A finite element model for seismic response ... - Wiley Online Library

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2016 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2799

A finite element model for seismic response analysis of deformable rocking frames Michalis F. Vassiliou1,*,†, Kevin R. Mackie2 and Božidar Stojadinović1 1

Institute of Structural Engineering (IBK), Swiss Federal Institute of Technology (ETHZ), Stefano-Franscini-Platz 58093 Zürich, Switzerland 2 Civil Environmental and Construction Engineering Department, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-2450, USA

SUMMARY A new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero-length fiber cross-section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti-symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd. Received 4 February 2016; Revised 8 July 2016; Accepted 12 July 2016 KEY WORDS:

rocking structures; uplifting structures; seismic isolation; overturning stability; FE modeling

1. INTRODUCTION The response of rigid blocks allowed to uplift and rock on a rigid foundation under seismic ground motion excitation has been studied for more than a century [1]. Housner [2] demonstrated that large rigid blocks are difficult to overturn dynamically. The same holds for large rocking rigid frames [3, 4] as well as for a class of rigid rocking structures [5]. This explains the survival of ancient Greek and Roman top-heavy temple structures in regions of high seismicity [6], despite the lack of historical evidence that ancient engineers were aware of the size effect of rocking structures. This size effect has lead researchers to propose rocking as a seismic response modification technique. In addition, structures designed to uplift at the base and sustain rocking motion are characterized by small residual displacements and small forces transmitted to foundations [7–13]. Large structures, such as tall chimneys and tall bridges are ideal candidates for rocking seismic response modification due to the aforementioned size effect. A 60-m-tall bridge designed to rock has already been built across the Rangitikei River in New Zealand in 1981 [14, 15]. Moreover, a 33-m-tall *Correspondence to: Michalis F. Vassiliou, Institute of Structural Engineering (IBK), Swiss Federal Institute of Technology (ETHZ), Stefano-Franscini-Platz 5, 8093, Zürich, Switzerland, † E-mail: [email protected] Copyright © 2016 John Wiley & Sons, Ltd.

M. F. VASSILIOU, K. R. MACKIE AND B. STOJADINOVIĆ

chimney at the Christchurch New Zealand airport has been designed to uplift [16]. Furthermore, three 30 to 38-m-tall chimneys in Piraeus, Greece, have been retrofitted by letting them uplift in case of an earthquake [17]. The term ‘rocking structures’ is used to describe moment-resisting frame or shear wall structures that uplift and rock. There exist two classes of such structures: those that utilize post-tensioning cables and special purpose energy dissipation elements installed at the rocking connections to control displacements [18–23] and thereby typically have positive post-uplift stiffness and those that have negative post-uplift stiffness because their elements are free to move at rocking interfaces. While the response spectrum methods can be used for positive post-uplift stiffness rocking structures, rocking structures with negative post-uplift stiffness cannot be adequately described by an equivalent SDOF elastic system [24–26]. Therefore, the models and the analysis methods used to describe the seismic response of these two classes of rocking structures are very different. Similarly, the simplified models of rigid and deformable rocking structures, especially of the negative stiffness ones, are not adequate for seismic design of such structures [24]. Tools to quantify the seismic response of rocking structures, both rigid and deformable, simultaneously at the level of rocking connections, structural elements and the entire rocking structure are needed. It would be advantageous if these tools were consistent with existing finite element simulation frameworks. The existing finite element models usually use continuous viscous damping (either classical or non-classical) to describe the energy dissipated by the structure during its motion. The use of Rayleigh damping in strongly non-linear problems, such as rocking, creates difficulties extensively described in [27–29]. The damping parameters for a rocking body, especially when classical damping is used, cannot be predicted a priori [30]. The main reason is that the principal mechanism through which a rocking body dissipates energy is impact, an instantaneous and discontinuous energy dissipation mechanism. A finite element approach to model and quantify the in-plane rocking response of systems consisting of either rigid or deformable rocking bodies is presented in this paper. This relatively straightforward modeling approach aims to overcome the aforementioned difficulties by providing a robust way to model the damping in rocking motion. The proposed finite element model utilizes a fiber cross section at the base of an elastic column that undergoes large displacements. One advantage of this model is that it couples directly to existing finite element libraries and simulation frameworks. The proposed model is validated against the Housner model of the response of a rigid rocking column and a rigid rocking frame structure to ground motion excitations. Then, the proposed model is augmented by a model of the ground under the rocking surface to facilitate computationally efficient modeling of the response of deformable rocking structures by overcoming numerical issues related to the propagation of impact-induced high-frequency vibrations. This important new capability is illustrated in an investigation of the seismic response of a deformable rocking frame structure where the uplift and overturning rocking response spectra to symmetric and anti-symmetric Ricker pulse ground motion excitation are computed.

2. DEFORMABLE ROCKING BODY MODEL The proposed model is intended to facilitate a numerical time history analysis of the in-plane response of rocking structures to earthquake ground motion excitation. Typical rocking structures are comprised of either solitary deformable rocking bodies or assemblies of similar deformable rocking bodies. The rocking motion is unrestrained, in the sense that the motion at the rocking interface is free, that is, no additional devices (fuses, connectors, dissipaters, post-tensioned cables, etc.) are used. Sliding on the rocking surfaces is not permitted. The proposed deformable rocking body (DRB) model has two components: (1) a finite element model of the solitary DRB and (2) a set of requirements for conducting the time step integration and the geometric transformations during the time history solution process. The model of a solitary DRB consists of a beam–column element representing the body itself and zero-length fiber cross-section element representing the rocking surface (Figure 1). The proposed DRB model was implemented Copyright © 2016 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. (2016) DOI: 10.1002/eqe

FE MODELING AND BEHAVIOR OF A DEFORMABLE ROCKING FRAME

Figure 1. A solitary rigid block in rocking motion (left) and the proposed DRB model (right).

