shape, overturning probability, and either the PGA normalized peak ground velocity (PGV/PGA), spectral acceleration at 1 s (S_a(1)/PGA), or spectral acceleration at 2 s (S_a(2)/PGA). These later intensity measures are correlated with the duration of the predominant acceleration pulse. The overturning fragilities are compared with shake table experiments using blocks ranging in height from ～0.2 to 1.2 m and with height‐to‐width ratios ranging from ～2 to 10. Excitations utilized in the shake table experiments include recordings of the 1979 Imperial Valley, 1985 Michoacan, 1999 Duzce, 1999 Chi‐Chi, and 2002 Denali Earthquakes along with synthetic waveforms. The overturning fragilities accurately represent the overturning responses of blocks with simple basal contact conditions. Objects with multiple rocking points, such as precariously balanced rocks, are more fragile than predicted. Nondestructive tilting tests are used to account for blocks with complex basal contact conditions, demonstrating that these blocks overturn similarly to more slender blocks with simple contact conditions. Copyright © 2008 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.     

Neural network analysis of overturning response under near-fault type excitation

Under strong seismic excitation, a rigid block will uplift from its support and undergo rocking oscillations which may lead to （complete） overturning. Numerical and analytical solutions to this highly nonlinear vibration problem are first highlighted in the paper and then utilized to demonstrate how sensitive the overturning behavior is not only to the intensity and frequency content of the base motion, but also to thc presence of strong pulses, to their detailed sequence, and even to their asymnletry. Five idealised pulses capable of representing ＂rupture-directivity＂ and ＂fling＂ affected ground motions near the fault, are utilized to this end ： the one-cycle sinus, the one-cycle cosinus, the Ricker wavelet, the truncated （T）-Ricker wavelet, and the rectangular pulse ＂Overturning-Acceleration Amplification＂ and ＂Rotation＂ spectra are introduced and presented. Artificial neural network modeling is then developed as an alternative numerical solution. The neural network analysis leads to closed-form expressions for predicting the overturning failure or survival of a rigid block, as a function of its geometric properties and the characteristics of the excitation time history. The capability of the developed neural network modeling is validated through comparisons with the numerical solution. The derived analytical expressions could also serve as a tool for assessing the destructiveness of near-fault ground motions, for structures sensitive to rocking with foundation uplift.     

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 protection of rocking structures with inerters

The seismic behaviour of a wide variety of structures can be characterized by the rocking response of rigid blocks. Nevertheless, suitable seismic control strategies are presently limited and consist mostly on preventing rocking motion all together, which may induce undesirable stress concentrations and lead to impractical interventions. In this paper, we investigate the potential advantages of using supplemental rotational inertia to mitigate the effects of earthquakes on rocking structures. The newly proposed strategy employs inerters, which are mechanical devices that develop resisting forces proportional to the relative acceleration between their terminals and can be combined with a clutch to ensure their rotational inertia is only employed to oppose the motion. We demonstrate that the inclusion of the inerter effectively reduces the frequency parameter of the block, resulting in lower rotation seismic demands and enhanced stability due to the well-known size effects of the rocking behaviour. The effects of the inerter and inerter-clutch devices on the response scaling and similarity are also studied. An examination of their overturning fragility functions reveals that inerter-equipped structures experience reduced probabilities of overturning in comparison with uncontrolled bodies, while the addition of a clutch further improves their seismic stability. The concept advanced in this paper is particularly attractive for the protection of rocking bodies as it opens the possibility of nonlocally modifying the dynamic response of rocking structures without altering their geometry.     

The rocking spectrum and the limitations of practical design methodologies

This paper is concerned with the superficial similarities and fundamental differences between the oscillatory response of a single‐degree‐of‐freedom (SDOF) oscillator (regular pendulum) and the rocking response of a slender rigid block (inverted pendulum). The study examines the distinct characteristics of the rocking spectrum and compares the observed trends with those of the response spectrum. It is shown that the rocking spectrum reflects kinematic characteristics of the ground motions that are not identifiable by the response spectrum. The paper investigates systematically the fundamental differences in the dynamical structure of the two systems of interest and concludes that rocking structures cannot be replaced by ‘equivalent’ SDOF oscillators. The study proceeds by examining the validity of a simple, approximate design methodology, initially proposed in the late 1970s and now recommended in design guidelines to compute rotations of slender structures by performing iteration either on the true displacement response spectrum or design spectrum. This paper shows that the simple design approach is inherently flawed and should be abandoned, in particular for smaller, less‐slender blocks. The study concludes that the exact rocking spectrum emerges as a distinct intensity measure of ground motions. Copyright © 2002 John Wiley & Sons, Ltd.     

