Simplified discrete systems for dynamic analysis of structures on footings and piles Andreas Maravas a,n , George Mylonakis b,c , Dimitris L. Karabalis d a Department of Engineering, University of Patras, Patras 26500, Greece b Department of Civil Engineering, University of Patras, Patras 26500, Greece c University of California at Los Angeles, Los Angeles, CA 90095-1593, USA d Department of Civil Engineering, University of Patras, Patras 26500, Greece article info Article history: Received 11 January 2014 Received in revised form 18 January 2014 Accepted 20 January 2014 Available online 18 February 2014 Keywords: Soil–structure interaction Piles Footings Natural period Radiation damping Foundation impedance abstract A simplified discrete system in the form of a simple oscillator is developed to simulate the dynamic behavior of a structure founded through footings or piles on compliant ground, under harmonic excitation. Exact analytical expressions for the fundamental natural period and the corresponding damping coefficients of the above system are derived, as function of geometry and the frequency-dependent foundation impedances. In an effort to quantify the coupling between swaying and rocking oscillations in embedded foundations such as piles, the reference system is translated from the footing–soil interface to the depth where the resultant soil reaction is applied, to ensure a diagonal impedance matrix. The resulting eccentricity is a measure of the coupling effect between the two oscillation modes. The amounts of radiation damping generated from a single pile and a surface footing are evaluated. In order to compare the damping of a structure on a surface footing and a pile, the notion of static and geometric equivalence is introduced. It is shown that a pile may generate significantly higher radiation damping than an equivalent footing, thus acting as an elementary protective system against seismic action. & 2014 Published by Elsevier Ltd. 1. Introduction During the last fifty years, the problem of dynamic soil–structure interaction (SSI) has received considerable research attention. Knowl- edge on the subject has been derived mainly from studies on the dynamic behavior of structures resting on surface foundations. Some of these studies have become standard references in the area of foundation dynamics, e.g., Richart et al. [1], Parmelee [2], Bielak [3], Veletsos and Wei [4], Veletsos and Verbic [5], Veletsos [6], Luco and Westmann [7,8], Vaish and Chopra [9], Luco [10], Wong and Luco [11,12] and Wolf [13]. It is established that SSI causes significant alterations to the dynamic response of structures supported on deformable soils in comparison to the response of the same structures when considered fixed at their base, mainly by increasing the natural period and damping of the fundamental mode [6,13,14]. The analysis of soil–foundation–structure interaction can be accomplished by various methods, depending on the part of the system that is analyzed. These methods can be classified as: (a) analytical, which usually refer to simple foundation geometries lying on elastic half-space, e.g., Triantafyllidis [15], (b) semi-analytical, that combine analytical formulations for the half-space with numerical procedures e.g., the subdivision of the soil–foundation contact area for solving the mixed boundary value problem by integration of the corresponding surface-to-surface Green's functions used by Vrettos [16] to determine the vertical and rocking response of rigid founda- tions resting on a non-homogeneous half-space, or usually Finite Element Methods (FEM) for the discretization of the foundation and the structure, e.g., Wong and Luco [12], (c) numerical, usually FEM, e.g., Lysmer et al. [17], Boundary Element Methods (BEM) e.g., Ahmad and Banerjee [18] or a combination of FEM and BEM, e.g., Karabalis and Beskos [19], Gaitanaros and Karabalis [20] employed for the discretiza- tion of the soil medium, the foundation and the structure, and (d) simplified discrete models, which allow fast calculation of the founda- tion–soil–structure system properties, e.g., Veletsos and Meek [21], Dobry and Gazetas [22]. More information on the subject can be found in various sources, e.g., Gazetas [23,24], Karabalis [25], Beskos [26], Wolf [27,28], Mylonakis et al. [29], Stewart et al. [30,31], among others. This work is focused on the development and use of discrete models, for estimating the dynamic behavior of simple structures resting on an elastic half-space. Discrete models for the analysis of soil–foundation–structure system have been developed by various researchers. One of the earliest such attempts is the one by Lysmer and Richart [32] who derived a single-degree-of-freedom model for computing the vertical dynamic response of foundations connected to an elastic half-space. Luco [10] derived analytical expressions for the dynamic stiffness of circular, long strip and square foundations laying on the surface of a layered elastic half-space. Parmelee [2], Veletsos Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/soildyn Soil Dynamics and Earthquake Engineering 0267-7261/$ - see front matter & 2014 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.soildyn.2014.01.016 n Corresponding author. Tel.: þ302610996557; fax: þ30 2610997660. E-mail addresses: anmaravs@upatras.gr, andreasmaravas@gmail.com (A. Maravas), g.mylonakis@bristol.ac.uk (G. Mylonakis), karabali@upatras.gr (D.L. Karabalis). Soil Dynamics and Earthquake Engineering 61-62 (2014) 29–39