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.     

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.     

Three‐storey building retrofit: rocking isolation versus conventional design

Although modern seismic codes have undoubtedly led to safer structures, the seismic vulnerability of metropolitan areas is unavoidably governed by that of older buildings, which constitute the vast majority of the current building stock. Quite alarmingly, even relatively moderate intensity earthquakes have been proven capable of challenging their structural integrity, leading to severe damage or collapse. Therefore, there is an urgent need to assess the vulnerability of existing structures and to evaluate the efficiency of novel retrofit techniques. This paper studies experimentally the seismic performance of an existing three‐storey building, retrofitted through addition of shear walls. Emphasis is placed on the foundation of the shear walls, and two design alternatives are comparatively assessed: (a) conventional design according to current seismic codes and (b) ‘rocking isolation’ by reducing the size of the foundation. A series of reduced‐scale shaking table tests are conducted at the Laboratory of Soil Mechanics of the National Technical University of Athens. The physical model encompasses the structural system, along with the foundations, and the soil. The nonlinearity of structural members is simulated through specially designed and carefully calibrated artificial plastic hinges. The vulnerability of the original structure is confirmed, as it is found to collapse with a soft‐storey mechanism when subjected to moderate intensity shaking. The conventionally retrofitted structure is proven capable of sustaining larger intensity shaking, and the rocking‐isolated structure is shown to offer increased safety margins. Thanks to its inherent self‐centering mechanism, the rocking system is characterized by reduced permanent drifts. Copyright © 2014 John Wiley & Sons, Ltd.     

An analytical model of a deformable cantilever structure rocking on a rigid surface: experimental validation

This paper describes an experimental program to examine the dynamic response of deformable cantilevers rocking on a rigid surface. The primary goal of the tests is to verify and validate a dynamic rocking model that describes the behavior of these structures. The benchmark response data was obtained from shaking‐table tests on deformable rocking specimens with different natural vibration frequencies and different aspect ratios excited by analytical pulses and recorded ground motions. The responses computed using the model are found to be in good agreement with the benchmark test results. Widely used impact, restitution and damping assumptions are revisited based on the experiment results and the analytical model findings. Copyright © 2015 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.     

Analysis of the rocking response of rigid blocks standing free on a seismically isolated base

This paper examines the rocking response and stability of rigid blocks standing free on an isolated base supported: (a) on linear viscoelastic bearings, (b) on single concave and (c) on double concave spherical sliding bearings. The investigation concludes that seismic isolation is beneficial to improve the stability only of small blocks. This happens because while seismic isolation increase the ‘static’ value of the minimum overturning acceleration, this value remains nearly constant as we move to larger blocks or higher frequency pulses; therefore, seismic isolation removes appreciably from the dynamics of rocking blocks the beneficial property of increasing stability as their size increases or as the excitation pulse period decreases. This remarkable result suggests that free‐ standing ancient classical columns exhibit superior stability as they are built (standing free on a rigid foundation) rather than if they were seismically isolated even with isolation system with long isolation periods. The study further confirms this finding by examining the seismic response of the columns from the peristyle of two ancient Greek temples when subjected to historic records. Copyright © 2011 John Wiley & Sons, Ltd.     

Structural performance assessment under near‐source pulse‐like ground motions using advanced ground motion intensity measures

This paper demonstrates the effectiveness of utilizing advanced ground motion intensity measures (IMs) to evaluate the seismic performance of a structure subject to near‐source ground motions. Ordinary records are, in addition, utilized to demonstrate the robustness of the advanced IM with respect to record selection and scaling. To perform nonlinear dynamic analyses (NDAs), ground motions need to be selected; as a result, choosing records that are not representative of the site hazard can alter the seismic performance of structures. The median collapse capacity (in terms of IM), for example, can be systematically dictated by including a few aggressive or benign pulse‐like records into the record set used for analyses. In this paper, the elastic‐based IM such as the pseudo‐spectral acceleration (S_a) or a vector of S_a and epsilon has been demonstrated to be deficient to assess the structural responses subject to pulse‐like motions. Using advanced IMs can be, however, more accurate in terms of probabilistic response prediction. Scaling earthquake records using advanced IMs (e.g. inelastic spectral displacement, S_di, and IM _1I&2E; the latter is for the significant higher‐mode contribution structures) subject to ordinary and/or pulse‐like records is efficient, sufficient, and robust relative to record selection and scaling. As a result, detailed record selection is not necessary, and records with virtually any magnitude, distance, epsilon and pulse period can be selected for NDAs. Copyright © 2008 John Wiley & Sons, Ltd.     

