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.     

Modeling of energy dissipation in structural devices and foundation soil during seismic loading

Though rocking shallow foundations could be designed to possess many desirable characteristics such as energy dissipation, isolation, and self-centering, current seismic design codes often avoid nonlinear behavior of soil and energy dissipation beneath foundations. This paper compares the effectiveness of energy dissipation in foundation soil (during rocking) with the effectiveness of structural energy dissipation devices during seismic loading. Numerical simulations were carried out to systematically study the seismic energy dissipation in structural elements and passive controlled energy dissipation devices inserted into the structure. The numerical model was validated using shaking table experimental results on model frame structures with and without energy dissipation devices. The energy dissipation in the structure, drift ratio, and the force and displacement demands on the structure are compared with energy dissipation characteristics of rocking shallow foundations as observed in centrifuge experiments, where shallow foundations were allowed to rock on dry sandy soil stratum during dynamic loading. For the structures with energy dissipating devices, about 70–90% of the seismic input energy is dissipated by energy dissipating devices, while foundation rocking dissipates about 30–90% of the total seismic input energy in foundation soil (depending on the static factor of safety). Results indicate that, if properly designed (with reliable capacity and tolerable settlements), adverse effects of foundation rocking can be minimized, while taking advantage of the favorable features of foundation rocking and hence they can be used as efficient and economical seismic energy dissipation mechanisms in buildings and bridges.     

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.     

Dynamic response and impact energy loss in controlled rocking members

Unbonded posttensioning anchors a rocking structural member to its foundation and produces its controlled rocking response when the member undergoes seismic action. Unlike rocking of free-standing bodies, little attention has been given to the dynamic behavior of these controlled rocking members. This paper utilizes experiments of concrete structural members with unbonded posttensioning, varying member geometries, and levels of initial posttensioning force to (a) characterize the associated impact energy loss and (b) improve modeling of controlled rocking motions. Experimental results show that impact energy loss in controlled rocking members can be captured accurately using the coefficient of restitution (r) approach of the modified simple rocking model (MSRM). Based on the MSRM, a controlled rocking model (CRM) is developed that additionally accounts for the variations in contact length at the member-to-foundation (rocking) interface. The CRM reproduces the experimental responses of controlled rocking members with good accuracy and is used to investigate controlled rocking motions under horizontal base excitations.     

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 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.     

Principal axes of M-DOF structures Part II: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

Principal axes of M-DOF structures Part Ⅱ: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

THE THEORY OF RESPONSE ANALYSIS OF FUZZY STOCHASTIC DYNAMICAL SYSTEMS WITH A SINGLE DEGREE OF FREEDOM

Most real-life structural/mechanical systems have complex geometrical and material properties and operate under complex fuzzy environmental conditions. These systems are certainly subjected to fuzzy random excitations induced by the environment. For an analytical treatment of such a system subjected to fuzzy random excitations, it becomes necessary to establish the general theory of dynamic response of a system to fuzzy random excitations. In this paper, the theory of response, fuzzy mean response and fuzzy covariance response of a single-degree-of-freedom (sdf) system to fuzzy random excitations in the time domain and frequency domain is put forward. The theory of response analysis of an sdf system to both stationary and non-stationary fuzzy random excitations in the time domain and frequency domain is established. Two examples are considered in order to demonstrate the rationality and validity of the theory, and the models of stationary filtered white noise and non-stationary filtered white noise fuzzy stochastic processes of the earthquake ground motion are set up. Methods of analysis for fuzzy random seismic response of sdf systems are put forward using the principles of response analysis of an sdf fuzzy random dynamic system.     