using the finite elements, and the solution processes already available in the OpenSees finite element modeling and simulation framework [31]. 2.1. Model of the rocking surface A simplified version of the model proposed by Barthes [32] is used to model the rocking surface at either end of the rocking body. Namely, a rocking surface is modeled using the OpenSees zero-length fiber cross-section element placed between nodes i and j (Figure 1). The cross section consists of one row of fibers, sufficient to simulate the in-plane rocking motion. The fiber material is non-linear, with no resistance in tension and an elastic response in compression, defined using a stress/displacement (σ-Δl) relation, where Δl stands for the elongation of the fibers. The material constant has units of [force]/[length]3 and is, in essence, equivalent to a Winkler soil spring (but is implemented more conveniently to model the rocking surface). The fiber material is non-dissipative (consumes no energy). No dampers (e.g. viscous dashpots) are used. The OpenSees zero-length element computes the relative displacement between nodes i and j (Figure 1, right) assuming that the displacements and the rotations are small. Namely, the vertical and the horizontal components of the relative displacement are bsinθ ≈ bθ and b(1 cosθ) ≈ 0, respectively. A comparison of the relative horizontal and vertical displacements of the nodes computed using the OpenSees zero-length element and a large-displacement analytical model is shown in Figure 2. The displacements are normalized with respect to the size of the rocking body, 2R, while the rotations are normalized with respect to the aspect ratio of the rocking body, α. The rocking body (between nodes j and k) is assumed to be rigid. The error in modeling the horizontal displacement of the bottom node j is evident but will be shown to be insignificant when this model is used to simulate rocking response. Note that, contrary to the concentrated spring model [33], the proposed zero-length element captures the horizontal and the vertical displacements of node k (the top of the rocking body) with sufficient accuracy up to large rotation angles (1.5 times larger than the aspect ratio of the rocking body, well into the overturning response phase). Hence, the zero-length element model can be used to model rocking bodies post-tensioned using vertical prestressing tendons, where the relative displacement between nodes i and k is used to determine the tendon force. 2.2. Model of the rocking body The rocking body is modeled using beam–column finite elements. The current implementation of the DRB model in OpenSees utilizes linear elastic beam–column elements; however, this does not preclude the use of other beam–column finite elements. The stiffness of the fibers used to model the rocking surface, Ef, and of the beam element material used to model the rocking body, Eb, is not numerically comparable: The former has units of [force]/[length]3, while the latter has units of [force]/[length]2. To represent rocking of a rigid body on a rigid surface, the values of both of these parameters should be set to sufficiently large values and selected such that impact forces deform the Copyright © 2016 John Wiley & Sons, Ltd.



Figure 2. Comparison of the motion of the center points of the top and bottom cross sections of the rocking body computed using an analytical and the zero-length element models.

rocking body, not the rocking surface. The rocking surface fiber stiffness should be selected such that increasing their stiffness further does not result in appreciable changes in the computed response. No damping, classical or non-classical, is associated with the rocking body model or the rocking surface model. This is in accordance to the experiments presented in [44], where it was shown that if one adds some form of viscous damping to the model, the energy dissipation in the uplifted state of the structure will grossly increase, a behavior that was not observed and confirmed experimentally. The moment of inertia of a rigid body rotating around its pivot point as it rocks is 43 mR2. If the beam–column finite element used to model the rocking body had only translational masses, its moment of inertia would be 13 mR2 cos2 α þ mR2 . Therefore, the difference ΔI o ¼ 13 mR2 sin2 a was evenly distributed among the rotational degrees of freedom of the nodes connecting the beam–column elements used to model the rocking body, following an approach similar to the one described in [33]. The sensitivity of the proposed DRB model to the selection of material characteristics and the number of nodes and fibers used to model the rocking body and the rocking surface will be examined in a later section. 2.3. Response analysis process The state of the DRB model at each time step was computed using the corotational formulation [34] to account for the effect of large displacements and rotations that may occur during the motion of a rocking body. The model for the energy dissipated during the rocking motion is discussed next. Housner [2] model assumes that a rigid body is rocking on a rigid surface. This assumption enables modeling of the rocking part of the motion but fails to capture the instant of impact and the consequent loss of energy experienced by the rocking body. To model energy dissipation, Housner adopts an ad hoc coefficient of restitution that is exact only if the block and the ground are ideally rigid and the impact forces are concentrated at the impact points. This drawback is reflected in the lack of a perfect experimental confirmation [35–44] of Housner’s restitution coefficient model: Rocking body specimens and the surfaces they rock on are not ideally rigid, and the impact forces are not concentrated exactly at the pivot point. It should be stated, however, that even if experimental Copyright © 2016 John Wiley & Sons, Ltd.



testing shows that the material properties and impact velocity do influence the energy dissipation, it also shows that the dominant parameter is still the geometry of the columns, its slenderness, as originally assumed by Housner. Housner’s model cannot be easily extended to explicitly incorporate inelastic deformation, radiation damping into the ground or shock waves propagating into the rocking body, that is, it determines the amount of energy dissipation, not the mechanism of such energy dissipation. Nevertheless, the Housner model can serve as a well-established analytical benchmark problem to test the proposed DRB model in the limit case where both the rocking body and the rocking surface are infinitely stiff. In that case, and if the fibers of the zero-length element are set stiff enough, the impact at pivot points induces elastic axial and flexural shock waves that propagate into the rocking body. Modeling the origination and propagation of these high-frequency small-amplitude shock waves requires a very small integration time step and a very fine finite element mesh [45]. To facilitate the proposed DRB model, it was assumed that the shock waves do not affect the motion of the rocking body between the two consecutive rocking impacts. This justifies using a dissipative time-stepping integration procedure to numerically damp out the shock waves in the beam–column element used to model the rocking body and neglecting the propagation of shock waves into the rocking surface. One of the most widely used dissipative time-stepping integration procedures is the Hilber–Hughes–Taylor (HHT) [46] algorithm. This algorithm damps out high-frequency components of the computed response without markedly affecting the low-frequency response components, thus affecting the response in higher vibration modes of the structure. Numerical damping in the HHT algorithm is controlled by a dissipation factor ad ′ and the time integration step dt. Note that the dissipation factor ad ′ = ad 1, where ad is the dissipation factor used in the OpenSees implementation of the HHT integration algorithm.