Near‐collapse response of existing RC building under severe pulse‐type ground motion using hybrid simulation

Column shear‐axial failure is a complex response, which lends itself to physical experimentation. Reinforced concrete structures built prior to the mid‐1970s are particularly susceptible to such failure. Shear‐axial column failure has been examined and studied at the element level, but current rehabilitation practice equates such a column failure with structural collapse, neglecting the collapse resistance of the full structural system following column failure. This system‐level response can prevent a column failure from leading to progressive collapse of the entire structure. In this study, a hybrid simulation was conducted on a representative pre‐1970s reinforced concrete frame structure under severe seismic ground motion, in which three full‐scale reinforced concrete columns were tested at the University of Illinois at Urbana Champaign. The analytical portion of the model was represented in the computer program OpenSees. Failure occurred in multiple physical specimens as a result of the ground motion, and the hybrid nature of the test allowed for observation of the system‐level response of the tested columns and the remaining structural system. The behavior of the system accounting for multiple column shear‐axial failure is discussed and characterized. Copyright © 2016 John Wiley & Sons, Ltd.     

Linearization and first‐order expansion of the rocking motion of rigid blocks stepping on viscoelastic foundation

In structural mechanics there are several occasions where a linearized formulation of the original non‐linear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first‐order expansion of the dynamic equilibrium of the system about a ‘static’ configuration; yet caution should be exercised when identifying the ‘correct’ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non‐linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first‐order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates. Copyright © 2008 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.     

Wavelet‐based non‐stationary seismic rocking response of flexibly supported tanks

The non‐stationary rocking response of liquid storage tanks under seismic base excitations including soil interaction has been developed based on the wavelet domain random vibration theory. The ground motion has been characterized through statistical functionals of wavelet coefficients of the ground acceleration history. The tank–liquid–foundation system is modelled as a multi‐degree‐of‐freedom (MDOF) system with both lateral and rocking motions of vibration of the foundation. The impulsive and convective modes of vibration of the liquid in the tank have been considered. The wavelet domain coupled dynamic equations are formulated and then solved to get the expressions of instantaneous power spectral density function (PSDF) in terms of functionals of input wavelet coefficients. The moments of the instantaneous PSDF are used to obtain the stochastic responses of the tank in the form of coefficients of hydrodynamic pressure, base shear and overturning base moment for the largest expected peak responses. Parametric variations are carried out to study the effects of various governing parameters like height of liquid in the tank, height–radius ratio of the tank, ratio of total liquid mass to mass of foundation, and shear wave velocity in the soil medium, on the responses of the tank. Copyright © 2003 John Wiley & Sons, Ltd.     

Planar rocking response and stability analysis of an array of free‐standing columns capped with a freely supported rigid beam

This paper investigates the planar rocking response of an array of free‐standing columns capped with a freely supported rigid beam in an effort to explain the appreciable seismic stability of ancient free‐standing columns that support heavy epistyles together with the even heavier frieze atop. Following a variational formulation, the paper concludes to the remarkable result that the dynamic rocking response of an array of free‐standing columns capped with a rigid beam is identical to the rocking response of a single free‐standing column with the same slenderness yet with larger size, that is a more stable configuration. Most importantly, the study shows that the heavier the freely supported cap beam is (epistyles with frieze atop), the more stable is the rocking frame regardless of the rise of the center of gravity of the cap beam, concluding that top‐heavy rocking frames are more stable than when they are top light. This ‘counter intuitive’ finding renders rocking isolation a most attractive alternative for the seismic protection of bridges with tall piers, whereas its potential implementation shall remove several of the concerns associated with the seismic connections of prefabricated bridges. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of structural response under high‐frequency ground excitation

