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The three‐dimensional behavior of inverted pendulum cylindrical structures during earthquakes
Abstract and Figures
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
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... 44,45 A refined FEM model of a real bridge structure was established to examine its seismic performance under fixed-base pier, rocking pier, and rocking foundation conditions. 18 Recently, the bounded cylinders and wobbling frames, for which the 3-D rocking motions were restrained in the initial position without rolling-out motion, were numerically and experimentally studied 31,46,47 to demonstrate their expected seismic performance. ...
... Since the motion of the bearing is assumed to be constrained in its initial position in the horizontal plane by the restrainers, it can be proven that ψ = −φ. 46 On the other hand, the bearing has a size of r = √ b 2 + h 2 , which is the diagonal of the Type 1 bearing (cylinder-shaped) or the slant height of the Type 2 bearing (cone-shaped). The radius of the bearing's circular bottom plate b and the height of the bearing h, which describe the inclined angle of the bearing α = arctan b∕h. ...
... For example, if either the X or Y components of the motion of the model are assumed to be zero (i.e., at rest in that direction), the proposed model reduces to the model of the Uni-RIBS that we proposed in our previous study. 48 If the substructure piers are assumed to be a rigid body, the proposed model reduces to a special case of the 3-D wobbling frame structures, 46,47 where the mass of the 3-D wobbling columns is neglected. ...
A bidirectional rocking isolation bearing system (Bi‐RIBS) is proposed to provide seismic protection for bridge structures. By using the 3‐D rocking motion of the Bi‐RIBS, this system acts as a mechanical fuse to limit the maximum force transmitted to the bridge piers and as a restoring component to control an excessive girder response. Possible applications in bridge structures were discussed. A simple analytical model was established to characterize the dynamics of an example bridge featuring such a Bi‐RIBS. A series of dynamic analyses were performed by using the proposed model to investigate the effects of several factors on controlling the seismic responses of the bridge, for example, the design parameters of Bi‐RIBS including inclined angle and size, the damping property at the support interface, and the mass ratio. The peak ground accelerations of the bidirectional ground motion record were scaled to various levels to evaluate the maximum performance indices of the bridge structure and the response control effectiveness of the Bi‐RIBS compared to the uncontrolled counterparts. The simulation results demonstrated that the proposed Bi‐RIBS could effectively control the maximum pier displacement while keeping the bearing from overturning if suitable parameters were selected. In particular, the control effectiveness on the maximum pier response becomes more significant as the seismic intensity increases, due to its distinctive negative stiffness property.
... Recently, a 3D dynamic model of a wobbling rigid cylindrical column was presented. 62 The model was developed to describe the motion of cylinders that are constrained to wobble above their initial position (Figures 1 and 2), without sliding. Therefore, it is not applicable to rocking equipment, [63][64][65][66][67][68][69][70][71][72][73] but to structures designed to rock and return to their initial position (such as rocking bridges). ...
... The assumptions are discussed in ref. 62 but are repeated in this paper for reasons of completeness: • The cylinder is considered rigid and homogeneous. ...
... For γ m = 0 the equations yield the solitary wobbling rigid cylindrical column equations derived in. 62 It can also be seen that, unlike the case of a planar rocking frame, for which there is a formal and elegant equivalence with a geometrically similar, yet larger, column, in the 3D case there is no "equivalent solitary 3D column." However, if one keeps only the linear term of Equations (5) and (6) the size of a quasi-equivalent solitary 3D column is: ...
The simplest 3D extension of Housner's planar rocking model is a rocking (wobbling) cylinder allowed to uplift and roll on its circumference, but constrained not to roll out of its initial position. The model is useful for the description of bridges that use rocking as a seismic isolation technique, in an effort to save material by reducing the design moment and the size of the foundations. This paper shows that describing wobbling motion in terms of displacements rather than rotations is more useful. It unveils that a remarkable property of planar rocking bodies extends to 3D motion: A small and a large wobbling cylinder of the same slenderness will sustain roughly equal top displacement, as long as they are not close to overturning. This allows for using the response of an infinitely large wobbling cylinder of slenderness α as a proxy to compute the response of all cylinders having the same slenderness, irrespectively of their size. Thus, the dimensionality of the problem is reduced by one. Moreover, this paper shows that the median wobbling response to sets of ground motions can be described as an approximate function of only two non‐dimensional parameters, namely or where u is the top displacement of the wobbling body.
... However, planar rocking is notoriously unpredictable by analytical or numerical models, while shake table tests of such systems have often proven to be non-repeatable Anagnostopoulos et al., 2019 among others). Wobbling, a three-dimensional (3D) motion on a horizontal plane characterized by simultaneous uplift from the ground (rocking) and change of the contact point with the ground (nutation) without spinning or sliding out of its original position, like Euler's disk (van den Engh et al., 2000;Vassiliou, 2018;Vassiliou et al., 2017a) is even harder to predict than planar rocking. Thus, developing seismic wobbling models is challenging, yet useful to further develop seismic design of realistic rocking structures expected to respond in all three dimensions, as well as to understand the seismic behavior of nonstructural elements such as unanchored equipment (Bao and Konstantinidis, 2020;Dar et al., 2016Dar et al., , 2018Di Egidio et al., 2015;Di Sarno et al., 2019;Konstantinidis and Makris, 2010;Sextos et al., 2017;Voyagaki et al., 2018;Wittich and Hutchinson, 2015). ...
... Given that planar and 3D rocking response is size-dependent, the ground motion excitations used in the shake table experiments were scaled to preserve acceleration scaling (as discussed later in this data paper) so that the model represents a structure with a height similar to a real bridge. The intention of wobbling model design was to mimic the assumptions of the ''wobbling bridge'' analytical model developed by Vassiliou et al. (2017a;Vassiliou, 2018), namely, that the structure is rigid and that the columns wobble without sliding or spinning about their longitudinal axis. To achieve this, the ends of the columns were machined (lathed) to ensure they are right angled to their longitudinal axis. ...
... The main purpose of the generated dataset is to enable the validation of analytical and numerical models for seismic response of 3D uplifting structures that wobble, that is, 3D motion characterized by simultaneous uplift from the ground (rocking) and change of the contact point with the ground (nutation) without spinning or sliding out of its original position (van den Engh et al., 2000;Vassiliou, 2018;Vassiliou et al., 2017a). Improving the numerical models will reduce the epistemic uncertainty and will allow for better prediction of the response of wobbling structures that can in turn enhance their applicability in earthquake engineering applications. ...
Conventional validation of analytical and numerical models in Earthquake Engineering involves the comparison of numerically simulated response time histories to experimentally obtained benchmark responses to the same earthquake excitations. As the seismic design problem is inherently stochastic, an alternative, statistical and easier-to-pass validation procedure has been suggested. As an example, numerical and analytical models may fail to predict the planar rocking response of a rigid block to a specific ground motion, but they can be proven quite successful in predicting the statistical distribution of the maxima of that response to an ensemble of ground motions. This paper describes the publicly available data obtained from a series of 226 shake table tests of a 3D rocking podium structure, designed at ETH Zurich and carried out at EQUALS Lab, University of Bristol. This well documented dataset is the largest one involving a shake table and can be used to statistically validate analytical and numerical models of rocking structures.
... However, the tests unveiled that the slab sustained significant torsional motion (even though it was nominally symmetric). This torsional motion was neither explained nor captured by the simplified numerical model 61,62 used in Ref. 54 This paper attempts to understand the source of this torsion following two approaches: a) By performing shake table tests of the same elements tested in Ref. 54, but with a stiffer restraining system (that provides positive post-uplift stiffness) to observe whether the torsion will be reduced or even eliminated; b) By developing a detailed finite element (FE) model that can account for imperfections of the columns that lead to break of symmetry and torsion. In parallel, as it has been argued that rocking systems can rely on their inherent impact and friction damping 63 and that no extra dissipators should be provided, this paper uses the validated FE model to evaluate whether adding yielding rebars to the columns to create a flag shaped system improves its behavior. ...
... The structure has finite stiffness in the pre-uplift phase, contrary to what rigid body models assume. 61,62 For each direction (x or y), the model uplifts at approximately 0.09-0.10 g of acceleration. ...
This paper presents the shake table testing and finite element (FE) modeling of a modular prefabricated concrete bridge‐like specimen. The specimen comprised four equal‐height cylindrical reinforced concrete (RC) columns capped with an RC slab. The structural connections were non‐monolithic. Hence, controlled relative motion of the members, including rocking (uplift) of the piers, was allowed. The columns were connected to the slab with stiff tendons that provided positive post‐uplift stiffness. The specimen was subjected to 184 triaxial shake table tests, so that a statistical validation of numerical models can be performed. Subsequently, a detailed three‐dimensional FE model of the bridge was developed. The objectives of the present study were to: i) investigate the shake table response of a modular bridge with positive post‐uplift stiffness under multiple ground motions, ii) develop an FE model of the proposed structural system, iii) investigate the influence of geometrical imperfections on rocking bridges, and iv) evaluate the efficiency of using additional dissipative rebars. After being subjected to 184 shake table tests, the specimen showed zero damage, moderate displacements and tendon forces ( TFs ), low slab torsion, and zero residual displacements. The shake table tests were practically repeatable. The proposed FE model accurately captured the experimental results. Geometrical imperfections heavily affect the response of negative stiffness systems. However, they have a marginal influence on positive stiffness systems. When comparing systems with equivalent uplift resistance and post‐uplift stiffness, the use of additional dissipative rebars results in lower slab torsion and TFs , provided that the rebars do not fracture.
... Vassiliou et al. [46] studied a 2DOF rigid cylinder constrained to wobble above its initial position without twisting or sliding, i.e. without stepping out of its original position (Fig. 2, left). Subsequently, Vassiliou [47] extended this model to include a slab on top of a set of cylindrical columns (Fig. 2, right). ...
... Cylinder allowed to wobble without twist[46] (Left); Wobbling frame[58] (Right). ...
Rocking motion is sensitive to the boundary and initial conditions of a rocking structure. Thus, the claims that numerical rocking motion models are not only inaccurate, but that all rocking structures behave unpredictably. Hence, rocking is not used as a seismic design approach. This paper revisits the issue of rocking motion unpredictability. Seismic behavior of structures is inherently stochastic, because the loading is stochastic. Therefore, the question of interest is not whether models can predict the seismic response to a single ground motion, but if the statistical characteristics of the ensemble of responses to a set of ground motions that define the seismic hazard can be predicted. For this purpose, a rocking podium, which is a three-dimensional structure comprising an aluminum slab supported by 4 tubular steel columns, was tested
on a shake table excited by two sets of 100 consistently generated ground motions. It was found that the Cumulative Distribution Function (CDF) of the experimentally obtained displacements is statistically stable. Next, a blind prediction contest was organized. The contestants were invited to predict the CDFs of the slab lateral displacement. They were able to predict the slab displacement CDF relatively well. Both finite element and discrete element modeling approaches were used, but no clear pattern emerged as it was found
that the performance of either approach depends on the input parameters used and the assumptions made.