Centrifuge modeling of rocking‐isolated inelastic RC bridge piers

Experimental proof is provided of an unconventional seismic design concept, which is based on deliberately underdesigning shallow foundations to promote intense rocking oscillations and thereby to dramatically improve the seismic resilience of structures. Termed rocking isolation, this new seismic design philosophy is investigated through a series of dynamic centrifuge experiments on properly scaled models of a modern reinforced concrete (RC) bridge pier. The experimental method reproduces the nonlinear and inelastic response of both the soil‐footing interface and the structure. To this end, a novel scale model RC (1:50 scale) that simulates reasonably well the elastic response and the failure of prototype RC elements is utilized, along with realistic representation of the soil behavior in a geotechnical centrifuge. A variety of seismic ground motions are considered as excitations. They result in consistent demonstrably beneficial performance of the rocking‐isolated pier in comparison with the one designed conventionally. Seismic demand is reduced in terms of both inertial load and deck drift. Furthermore, foundation uplifting has a self‐centering potential, whereas soil yielding is shown to provide a particularly effective energy dissipation mechanism, exhibiting significant resistance to cumulative damage. Thanks to such mechanisms, the rocking pier survived, with no signs of structural distress, a deleterious sequence of seismic motions that caused collapse of the conventionally designed pier. © 2014 The Authors Earthquake Engineering & Structural Dynamics Published by John Wiley & Sons Ltd.     

Seismic response of large-span spatial structures under multi-support and multidimensional excitations including rotational components

To achieve rational and precise seismic response predictions of large span spatial structures(LSSSs),the inherent non-uniformity and multidimensionality characteristics of earthquake ground motions should be properly taken into consideration.However,due to the limitations of available earthquake stations to record seismic rotational components,the effects of rocking and torsional earthquake components are commonly neglected in the seismic analyses of LSSSs.In this study,a newly developed method to extract the rocking and torsion components at any point along the area of a deployed dense array from the translational earthquake recordings is applied to obtain the rotational seismic inputs for a LSSS.The numerical model of an actual LSSS,the Dalian International Conference Center(DICC),is developed to study the influences of multi-support and multidimensional excitations on the seismic responses of LSSSs.The numerical results reveal that the non-uniformity and multidimensionality of ground motion input can considerably affect the dynamic response of the DICC.The specific degree of influence on the overall and local structural displacements,deformations and forces are comprehensively investigated and discussed.     

罕遇地震作用下的框架-摇摆刚架结构体系动力可靠度研究

针对钢筋混凝土框架结构的受力特点,采用增设摇摆刚架的抗震设计方法,以提高罕遇地震下建筑结构的安全性。建立了框架一摇摆刚架结构体系的计算模型,结合状态空间法与虚拟激励法,求解结构的平稳随机响应,并根据计算所得随机响应对框架一摇摆刚架体系的动力可靠度进行分析。以西部地区某已建成的6层框架结构为算例,探讨了罕遇地震作用下不同刚度比的摇摆刚架对新结构体系动力可靠度的影响。结果表明,通过增设不同刚度比的摇摆刚架,可以有效协调结构体系的变形模式,充分发挥结构的耗能能力,降低整体结构的条件失效概率。     

Dynamic response analysis of nonlinear secondary oscillators to idealised seismic pulses

The paper deals with the seismic response analysis of nonlinear secondary oscillators. Bilinear, sliding and rocking single-degree-of-freedom dynamic systems are analysed as representative of a wide spectrum of secondary structures and nonstructural components. In the first stage, the equations governing their full dynamic interaction with linear multi-degree-of-freedom primary structures are formulated, and then conveniently simplified using primary-secondary two-degree-of-freedom systems and dimensionless coefficients. In the second stage, the cascade approximation is applied, whereby the feedback action of the secondary oscillator on the primary structure is neglected. Owing to the piecewise linearity of the secondary systems being considered, efficient semi-analytical and step-by-step numerical solutions are presented. The semi-analytical solutions allow the direct evaluation of the seismic response under pulse-type ground excitations and are also used to validate step-by-step numerical schemes, which in turn can be used for general-type seismic excitations. In the third stage, a set of decoupling criteria are proposed for the pulse-type base excitations, identifying the conditions under which a cascade analysis is admissible from an engineering standpoint. Finally, the influence and relative dependencies between the input parameters of the ground motion and the primary-secondary assembly are quantified on the response of the secondary systems through nonlinear floor response spectra, and general trends are identified and discussed.     