3. DEFORMABLE ROCKING BODY MODEL VALIDATION FOR RIGID ROCKING STRUCTURES The proposed DRB model is validated against the Housner model (and its impact assumptions) for computing the rocking response of rigid structures to analytical pulse and recorded ground motion excitations. The response of a solitary rigid rocking body is examined first, followed by the response of epistyle frame structures with rigid columns. 3.1. Solitary rigid rocking body A solitary rigid rocking body (Figure 1) with dimensions 2b × 2h = 4 × 12 m (aspect ratio 0.33) and material density of 25,000 kg/m3 (=25 tn/m3) was used as the prototype. The Housner model was implemented in MATLAB, and the DRB model was implemented in OpenSees. The Young’s modulus of the rocking body model was Eb = 30 × 109 kPa, while the elastic modulus of the fibers used to model the rocking surface was Ef = 30 × 109 kN/m3. The rocking body was modeled using 20 linear elastic beam–column elements, while the rocking surface was modeled with a varying number of fibers. The HHT integration time step dt was 104 s. Parameter αd of the OpenSees HHT algorithm was set to 2/3 (ad ′ ¼ ad 1 ¼ 1 3 in the original notation [46]), the value that provides maximum numerical dissipation. The tilt angle, θ, (Figure 1) responses of the Housner and two DRB models (with 2048 and 2 fibers) in free rocking starting from an initially tilted position are shown in Figure 3(a). The distribution of the fibers in the DRB model with 2048 fibers was uniform. The centroids of the fibers in the two-fiber DRB model were placed at the outer edges of the cross section, with each fiber having an area equal to half of the rocking surface area. Both DRB model solutions are virtually indistinguishable from the Housner solution. Hence, the DRB model is capable of numerically reproducing the instantaneous change of the pivot point inherent to the Housner model. Furthermore, the DRB model is not sensitive to the number of fibers used to model the rocking surface when both the surface and the rocking body are rigid (i.e. numerically very stiff). Thus, in the remainder of this study, the rocking surface was modeled using only two fibers. Copyright © 2016 John Wiley & Sons, Ltd.



Figure 3. (a) Comparison of rigid body free rocking responses computed using the 2048-fiber and 2-fiber DRB models to the Housner model. (b, c) Sensitivity of the DRB model response to the number of beam–column elements used to model the rocking body and to the length of the integration time step. (d) Sensitivity of the DRB model response to different values of OpenSees HHT numerical dissipation factor αd.

It is important to note that the sole source of energy dissipation in the proposed DRB model is the numerical damping in the HHT numerical integration algorithm. No other damping models, such as the classical or non-classical damping matrices or the various discrete dampers, were used. Because the rocking motion is of relatively low frequency, the numerical damping does not explicitly affect it. Instead, on every impact, the kinetic energy from the rocking mode of motion is transformed to kinetic energy of high-frequency impact waves in the rocking body. The kinetic energy of the high-frequency impact waves is instantaneously numerically damped: This is the only damping in the model. The matches between the rocking response time histories produced using the Housner and the DRB model are excellent. In the next sections, it will be shown that this is not a coincidence and that it holds across a range of rigid body size and slenderness parameter values. Achieving such a good match in the response time history requires that the energy dissipation in the Housner and the DRB models matches perfectly; otherwise, the interdependence between the period and the amplitude of the response motion would lead to both phase and amplitude errors. Note that these simulations are conducted for rigid (numerically very stiff) rocking bodies and rocking surfaces: Thus, a good agreement between the numerical dissipation obtained using the HHT integration algorithm in the DRB model and the restitution model used in the Housner model can be obtained only for rigid rocking bodies. The energy dissipation model needs to be augmented to model the energy dissipated by DRBs and deformable rocking surfaces, as discussed later. 3.1.1. Sensitivity of the deformable rocking body model. The free rocking responses of a 4 × 12 m rigid body with material density of 25 tn/m3 computed using the Housner model and DRB models with OpenSees HHT numerical dissipation factor αd = 2/3 and different mesh sizes and integration time steps are shown in Figure 3(b) and (c). The DRB model is not sensitive to the number of beam–column elements used to model the rocking body: Using five or more elements produces an essentially identical response. The DRB model is also not sensitive to the variation of the integration time step dt as long as it is reasonably short with respect to the periods of the dominant motion components. Copyright © 2016 John Wiley & Sons, Ltd.



The outcomes of the same free rocking response simulations computed using DRB models with different OpenSees HHT numerical dissipation factors αd, a 20-beam element rigid body model and the integration time step dt = 104 s, are shown in Figure 3(d). Note that the HHT algorithm numerical damping is maximum for αd = 2/3 and zero for αd = 1. Evidently, the DRB model is not sensitive to the selected OpenSees numerical dissipation factor values as long as they are below 0.95. Note that the simulation with αd = 0.99 is also quite good. Figure 4 demonstrates that convergence is also achieved when the previously mentioned block is excited by the Takatori 090 ground motion from the 1995 Kobe earthquake. For that case, convergence is also demonstrated when the Newmark time integration method [47] is used. 3.1.2. Parametric analysis. Responses of six different rigid rocking bodies, the dimensions of which are given in Table I, to analytical pulse and recorded ground motion excitations were investigated next. The responses computed using the Housner model are compared with the responses computed using a two-fiber 20-beam element DRB model with Ef = 30 × 109 kN/m3, Eb = 30 × 109 kPa, mass density of 25 tn/m3, integrated using dt = 104 s and ad = 2/3. The responses of the Housner model and the DRB model to support motion specified as a symmetric Ricker pulse [48–51] with characteristic period [52] Tp = 0.5 s and amplitude ap = 0.2 g for the slender rigid bodies and Tp = 0.5 s and ap = 0.6 g for the squat rigid bodies are shown in Figure 5. Figure 6 shows the response of the rigid bodies to the ground motion recorded in the OTE building during the 1995 Aigion earthquake. The agreement is again excellent for all cases.

3.2. Rigid rocking frame A rigid frame consisting of four rigid bodies, three identical columns and a cap beam is shown in Figure 7. The columns rest on rigid ground, and the cap beam rests on the columns. The columns can freely uplift and rock, but sliding of the column on the rocking surface or sliding of the beam on the columns is not permitted. The columns and the cap beam have the same mass (mc = mb)

Figure 4. Demonstration of convergence for a recorded ground motion excitation (Takatori 090, Kobe 1995) – dt = 104 s for the HHT and the Newmark method β = γ/2. Table I. Dimensions of the solitary rocking bodies.

Slender Squat

Width, 2b (m)

Height, 2h (m)

Slenderness, tanα

Size parameter, R (m)

0.2 1 4 0.2 1 4

2 10 40 0.6 3 12

0.1 0.1 0.1 0.33 0.33 0.33

1.00 5.02 20.10 0.32 1.58 6.32

Copyright © 2016 John Wiley & Sons, Ltd.