... Vassiliou et al. [56] studied a 2DOF rigid cylinder constrained to wobble above its initial position without twisting or sliding, i.e. without stepping out of its original position (Figure 2, left). Subsequently, Vassiliou [57] extended this model to include a slab on top of a set of cylindrical columns (Figure 2, right). ...
... Cylinder allowed to wobble without twist[56] (Left); Wobbling frame[68] (Right). ...
... Rocking has been proposed as a seismic isolation method for both bridges [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and buildings [21][22][23], because uplift works as a mechanical fuse and limits the design forces of both the superstructure and the foundation. Unlike structures designed to yield, the free rocking rigid block exhibits negative post-uplift stiffness [24]. ...
... The columns have a height of 9.6 m and a diameter of 1.6 m, whereas the deck is much heavier than the columns (γ m →∞). Planar rocking (i.e., one directional excitation) is assumed as a first approximation (even though this has been proven unconservative [10,11]). Then, the proposed design steps are: 1. Calculate the normalized yielding strength of the system (f up /(mg)). ...
... Accordingly, the seismic behavior of the combined objects can be evaluated using approaches developed in the literature, for example, in Housner (1963) [19] and Ishiyama (1982). [20] To investigate the rocking behavior of a freestanding cylinder, Vassiliou et al. (2017) [26] developed a three-dimensional (3D) analytical rocking model. When three degrees-of-freedom were considered, including damping, it was found that the 3D model could more accurately represent the rocking behavior than the two-dimensional (2D) model. ...
... Accordingly, the seismic behavior of the combined objects can be evaluated using approaches developed in the literature, for example, in Housner (1963) [19] and Ishiyama (1982). [20] To investigate the rocking behavior of a freestanding cylinder, Vassiliou et al. (2017) [26] developed a three-dimensional (3D) analytical rocking model. When three degrees-of-freedom were considered, including damping, it was found that the 3D model could more accurately represent the rocking behavior than the two-dimensional (2D) model. ...
Freestanding nonstructural objects are very common in various infrastructure systems, but their dynamic properties with variations in the center of gravity (CG) are not yet fully understood. To investigate the dynamic properties of a freestanding object as its CG is varied, a steel frame model was fabricated using movable steel plates. Two scenarios, with varying and fixed CG, were considered in the free rocking tests. The experimental rotation responses generally matched the analytical responses in the initial cycles, and the discrepancy increased as the height of the CG (hcg) increased. The magnitude of the quarter period exhibited a similar trend. When the CG was fixed, the resulting magnitude of the quarter period varied slightly with hcg, indicating that the CG is one of the primary parameters affecting the quarter period. When the CG was varied, the experimental energy reduction factor was generally larger than the analytical results computed using the methods suggested in the literature, and it was scattered around the fixed CG value. The magnitude of the damping ratio was also scattered as the quarter cycle number increased, and it slightly decreased as hcg increased. An equivalent rectangular rigid block model with an equivalent height (heq) was employed to compute the geometric and dynamic parameters of the steel frame. The accuracy of this measure was investigated in terms of the quarter period and energy reduction factor. This work is beneficial for understanding the rocking behavior of freestanding objects.
... Following Housner's derivation of the equation of motion of the block at smooth time instants and of the kinetic energy dissipation due to instantaneous and perfectly inelastic impacts, the single-block rocking problem has been extensively explored (e.g., see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). Generalizations have been later on considered to describe the dynamic behavior of multi-block 2D masonry structures, either simplified as single-degree-of-freedom systems (e.g., see [22][23][24][25][26][27][28][29]) or treated as multi-degree-of-freedom systems (e.g., see [30][31][32][33][34][35]). In addition, the spatial rocking motion of a single 3D rigid block has been investigated (e.g., see [36][37][38][39][40][41]). ...
... Moreover, uplift works as a fuse, capping the accelerations transmitted to the structure. Hence, rocking can be used as a seismic design (seismic isolation) methodology, both for buildings [30][31][32][33][34][35][36] and bridges [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. ...
... The rocking and wobbling (unsteady rolling) motion of cylindrical bodies was studied elsewhere. 12 Very recently, Pradhan et al. 13 developed and validated a general-purpose 3D rocking model applicable for both unidirectional and bidirectional shaking. This study has also made a stability analysis considering the nonlinear nature of the phenomena. ...
The dynamics of three-dimensional (3D) slender systems such as statues, obelisks, monuments, museums, and household objects are highly nonlinear. In this study, a numerical model for rigid blocks in 3D, referred to as the physical model (PM), is developed to explore and view the motion of the slender blocks to base excitations. The model can represent the dynamics of both symmetric and mass-eccentric 3D blocks for unidirectional and bidirectional excitations. Keeping the kinematics of the system in view, the PM is developed as an assemblage of different subsystems in the Simscape Multibody Library. The rigid block supported by four tiny spheres at the corners is resting on a rigid table. Physical interaction between the table and the rigid body is simulated by introducing a virtual plane characterized by a spring and a damper that follows the visco-elastic Kelvin model. The values of the depth of the virtual plane, parameters of the Kelvin model, and the dimension of the tiny spheres are suitably chosen for such interaction. Adequately large values for kinetic and static friction are used in the virtual plane to prevent the body from sliding during its rocking motion. The motion of the table is described by choosing the inertial reference frame with origin at a fixed point and another parallel frame fixed with the tabletop. The response of the rigid body is described by the asymmetric Euler sequence. A third set of the frame, i.e. the body-fixed reference with the origin at the centre of mass of the corresponding symmetric body is introduced by a Bushing joint. A calculated amount of mass-eccentricity is generated by adding very small square blocks to the side-faces of the 3D symmetric block. The response of mass eccentric systems to trigonometric base excitations applied in-plane and out-of-plane of the eccentricity is computed using the PM, and a comparison of the same to the results derived analytically confirms the adequacy of the model. In the sequel, representative case studies are presented and interpreted to uncover the complex dynamics of the mass eccentric rocking systems. The PM can also serve as the basis of analyzing and viewing the motion of blocks with arbitrary geometry.
... Moreover, uplift works as a fuse, capping the accelerations transmitted to the structure. Hence, rocking can be used as a seismic design (seismic isolation) methodology, both for buildings [30][31][32][33][34][35][36] and bridges [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. ...
This paper presents a three-dimensional finite element model to predict the shake-table response of a rocking bridge-like specimen. The numerical model is statistically validated against experimental results, which involved testing of the system under 169 three-directional ground motions. The model comprises four cylindrical rocking columns capped with a concrete slab. The columns are connected to the slab with flexible tendons and they are allowed to uplift and wobble. The use of flexible tendons allows for large displacements of the system and negative post-uplift stiffness. This mechanism acts as a form of seismic isolation, limiting the accelerations transmitted to the superstructure.
The rocking columns, the slab and the shake table are modeled using elastic elements. The tangential behavior of the contact surfaces is modeled with Coulomb friction, which is the main energy dissipation mechanism. The tendons are modeled with equivalent elastic springs. The numerical analysis accounts for the geometric non-linearity of the response.
Rocking motion is sensitive to the parameters that define it and experimental tests are often non-repeatable. Hence, this study employs a statistical approach to validate the proposed numerical model. The cumulative distribution function (CDF) of the response quantity of interest (e.g., maximum displacement at the center of the slab) is employed, instead of comparing the numerical and experimental results one-by-one for each test (deterministic comparison). The deterministic validation of the model shows a moderate correlation of the experimental and the numerical results. However, the model can accurately predict the statistical response for both parameters of interest (i.e., maximum displacement and maximum rotation) of the system under 169 ground motion excitations.
... The system contains a range of dynamical behaviour including period-doubling cascades. Rigid cylinders in three dimensions, such as classical columns or grain silos, may experience the same types of behaviour and have been explored in the structural engineering literature [21,22]. Such work may benefit from the approach taken in this paper. ...
The rocking can problem consists of a empty drinks can standing upright on a horizontal plane which, when tipped back to a single contact point and released, rocks down towards the flat and level state. At the bottom of the motion, the contact point moves quickly around the rim of the can. The can then rises up again, having rotated through some finite angle of turn . We recast the problem as a second order ODE and find a Frobenius solution. We then use this Frobenius solution to derive a reduced equation of motion. The rocking can exhibits two distinct phenomena: behaviour very similar to an inverted pendulum, and dynamics with the angle of turn. This distinction allows us to use matched asymptotic expansions to derive a uniformly valid solution that is in excellent agreement with numerical calculations of the reduced equation of motion. The solution of the inner problem was used to investigate of the angle of turn phenomenon. We also examine the motion of the contact locus and see a range of different trajectories, from circular to petaloid motion and even cusp-like behaviour. Finally, we obtain an approximate lower bound for the required coefficient of friction to avoid slip.
... Such a rocking mechanism limits the design loads and moments on both the structure and the foundation, offering great optimization potential. The effectiveness of rocking as an earthquake hazard mitigation strategy has been explored for various classes of structures, including rigid blocks, [1][2][3][4][5][6][7][8] columns of ancient temples [9][10][11][12][13][14][15][16][17][18] rocking frames, [19][20][21][22][23][24][25][26][27] rocking bridges [28][29][30][31][32][33][34][35][36] and rocking buildings. [36][37][38][39][40][41][42][43][44] Makris and ...
Problems for which it is impossible to make a precise causal prediction are commonly tackled with statistical analysis. Although fairly simple, the problem of a rocking block on a rigid base subjected to seismic excitation exhibits a fascinating, complex response, making it extremely difficult to validate numerical models against experimental results, thus calling for a statistical approach. In this context, this paper statistically studies the rocking behaviour of rigid blocks, excited by synthetic far‐field ground motions. A total of 50 million analyses are performed, considering rocking blocks of height ranging from 1 to 20 m and slenderness angle ranging from 0.1 to 0.35 rad. The results are used to explore the performance of different ground motion intensity measures (IMs), in terms of their ability to predict the maximum rocking rotation. By comparing the efficiency, sufficiency and proficiency of the IMs, it is found that the peak ground velocity (PGV) performs optimally. Then, fragility curves are constructed using different IMs, concluding again that the PGV is the most efficient IM. Impressively, the fragility curves for different block sizes collapse to a single curve, if a non‐dimensional IM that involves PGV and the block geometry is used. Finally, the results produced on the basis of far‐field synthetic motions are compared to results based on recorded ground motions.
... As the starting point of this problem, theoretical studies on rocking response of the rigid blocks with two-dimensional (2-D) geometries subjected to specific ground motions of the rigid bases were conducted [1][2][3][4][5][6][7][8]. Theoretical models were then proposed for the 3-D rocking problem [9][10][11][12][13]. For example, a 3-D rigid block model was developed, which can rock around a side or a vertex of the rectangular base. ...