The structural response to high‐frequency ground motions is complicated due to the involvement of local‐mode vibration. At present such a characteristic is not well recognized and this can cause confusion over the analytical and experimental modelling of the corresponding response and damage. The fact that most existing regulatory guides for limits on allowable construction vibrations are necessarily simplified for administrative reasons calls upon the derivation of more sophisticated approaches for special cases. This requires accumulation of pertinent experimental evidence. This paper attempts to provide some insights into the local‐mode dynamic response characteristics, with emphasis on appropriate modelling techniques and experimental measurements. A preliminary testing program is reported, in which efforts were made to reproduce high‐frequency response with a reduced scale reinforced concrete model with shaking table facilities. The results demonstrate the dependence of the response amplitudes with the excitation frequency. On a ppv‐basis, the current test results indicate that a substantial increase of the allowable ppv value from those specified by various standards may be considered for structural damage to reinforced concrete building structures. More analytical and experimental data are needed for further evaluation of the local‐mode effects and to quantify their impact on the structural damage process. Copyright © 2001 John Wiley & Sons, Ltd.     

Dimensional analysis of structures with translating and rocking foundations under near-fault ground motions

This paper evaluates the inertial soil–structure interaction (SSI) effects on linear and bilinear structures supported on foundation that is able to translate and rock when subject to near-fault ground motions. Through rigorous dimensional analysis, the peak structural responses (e.g. structural drift and total acceleration) of the soil–foundation–structure interacting (SFSI) systems are characterized by a set of dimensionless Π-parameters, which can decisively describe the interactive behavior of SFSI systems. By comparing the normalized structural responses of various structure–foundation systems with their fixed-base counterparts, the study reveals that SSI effects highly depend on the structure-to-pulse frequency ratio, Π_ω, the foundation-to-structure stiffness ratio, Π_k, damping coefficient of foundation impedance, Π_c, the foundation rocking, and the development of nonlinearity in structures. For linear structures, the SSI effects are insignificant when the structure-to-pulse frequency ratio (Π_ω) is smaller than 1.5 and can amplify the structural responses when Π_ω is higher than 1.5. Foundation rocking can shift and enlarge the response amplification zone of SSI. For nonlinear structures, SSI tends to reduce the structural responses for Π_ω<3 while can increase the ductility demands for Π_ω≥3. The bilinear structures may experience more significant SSI effects than linear structures in certain frequency ranges. The numerical simulations on ten real building cases exhibiting significant rocking and a detailed case study on a nine-story frame structure demonstrate the applicability of dimensional analysis results to predict the SSI effects on realistic building structures. The study demonstrates that the dimensional analysis provides a concise and systematic way of evaluating SSI effects.     

Near‐source seismic demand and pulse‐like records: A discussion for L'Aquila earthquake

Rupture directivity effects in ground motion are known since many years to both seismologists and earthquake engineers, i.e. in sites that are in a particular geometrical configuration with respect to the rupture, the velocity fault‐normal signals may show a large pulse which occurs at the beginning of the record and contains the most of energy. The results are waveforms different from ordinary ground motions recorded in the far field or in geometrical conditions not favorable with respect to directivity. Current attenuation laws are not able to capture such effect well, if at all, and current probabilistic seismic hazard analysis is not able to predict the resulting peculiar spectral shape. Moreover, it is believed that structures with dynamic behavior in a range of periods related to the pulse period may be subjected to underestimated seismic demand. In the paper this is investigated and increments in both elastic and inelastic seismic actions are quantified using a large dataset of records, from the next generation attenuation project (NGA), in which a fraction is comprised of velocity pulses identified in other studies. These analyses employ recently developed tools and procedures to assess directivity effects and to quantify the associated threat in terms of seismic action on structures. Subsequently, the same tools are used in one of the first attempts to identify near‐source effects in the data recorded during a normal faulting earthquake, the mainshock of the recent Abruzzo (central Italy) sequence, leading to conclude that pulse‐like effects are likely to have occurred in the event, that is (1) observation of pulse‐like records in some near‐source stations is in fair agreement with existing predictive models, (2) the increment in seismic demand shown by pulse‐like ground motion components complies with the results of the analysis of the NGA data, and (3) seismic demand in non‐impulsive recordings is generally similar to what expected for ordinary records. The results may be useful as a benchmark for inclusion of near‐source effect in design values of seismic action and structural risk analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