Collapse risk of controlled rocking steel braced frames with different post‐tensioning and energy dissipation designs

Controlled rocking steel braced frames (CRSBFs) are low‐damage self‐centring lateral force resisting systems. Previous studies have shown that designing the energy dissipation (ED) and post‐tensioning (PT) in CRSBFs using a response modification factor of R=8 can prevent collapse of structures during earthquakes beyond the design level. However, designers have unique control over the hysteretic behaviour of the system, even after the response modification factor is selected. Additionally, recent studies have suggested that CRSBFs could also be designed using R>8 while still satisfying performance limits. This paper examines how the response modification factor and the design of the ED and PT influence the collapse performance of CRSBFs with three and six storeys where collapse occurs because of over‐rotation of the base rocking joint. In addition, the influence of using an additional rocking joint above the base to mitigate higher‐mode forces is evaluated for a 12‐storey frame. A total of 18 different designs are considered for the three buildings using different ED and PT design parameters, including different response modification factors. A suite of 44 ground motions is scaled until at least 50% of the records cause collapse, and fragility curves are generated using the truncated incremental dynamic analysis curves. The results from two different assessment methodologies show that the parameters selected have a marked influence on the collapse performance of a CRSBF. Nevertheless, even CRSBFs designed using R>8 or without supplemental ED can have acceptably low probabilities of collapse, provided that the frame members are designed to remain elastic. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimating bearing response in symmetric and asymmetric‐plan isolated buildings with rocking and torsion

A comprehensive approach is developed to estimate relevant design quantities—lateral deformations and axial forces—in isolation systems composed of lead–rubber bearings. The approach, applicable to symmetric and asymmetric‐plan systems, includes the effects of bidirectional excitation, rocking, and torsion; and is the culmination of previous work on this topic. The approach is based on nonlinear response history analysis of an isolated block using an advanced bearing model that incorporates the interaction between axial force and lateral response of the bearing, known as axial‐load effects. The rocking response of the system and peak axial forces are shown to depend on the isolation period, the normalized strength—or yield strength normalized by peak ground velocity, the ratios of rocking frequency about each horizontal axis to vertical frequency, and the normalized stiffness eccentricity. In an attempt to develop results widely applicable to asymmetric‐plan systems, eccentricity is introduced by varying the stiffnesses and strengths of individual bearings in an idealized, rectangular plan. This idealized system approach is shown to have limited success; when applied to actual asymmetric‐plan systems the design equations to estimate response are accurate for lateral deformations but err by up to 25% for axial forces. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic response of multistoried buildings including the effects of soil-structure interaction

A simplified response spectrum superposition method has been generalized for the dynamic analysis of the multistoried building-soil response to earthquake ground motions via Fourier-transformed frequency domain. It involves the “scaling” of the Fourier amplitudes of the freefield translational and rocking motions to account for the soil-structure interaction effects, and then analyzing the building as fixed at the base. Envelopes of peak displacements, shear forces and overturning moments in the building are illustrated in terms of the order statistics of the response peaks.     

The rocking response of large flexible structures to earthquakes

The rocking response of structures subjected to strong ground motions is a problem of ‘several scales’. While small structures are sensitive to acceleration pulses acting successively, large structures are more significantly affected by coherent low frequency components of ground motion. As a result, the rocking response of large structures is more stable and orderly, allowing effective isolation from the ground without imminent danger of overturning. This paper aims to characterize and predict the maximum rocking response of large and flexible structures to earthquakes using an idealized structural model. To achieve this, the maximum rocking demand caused by different earthquake records was evaluated using several ground motion intensity measures. Pulse-type records which typically have high peak ground velocity and lower frequency content caused large rocking amplitudes, whereas non-pulse type records caused random rocking motion confined to small rocking amplitudes. Coherent velocity pulses were therefore identified as the primary cause of significant rocking motion. Using a suite of pulse-type ground motions, it was observed that idealized wavelets fitted to velocity pulses can adequately describe the rocking response of large structures. Further, a parametric analysis demonstrates that pulse shape parameters affect the maximum rocking response significantly. Based on these two findings, a probabilistic analysis method is proposed for estimating the maximum rocking demand to pulse-type earthquakes. The dimensionless demand maps, produced using these methods, have predictive power in the near-field provided that pulse period and amplitude can be estimated a priori. Use of this method within a probabilistic seismic demand analysis framework is briefly discussed.     

Mainshock-aftershock state-dependent fragility curves: A case of wood-frame houses in British Columbia,Canada

During a mainshock-aftershock (MSAS) sequence, there is no time to retrofit structures that are damaged by a mainshock; therefore, aftershocks could cause additional damage. This study proposes a new approach to develop state-dependent fragility curves using real MSAS records. Specifically, structural responses before and after each event of MSAS sequences are used to obtain statistical relationships among the engineering demand parameter prior to the seismic event (pre-EDP), the intensity measure of the seismic event (IM), and the engineering demand parameter after the seismic event (post-EDP). The developed fragility curves account for damage accumulation, providing the exceeding probability of damage state (DS) given the IM of the event and the DS of the structure prior to the seismic excitation. The UBC-SAWS model, which was developed for wood-frame houses in British Columbia, Canada, is considered as a case study application. Results indicate that for the examined structural typology, state-dependent fragility curves based on residual interstorey drift ratio (pre-EDP), peak ground velocity (IM), and maximum inter-storey drift ratio (post-EDP) are the best choice to characterise the cumulative damage effect. An illustration of the developed fragility curves is provided by considering a hypothetical MSAS scenario of a M_w 9.0 Cascadia mainshock triggering a M_w 6.0 crustal event in the Leech River fault, affecting wooden houses in Victoria, Canada. The MSAS scenario increases Yellow tags (restricted access) by 12.3% and Red tags (no access) by 4.8%.     