The rocking response of freestanding building contents simplified as rigid bodies was usually investigated in two dimensions with ground motions as the input, while their three-dimensional (3-D) rocking response subjected to horizontal bidirectional floor motions needs further investigation in both deterministic and probabilistic views. In this paper, the rocking response of freestanding 3-D rectangular blocks subjected to seismic excitations with two horizontal components are first investigated by shake table tests and finite element (FE) numerical simulation. An incremental dynamic analysis-based overturning fragility assessment method for the blocks within a building subjected to bidirectional ground motions is then proposed, using uncoupled nonlinear FE models of the block-building systems. The overturning fragility of six blocks within a four-story reinforced concrete frame structure is assessed. The influence of excitation directions, floor levels, block slenderness ratios (2.5, 3, 3.5 and 4), block sizes (size parameters of 91 mm, 182 mm and 364 mm) and intensity measurements (IMs) on the overturning fragility of the block is characterized. The experimental overturning fragility curves of three blocks in real conditions under bidirectional seismic excitations are finally obtained from the shake table tests. Rocking-torsional rotations are observed for the block with sliding not allowed. The block rocks around a side or a vertex while overturns around a side. Except the largest block, the block under bidirectional excitations is usually more and sometimes equally vulnerable to overturn in comparison to that under unidirectional excitations. The block on a higher floor is found to be more vulnerable to overturn than that on a lower floor considering the peak ground acceleration-based IM. These observations highlight on the significance of 3-D analysis for the rocking blocks considering floor acceleration amplification. This paper also enlarges the experimental database of seismic rocking response of rectangular blocks under bidirectional horizontal motions.
... Under real seismic excitation, rocking motion of even a symmetric body is three-dimensional (3D) in nature particularly when base aspect ratio is close to unity. Constraining the rocking body to restore its original position, (sliding and rolling-out phenomena locked), Vassiliou et al. [51] studied rocking and wobbling (unsteady rolling) of cylindrical bodies. Employing multi-degree-of-freedom (MDOF) models, investigation on the dynamics of non-cylindrical bodies such as 3D prisms [50,[52][53][54][55] have noticed permanent displacement of the body. ...
Highly nonlinear behaviour of slender rigid blocks subjected to base excitations is intended to be explored for a 3D prism. To this end, a physical model (PM) capable of handling the intricate dynamics and of viewing the motion of a rigid body during rocking/ overturning has been developed. Limited case studies show satisfactory performance of the PM to represent dynamics of such blocks both in unidirectional and bidirectional excitations. Subsequently, the dynamics of the3D prism has been studied under trigonometric pulses. Interestingly, it appears that, under bidirectional shaking, the body may rotate about its own longitudinal axis and, due to this spin, the body may ‘walk’ from its original position even when sufficient friction at the interface prevents sliding. This residual displacement of the body under no sliding condition could not be recognized from 2D analysis. Using principles of dimensional analysis, self-similarity in response for complex nonlinear motions of the rocking oscillators has been established. This follows a comparative account on overturning spectra for 2D and 3D motions constructed in non-dimensional formats. A prognosis to chaos is uncovered from a scrutiny of long-term behavior of rocking oscillators in 3D using a set of non-dimensional parameters.
... Inverted pendulum (IP) system was firstly introduced as cart inverted pendulum which has limitation related to its length (Huang et al., 2019b;Roose et al., 2017;Vassiliou et al., 2017;Huang et al., 2019a;Huang et al., 2010). For this reason, IP system was extended by introducing the Rotary-IP (RIP) system which has two main components including arm which can rotate in horizontal plane and pendulum connected to the arm which can rotate in vertical plane (Fukushima et al., 2014;Dwivedi et al., 2017;Nath and Dewan, 2017;Du et al., 2019;Watson et al., 2019). ...
In this paper, the finite-time stabilization of the disturbed and uncertain rotary-inverted-pendulum system is studied based on the adaptive backstepping sliding mode control procedure. For this purpose, first of all, the dynamical equation of the rotary-inverted-pendulum system is obtained in the state-space form in the existence of external disturbances and model uncertainties with unknown bound. Afterward, a novel command filter is defined to enhance the control strategy by consideration of a virtual control input. Therefore, the differential signal is replaced by the output of the command filter to reduce the complicated computing in the control process. Hence, the finite-time convergence of the sliding surface to the origin is attested by using the backstepping sliding mode control scheme according to the Lyapunov theory. Besides, the unknown upper bound of the exterior perturbation and uncertainty is approximated providing the adaptive control technique. Finally, simulations and experimental results are done to demonstrate the impression and proficiency of the suggested method.
... The rocking motion of a rigid cylinder has been known for long time to be inherently three dimensional [34]. Rocking cylindrical columns have been studied numerically by using rigid models [25,37] as well as the discrete element method [1]. Some experimental investigations focusing on the spatial rocking behavior of ancient cylindrical columns have also been reported [19,30]. ...
Studies of rocking motion aim to explain the remarkable earthquake resistance of rocking structures. State-of-the-art assessment methods are mostly based on planar models, despite ongoing efforts to understand the significance of three-dimensionality. Impacts are essential components of rocking motion. We present experimental measurements of free-rocking blocks on a rigid surface, focusing on extreme sensitivity of impacts to geometric imperfections, unpredictability, and the emergence of three-dimensional motion via spontaneous symmetry breaking. These results inspire the development of new impact models of three-dimensional facet and edge impacts of polyhedral objects. Our model is a natural generalization of existing planar models based on the seminal work of George W. Housner. Model parameters are estimated empirically for rectangular blocks. Finally, new perspectives in earthquake assessment of rocking structures are discussed.
... The foundation often comprises up to 50% of the total Reinforced Concrete (RC) used in the project. To address this waste of material, [45][46][47][48][49][50][51][52][53][54][55] have proposed that a rocking system without a tendon (a "free rocking" system - Fig. 1) is stable enough and leads to much smaller design moments for the foundation. The analysis of such a system is fundamentally different to elastic and positive stiffness elastoplastic or flag-shaped systems, because it exhibits a negative post uplift stiffnessthat is, after uplift the restoring force decreases as displacement increases. ...
Letting a column uplift and sustain rocking motion has been suggested as a seismic design method for bridges. In an effort to increase the redundancy of a rocking bridge, most researchers use ungrouted restraining tendons passing through the columns. However, it has been argued that these tendons unnecessarily increase the design forces of the superstructure and of the foundation, and that rocking systems should be designed to be unrestrained.
In an effort to combine the benefits of both approaches this paper suggests the use of flexible restraining systems comprising a tendon in series with disc springs, essentially forming a seismic isolation method for precast structures. It presents cyclic tests of two 1:5 scale RC columns with ends protected either with steel jackets or with steel discs. The columns are able to sustain drifts of more than 15% (and in some cases 30%) without any significant damage – hence they are resilient. The behavior of the disc springs is well predicted by analytical models. The strength (i.e. uplift force) and post uplift stiffness of the system can be predicted with a reasonable accuracy using a rigid body model. However, the rigid body model does not predict well the pre-uplift behavior. As the tendon is anchored within the column, the design moment of the foundation drastically decreases, therefore costly and material intensive pile foundations could be avoided – hence the design concept contributes to sustainability.
... Figure 5 shows a 3D extension of the rocking frame model. 62,63 The assumptions made for the planar frame (rigid bodies, no sliding or "flying" allowed, pointwise contact) are extended to include the following: ...
This paper presents the shake table test results of a novel system for the design of precast reinforced concrete bridges. The specimen comprises a slab and four precast columns. The connections are dry and the columns are connected to the slab by an ungrouted tendon. One of the tendon ends is anchored above the slab, in series with a stack of washer springs, while the other end is anchored at the bottom of the column. The addition of such a flexible restraining system increases the stability of the system, while keeping it relatively flexible allowing it to experience negative post‐uplift stiffness. It is a form of seismic isolation. Anchoring the tendon within the column, caps the design moment of the foundation, and reduces its size. One hundred and eighty‐one shake table tests were performed. The first 180 caused negligible damage to the specimen, mainly abrasion at the perimeter of the column top ends. Hence, the system proved resilient. The 181st excitation caused collapse, because the tendons unexpectedly failed at a load less than 50% of their capacity (provided by the manufacturer), due to the failure of their end socket. This highlights the importance of properly designing the tendons. The tests were used to statistically validate a rigid body model. The model performed reasonably well never underestimating the median displacement response of the center of mass of the slab by more than 30%. However, the model cannot predict the torsion rotation of the slab that was observed in the tests and is due to imperfections.
... In rocking structures, uplift works as a mechanical fuse and limits the design forces of the foundation. Therefore, several researchers have proposed that bridges can be designed as rocking frames [36][37][38][39][40][41][42][43][44][45][46]. They have claimed that restraining the rocking motion, as the PRESSS idea suggests, is not always needed in terms of stability and unnecessarily increases the design moment of the foundation [47,48]. ...
... These design moments often require huge pile foundations which can comprise up to 50% of the total Reinforced Concrete used in the whole project. Aiming at decreasing the foundation design moments and obtaining a more sustainable system, it has been suggested either to not restrain the bridge columns at all [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] or to restrain them with flexible restraining systems [43][44][45][46]. In an effort to bring the concept of negative stiffness closer to practice, this paper presents quasi-static cyclic tests of a restrained RC concrete column exhibiting negative post-uplift stiffness. ...
... The analytical model proposed by Housner describes the planar response of a rigid rocking body when subjected to one-directional excitation. However, under seismic excitation, rocking structures are subjected to bidirectional (or three-directional when the vertical acceleration is considered) excitation [66][67][68][69]. Under these conditions, an unanchored body may rock in 3D dimensions (wobble). ...
... Rocking has been proposed as an alternative design method in seismic prone regions because it reduces foundation moments and results in resilient structures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Differently from structures designed to yield, rocking systems present negative lateral stiffness after uplift. ...
... More recently, Vassiliou et al. (2017) derived a twodegree-of-freedom model that describes the 3D rocking and wobbling (unsteady rolling) motion of a rigid cylinder with the constraint that it does not slide or roll-out of its original position. The model is the simplest 3D extension to a cylinder of the 2D Housner (1963) rigid block rocking model. ...
A three dimensional distinct element model was developed for the multiblock Baalbek columns to investigate the response of the existing columns as well as 0.5m and 1m scaled models used in shaking table experiments. Periodic pulses of varying shapes, periods, and accelerations were applied and the critical combinations of acceleration and period that caused collapse were noted. The response and mode of collapse are shown to be dependent on the period and the amplitude of the pulse. Depending on the size of the column and the period of the applied critical pulse three modes of collapse are observed: (1) multiblock rocking/wobbling followed by toppling of one or more blocks; (2) rock and/or wobble then overturn along or outside the applied pulse plane; (3) overturn after one or two impacts along the plane of the applied pulse. The displacement history of the column was plotted alongside the ground motion to visualize and quantify the rocking and /or wobbling. All column sizes showed the same response qualitatively in the acceleration-period space and the wobbling was shown to be confined to a well-defined region. Stability charts delineating the safe-unsafe boundaries showed bi-modal behavior for all pulses tested but with a shift due to the different pulse energy contents. Numerical results compared well with experimental data from shaking table experiments on the scaled columns. The capabilities of the three dimensional numerical modeling approach are confirmed in this work for use as a predictive tool or an archaeoseismic analysis tool for columns and large masonry structures.