Figure 5. Rocking responses of rigid bodies to symmetric Ricker pulse support motion: slender rigid bodies (left) and squat rigid bodies (right).

Figure 6. Response of rigid bodies to the OTE ground motion recorded during the 1995 Aigion earthquake: slender rigid bodies (left) and squat rigid bodies (right).

Figure 7. Rigid three-column rocking frame. Copyright © 2016 John Wiley & Sons, Ltd.



giving a characteristic mass ratio γm = mc/3mb = 1/3. The response of this frame has been investigated in [3, 4] using a Housner-like approach. Rocking responses of six different frames were analyzed, the columns of each frame having one of the six column geometries listed in Table I. A two-fiber 20-beam DRB model with Ef = 30 × 109 kN/m3, Eb = 30 × 1015 kPa, mass density of 25 tn/m3, integrated using dt = 104 s and αd = 2/3, was used to model the columns and the cap beam. Each column model had a zero-length fiber element at either end to model the interaction with the rocking surface (on the bottom) and the cap beam (on the top). The responses to the symmetric Ricker pulse ground motion excitation (used in previous section) are shown in Figure 8, while the response to the OTE ground motion recorded during the 1995 Aigion earthquake is shown in Figure 9. Evidently, the agreement between the Housner and the DRB

Figure 8. Response of rigid rocking frames to a symmetric Ricker pulse: slender columns (right) and squat columns (left).

Figure 9. Response of rigid rocking frames to the OTE record of the 1995 Aigion earthquake: slender columns (left) and squat columns (right). Copyright © 2016 John Wiley & Sons, Ltd.



models is excellent. The loss of response phasing in some cases originates from the amplitude-period dependence but can be considered negligible. Again, the DRB model utilized only the HHT algorithm numerical dissipation to model the energy dissipated during rocking impacts.

4. AUGMENTED DEFORMABLE ROCKING BODY MODEL FOR DEFORMABLE ROCKING STRUCTURES The results presented in the previous section show that the proposed DRB model can be used to model the rocking response of rigid (numerically very stiff) rocking structures and that its output matches that of the analytical model based on Housner’s approach. One advantage of the DRB model and its implementation using the OpenSees finite element modeling and simulation framework is that it can be used to simulate the rocking response of complex assemblies of rigid bodies (such as the non-symmetric rocking frame studied in [53]) where an analytical solution could become cumbersome. Another advantage of the DRB model and its OpenSees implementation is that it can be used to investigate the rocking response of solitary DRBs and of rocking frames comprised of deformable bodies. 4.1. Solitary deformable rocking body The DRB model of a rigid rocking body utilized a high Young’s modulus of the rocking body Eb and stiffness of the fibers Ef to ensure that the deformations of the body and the surface were negligible for the overall rocking response. The high-frequency shock waves generated in the rocking body at impact were successfully attenuated using the numerical damping property of the HHT integration algorithm. Note that no impact wave energy was transmitted to the rocking surface. The rocking surface was modeled with a zero-length fiber element implemented without any dampers using a non-dissipative material. A DRB is softer, causing the frequency of the generated shock waves to decrease. Thus, as the rocking body becomes more deformable, the amount of energy numerically dissipated by the HHT algorithm diminishes. While the deformations of the rocking body are still small enough such that they do not appreciably affect the accuracy of the computed rocking response, the presence of undamped high-frequency components of motion causes the solution to diverge, that is, decreasing the integration time step and increasing the number of elements do not lead to a unique solution. This divergent behavior is demonstrated in Figure 10, which reproduces the free rocking response of a solitary rocking body (Figure 3) that has a realistic Young’s modulus Eb = 30 GPa. Handling such behavior may require mesh refinement and smaller time steps or use of solution techniques suitable

Figure 10. Sensitivity of the DRB model response to the number of beam–column elements used to model the rocking body and to the length of the integration time step (Eb = 30 GPa). Copyright © 2016 John Wiley & Sons, Ltd.



for stiff ordinary differential equation problems that may not be readily available in common finite element modeling software. To facilitate the analysis of the rocking response of deformable rocking structures using finite element simulation frameworks such as OpenSees, the DRB model is augmented by taking into account the energy radiated into the ground under the rocking surface. The model of the rocking body is preserved: No energy dissipation mechanisms are added there. Instead, the ground on which the body is rocking is modeled as a rigid massless foundation slab that rests on a deformable and dissipative ground (soil) surface represented by a vertical, a horizontal and a rotational support (Figure 11, left). An elastic spring and a viscous dashpot in each support are arranged in parallel. The rigid foundation slab does not uplift. It is important to clarify that modeling the response of a rocking structure on soft soil lies beyond the scope of this study. The supporting surface is still so stiff that the pre-uplift eigenmodes of the rocking body are not influenced by its deformability. The DRB model is augmented such that the stiff rocking surface is modeled numerically using dashpots and stiff springs rather than using a fixed finite element model node in an effort to avoid the generation of shock waves in the rocking body and, thus, creating an alternate energy dissipation path. It is well known from machine vibration theory [54] that the spring stiffness K and dashpot damping coefficient C characteristics that represent the vibration properties of an elastic halfspace under the foundation are frequency dependent. The steady-state solutions for harmonic excitation have been derived and are summarized in [54]. A rocking body, however, transmits instantaneous impact forces (which could be modeled by Dirac delta functions) and alternating moments with amplitude-dependent frequency (which could be approximated by variable frequency square wave functions) to the supporting ground. Therefore, the selected spring–dashpot support model, whose properties are derived using the harmonic load solution, cannot accurately reproduce the response of the ground supporting the rigid foundation slab on which rocking occurs. Instead, a more detailed and computationally expensive model [54, 55] is required. Nevertheless, it is useful to explore how sensitive the response of the proposed simple ground model is to the choice of the spring and dashpot parameter values. The OpenSees implementation of the modified DRB model features an additional set of three zero-length elements that model the ground (Figure 11, right) placed under the zero-length fiber element that models the rocking surface. Each one of the three zero-length elements has an elastic spring and a viscous dashpot arranged in parallel. Their mechanical properties are determined according to [54], assuming that the rocking body is supported on a massless circular surface foundation of radius Rf. The stiffness of the springs is K = Kst × k, and the damping R coefficient is C ¼ K st Vfs c , where Kst is the static stiffness, Vs is the shear wave velocity of the ωR

supporting ground, k and c are dynamic amplification factors given in [54] as a function of αo ¼ V sf , and ω is the cyclic frequency of the harmonic excitation. Poisson’s ratio of the supporting ground is assumed to be ν = 1/3. According to [54], for the range of frequencies considered, the dynamic

