Analysis of block random rocking on nonlinear flexible foundation

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Abstract

In this paper the rocking response of a rigid block randomly excited at its foundation is examined. A nonlinear flexible foundation model is considered accounting for the possibility of uplifting in the case of strong excitation. Specifically, based on an appropriate nonlinear impact force model, the foundation is treated as a bed of continuously distributed springs in parallel with nonlinear dampers. The statistics of the rocking response is examined by an analytical procedure which involves a combination of static condensation and stochastic linearization methods. In this manner, repeated numerical integration of the highly nonlinear differential equations of motion is circumvented, and a computationally efficient semi-analytical solution is obtained. Comparisons with pertinent Monte Carlo simulations demonstrate the efficiency and reliability of the proposed approach. Finally, the survival probability related to toppling of the blocks subject to filtered white noise excitation is investigated, for different kinds of block geometries, foundation materials, and filter parameters.

Introduction

Rocking motion, a complex phenomenon related to the behavior of block-like structures allowed to rock due to base excitation has been extensively studied over a period of several decades. In this regard, considerable research efforts have been devoted to determining the rocking response of structures not rigidly connected to their foundations. Representative examples are oil tanks [1]; pieces of machinery [2], [3]; and museum exhibits [4] of marble ancient structures [5], [6]. Nevertheless, the problem remains technically challenging due to, inter alia, the potential beneficial effects induced by base uplifting occurring during rocking motion, as exploited for the design of several bridges [7], [8], [9], [10].

Several alternative analytical models have been proposed to study rocking dynamics. However, two models are primarily used to describe the rocking of rigid bodies subjected to ground motion; they are two-dimensional, and afford a reasonable representation of the phenomenon. The first model, henceforth referred to as the Housner model (HM) [1], deals with the motion of a rigid block rocking about its base corners on a rigid foundation [7], [11], [12], [13], [14]. In this regard, piecewise analytical solutions have been obtained both for the free-rocking case [15], and for harmonic base excitation [16]. The second model, known as the Winkler foundation model (WFM), deals with the motion of a rigid block rocking on a flexible foundation of distributed linear vertical springs and dashpots [17]. In this regard, several authors have analyzed several aspects of this model, providing results for the case of harmonic and earthquake base excitations [17], [18], [19], as well, insight on the coupled effect of sliding and uplifting [20].

Note that, considering the inherent stochastic nature of the seismic excitation, several authors have analyzed the rocking response in case of random base motion, although to a lesser degree with respect to the case of deterministic excitation. Specifically, some analyses considering the HM are reported in [21], [22], [23], [24], [25], [26], [27], while some studies considering the WFM can be found in [28], [29].

Further, recent studies have pointed out the strong effect of different foundation materials on the rocking dynamics, which may lead in some cases to counterintuitive responses [11], [30], especially when flexible foundation materials are employed. In this regard, to better understand this complex phenomenon, as well as to take into account aspects which may arise during the rocking motion of rigid blocks on flexible foundations, recently in [31] a nonlinear model has been used for the base-foundation interaction. Specifically, the Hunt and Crossley nonlinear impact force model [32] has been adopted, and the foundation has been treated as a bed of continuously distributed linear tensionless springs in parallel with nonlinear dampers.

In this paper, based on the aforementioned study, the random rocking response of rigid blocks on the proposed nonlinear foundation model is considered. Specifically, both the cases of time-modulated white noise and filtered white noise excitations, possessing the commonly employed Tajimi–Kanai power spectral density (PSD), are considered for the horizontal ground acceleration. In this manner a combination of the statistical linearization, and of the static condensation approaches is employed to derive a computationally treatable equation governing the evolution of the rocking response statistics. The accuracy of the procedure is assessed via pertinent Monte Carlo simulation data using as model parameter values those obtained from previous experimental studies [31]. Finally, the survival probability related to toppling of blocks with different dynamic characteristics (geometry and foundation materials) is examined via Monte Carlo analyses. In this case, time-modulated filtered white noise excitations with PSD of the Tajimi–Kanai type are used to examine whether spectra with the same variance but different peak frequencies lead to variations of the survival probabilities of the blocks depending on their dynamic characteristics.

Section snippets

Mathematical formulation

Consider a rectangular rigid block with mass m, and polar moment of inertia Icm about the center of mass cm, as shown in Fig. 1(a). The variable R denotes the distance of the base corners from the center of mass, situated at height h above the base of width 2b. Further, let θcr be the critical tilt-angle, that is, the maximum angle to which the block can be tilted without overturning under the action of gravity, g, alone.

To simplify the ensuing analysis, the center of the base cb is restricted

Stochastic linearization

Let the horizontal ground acceleration be modeled as a time-modulated random excitation. That is, ẍg=qtvt,where vt is a stationary broad-band random process with double-sided power spectral density (PSD) Svω, and qt is a deterministic function used to capture the typical intensity variation of classical earthquake records. Next, the method of stochastic linearization can be applied to estimate block response to moderate level random base excitation. To this end, firstly the piecewise variable

Numerical results

In this section the proposed approach is applied to the case of rigid blocks on nonlinear foundation model considering both the cases of time-modulated white noise, and filtered white noise excitation employing the modulating function [35] qt=4expt4expt2,which rises rapidly from zero to a unitary peak and then decays exponentially to zero.

To assess the accuracy of the proposed procedure, stochastic linearization based statistical moments are juxtaposed with the results of pertinent Monte

References (35)

  • AgbabianM.S. et al.

    Evaluation of earthquake damage mitigation methods for museum objects

    Stud. Conserv.

    (1991)
  • KounadisA.N.

    On the rocking complex response of ancient multispondyle columns: a genious and challenging structural system requiring reliable solution

    Meccanica

    (2015)
  • YimC.S. et al.

    Rocking response of rigid blocks to earthquakes

    Earthq. Eng. Struct. Dyn.

    (1980)
  • KohA.S. et al.

    Base isolation benefits of 3-D rocking and uplift I: theory

    J. Eng. Mech.

    (1991)
  • AhmadiE. et al.

    On the use of entangled wire materials in pre-tensioned rocking columns

    J. Phys. Conf. Ser.

    (2019)
  • PalmeriA. et al.

    Response of rigid structures rocking on viscoelastic foundation

    Earthq. Eng. Struct. Dyn.

    (2008)
  • GesualdoA. et al.

    Rocking of freestanding objects: theoretical and experimental comparisons

    J. Theoret. Appl. Mech.

    (2018)
  • Cited by (0)

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