... The analytical model proposed by Housner describes the planar rocking response of a rigid body when subjected to one-directional excitation. However, real rocking structures are subjected to bidirectional (or three-directional when the vertical acceleration is considered) excitation [43][44][45][46]. Under these conditions, an unanchored body may rock, uplift, translate with the ground, and/or wobble. ...
... Rocking structures are not fixed to the ground, but can uplift and sustain a strongly nonlinear motion. It has been suggested that this uplift can be used as a form of seismic isolation (sometimes referred to as ''rocking isolation'') for both buildings (Bachmann, 2018;Bachmann et al., 2018bBachmann et al., , 2019Cherepinskiy, 2004;Rı´os-Garcı´a and Benavent-Climent, 2020;Uzdin et al., 2009) and bridges (Agalianos et al., 2017;Dimitrakopoulos and Giouvanidis, 2015;Kashani et al., 2018;Makris and Vassiliou, 2013;Reggiani Manzo and Vassiliou, 2021;Sideris et al., 2014;Thiers-Moggia and Ma´laga-Chuquitaype, 2020;Vassiliou, 2018;Vassiliou et al., 2017). However, until recently, it was believed that rocking motion is unpredictable by analytical or numerical models, because numerical models often failed to reproduce the experimental data available in the literature (Aslam et al., 1980;Drosos and Anastasopoulos, 2014;ElGawady et al., 2011;Kalliontzis and Sritharan, 2018;Lipscombe and Pellegrino, 1993;Ma, 2010;Mouzakis et al., 2002;Pen˜a et al., 2007;Priestley et al., 1978), especially when it comes to seismic response. ...
Conventional validation of analytical and numerical models in Earthquake Engineering
involves the comparison of numerically simulated response time histories to experimentally obtained benchmark responses to the same earthquake excitations. As the seismic design problem is inherently stochastic, an alternative, statistical, and easier to-pass validation procedure has been suggested. As an example, numerical and analytical models may fail to predict the planar rocking response of a rigid block to a specific ground motion, but they can be proven quite successful in predicting the statistical distribution of the maxima of that response to an ensemble of ground motions. This article describes the publicly available data obtained from a series of 226 shake table tests of a 3D rocking podium structure, designed at ETH Zurich and carried out at EQUALS Lab, University of Bristol. This well-documented dataset is the largest one involving a shake table and can be used to statistically validate analytical and numerical models of rocking structures.
... The rocking motion of a rigid cylinder has been known for long time to be inherently three-dimensional [32]. Rocking cylindrical columns have been studied numerically by using rigid models [23,35] as well as the Discrete Element Method [1]. Some experimental investigations focusing on the spatial rocking behaviour of ancient cylindrical columns have also been reported [29,17]. ...
Studies of rocking motion aim to explain the remarkable earthquake resistence of rocking structures . State - of - the -art assessment methods are mostly based on planar models , despite ongoing efforts to understand the significance of three - dimensionality . Impacts are essential components of rocking motion . We present experimental measurements of free -rocking blocks , focusing on extreme sensitivity of impacts to geometric imperfections , unpredictability , and the emergence of three - dimensional motion via spontaneous symmetry breaking . These results inspire the development of new impact models of three dimensional facet and edge impacts of polyhedral objects . Our model is a natural generalization of existing planar models based on the seminal work of George W. Housner . Model parameters are estimated empirically for rectangular blocks . Finally , new perspectives in earthquake assessment of rocking structures are discussed .
... Note that, the column-cap-beam connection could be designed in a similar way. For the general case of nonplanar motion though, the reader is referred to [61,102,132,133], while sliding is thoroughly examined in [112]. The present section investigates the seismic performance of the symmetric rocking frame ( Fig. 11) and evaluates the effects of basic design parameters within the context of potential future applications. ...
Conventional (code-based) seismic design aims to provide the structure the necessary strength and ductility to withstand seismic forces. A strong earthquake, though, might be accompanied by irreversible structural deformation and therefore damage. The aftermath of recent severe earthquakes (e.g., the 2010 Canterbury, the 2011 Tohoku, as well as, the 2016 Central Italy and Kaikoura among others) highlighted the need for alternative design methodologies that can limit structural damage and guarantee post-earthquake serviceability. Rocking behavior isolates the structure from the ground to withstand strong seismic forces. Rocking allows the structure to uplift and pivot around predefined points relieving it from deformation, stresses and ultimately damage. Consequently, rocking behavior is currently resurging.
Due to the increasing demand to predict the response of various rocking configurations, the present study sheds light to major challenges of rocking dynamics. As a first approach, it investigates, analytically and numerically, the seismic performance of a rocking frame which is either freestanding or hybrid i.e., supplemented with re-centering and energy dissipation devices exhibiting flag-shaped hysteretic behavior. This work establishes the equations of motion for the hybrid (and the freestanding) rocking frame following principles of analytical dynamics and examines its seismic behavior under both mathematical (pulse-type) and historic ground excitations. Throughout the analysis, the deformation of the structural members and sliding between the contacting bodies are ignored. The study introduces dimensionless design parameters which control the flag-shaped hysteretic loop and reveals their influence on the seismic performance of the hybrid rocking frame. The results unveil that prestressing the tendons could be beneficial mostly for small rocking rotations. For large rocking rotations, the prestressing force becomes progressively detrimental as the size of the columns increases. This study also compares hybrid rocking frames with negative, zero and/or positive post-uplift lateral stiffness. The analysis shows that increasing the stiffness of the frame does not necessarily lead to superior performance. Further, (hybrid) rocking frames with negative stiffness yield the lowest hysteretic energy demands, whereas positive stiffness frames might mitigate the response but at the expense of higher energy demands by the dissipaters; making the examined rocking frames sensitive to the characteristics of the ground motion. Consequently, the hybrid rocking frame could outperform the freestanding or the opposite.
Further, this study thoroughly investigates the contact phenomenon during rocking motion. Specifically, it revisits analytically and numerically the contact process adopting a nonsmooth dynamics approach assuming impacts behave as unilateral contacts. To validate the proposed methodology, it examines both rigid and flexible structures, specifically, the archetypal rigid rocking block and the flexible rocking oscillator. The present work treats impact and uplifting events by introducing a system of inequalities, which is known as the linear complementarity problem. Impact is considered to be instantaneous and is described by contact laws (i.e., Newton’s and/or Poisson’s). These set-valued laws capture the behavior in the normal direction of the (unilateral) contact. In the tangential direction, sliding is prevented either by designing the contact points to act as shear keys or by sufficient friction coefficient. This study demonstrates the ability of the proposed methodology to capture the impact behavior during rocking and liberates from the need for additional ad-hoc assumptions. The analysis further verifies pertinent analytical results from other methodologies for both examined structures, and proves that a given rocking oscillator might display different post-impact state i.e., bouncing, full contact, or immediate rocking, depending on its flexural deformation at the time of impact. In addition, the proposed response-history analysis of the flexible rocking oscillator is also validated with numerical and experimental results from literature.
Finally, due to the sensitivity of the rocking behavior to the characteristics of the ground motion, this work investigates the ability of (natural) earthquake records to induce rocking
demands on different rigid structures. Focusing on freestanding rocking configurations of
different size and slenderness subjected to a large number of historic earthquake records, this study unveils the predominant importance of the strong-motion duration to rocking amplification (i.e., rocking 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 scalar IMs which hinge on intensity, frequency-content, duration and/or energy-input characteristics of the ground excitation 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.
... Rocking design, on the other hand, relieves the structure from seismic damage by allowing structural components to uplift and pivot during an earthquake. After the Chilean earthquake in 1960, the seminal work of Housner [1] revealed the benefits of rocking design and triggered a thorough investigation on the seismic performance [2,3,4,5,6,7,8,9,10,11,12] of various rocking structural systems [13,14,15,16,17,18]. ...
Conventionally designed bridges, i.e. when the column is monolithically connected with the ground (fixed-base), sustain considerable damage at the column ends after severe earthquakes. Seismic damage often determines whether or not the bridge remains functional after an earthquake event. Rocking isolation, on the other hand, allows structural components to uplift and pivot; thus, in principle, it relieves the structure from excessive deformations and damage. However, rocking isolation is still rarely applied in engineering practice, mainly due to the lack of thorough understanding of its dynamic (seismic) performance and its post-earthquake financial benefits. This paper redirects our attention to the main benefits of rocking design, and conducts a thorough seismic loss assessment adopting two different rocking bridge configurations in terms of their post-earthquake economic losses and resilience. In particular, this work extends the well-established performance-based earthquake engineering framework to evaluate the seismic losses of the examined rocking structures accumulated following severe seismic events and quantify their post-earthquake resilience. The analysis reveals the considerably mitigated seismic losses and the remarkable post-earthquake resilience that a rocking bridge offers when carefully designed. In particular, even a slight modification of the slenderness of the structure leads to a substantial enhancement of its seismic performance; reinforcing its potential as an alternative seismic design paradigm for bridges. The above findings illustrate the considerable financial benefits of such innovative seismic-resistant structural systems, which can serve as an efficient seismic design paradigm for future bridge engineering applications.
In recent years, cylindrical structures free to rock have been exploited in practical engineering. However, their seismic response in three dimensions (3D), greatly sensitive to the parameters that define it, is difficult and time‐consuming to predict. To this end, this study focuses on developing a rocking spectrum, an efficient graphical tool linking seismic rocking response to structural parameters, for seismic response prediction and performance‐based seismic design of cylindrical structures. The development of the rocking spectrum is based on the numerical rocking response of 2500 idealized rigid cylinders excited by 100 sets of synthetic bidirectional pulse‐like ground motions. The minimum Redundancy Maximum Relevance (mRMR) algorithm is first employed to reveal that the rocking response is more related to ground acceleration, ground velocity, and ground displacement when the response is small (close to uplift), intermediate, and large (close to overturning), respectively. Following these relations, the support vector machine (SVM) algorithm is employed to develop the rocking spectrum. The obtained rocking spectrum can reliably predict the rocking response of cylinders subjected to the synthetic pulse‐like ground motions. The applicability of the spectrum is also discussed for as‐recorded pulse‐like ground motions.
This study established a numerical model for soil-structure interaction (SSI) to examine the effects of the spatial incidence angle of SV waves and soil nonlinearity, utilizing viscoelastic artificial boundaries (VAB) and equivalent nodal force (ENF) method. Both the foundation's and superstructure's torsion and rocking responses were then analyzed. The findings indicate that subjected to spatially oblique incident SV waves, the rectangular foundation primarily has the rocking response while the torsional response is negligible. Furthermore, the maximum torsional and rocking angles about the x-axis at each frame floor are significantly enlarged by comparison with the perpendicular incident case. Moreover, the soil nonlinearity could increase the foundation's rocking angle and enlarge the maximum torsion and rocking responses of the structure's floors. Consequently, structural seismic damage assessment requires considering both the soil nonlinearity and incident seismic wave angles.