Figure 11. A rigid or deformable body rocking on a massless foundation supported by spring–dashpot support (left) and the augmented DRB model (right). Copyright © 2016 John Wiley & Sons, Ltd.



stiffness K of the three springs (vertical, horizontal and rotational) varies between the static value Kst and 0.5Kst. Similarly, the damping amplification factors for the vertical, horizontal and rotational dashpots, respectively, are as follows: cv is approximately 1.0, ch is approximately 0.5, while cr varies from 0 to 0.5. The values of dynamic amplification factors kv = 1, kh = 0.5, kr = 1, cv = 1, ch = 0.5 and cr = 0 are adopted for the three supports in the proposed model of the ground. In a later section, it will be shown that for stiff soils, the response of rocking frames is not sensitive to the values of k and c selected for the three supports in the ground model. The models of the rocking body and the rocking surface developed in Section 2 and examined in Section 3 of this paper remain the same. Note that local deformations and damage of the rocking body and the rocking surface at the pivot points are neglected in this model. Strictly, this limits the use of the proposed DRB model to structures with resilient rocking surfaces such as those reinforced with steel plates [18] or high-strength concrete. Note also that the proposed DRB model represents bodies rocking on a foundation that itself does not uplift, as opposed to a column monolithically connected to an uplifting foundation as in [7]. The seismic response results for a solitary DRB analogous to those presented in Figures 5 and 6 are omitted due to space limitations. The use of the augmented DRB model to compute the seismic response of a deformable rocking frame is presented next. 4.2. Deformable rocking frame It has been shown [3, 4, 56] that the rocking response of a rigid rocking frame is more stable than the rocking response of one of its columns alone. 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 system. This counter intuitive result explains the survival of ancient Greek and Roman monuments for more than 2.5 millennia and has led researchers to propose the rocking frame as a seismic isolation method for bridges. Modern rocking structures may have a similar configuration but will have elements that are more flexible than the elements of ancient temples. Therefore, deformability of columns and girders cannot be neglected in the analysis of the seismic rocking response. Researchers have used simple cantilever models [57–61] to conceptually show that the deformability of a structure does not dramatically change its rocking stability. This section aims to explore the rocking stability of the deformable rocking frame shown in Figure 12 using the augmented DRB model. Again, sliding on the rocking surfaces is not permitted. 4.2.1. Description of the structure and the model sensitivity analysis. The rocking frame consists of three identical deformable columns, each with a cross section that is constant along its height, and a rigid cap beam. The columns are free to uplift (but not slide), as shown in Figure 12. The exact solution of the problem involves the moments of inertia of the columns around their pivot points. Hence, even in the rigid column case, the exact distribution of the mass (both along the column height and in the cross section) is needed. However, it has been shown [33] that the error introduced if one considers that all the mass is concentrated at the column centerline is small. Therefore, in order to keep the parameters of the problem to as few as possible in this study, the mass has been concentrated

Figure 12. Deformable three-column rocking frame: The columns are deformable in bending, but axially rigid, while the cap beam is rigid. Copyright © 2016 John Wiley & Sons, Ltd.



along the centerline of the columns. The axial deformability of the columns is neglected as they are assumed to be axially rigid. Note that this still allows for in-plane transverse vibrations of the columns. The underlying ground is considered to be so stiff that increasing its stiffness would not change the natural vibration periods and modes of the structure. In a simple, non-rocking model, one could merely use fixed nodes at the bottom of the columns. However, to suppress the propagation of impact-induced vibration into the deformable columns, a DRB model augmented by the three-support model of the ground under the rocking surface is used here. Figure 13 plots the in-plane response of a three-column frame structure with concrete (E = 30 GPa, ρ = 2.5 tn/m3) 80 m (h = 40 m) tall and 8 m (b = 4 m) wide columns and a mass ratio γm = 1 to a symmetric Ricker pulse with ap = 1.5 g and Tp = 1.74 s in terms of the base rotation, θ (Figure 12). The columns have a square (2b × 2b) cross section. The foundation is assumed to be circular with radius Rf = 2b. Unless otherwise stated, 20 elements per column, time step dt = 103 s, Vs = 1000 m/s (which represents a Eurocode 8 class A soil) and kv = 1, kh = 0.5, kr = 1, cv = 1, ch = 0.5 and cr = 0 ground parameters are used. Convergence of the base rotation angle response time history with respect to the mesh size and the time step is demonstrated in Figure 13 (top), showing that the 20-element dt = 103 s model is acceptable. The bottom-left plot of Figure 13 shows the response for different damping coefficient values equal to 0.1, 10 and 100 times the reference value of the values selected for the horizontal and the vertical ground dashpots. It also shows the response in the case where damping is added to the rocking degree of freedom of the ground (cr = 0.5). It is evident that, for hard soils and for structures as tall as 80 m, the computed rocking response is insensitive to the value of the ground dashpot damping coefficients. This is in contrast to problems where a body impacts the ground centrally and the energy dissipated at impact is proportional to the value of the damping coefficient of the vertical spring in the ground model (and, where, if the damping is larger than a critical value, all the energy is dissipated). Figure 13 bottom-right shows that the selected shear wave velocity of the supporting ground, Vs = 1000 m/s, is sufficiently large to represent the high stiffness of the ground under the rocking surface. It should be noted that while increasing the ground stiffness by increasing Vs leads to convergence of the column rotation time history, the same does not hold for the time history of the column base moment response. On the contrary, high-frequency oscillations of the base moment appear in the solution. They are caused by the impact and depend on the selected ground stiffness.

Figure 13. Convergence study of the response of a deformable rocking frame with three concrete (E = 30 GPa, ρ = 2.5 tn/m3) 80 × 8 m columns and mass ratio γm = 1 to a symmetric Ricker pulse with ap = 1.5 g and Tp = 1.74 s. Copyright © 2016 John Wiley & Sons, Ltd.