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.     

A nonlinear macroelement model for the seismic analysis of masonry buildings

The macroelement technique for modelling the nonlinear response of masonry panels is particularly efficient and suitable for the analysis of the seismic behaviour of complex walls and buildings. The paper presents a macroelement model specifically developed for simulating the cyclic in‐plane response of masonry walls, with possible applications in nonlinear static and dynamic analysis of masonry structures. The model, starting from a previously developed macroelement model, has been refined in the representation of flexural–rocking and shear damage modes, and it is capable of fairly simulating the experimental response of cyclic tests performed on masonry piers. By means of two internal degrees of freedom, the two‐node macroelement permits to represent the coupling of axial and flexural response as well as the interaction of shear and flexural damage. Copyright © 2013 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.     

Response of seismically isolated structures to rocking‐type excitations

In this paper, the origin of rocking‐type excitations and their effects on the response of base isolated structures are studied. In particular, the role of kinematic interaction in the determination of the rocking excitation is highlighted. The cases of surface foundations subjected to horizontally propagating waves, as well as of embedded foundations under vertically incident shear waves are examined. The validity of the kinematic interaction based on the rigid base mat assumption is discussed. It is shown that, in the case of classical horizontal isolation, rocking input may amplify significantly the response of the lower non‐isolated modes. The examination of full three‐dimensional isolation and active and semi‐active control methods demonstrates the efficacy of these methods to improve the performance of seismically isolated structures subjected to rocking excitations. Copyright © 2009 John Wiley & Sons, Ltd.     

Performance evaluation of a shallow foundation built on structured clays under seismic loading

Due to the increased need of storage, larger and higher structures are being built all over the world, thus requiring a more careful evaluation of the mechanical performance of their foundation deposits both in terms of bearing capacity and compressibility behaviour. The design of such structures and their serviceability and stability is largely governed by the effects of the dynamic loading conditions principally because of their significantly elevated risk in seismic prone zones. In this paper, numerical analyses using an advanced constitutive model, able to account for the initial soil structure and its progressive degradation, have been performed to investigate the seismic response of a silo foundation built on structured clays. The proposed analyses involve the use of a fully-coupled finite element approach. For the dynamic simulations, three different input motions have been selected form earthquake databases according to the seismic hazard study of the specific site. The results of the silo dynamic response are illustrated in terms of signal amplification, permanent excess pore water pressures, accumulated displacements and structure induced degradation during and after the seismic loading. The dynamic behaviour of the footing indicates that extreme earthquake events can induce large destructuration in natural clays, leading to ground settlements up to twice the observed ones under static loads, which need to be properly accounted for in the design. This suggests that there are significant advantages in using advanced models which recognise the existence of initial soil structure and its subsequent damage due to the applied dynamic loads.     

Effect of earthquake‐induced axial load fluctuations on asymmetric frame–wall–rocking foundation systems

Fluctuations in axial load imposed on a rocking footing will affect its moment capacity, the shape of its moment–rotation hysteresis, and potentially the system's seismic performance. Structural asymmetry increases the likelihood of axial load variation during earthquake excitations. To investigate this issue, a unique centrifuge testing program was carried out on low‐rise frame–wall–rocking foundation systems. In this paper, the seismic behaviors of asymmetric and symmetric models from this test program are systematically compared. Experimental results reveal that placing the lateral force resisting shear wall outboard produces significant axial load fluctuation, which in turn greatly deteriorate the lateral load‐carrying capacity of a foundation rocking dominated frame–wall system, particularly in its weak direction. However, it strengthens the system when loading is towards the shear wall, leading to a highly asymmetric hysteretic response. During earthquake loading, all asymmetric rocking foundation systems observe smaller peak roof accelerations, but larger peak and permanent roof drifts compared with the symmetric systems. Despite these differences in response, the axial load fluctuation and structural asymmetry do not significantly change the relative energy dissipated by the rocking foundations and inelastic structural components within each frame–wall–rocking foundation model. Copyright © 2015 John Wiley & Sons, Ltd.