This dissertation investigates response modification techniques for the seismic protection of structures. The proposed methodologies include the use of: i) rolling-based seismic isolators for the protection of low-rise buildings and ii) rocking piers for the protection of prefabricated bridges. The two methodologies stand out due to their geometry- and gravity-driven stability under lateral loads. The geometry-based restoring force of the two systems provides recentering after earthquake loading. Moreover, the reduced cost of the proposed systems could allow for their implementation both in the developed and in the developing world.
The proposed seismic isolators comprise a sphere rolling between concrete plates. The use of concave concrete plates provides gravitational restoring force. Three different types of spheres are considered: i) solid spheres made of natural rubber (NR), ii) spheres made of polyurethane (PU) (solid or with a steel core), and iii) grout-filled tennis balls. The tennis balls are used as a permanent spherical mold, with their shell providing energy dissipation and distribution of the contact stresses.
This dissertation starts with the experimental and numerical investigation of the rolling-based isolators. Parameters of investigation included the sphere type, the geometry of the concrete plates, the testing velocity, the bearing load, the degradation due to consecutive loading, the bearing temperature, and the specimen-to-specimen variability. Different combinations of the aforementioned parameters were tested, leading to 30 different setups. The isolators were subjected to compression, sustained compression (creep test), lateral cyclic, and shake-table testing. In total, 238 lateral cyclic and 1658 shake-table tests were performed. The tests of the NR and the PU spheres were performed in 1:2 scale. The tests of the grout-filled tennis balls were in full scale.
Results showed that the deformability of the NR or PU spheres increases the energy dissipation and induces a flag-shaped lateral response. Contrariwise, the response of the grout-filled tennis ball isolators is bilinear and can be approximated by a rigid-body model. The use of concave plates provides gravitational restoring force. The energy dissipation of rolling systems is quantified by the rolling friction coefficient (the lateral-to-vertical force ratio at zero lateral displacement). The use of small spheres, high compressive loads, or softer spheres amplifies the deformability-related effects and increases the rolling friction coefficient. The rolling friction coefficient of PU spheres ranged between 4% and 7% (velocity-dependent). The same magnitude ranges between 5% and 20% for the NR spheres and between 4% and 8% for the grout-filled tennis balls (velocity-independent). Analytical equations are provided to describe the weight- and velocity-dependent response. The isolators that used NR or PU spheres were shake-table tested and limited the acceleration transmitted to the superstructure to 0.1-0.3 g (depending on the configuration), while maintaining moderate peak and practically zero residual displacements. The investigated isolators did not deteriorate after 65 ground motions, and the shake-table tests were repeatable.
Moving to the seismic protection of prefabricated bridges, this dissertation proposes the use of rocking piers, whose seismic stability originates from their geometry and their rotational inertia. This dissertation presents a three-dimensional (3D) finite element (FE) model to predict the seismic response of freestanding cylindrical columns. Different columns with varying slenderness and size were examined. The columns were able to slide, rock, and wobble. The FE results were statistically validated against a large database of experimental tests. The influence of all modeling and physical parameters was elucidated. Results show that, for the range considered, the numerical parameters do not influence the statistics of the response, even though they influence each individual oscillation. The friction coefficient between the interfaces (physical parameter) can influence the statistics of the response and should be carefully selected. Energy dissipation should be modeled explicitly, following the physics of the problem.
Finally, this dissertation presents the shake-table testing and FE modeling of a modular prefabricated concrete bridge. The specimen comprised four cylindrical reinforced concrete (RC) columns capped with a RC slab. The structural connections were non-monolithic and the controlled relative motion of the members (rocking of the piers) was allowed. The columns were connected to the slab with stiff tendons that provided positive post-uplift stiffness. The specimen was subjected to 184 triaxial shake table tests. Subsequently, a detailed three-dimensional finite element (FE) model of the bridge was developed. The objectives of the present study were to: i) investigate the shake table response of a modular bridge with positive post-uplift stiffness under multiple ground motions, ii) develop a FE model of the proposed structural system, iii) investigate the influence of geometrical imperfections on rocking bridges, and iv) evaluate the efficiency of using additional dissipative rebars in rocking systems. After being subjected to 184 shake table tests, the specimen showed zero damage, moderate displacements and tendon forces, low slab rotations, and zero residual displacements. The shake table tests were practically repeatable. The proposed FE model accurately captured the experimental results. Geometrical imperfections heavily affect the response of negative post-uplift stiffness systems. However, they have a marginal influence on positive post-uplift stiffness systems. The use of additional dissipative rebars results in lower slab rotations and tendon forces but may result in larger displacements due to rebar fracture.
p>This paper focuses on predicting the seismic overturning of freestanding cylindrical structures. Idealized cylinders of different sizes and slenderness are excited by synthetic pulse-like ground motions. A total of 245000 results are summarized in the form of the overturning spectrum. The obtained spectrum, however, shows large motion-to-motion variability. To reduce the variability, the support vector machine (SVM) algorithm is employed subsequently. Three geometry-related parameters of cylinders and twenty-five intensity measures characterizing ground motions are selected as candidate features. Using the minimum Redundancy Maximum Relevance (mRMR) algorithm and forward stepwise feature selection method, the optimal SVM model is determined by which model makes the least false-negative misclassification cases, that is, wrongly predicting actual overturning as non-overturning.</p
Numerical studies on freestanding structures typically leverage the Distinct Element Method where the response is dictated by joints; however, the response carries significant uncertainty due to joint stiffness parameters. This study aims to quantify the influence of joint stiffness parameters on the global response of freestanding structures subjected to seismic excitation in a probabilistic formulation. By using a range of joint stiffness values with rectangular rigid blocks of different aspect ratios, the impact of joint stiffness parameters is evaluated against a large suite of historical ground motions. Results indicate that increasing contact stiffness results in increased stability against overturning.
The relatively uncontrolled dynamic behavior of a rigid block resting on a flat surface under ground motion has been well studied. In contrast, a block with a gently inclined V- or W-shaped sliding key would ensure dynamic self-centering or resetting performance under various seismic conditions. To better understand the nonlinear responses of rigid blocks having this type of interface where both rocking and sliding movements are possible, an event-based algorithm is put forward to cover various types of motions, including the transition stages between different types of motions. A comprehensive study is carried out to examine the dynamic responses of blocks of different sizes and aspect ratios. Moreover, the simplified pure rocking model and pure sliding model are also adopted to obtain rough estimates of dynamic response for comparison. The results show that the simplified models are reliable for some special cases only, e.g. squat blocks and slender blocks. The study also evaluated the influence of the coefficient of friction and slope inclination on the resetting capabilities of typical sliding-prone and rocking-prone systems. For those cases that are prone to toppling or excessive sliding under strong earthquakes, the provision of vertical post-tensioning can effectively ensure stability and resetting performance. Such findings provide insight into the performance of precast segmental bridge columns with resettable sliding joints.
In recent years, there has been a growing recognition that the rocking motion of structures, especially cylindrical ones, should be evaluated with three‐dimensional (3D) models. However, dissipation mechanisms in 3D models are complex and have yet to be well established. This paper presents an analytical study on an inherent dissipation mechanism based on the rolling friction model and an external dissipation mechanism based on energy dissipators (EDs) for 3D rocking cylindrical structures. Through a variational formulation, the nonlinear equations of motion are derived considering the rolling friction model and the Bouc–Wen model. Compared with the existing inherent dissipation model in literature, the rolling friction model has a noticeable advantage that it can capture the energy dissipation behavior during free vibration under various initial conditions. Additionally, the rolling friction model is more versatile since the contact coefficient is adjustable. On the other hand, external EDs further enhance the energy dissipation of the structures. Roles of the inherent and external mechanisms in the 3D rocking motion of cylindrical structures against earthquakes are assessed using a series of near‐fault pulse‐like ground motions. Results indicate the inherent dissipation mechanism, that is, the rolling friction model, cannot avoid overturning under near‐fault pulse‐like ground motions, leading to unstable structures. This is because an uplifted cylinder will continuously absorb energy from earthquake excitations, while the amount of energy dissipated by the rolling friction model is small. Comparatively, adding external EDs is effective in mitigating seismic responses and thereby avoiding overturning.
To provide an insight for reliable seismic risk assessment of small building contents, the rocking response of freestanding wooden blocks subjected to 11 bidirectional horizontal ground motion records scaled to various intensity levels is investigated via shaking table tests. Three prisms and three cylinders with slenderness ratios of 2, 3 and 4, and three prisms having the same slenderness ratio of 3 but different size parameters of 158 mm, 237 mm and 316 mm, are considered. The peak ground acceleration statistics corresponding to rocking and overturning of the blocks are obtained. Shaking table tests to investigate the rocking response of the prisms located on different floors of a full‐scale three‐story reinforced concrete frame structure were also conducted. Experimental fragility curves corresponding to rocking and overturning occurrence of the blocks are finally generated. The experimental results highlight that the minimum rocking accelerations are different for the prisms with the same slenderness ratio but different sizes. Moreover, compared with the prism on the ground, the prism on the top floor has significantly higher rocking demand, while the prism on the second floor may have lower rocking demand. Furthermore, the cylinder has higher rocking and overturning fragility than the prism with the same height and width. The rocking and overturning fragility increases as the block becomes slenderer or smaller. The proposed critical rocking acceleration under bidirectional seismic excitation in real conditions is a half of the theoretical one under unidirectional excitation.
With growing interest in rocking structures, it is important to quantify the effects of ground motion processing schemes on rocking response. To this end, this paper studied the influence of different ground motion correction schemes and parameter values on the rocking displacement spectrum. When the problem is treated on an individual ground motion basis, then it seems that ground motion correction does have an influence on the rocking response, especially for more slender blocks. This influence is larger for causal filtering. However, no specific trend related to the cut off period of the filter can be observed (at least for cut off periods longer than or equal to 10 s) or on the type of the recording device (analog or digital). On the other hand, when treating the problem statistically (by comparing the statistics of the response due to sets of ground motion, not a single ground motion), the effect is significantly reduced: The processing schemes do not induce any significant bias to the rocking response and the motion‐to‐motion variability seems more important than the ground motion correction method. This conclusion applies to both analog and digital records, both causal and acausal filters, and to all near field pulse like, near field no pulse like, and far field records.
This paper presents uniform risk spectra for zero stiffness bilinear elastic (ZSBE) systems. The ZSBE oscillator is a bilinear elastic system with zero post‐“yield” stiffness that satisfactorily predicts the response of different systems with negative lateral stiffness (e.g., free‐standing or restrained rocking blocks). It can be described by a single parameter; thus, it is simpler to produce its spectrum. Using the ZSBE proxy, this paper provides the uniform risk spectra for sites in six locations in Europe. The spectra are constructed using two distinct intensity measures (IMs): peak ground velocity (PGV) and peak ground acceleration (PGA). The efficiency of both IMs at different ranges of displacement demands is discussed and analytical approximations of the spectra are proposed.