The high-frequency components of the column base moment response were observed both in small-scale experiments [39, 41, 44] and in analytical models [39, 57–61]. The analytical models are, generally, not able to accurately capture the high-frequency components of the column base moment response. More experimental and analytical work is needed to model the variation of impact-related forces in the rocking columns. Therefore, the augmented DRB model presented here can be used to predict the overturning stability of DRB and frames, but the ability of the proposed model to accurately predict the internal forces in DRBs has not been proven. 4.2.2. Overturning spectra for a deformable three-column rocking frame under pulse excitation. The augmented DRB model is used to compute the overturning spectra for a three-column deformable rocking fame shown in Figure 12 to symmetric and anti-symmetric Ricker pulse ground motion

Figure 14. Rocking spectra of a deformable three-column rocking frame with column slenderness equal to tanα = 0.2 for a symmetric Ricker pulse excitation. Copyright © 2016 John Wiley & Sons, Ltd.



excitation. Because the axial stiffness of the columns is neglected, the base rotation is a function of nine variables: θ ¼ f α; R; mc ; ωn ; mb ; ap ; ωp ; g

(1)

where ωn is the fundamental natural cyclic vibration frequency of the frame before it uplifts and ωp is derived from the excitation pulse period Tp. According to Buckingham’s Π theorem, the previous equation can be written as ωp ωn a p θ ¼ f α; ; γm ; ; p p gtanα where p ¼

qffiffiffiffi 3g 4R

(2)

mb and γm ¼ 3m . c

Figure 14 shows the symmetric Ricker pulse overturning spectra for the three-column deformable rocking frame with column slenderness tanα = 0.2 and different values of normalized column stiffness ωn/p and mass ratio γm. As ωn/p goes to infinity, the rigid frame case from Section 3.2 is recovered, and the symmetric Ricker pulse rocking spectra trends are confirmed [13]. Namely, large rocking structures require high acceleration or long-period pulses to overturn. Moreover, there are multiple modes of overturning: after zero, one or more rocking impacts. In Figure 14, the black regions correspond to overturning in the negative drift θ direction, while the gray region corresponds to overturning in the positive drift θ direction. Figure 15 (left) plots the minimum non-dimensional symmetric Ricker pulse acceleration (ap/gtanα) needed to overturn a three-column deformable rocking frame with a given size–frequency parameter ωp/p. The minimum non-dimensional symmetric Ricker pulse acceleration needed to uplift the same frame is plotted in Figure 15 (middle). It can be observed that 1. Relative to the rigid case, the region representing the overturning-with-two-impacts response mode shrinks, and the overturning non-dimensional symmetric Ricker pulse accelerations decrease, as was observed for cantilever structures in [60]; 2. A new region of stability (obviously not relevant to design) between the overturning-with-noimpact response mode and overturning-with-one-impact response mode appears; 3. Stability of the frame depends on its size and deformability, as well as on the excitation pulse period. For smaller frames or for long pulses, that is, when ωp/p < 4, the rigid three-column frame is slightly more stable than the deformable three-column frame. However, for ωp/p > 4 (large frames or short-period pulses), a deformable frame of the same size is more stable than a corresponding rigid frame, showing that the deformability of the columns does not jeopardize the stability of a rocking frame. In any case, the differences between the minimum non-dimensional overturning pulse acceleration spectra of the deformable and the rigid frames are not large;

Figure 15. Minimum overturning acceleration spectra of a deformable three-column rocking frame with column slenderness equal to tanα = 0.2 for a symmetric Ricker pulse excitation. Copyright © 2016 John Wiley & Sons, Ltd.



Figure 16. Rocking spectra of a deformable three-column rocking frame with column slenderness equal to tanα = 0.2 for an anti-symmetric Ricker pulse excitation.

4. Increasing the weight of the cap beam three times makes the frame slightly more stable; 5. As expected, the value of the non-dimensional pulse acceleration needed to uplift the frame is strongly influenced by the deformability (stiffness) of the columns. The quantity gtanα used to normalize the overturning acceleration represents the ground acceleration that causes uplift of a solitary rigid rocking body. When it comes to deformable rocking structures, the pulse that is capable of barely uplifting the structure does not generally have an acceleration amplitude equal to gtanα; instead, the uplifting pulse acceleration depends on the eigenfrequencies of the structure. Therefore, normalizing the pulse acceleration, ap, with the pulse acceleration that barely uplifts the structure, apuplifting, leads to a more meaningful excitation magnitude comparison for deformable rocking structures. Copyright © 2016 John Wiley & Sons, Ltd.



Figure 17. Minimum overturning acceleration spectra of a deformable three-column rocking frame with column slenderness equal to tanα = 0.2 for an anti-symmetric Ricker pulse excitation.

The ratio

aoverturning p auplifting p

represents the safety factor against overturning of a deformable rocking frame

assuming that the frame was designed against uplift with a safety factor equal to 1. It shows how many times stronger the ‘overturning pulse’ is than the ‘uplifting pulse’. The plot of this safety factor for a three-column deformable rocking frame under the symmetric Ricker pulse ground motion excitation (Figure 15, right) shows that this safety factor increases as the deformability of the column increases. Consequently, when a flexible and a rigid frame of the same geometry barely uplift for the design earthquake, the flexible one has a larger safety margin against overturning. Figures 16 and 17 plot the same overturning spectra but for an anti-symmetric Ricker pulse ground motion excitation. The results and the observed trends are similar to those for the symmetric Ricker pulse ground motion excitation.

CONCLUSIONS Analytical models of rocking response of rigid or deformable bodies and simple structures comprised of rigid or deformable bodies exist. Analytical models of more complex rocking structures are cumbersome, while numerical models of such rocking structures, particularly if they comprise deformable elements, are difficult to implement in conventional finite element modeling frameworks and are computationally demanding. A new finite element model to analyze the response of DRBs and structures, assembled of rigid and DRBs and deformable structural elements, to ground motion excitation was presented in this paper. The DRB model comprises a set of beam elements to represent the DRB and a pair of zero-length fiber cross-section elements at the ends of the rocking body to represent the rocking surfaces. As such, the DRB model integrates into existing finite element simulation frameworks without difficulties. The amplitude decay of the rocking motion only comes from numerically damping out the impact waves generated at every impact: No viscous Rayleigh or concentrated damping models were used. This novel approach is a step toward solving the long-standing problem of accurately modeling the instantaneous impact-induced energy dissipation that occurs during rocking motion. The proposed DRB model was first verified by correctly predicting the horizontal and the vertical displacements of a rigid rocking block. Then, the proposed model was validated against the analytical Housner model for predicting the rocking response of rigid bodies to ground motion excitation. The proposed DRB model was augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. This augmentation was necessary to minimize the high-frequency vibration of the DRB due to pulse loads occurring at rocking impacts and, thus, facilitate numerical integration of the response time history using conventional numerically dissipative algorithms. The sensitivity of the augmented DRB model with respect to the stiffness and damping parameters of the ground model was Copyright © 2016 John Wiley & Sons, Ltd.