The existence of ‘complete similarities’ in the response of seismic isolated structures subjected to pulse‐like ground motions and their implications in analysis

In this paper the seismic response of isolated structures supported on bearings with bilinear and trilinear behavior is revisited with dimensional analysis in an effort to better understand the relative significance of the various parameters that control the mechanical behavior of isolation systems. An isolation system that consists of lead rubber bearings or of single concave spherical sliding bearings exhibits bilinear behavior; whereas, when a double concave configuration is used the behavior is trilinear. For the case of bilinear behavior it is well known that the value of the normalized yield displacement is immaterial to the response of the isolated superstructure—or, in mathematical terms, that the response of the bilinear oscillator exhibits complete similarity in the dimensionless yield displacement. Similarly, for the case of trilinear behavior the paper shows that the presence of the intermediate slope is immaterial to the peak response of most isolated structures—a finding that shows the response of the trilinear oscillator exhibits a complete similarity in the difference between the coefficients of friction along the two sliding surfaces as well as in the ratio of the intermediate to the final slope. This finding implies that even when the coefficients of friction of the two sliding surfaces are different, the response of isolated structures for most practical configurations can be computed with confidence by replacing the double concave spherical bearings with single concave spherical bearings with an effective radius of curvature and an effective coefficient of friction. Copyright © 2010 John Wiley & Sons, Ltd.     

Seismic behavior of single‐story asymmetric‐plan buildings under uniaxial excitation

The critical parameters that influence the nonlinear seismic response of asymmetric‐plan buildings are identified by evaluating the effects of different asymmetries that may characterize the structure of a building as well as exploring the influence of the ground motion features. First, the main findings reported in the literature on both the linear and nonlinear dynamic response of asymmetric‐plan buildings are presented. The common findings and the conflicting conclusions reached in different investigations are pointed out. Then, the results of comprehensive nonlinear dynamic analyses performed for evaluating the seismic response of systems characterized by different strength and stiffness configurations, representative of a large class of asymmetric‐plan buildings, are reported. Findings from the study indicate that the building response changes when moving from the linear to the nonlinear range, so that the seismic behavior of asymmetric‐plan buildings, apart from the source of asymmetry, can be always classified as irregular. Additionally, it was observed that as the seismic demands cause amplification of system nonlinearity with increasing earthquake intensity, the maximum displacement demand in the different resisting elements tends to be reached with the same deformed configuration of the system. The resultant of the seismic forces producing such a maximum demand is located at the center of resistance and corresponds to the collapse mechanism of the system that provides the maximum lateral strength in the exciting direction of the seismic action. Copyright © 2008 John Wiley & Sons, Ltd.     

Comparison between rocking analysis and kinematic analysis for the dynamic out‐of‐plane behavior of masonry walls

This paper provides a contribution to the rocking analysis of masonry walls by making a comparison with the kinematic analysis suggested by the Italian code. It is shown that the latter approach is generally over‐conservative and therefore potentially inappropriate for historic buildings, where rehabilitation can be expensive and can affect their cultural value. The equation of motion given by the Housner formulation, corresponding to the movement of a rigid block, is here modified to account for different boundary conditions at different heights of the wall. These boundary conditions or horizontal restraints can represent vaults, transverse walls, or retrofitting devices such as steel tie‐rods. A systemic analysis of walls having different dimensions and slenderness is performed, and the results from the Italian code and rocking analysis are compared. Finally, the improvement in the response offered by retrofitting devices is discussed in terms of reduction of amplitude ratio. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic analysis of flexible rectangular fluid containers subjected to horizontal ground motion

In this paper, an analytical method is proposed to determine the dynamic response of 3‐D rectangular liquid storage tanks with four flexible walls, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The velocity potential in which boundary conditions are satisfied is solved by the method of separation of variables using the principle of superposition. The impulsive pressure distribution is then computed. Solutions based on 3‐D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that evaluates convective pressure and surface displacements in a similar rigid tank. The variation of dynamic response characteristics with respect to different tank parameters is investigated. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts, and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a simplified but sufficiently accurate design procedure is developed to improve code formulas for the seismic design of liquid storage tanks. Copyright © 2013 John Wiley & Sons, Ltd.