Rocking motion is notoriously sensitive to the parameters that define it, with experimental tests oftentimes being non‐repeatable. Therefore, validating numerical models using a deterministic approach is impossible, since the consistency of any benchmark experimental test is dubious. Three‐dimensional rocking is even harder to predict than planar rocking. This paper presents a three‐dimensional finite element model to predict the statistics of the rocking/sliding response of free‐standing cylindrical columns. The response parameters of interest were the maximum displacement at the top of the columns and the residual displacement. Three different columns with varying slenderness and size were examined. The columns were able to slide, rock, and wobble in all directions, with this behavior being representative of building components and monumental structures. The numerical results were statistically compared to a large database of experimental tests, proving the accuracy of the proposed model. The influence of all modeling and physical parameters was elucidated, employing a large number of non‐linear time‐history analyses. It is shown that, when the numerical parameters are varied within a reasonable range, they do not influence the statistics of the response, even though they influence each individual oscillation. The friction coefficient between the interfaces (physical parameter) can influence the statistics of the response and should be carefully selected. Energy dissipation should be modeled explicitly, following the physics of the problem.
This paper presents cyclic tests of 1:5 scale restrained rocking RC columns exhibiting negative stiffness. The proposed recentering system can be used for the seismic design of resilient bridges. Negative stiffness leads to a reduction of the design moments of both the superstructure and the foundation, compared to fixed base or restrained rocking systems of positive stiffness. Moreover, the suggested design method, allows for the precasting of the substructure and easy assemblage on-site, therefore reducing on-site construction time, and consequently traffic impact. Cyclic tests were carried out on a reinforced concrete column allowed to rock on its interface with the foundation and cap-beam. An unbonded tendon was used to restrain the column. The tendon was connected in series to a disc ("Belleville") spring system so that the overall postuplift stiffness of the system stays negative. Steel plates were introduced at the ends of the columns to protect them from stress concentrations. The tests results show that the column returned to its original position with negligible to no residual displacement, even when tilted to 15% drift ratio.
To improve seismic performance of double-deck frame bridge system, a novel precast double-deck rocking frame bridge pier system is developed based on ductile and rocking seismic conception design in this paper. In this novel system, the upper columns are allowed to rock between the precast cap beams to form the rocking story, while the lower columns are connected to precast cap beam and footings by grouted splice sleeve to form emulative cast-in-place story. The seismic response of this novel frame bridge pier system is investigated. Nonlinear equations of motion of the system are established following Lagrange's equation and conservation of angular momentum, and the analytical model is used to conduct case study, parametric study and stability analysis. The results show that seismic responses of the lower columns of the proposed structural system are smaller than cast-in-place bridge pier, while it has an opposite trend for upper columns. The effect of deformability of the lower story on the stability of the system is marginal for low-frequency pulses or small column size, while for high-frequency pulses or large column size the deformability jeopardizes the stability of the system.
Rocking modifies the seismic response of structures, because uplifting works as a mechanical fuse and limits the forces transmitted to the structure. However, the engineering community is in general reluctant to let a structure uplift because it can overturn, and, more important, an unanchored structure has no redundancy against this failure mode. Using a safety factor for the design of a flat rocking foundation (i.e. designing it larger than minimum required to prevent overturning) goes against the essence of the rocking seismic isolation method, as the structure would end up behaving as fixed to the ground. To protect against overturning but preserve the ability to uplift we propose to extend the flat rocking foundation using curved wedges at its ends. This paper presents the results of dynamic tests of small bodies rocking on curved foundations. The results compare relatively well with the analytical solutions , but they are shown to be very sensitive to the coefficient of restitution.
This paper deals with the dynamic response of a free-standing ancient column in the Roman Agora of Thessaloniki, Greece as a means to shed more light on the complex behaviour of rocking bodies under seismic excitation. Discrete Element Method numerical analyses were carried out with the use of multiple seismic records based on the disaggregation of the seismic hazard for the area of interest. To identify their impact on structural performance, alternative earthquake Intensity Measures, such as Peak Ground Acceleration and Peak Ground Velocity are comparatively examined for the case of a column that sustained no visible permanent deformations during the Ms=6.5 Thessaloniki earthquake of 1978. The analysis revealed a weak correlation of PGA and PGV with the response results and a significant influence of the mean frequency (fm) of the seismic motion. No coupling was found between the maximum displacement of the top during the oscillation and the permanent post-seismic deformations. The complementarity of both earthquake Intensity Measures in the structural vulnerability assessment is also depicted.
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
When a free-standing column with a given base becomes taller and taller, there is a competition between the increase in its size (more stable) and the increase in its slenderness (less stable). This article investigates how these two competing phenomena affect the stability of tall, slender, free-standing columns when subjected to horizontal and vertical ground shaking. The main conclusion of the article is that the outcome of this competition is sensitive to local details of the ground shaking and the dominant frequency of a possible coherent, distinguishable pulse. The often-observed increase in stability due to increase in height (despite the increase in slenderness) may be further enhanced due to a sudden transition from the lower mode of overturning with impact to the higher mode of overturning without impact. The article proceeds by offering a simple mathematical explanation why the vertical ground acceleration has a marginal effect on the stability of a slender, free-standing column and concludes that the level of ground shaking that is needed to overturn a tall free-standing column of any size and any slenderness is a decreasing function of the length scale a(p)T(p)(2) of the dominant coherent acceleration pulse normalized to the base width of the column.
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.
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.
This paper investigates the rocking response of a slender column that is vertically restrained with an elastic tendon that passes through its centerline. Following a variational formulation, the nonlinear equation of motion is derived, in which the stiffness and the prestressing force of the tendon are treated separately. In this way, the post-uplift stiffness of the system can be anywhere from negative to positive depending on the axial stiffness of the vertical tendon. This paper shows that vertical tendons are effective in suppressing the response of smaller columns subjected to long-period excitations. As the size of the column or the frequency of the excitation increases, the effect of the vertical tendon becomes immaterial given that most of the seismic resistance of large rocking columns originates primarily from the mobilization of their rotational inertia.
A half century ago, Housner (1963) explained that there is a safety margin between uplifting and overturning of slender, free-standing columns and that as the size of the column or the frequency of the excitation increases, this safety margin increases appreciably to the extent that large, free-standing columns enjoy ample seismic stability. This paper revisits the important implications of this postuplift dynamic stability and explains that the enhanced seismic stability originates from the difficulty of mobilizing the rotational inertia of the free-standing column. As the size of the column increases, the seismic resistance (rotational inertia) increases with the square of the column size, whereas the seismic demand (overturning moment) increases linearly with size. Accordingly, in theory, a slender, free-standing column can survive any ground shaking provided that the column is sufficiently large, because a quadratic term eventually dominates over a linear term. The same result applies to the articulated rocking frame given that its dynamic rocking response is identical to the rocking response of a single free-standing column with the same slenderness but larger size.
This paper investigates the planar rocking response and stability analysis of an array of free-standing columns capped with a freely supported rigid beam. Part of the motivation for this study is the emerging seismic design concept of allowing framing systems to uplift and rock along their plane in order to limit bending moments and shear forces. Following a variational formulation the paper reaches 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.
The uplifting and rocking of slender, free-standing structures when subjected to ground shaking may limit appreciably the seismic moments and shears that develop at their base. This high-performance seismic behavior is inherent in the design of ancient temples with emblematic peristyles that consist of slender, free-standing columns which support freely heavy epistyles together with the even heavier frieze atop. While the ample seismic performance of rocking isolation has been documented with the through-the-centuries survival of several free-standing ancient temples; and careful post-earthquake observations in Japan during the 1940's suggested that the increasing size of slender free-standing tombstones enhances their seismic stability; it was George Housner who 50 years ago elucidated a size-frequency scale effect that explained the "counter intuitive" seismic stability of tall, slender rocking structures. Housner's 1963 seminal paper marks the beginning of a series of systematic studies on the dynamic response and stability of rocking structures which gradually led to the development of rocking isolation-an attractive practical alternative for the seismic protection of tall, slender structures. This paper builds upon selected contributions published during this last half-century in an effort to bring forward the major advances together with the unique advantages of rocking isolation. The paper concludes that the concept of rocking isolation by intentionally designing a hinging mechanism that its seismic resistance originates primarily from the mobilization of the rotational inertia of its members is a unique seismic protection strategy for large, slender structures not just at the limit-state but also at the operational state.
This paper investigates the rocking response and stability analysis of an array of slender columns caped with a rigid beam which are vertically restrained with elastic prestressed tendons that pass through the centerline of the columns while anchored at the foundation and the cap-beam. Following a variational formulation, the nonlinear equation of motion is derived in which the stiffness and the prestressing force of the tendons are treated separately. In this way, the postuplift stiffness of the vertically restrained rocking frame can be anywhere from negative to positive depending on the axial stiffness of the vertical tendons. The paper shows that the tendons are effective in suppressing the response of rocking frames with small columns subjected to long-period excitations. As the size of the columns, the frequency of the excitations, or the weight of the cap-beam increases, the vertical tendons become immaterial given that most of the seismic resistance of tall rocking frames originates primarily from the mobilization of the rotational inertia of their columns. The paper concludes with the presentation and validation of an equivalent rigid-linear system so that the rocking response of vertically restrained rocking frames can be computed with popular open-source or commercially available software simply by employing existing elastic-mutilinear elements.
This technical note investigates the dynamic response and stability of a rocking frame that consists of two identical free-standing slender columns capped with a freely supported rigid beam. Part of the motivation for this study is the emerging seismic design concept of allowing framing systems to uplift and rock along their plane in order to limit bending moments and shear forces-together with the need to stress that the rocking frame is more stable the more heavy is its cap-beam, a finding that may have significant implications in the prefabricated bridge technology. In this technical note, a direct approach is followed after taking dynamic force and moment equilibrium of the components of the rocking frame, and the remarkable results obtained in the past with a variational formulation (by the same authors) is confirmed-that the dynamics response of the rocking frame is identical to the rocking response of a solitary, free-standing column with the same slenderness, yet with larger size, which produces a more stable configuration. The motivation for reworking this problem by following a direct approach is to show, in the simplest possible way, that the heavier the freely supported cap beam, the more stable is the rocking frame, regardless of the rise of the center of gravity of the cap beam. The conclusion is that top-heavy rocking frames are more stable that when they are top-light. (C) 2014 American Society of Civil Engineers.
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.
SUMMARY 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 methodology for the performance-based seismic risk assessment of classical
columns is presented. Despite their apparent instability, classical columns
are, in general, earthquake resistant, as proven from the fact that many
classical monuments have survived many strong earthquakes over the centuries.
Nevertheless, the quantitative assessment of their reliability and the
understanding of their dynamic behavior are not easy, because of the
fundamental nonlinear character and the sensitivity of their response. In this
paper, a seismic risk assessment is performed for a multidrum column using
Monte Carlo simulation with synthetic ground motions. The ground motions
adopted contain a high- and low-frequency component, combining the stochastic
method, and a simple analytical pulse model to simulate the directivity pulse
contained in near source ground motions. The deterministic model for the
numerical analysis of the system is three-dimensional and is based on the
Discrete Element Method. Fragility curves are produced conditional on magnitude
and distance from the fault and also on scalar intensity measures for two
engineering demand parameters, one concerning the intensity of the response
during the ground shaking and the other the residual deformation of the column.