investigated to conclude that the model is not sensitive to the choice of parameter values as long as these values correspond to stiff soils and/or rock. To illustrate the new analysis options made possible by the augmented DRB model, it was used to compute the overturning and uplift rocking response spectra for a deformable three-column 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 that its rigid counterpart. Out of the two similar flexible and a rigid rocking frames that have the same safety factor against uplift, the flexible frame has a larger safety margin against overturning. While the proposed augmented DRB model can be used to investigate overturning of DRBs and rocking frame structures comprised of such deformable bodies under ground motion excitation, more work is needed to (1) to capture the impact-induced high-frequency vibration of the rocking bodies and excitation of their higher vibration modes and (2) to accurately compute the internal forces occurring in the DRBs and rocking frame elements. Benchmark experiments on large-scale deformable bodies are needed to verify and validate the internal force and high-frequency vibration predictions obtained using such improved analytical and finite element models. REFERENCES 1. Milne J. Seismic experiments. Trans. Seism. Soc. Japan 1885; 8:1–82. 2. Housner GW. The behaviour of inverted pendulum structures during earthquakes. Bulletin of the Seismological Society of America 1963; 53(2):404–417. 3. Makris N, Vassiliou MF. Planar rocking response and stability analysis of an array of free standing columns capped with a freely supported rigid beam. Earthquake Engineering & Structural Dynamics 2013; 42(3):431–449. 4. Makris N, Vassiliou MF. Are some top-heavy structures more stable? Journal of Structural Engineering 2014; 140(5):. 5. DeJong MJ, Dimitrakopoulos EG. Dynamically equivalent rocking structures. Earthquake Engineering & Structural Dynamics 2014; 43:1543–1563. 6. Stiros SC. Identification of earthquakes from archaeological data: methodology, criteria and limitations. Archaeoseismology, (Stiros S, Jones RE Eds.). British School at Athens: Oxford, 1996. p 129–152 Occasional Paper no. 7 of the Fitch Laboratory. 7. Anastasopoulos I, Gazetas G, Loli M, Apostolou M, Gerolymos N. Soil failure can be used for seismic protection of structures. Bulletin of Earthquake Engineering 2010; 8(2):309–326. 8. Hung HH, Liu KY, Ho TH, Chang KC. An experimental study on the rocking response of bridge piers with spread footing foundations. Earthquake Engineering & Structural Dynamics 2011; 40(7):749–769. 9. Gelagoti F, Kourkoulis R, Anastasopoulos I, Gazetas G. Rocking isolation of low-rise frame structures founded on isolated footings. Earthquake Engineering & Structural Dynamics 2012; 41:1177–1197. 10. Antonellis G, Panagiotou M. Seismic response of bridges with rocking foundations compared to fixed-base bridges at a near-fault site. Journal of Bridge Engineering 2013; 19(5):. 11. Makris N. The role of the rotational inertia on the seismic resistance of free-standing rocking columns and articulated frames. Bulletin of the Seismological Society of America. 2014; 104(5):2226–2239. 12. Makris, N., & Kampas, G. (2016). Size versus slenderness: two competing parameters in the seismic stability of free-standing rocking columns. Bulletin of the Seismological Society of America. 13. Makris N, Vassiliou MF. Sizing the slenderness of free-standing rocking columns to withstand earthquake shaking. Archive of Applied Mechanics 2012; 82(10-11):1497–1511. 14. Beck JL, Skinner RI. The seismic response of a reinforced concrete bridge pier designed to step. Earthquake Engineering & Structural Dynamics 1973; 2:343–358. 15. Ma, Q. T. and Khan, M. H. (2008). Free vibration tests of a scale model of the South Rangitikei Railway Bridge. In Proceedings of the New Zealand Society for Earthquake Engineering Annual Conference, Engineering an Earthquake Resilient NZ. 16. Sharpe RD, Skinner RI. The seismic design of an industrial chimney with rocking base. Bulletin of the New Zealand National Society for Earthquake Engineering 1983; 16(2):98–106. 17. Papadopoulos C. DOMOS Engineers, Athens, Greece, personal communication, September 19, 2014 18. Mander, J. B. and Cheng, C. T. (1997). Seismic resistance of bridge piers based on damage avoidance design. Technical Report NCEER-97-0014, SUNY, Buffalo, 1997. 19. Priestley MN, Sritharan S, Conley JR, Pampanin S. Preliminary results and conclusions from the PRESSS five-story precast concrete test building. PCI Journal 1999; 44(6):42–67. 20. Ricles JM, Sause R, Garlock MM, Zhao C. Posttensioned seismic-resistant connections for steel frames. Journal of Structural Engineering 2001; 127(2):113–121. Copyright © 2016 John Wiley & Sons, Ltd.