Robustness of simplified analysis methods for rocking structures on compliant soil

Recognizing the beneficial effect of nonlinear soil–foundation response has led to a novel design concept, termed ‘rocking isolation’. The analysis and design of such rocking structures require nonlinear dynamic time history analyses. Analyzing the entire soil–foundation–structure system is computationally demanding, impeding the application of rocking isolation in practice. Therefore, there is an urgent need to develop efficient simplified analysis methods. This paper assesses the robustness of two simplified analysis methods, using (i) a nonlinear and (ii) a bilinear rocking stiffness combined with linear viscous damping. The robustness of the simplified methods is assessed by (i) one-to-one comparison with a benchmark finite element (FE) analysis using a selection of ground motions and (ii) statistical comparison of probability distributions of response quantities, which characterize the time history response of rocking systems. A bridge pier (assumed rigid) supported on a square foundation, lying on a stiff clay stratum, is used as an illustrative example. Nonlinear dynamic FE time history analysis serves as a benchmark. Both methods yield reasonably accurate predictions of the maximum rotation θ_max. Their stochastic comparison with respect to the empirical cumulative distribution function of θ_max reveals that the nonlinear and the bilinear methods are not biased. Thus, both can be used to estimate probabilities of exceeding a certain threshold value of θ. Developed in this paper, the bilinear method is much easier to calibrate than the nonlinear, offering similar performance.     

The three‐dimensional behavior of inverted pendulum cylindrical structures during earthquakes

In order to use rocking as a seismic response modification strategy along both directions of seismic excitation, a three‐dimensional (3D) rocking model should be developed. Since stepping or rolling rocking structural members out of their initial position is not a desirable performance, a rocking design should not involve these modes of motion. To this end, a model that takes the aforementioned constraint into account needs to be developed. This paper examines the 3D motion of a bounded rigid cylinder that is allowed to uplift and sustain rocking and wobbling (unsteady rolling) motion without sliding or rolling out of its initial position (i.e., a 3D inverted pendulum). Thus, the cylinder is constrained to zero residual displacement at the end of its 3D motion. This 3D dynamic model of the rocking rigid cylinder has two DOFs (three when damping is included), making it the simplest 3D extension of Housner's classical two‐dimensional (2D) rocking model. The development of models with and without damping is presented first. They are simple enough to perform extensive parametric analyses. Modes of motion of the cylinder are identified and presented. Then, 3D rocking and wobbling earthquake response spectra are constructed and compared with the classical 2D rocking earthquake response spectra. The 3D bounded rocking earthquake response spectra for the ground motions considered seem to have a very simple linear form. Finally, it is shown that the use of a 2D rocking model may lead to unacceptably unconservative estimates of the 3D rocking and wobbling seismic response. Copyright © 2017 John Wiley & Sons, Ltd.     

An analytical model of a deformable cantilever structure rocking on a rigid surface: development and verification

This paper extends previously developed models to account for the influence of the column and the foundation masses on the behavior of top‐heavy deformable elastic cantilever columns rocking on a rigid support surface. Several models for energy dissipation at impact are examined and compared. A novel Vertical Velocity Energy Loss model is introduced. Rocking uplift and overturning spectra for the deformable elastic cantilever model excited by sinusoidal ground motions are constructed. The effects of non‐dimensional model parameter variations on the rocking spectra and the overturning stability of the model are presented. It is shown that the remarkable overturning stability of dynamically excited large cantilever columns is not jeopardized by their deformability. Copyright © 2015 John Wiley & Sons, Ltd.     

Planar investigation of the seismic response of ancient columns and colonnades with epistyles using a custom-made software

In this paper, the dynamic behavior of multi-drum columns and colonnades with epistyles under earthquake excitations is examined through planar numerical simulations. A specialized software application, developed utilizing the discrete element methods (DEM), is used to investigate the influence of certain parameters on the seismic response of such multi-body structural systems. First, this custom-made software is extensively validated by comparing the computed responses of various problems, such as sliding, rocking and free vibration dynamics of rigid bodies, with the corresponding analytical solutions. Then, the developed software is used to study the influence of the frequency content and amplitude of the ground motions on the columns and colonnades, as well as the geometric characteristics of these structures. Parameters such as the number of drums that assemble each column and the number of columns of a colonnade appear to be defining parameters that affect the seismic response of colonnades with epistyles. For ground motions with relatively low predominant frequencies, rocking is the dominant effect in the response, while with the increase of the excitation frequency the response becomes even more complex involving both sliding and rocking phenomena. The numerical simulations show that earthquakes with relatively low predominant frequencies seem to endanger both standalone columns and colonnades with epistyles more than earthquakes with higher predominant frequencies.