Three performance levels are assigned to each engineering demand parameter.
Fragility analysis demonstrated some of the salient features of these spinal
systems under near-fault seismic excitations, as for example, their decreased
vulnerability for very strong earthquakes of magnitude 7 or larger. The
analysis provides useful results regarding the seismic reliability of classical
monuments and decision making during restoration process.
In this paper, the dynamic response of the rocking block subjected to base excitation is revisited. The goal is to offer ne closed-form solutions and original similarity laws that shed light on the fundamental aspects of the rocking block. The focu is on the transient dynamics of the rocking block under finite-duration excitations. An alternative way to describe the respons of the rocking block, informative of the behaviour of rocking structures under excitations of different intensity, is offered.
In the process, limitations of standard dimensional analysis, related to the orientations of the involved physical quantities are revealed. The proposed dimensionless and orientationless groups condense the response and offer a lucid depiction of th rocking phenomenon. When expressed in the appropriate dimensionless–orientationless groups, the rocking response becomes perfectl self-similar for slender blocks (within the small rotations range) and practically self-similar for non-slender blocks (large rotations). Using this formulation, the nonlinear and non-smooth rocking response to pulse-type ground motion can be directl determined, and need only be scaled by the intensity and frequency of the excitation.
This paper investigates the problem of sizing the width of tall free-standing columns with a given height which are intended to rock, yet shall remain stable during the maximum expected earthquake shaking. The motivation for this study is the emerging seismic design concept of allowing tall rigid structures to uplift and rock in order to limit base moments and shears. The paper first discusses the mathematical characterization of pulse-like ground motions and the dimensionless products that govern the dynamics of the rocking response of a free-standing block and subsequently, using basic principles of dynamics, derives a closed-form expression that offers the minimum design slenderness that is sufficient for a free-standing column with a given size to survive a pulse-like motion with known acceleration amplitude and duration.
This paper follows two approaches to formulate the equations that describe the free rocking motion of a rigid block in three dimensions. In the first approach, the orientation of the block is parametrized using a 3-1-2 set of Euler angles, {, , } θφψ . Lagrange's equations of motion, where the Euler angles {, , } θφψ serve as generalized coordinates, yield a system χχ χ χ () ( , )
From the mechanical point of view, the particularity of masonry structures stems from the fact that the structural system is hard to be modeled by the classical Continuum Mechanics approach. The problem gets more complicated when imperfections, such as cracks are present. An example of a single multi-drum column, with fractured drums, is studied herein, using the Distinct Element Method (DEM). The purpose of the research is the investigation of the impact of the fractures to the overall stability of the structure. The 3D DEM numerical results are explained on the basis of simple 2D analytical considerations. The shear and normal crack deformation is monitored and the minimum required strength of the crack interface is quantified. An experimental program of direct shear tests is set in order to estimate the strength of the marble–cement interface. The experimental values are compared to the minimum required from the numerical analysis.
This paper examines in depth the transient rocking response of free-standing rigid blocks subjected to physically realizable trigonometric pulses. First, the expressions for the dynamic horizontal and vertical reactions at the pivot point of a rocking block are derived and it is shown that the coefficient of friction needed to sustain pure rocking motion is, in general, an increasing function of the acceleration level of the pulse. Subsequently, this paper shows that under cycloidal pulses a free-standing block can overturn with two distinct modes: (1) by exhibiting one or more impacts; and (2) without exhibiting any impact. The existence of the second mode results in a safe region that is located on the acceleration-frequency plane above the minimum overturning acceleration spectrum. The shape of this region depends on the coefficient of restitution and is sensitive to the nonlinear nature of the problem. This paper concludes that the sensitive nonlinear nature of the problem, in association with the presence of the safe region that embraces the minimum overturning acceleration spectrum, complicates further the task of estimating peak ground acceleration by only examining the geometry of free-standing objects that either overturned or survived a ground shaking.
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 validity of a simple, approximate design methodology, initially proposed in the late 70's 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.
Over the last few decades Accelerated Bridge Construction (ABC) has been developed to answer the growing number of societal needs, as well as advancing the bridge practice. There have been many applications of ABC in the United States, primarily in Texas, Utah, New Jersey, Florida, and Washington. However, application of ABC in regions with moderate-to-high seismicity requires in depth development, detailing consideration, experimental investigation, and analytical guidelines for the suitable connections between the precast members. This paper investigates the seismic performance of two types of connections through experimental testing for ABC in seismic regions. The research concludes that both connections have the potential to be used within the concept of ABC between the precast elements in a bridge substructure.
The present work investigates the influence of small geometrical defects on the behavior of slender rigid blocks. A comprehensive experimental campaign was carried out on one of the shake tables of CEA/Saclay in France. The tested model was a massive steel block with standard manufacturing quality. Release, free oscillations tests as well as shake table tests revealed a non-negligible out-of-plane motion even in the case of apparently plane initial conditions or excitations. This motion exhibits a highly reproducible part for a short duration that was used to calibrate a numerical geometrically asymmetrical model. The stability of this model when subjected to 2000 artificial seismic horizontal bidirectional signals was compared with the stability of a symmetrical one. This study showed that the geometrical imperfections slightly increase the rocking and overturning probabilities for earthquake signals in a narrow range of peak ground acceleration.
Numerous structures exhibit rocking behavior during earthquakes and there is a continuing need to retrofit these structures to prevent collapse. The behavior of stand-alone rocking structures has been thoroughly investigated, but there are relatively few theoretical studies on the response of retrofitted rocking structures. In practice, despite the benefits of allowing rocking motion, rocking behavior is typically prevented instead of optimized. This study characterizes the fundamental behavior of damped rocking motion through analytical modeling. A single rocking block analytical model is utilized to determine the optimal viscous damping characteristics which exploit the beneficial aspects of rocking motion while dissipating energy and preventing overturning collapse. To clarify the benefits of damping, overturning envelopes for the damped rocking block are presented and compared with the pertinent envelopes of the free rocking block. Preliminary experimental work to verify analytical modeling is also presented. Finally, the same principles of controlling rocking behavior with damping are extended to a particular class of rocking problems, the dynamics of masonry arches. A pilot application of the proposed approach to masonry arches is presented.
This paper presents a probability model to predict the maximum rotation of rocking bodies exposed to seismic excitations given particular earthquake intensity measures. After obtaining the nonlinear equations of motion and a clarification of the boundaries applied to a rocking body needed to avoid sliding, a complete discussion is provided for the estimation of the approximate period and equivalent damping ratio for the rocking motion. After that, instead of using an iterative solution, which has been proven defective, a new approximate technique is developed by finding the best representative ground motion intensities. Suitable transformations and normalizations are applied to these intensities, and the Bayesian updating approach is employed to construct a probability model. The proposed probability model is capable of accurately predicting the maximum rotation of a symmetric rocking block given the displacement design spectra, peak ground acceleration, peak ground velocity, and arias intensity of an earthquake. This probabilistic model along with the approximate capacity of rocking blocks are used to estimate the fragility curves for rocking blocks with specific geometrical parameters. In the end, a comprehensive and practical form of fragility curves are provided for design purposes along with numerical examples.
The rocking problem is applicable to a wide variety of structural and nonstructural elements. The current applications range from bridge pier and shallow footing design to hospital and data center equipment, even art preservation. Despite the increasing number of theoretical and simulation studies of rocking motion, few experimental studies exist. Of those that have been published, most are focused on a reduced version of the problem introducing modifications to the physical problem with the purpose of eliminating either sliding, uplift, or the three-dimensional (3D) response of the body. However, all of these phenomena may affect the response of an unrestrained rocking body. The intent of this work is to present a computer vision method that allows for the experimental measurement of the rigid body translation and rotation time histories in three dimensions. Experimental results obtained with this method will be presented to demonstrate that it obtains greater than 97% accuracy when compared against National Institute of Standards and Technology traceable displacement sensors. The work concludes with two example experimental studies of rigid body rocking measured with this method as proof of concept. The experimental results highlight important phenomena predicted in some state-of-the-art models for 3D rocking behavior. (C) 2015 American Society of Civil Engineers.
Unlike conventional seismic design, the columns of a rocking frame are designed to uplift and pivot during earthquake excitation. This paper investigates, analytically and numerically, the seismic response of a rocking frame with columns unequal in height (asymmetric), which are either freestanding or hybrid, i.e., enhanced with supplemental damping and recentering capacity. The paper establishes the equations of motion following the principles of analytical dynamics. Throughout the study, the deformation of the structural members is considered negligible. The analysis considers both pulse-type and non-pulse-type (historic) ground motions. It shows that the effect of asymmetry on the seismic stability of the rocking frame is marginal compared with the symmetric configuration, despite the very different kinematics of the corresponding rocking mechanisms. In contrast, the seismic stability of the hybrid rocking frame is very sensitive to fracture elongation of the supplemental restoring (tendons) and damping devices. The results confirm the high-performance seismic behavior of the planar rocking frame, thus illustrating its potential as an alternative seismic design paradigm.
For reasons more related to functionality than safety, it is not uncommon for heavy mechanical and electrical equipment to be placed on wheels. Examples of such devices are medical carts, mechanical equipment in hospitals, electrical transformers, and recently, even supercomputers. Although wheels facilitate the operation of these devices, they also affect the response of these objects during earthquakes, but not necessarily in a beneficial way. While a wheel rolling favors the translational displacement of the body in the horizontal direction parallel to its plane over rocking, rocking is still possible along the plane perpendicular to the plane of the wheel. Moreover, because the plane of the wheel is in most cases free to rotate with respect to the body, it is not easy to identify the directions that favor rocking or displacement at any time. The problem becomes even more complicated, if one considers that one of the wheels, which can swivel, may be locked. Thus, in the most general case, a body on wheels experiences three-dimensional displacements and rotations along with the three-dimensional dynamics of the wheels. In this work, a model for the dynamic behavior of bodies on wheels is presented and the corresponding response behavior is examined for several common cases.
This paper assesses the seismic fragility of single degree of freedom rocking structures within a probabilistic framework. The focus is on slender rigid structures that exhibit negative stiffness during rocking. The analysis considers ground motions with near-fault characteristics, either solely coherent pulses or synthetic ground motions that include, in addition, a stochastic high-frequency component. The study offers normalized fragility curves that estimate the overturning tendency, as well as the peak response rotation of a rocking structure. It shows that the use of bivariate intensity measures (IMs) can lead to superior fragility curves compared with conventional univariate IMs. Regardless, the study advocates the use of dimensionless–orientationless IMs that offer an approximately ‘universal’ description of rocking behavior/fragility, a normalized description almost indifferent to the amplitude and the predominant frequency of the excitation or the size and the slenderness of the rocking structure. Importantly, the analysis unveils hidden order in rocking response. There exists a critical peak ground acceleration, below and above which, peak rocking response scales differently. In particular, when the structure does not overturn, the peak rotation follows approximately a biplanar pattern with respect to the intensity and the predominant frequency of the excitation. Finally, the analysis verifies that rocking overturning depends primarily on the velocity characteristics of the ground motion. Copyright © 2015 John Wiley & Sons, Ltd.