FE MODELING AND BEHAVIOR OF A DEFORMABLE ROCKING FRAME 21. Palermo A, Mashal M. Accelerated Bridge Construction (ABC) and seismic damage resistant technology: a New Zealand challenge. Bulletin of the New Zealand Society for Earthquake Engineering 2012; 45(3):123–134. 22. Sideris, P. (2012). Seismic analysis and design of precast concrete segmental bridges. PhD Thesis, State University of New York at Buffalo. 23. Eatherton MR, Ma X, Krawinkler H, Mar D, Billington S, Hajjar JF, Deierlein GG. Design concepts for self-centering rocking steel braced frames. Journal of Structural Engineering 2014; 140(11):. 24. Makris N, Konstantinidis D. The rocking spectrum and the limitations of practical design methodologies. Earthquake Engineering & Structural Dynamics 2003; 32(2):265–289. 25. Makris N, Vassiliou MF. Dynamics of the rocking frame with vertical restrainers. Journal of Structural Engineering 2014; 141(10 04014245):. 26. Vassiliou MF, Makris N. The dynamics of the vertically restrained rocking column. Journal of Engineering Mechanics 2015; 141(12):. 27. Petrini L, Maggi C, Priestley MN, Calvi GM. Experimental verification of viscous damping modeling for inelastic time history analyzes. Journal of Earthquake Engineering 2008; 12(S1):125–145. 28. Hall JF. Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Engineering & Structural Dynamics 2006; 35(5):525–545. 29. Chopra AK, McKenna F. Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation. Earthquake Engineering & Structural Dynamics 2016; 45(2):193–211. 30. Wiebe, L., Christopoulos, C., Tremblay, R., Leclerc, M. (2012). Modelling inherent damping for rocking systems: results of large-scale shake table testing. In Proceedings of the 15th World Conference on Earthquake Engineering (pp. 24–28). 31. McKenna F, Fenves GL, Scott MH. Open system for earthquake engineering simulation. University of California: Berkeley, CA, 2000. 32. Barthes, C. B. (2012). Design of earthquake resistant bridges using rocking columns, PhD Thesis, UC Berkeley, USA. 33. Vassiliou MF, Mackie KR, Stojadinović B. Dynamic response analysis of solitary flexible rocking bodies: modeling and behavior under pulse-like ground excitation. Earthquake Engineering & Structural Dynamics 2014; 43 (10):1463–1481. 34. Argyris JH, Kelsey S, Kamel H. Matrix methods of structural analysis: a precis of recent developments. Pergamon Press: Oxford, 1964. 35. Lipscombe PR, Pellegrino S. Free rocking of prismatic blocks. Journal of Engineering Mechanics 1993; 119 (7):1387–1410. 36. Peña F, Prieto F, Lourenço PB, Campos Costa A, Lemos JV. On the dynamics of rocking motion of single rigidblock structures. Earthquake Engineering & Structural Dynamics 2007; 36(15):2383–2399. 37. Cheng CT. Shaking table tests of a self-centering designed bridge substructure. Engineering Structures 2008; 30 (12):3426–3433. 38. Konstantinidis D, Makris N. Experimental and analytical studies on the response of 1/4-scale models of freestanding laboratory equipment subjected to strong earthquake shaking. Bulletin of Earthquake Engineering 2010; 8 (6):1457–1477. 39. Ma, Q. T. M. (2010). The mechanics of rocking structures subjected to ground motion (Doctoral dissertation, ResearchSpace@ Auckland). 40. ElGawady MA, Ma Q, Butterworth JW, Ingham J. Effects of interface material on the performance of free rocking blocks. Earthquake Engineering & Structural Dynamics 2011; 40(4):375–392. 41. Baratta A, Corbi I, Corbi O. Towards a seismic worst scenario approach for rocking systems: analytical and experimental set-up for dynamic response. Acta Mechanica 2013; 224(4):691–705. 42. Zhang H, Brogliato B, Liu C. Dynamics of planar rocking-blocks with Coulomb friction and unilateral constraints: comparisons between experimental and numerical data. Multibody System Dynamics 2014; 32(1):1–25. 43. Truniger, R., Vassiliou, M. F., Stojadinovic, B. (2014). Experimental study on the interaction between elasticity and rocking. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014. 44. Truniger R, Vassiliou MF, Stojadinovic B. An analytical model to study deformable cantilever structures rocking on a rigid surface: experimental Verification. Earthquake Engineering & Structural Dynamics 2015; 44(15):2795–2815. 45. Bathe KJ. Finite element procedures. Prentice Hall: Englewood Cliffs, 2006. 46. Hilber HM, Hughes TJ, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering & Structural Dynamics 1977; 5(3):283–292. 47. Newmark NM. A method of computation for structural dynamics. Journal of the engineering mechanics division 1959; 85(3):67–94. 48. Mavroeidis GP, Papageorgiou AS. A mathematical representation of near-fault ground motions. Bull. Seismol. Soc. Am. 2003; 93(3):1099–1131. 49. Vassiliou MF, Makris N. Estimating time scales and length scales in pulselike earthquake acceleration records with wavelet analysis. Bulletin of the Seismological Society of America 2011; 101(2):596–618. 50. Lungu, A. & Giaralis, A. (2013). A non-separable stochastic model for pulse-like ground motions. Paper presented at the 11th International Conference on Structural Safety and Reliability for Integrating Structural Analysis, Risk and Reliability, 16–20 June 2013, New York, USA. Copyright © 2016 John Wiley & Sons, Ltd.


M. F. VASSILIOU, K. R. MACKIE AND B. STOJADINOVIĆ 51. Ricker N. Further developments in the wavelet theory of seismogram structure. Bulletin of the Seismological Society of America 1943; 33:197–228. 52. Vassiliou MF, Makris N. Analysis of the rocking response of rigid blocks standing free on a seismically isolated base. Earthquake Engineering & Structural Dynamics 2012; 41(2):177–196. 53. Dimitrakopoulos EG, Giouvanidis AI. Seismic response analysis of the planar rocking frame. Journal of Engineering Mechanics 2015; 141(7 04015003):. 54. Gazetas G. Analysis of machine foundation vibrations: state of the art. International Journal of Soil Dynamics and Earthquake Engineering 1983; 2(1):2–42. 55. Adamidis O, Gazetas G, Anastasopoulos I, Argyrou C. Equivalent-linear stiffness and damping in rocking of circular and strip foundations. Bulletin of Earthquake Engineering 2014; 12(3):1177–1200. 56. Papaloizou L, Komodromos P. Planar investigation of the seismic response of ancient columns and colonnades with epistyles using a custom-made software. Soil Dynamics and Earthquake Engineering 2009; 29(11):1437–1454. 57. Psycharis IN, Jennings PC. Rocking of slender rigid bodies allowed to uplift. Earthquake Engineering & Structural Dynamics 1983; 11:57–76. 58. Chopra AK, Yim SCS. Simplified earthquake analysis of structures with foundation uplift. Journal of Structural Engineering 1985; 111(4):906–930. 59. Oliveto G, Caliò I, Greco A. Large displacement behaviour of a structural model with foundation uplift under impulsive and earthquake excitations. Earthquake Engineering & Structural Dynamics 2003; 32:369–393. 60. Acikgoz S, DeJong MJ. The interaction of elasticity and rocking in flexible structures allowed to uplift. Earthquake Engineering & Structural Dynamics 2012; 41(15):2177–2194. 61. Vassiliou MF, Truniger R, Stojadinovic B. An analytical model of a deformable cantilever structure rocking on a rigid surface: development and verification. Earthquake Engineering & Structural Dynamics 2015; 44(15):2775–2794.

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