An approximation is developed for obtaining the nonlinear stiffness and damping of a shallow circular or strip footing undergoing rocking oscillation on a homogeneous but inelastic undrained clayey stratum. Based on the parametric results of 3-D and 2-D finite-element analyses, equivalent-linear and are expressed in readily usable dimensionless forms. , normalized by its linear elastic value, is shown to be a unique function of: (1) the vertical factor of safety against static bearing capacity failure, and (2) the angle of rotation normalized by a characteristic angle . The latter is approximately the angle for which uplifting usually initiates at one edge of the foundation. Three sources contribute to the value of the dimensionless damping ratio (derived from ): wave radiation, which is a function of frequency but is shown to amount to \(
Numerous existing structures exhibit rocking behavior during earthquakes, and there is a continuing need to retrofit these structures to prevent collapse. In addition, while rocking behavior is typically prevented instead of utilized, an increasing number of structures are being designed or retrofitted to allow rocking motion as a means of seismic isolation. This paper investigates the use of viscous damping to limit the rocking motion by characterizing the fundamental behavior of damped rocking structures through analytical modeling. A single rocking block analytical model is used to determine the viscous damping characteristics, which exploit the beneficial aspects of the rocking motion, while dissipating energy and preventing overturning collapse. To clarify the benefits of damping, overturning envelopes for pulse-type ground accelerations are presented and compared with the pertinent envelopes of the free rocking block. A semianalytical solution to the linearized equations of motion enables rapid generation of collapse diagrams for pulse excitations, which provide insight into the overturning mechanisms of the damped rocking block and the sensitivity of the response to the parameters involved. In addition, through solution of the nonlinear equations of motion, bilateral and unilateral linear viscous dampers are shown to provide similar benefit toward preventing overturning, while nonlinear damping is found to provide relatively little and inconsistent benefit with respect to linear damping.
During the Chilean earthquakes of May, 1960, a number of tall, slender structures survived the ground shaking whereas more stable appearing structures were severely damaged. An analysis is made of the rocking motion of structures of inverted pendulum type. It is shown that there is a scale effect which makes tall slender structures more stable against overturning than might have been expected, and, therefore, the survival of such structures during earth-quakes is not surprising.
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.
SUMMARYA method for generating an ensemble of orthogonal horizontal ground motion components with correlated parameters for specified earthquake and site characteristics is presented. The method employs a parameterized stochastic model that is based on a time‐modulated filtered white‐noise process with the filter having time‐varying characteristics. Whereas the input white‐noise excitation describes the stochastic nature of the ground motion, the forms of the modulating function and the filter and their parameters characterize the evolutionary intensity and nonstationary frequency content of the ground motion. The stochastic model is fitted to a database of recorded horizontal ground motion component pairs that are rotated into their principal axes, a set of orthogonal axes along which the components are statistically uncorrelated. Model parameters are identified for each ground motion component in the database. Using these data, predictive equations are developed for the model parameters in terms of earthquake and site characteristics and correlation coefficients between parameters of the two components are estimated. Given a design scenario specified in terms of earthquake and site characteristics, the results of this study allow one to generate realizations of correlated model parameters and use them along with simulated white‐noise processes to generate synthetic pairs of horizontal ground motion components along the principal axes. The proposed simulation method does not require any seed recorded ground motion and is ideal for use in performance‐based earthquake engineering. Copyright © 2011 John Wiley & Sons, Ltd.
SUMMARY A common type of ancient monuments around the Mediterranean is the ancient Greek temple. Unfortunately, very few remain intact; most of them surviving in the form of free-standing multidrum columns. Composed of stones resting on top of each other without any connection, such columns are considered vulnerable to earthquakes. The paper presents an experimental study of such structures, aiming to explore their seismic vulnerability and derive insights on the key factors affecting their response. Reduced scale models of a single multidrum column and of a portal were tested at the shaking table of the National Technical University of Athens Laboratory of Soil Mechanics. The models, constructed of marble just as the originals, were excited by idealized Ricker pulses and real seismic records. Single columns exhibit a remarkable earthquake resistance. Subjected to the strongest motions ever recorded in Greece, where many such monuments are situated, the columns hardly suffered any permanent deformation. Collapse is probable only for extremely harsh directivity-affected seismic motions. Portals proved even more robust, surviving extreme seismic excitations. Their superior performance is related to the beneficial role of the epistyle, which adds energy dissipation and restoring force to the system. Their performance is very sensitive to minor changes in geometry or input motion. The complexity increases exponentially with the number of drums, being directly associated with the number of drum-to-drum interfaces and the increased probability of interface imperfections. In contrast to PGA, the maximum spectral displacement SDmax and the length scale Lp have turned out to be effective intensity measures. Copyright © 2014 John Wiley & Sons, Ltd.
SUMMARY Predicting the rocking response of structures to ground motion is important for assessment of existing structures, which may be vulnerable to uplift and overturning, as well as for designs which employ rocking as a means of seismic isolation. However, the majority of studies utilize a single rocking block to characterize rocking motion. In this paper, a methodology is proposed to derive equivalence between the single rocking block and various rocking mechanisms, yielding a set of fundamental rocking parameters. Specific structures that have exact dynamic equivalence with a single rocking block, are first reviewed. Subsequently, approximate equivalence between single and multiple block mechanisms is achieved through local linearization of the relevant equations of motion. The approximation error associated with linearization is quantified for three essential mechanisms, providing a measure of the confidence with which the proposed methodology can be applied. Copyright © 2014 John Wiley & Sons, Ltd.
A model of 3D rigid body with a rectangular base, able to rock around a side or a vertex of the base is developed. Eccentricity of the center of mass with respect to the geometrical center of the body is also considered. The equations of motion are obtained through the general balance principle. A one-sine pulse base excitation is applied to the body in different directions. The analyses are conducted with the aim to highlight the role of the period, the amplitude and the direction of the external excitation. In significant ranges of the previous parameters, the results obtained with a bi-dimensional model, that does not consider the 3D rocking motions on a vertex of the base, are not in favor of safety. It is found, in fact, that in several conditions the overturning of the 3D block takes place for amplitudes of excitation smaller than those able to overturn the 3D block.
The rocking motion of a rigid rectangular prism on a moving base is a complex three dimensional phenomenon. Although, with very few exceptions, the previous models in the literature make the simplified assumption that this motion is planar, this is usually not true since a body will probably not be aligned with the direction of the ground motion. Thus, even in the case where the body is fully symmetric, the rocking motion involves three dimensional rotations and displacements.In this work, a three dimensional formulation is introduced for the rocking motion of a rigid rectangular prism on a deformable base. Two models are developed: the Concentrated Springs Model and the Winkler Model. Both sliding and uplift are taken into account and the fully non-linear equations of the problem are developed and solved numerically.The models developed are later used to examine the behavior of bodies subjected to general ground excitations. The contribution of phenomena neglected in previous models, such as twist, is stressed.
The motion of disks spun on tables has the well-known feature that the associated acoustic signal increases in frequency as the motion tends towards its abrupt halt. Recently, a commercial toy, known as Euler's disk, was designed to maximize the time before this abrupt ending. In this paper, we present and simulate a rigid body model for Euler's disk. Based on the nature of the contact force between the disk and the table revealed by the simulations, we conjecture a new mechanism for the abrupt halt of the disk and the increased acoustic frequency associated with the decline of the disk.
Block objects, standing on a shaking foundation, will tend to rock and possibly uplift. Rocking of buildings in earthquakes are of particular interest because human lives and high costs are at stake. Most previous studies on rocking are limited to two-dimensional motion for simplicity. The theory of a new model, which includes three-dimensional rocking, rolling, and uplift of a rigid cylinder subjected to ground motion, is put forth in the companion paper. This paper describes a numerical example of the new model. Computer simulations show that 3-D motion is significant under earthquake-like excitations. When rocking is predominantly 2-D, high spikes in accelerations and internal stresses are produced in the structure. The model that allows uplift is compared with the model that does not allow uplift. It is determined that restricting uplift can introduce higher stresses and accelerations inside the structure.
Rocking or toppling of a cylindrical structure or object on a shaking foundation is a common occurrence of engineering interest. The behavior is analyzed with a simple three-dimensional model which has an upright rigid cylinder standing on a rigid foundation. Given an initial side impulse, the cylinder will rock and rotate on the base under the influence of gravity alone. The constraint for the model is that no slipping occurs at the point of contact between the cylinder and the foundation. Otherwise, rolling, rocking, and toppling are not restricted. The governing equations are derived, and the response of the cylinder under free rocking is calculated for various initial conditions. Three types of response are observed: rocking, nutation, and toppling. Rocking is the side-to-side motion, while nutation is the wavy precession about the vertical. In addition, rocking and nutation both exhibit a stiff type and a smooth type. The regions for each type of response are mapped out in the domain formed by the initial conditions. From the plots of these regions, one can predict the type of response for a given set of initial conditions.
Block objects, standing on a shaking foundation, tend to rock and may uplift. Rocking of buildings in earthquakes is of particular interest because human lives and high costs are at stake. Most previous studies on rocking are limited to two-dimensional motion for simplicity. In this paper, a new model is studied, which includes the three-dimensional rocking, rolling, and uplift of a rigid cylinder when subjected to ground motion. The cylinder rests on a Winkler foundation of independent springs and dashpots. To simulate uplift, the springs and dashpots separate from the base of the cylinder when the springs are about to be in tension. The governing equations of this system are derived exactly using the Lagrange equation. Then numerical integration is employed to obtain the motion of any point within the structure, and simple beam theory is used to calculate the 3-D state of stresses within the cylinder. Computer simulations show that 3-D motion is significant under earthquake-like excitations. Near-2-D rocking also occurs and produces very high spikes in accelerations and internal stresses. Moreover, restricting uplift can introduce high stresses and accelerations inside the structure.
This paper describes the behavior of single rigid-block structures under dynamic loading. A comprehensive experimental investigation has been carried out to study the rocking response of four blue granite stones with different geometrical characteristics under free vibration, and harmonic and random motions of the base. In total, 275 tests on a shaking table were carried out in order to address the issues of repeatability of the results and stability of the rocking motion response. Two different tools for the numerical simulations of the rocking motion of rigid blocks are considered. The first tool is analytical and overcomes the usual limitations of the traditional piecewise equations of motion through a Lagrangian formalism. The second tool is based on the discrete element method (DEM), especially effective for the numerical modeling of rigid blocks. A new methodology is proposed for finding the parameters of the DEM by using the parameters of the classical theory. An extensive comparison between numerical and experimental data has been carried out to validate and define the limitations of the analytical tools under study.