Evaluation of soil-structure interaction effects on the damping ratios of buildings subjected to earthquakes

Soil Dynamics and Earthquake Engineering 100 (2017) 183–195

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Evaluation of soil-structure interaction effects on the damping ratios of buildings subjected to earthquakes

MARK

⁎

Cristian Cruz , Eduardo Miranda Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020, United States

A R T I C L E I N F O

A B S T R A C T

Keywords: Damping ratios Soil-structure interaction Radiation damping Damping of higher modes

This paper evaluates the effects of soil-structure interaction on damping ratios of buildings subjected to earthquake ground motions. Explicit expressions of transfer functions of absolute horizontal accelerations of multistory buildings on an elastic half-space subjected to horizontal ground motions are developed. An optimization procedure is then used to obtain the effective modal properties of a replacement or “equivalent” fixed-based multi-story building that minimize the difference in the ordinates of the absolute acceleration transfer functions in the building on flexible-base relative to those in the replacement fixed-base building. A parametric study was conducted to study the variation of effective modal damping ratios of the fundamental mode and of higher modes of vibration with changes on the wave parameter. Results indicate soil-structure interaction effects may either increase or reduce the effective modal damping ratio of the fundamental period. Typically, it reduces the effective damping ratios of slender long period structures such as tall buildings and increases the damping ratio of short and medium period structures. It is shown that the reduction of effective modal damping ratio of the fundamental mode with increasing height computed in this study follows a similar trend to that observed in instrumented buildings, indicating that the reduction in damping ratio observed in buildings as their height increases is primarily due to soil-structure interaction effects. Furthermore, it is shown that soil-structure interaction effects lead to an approximately linear trend in effective modal damping ratios with increasing modal frequency.

1. Introduction The viscous damping ratio is a simple mathematical representation of the energy dissipated by all the damping mechanisms in a structure other than those explicitly modeled through nonlinear hysteretic behavior. The idea is to group the contribution of all the sources of energy dissipation – whose modeling may be impractical, too complex, or not yet fully understood – into a simple set of modal parameters. Even when there is evidence that some of these sources do not behave in a viscous manner, assuming viscous damping has become the standard in current analysis methods (e.g., [1,2]) because it significantly simplifies the differential equations of motion [3] while producing acceptable results [4,5]. If the building is assumed to have a fixed base, i.e. considering the soil as an infinitely rigid material, then the damping ratios also include the effects of soil-structure interaction. It has been well documented that soil-structure interaction modifies the response of buildings. When sitting on flexible soil, the system considering the superstructure and the soil-foundation interface will have larger modal periods than its rigid counterpart [6–9]. Even when

⁎

Corresponding author. E-mail addresses: [email protected] (C. Cruz), [email protected] (E. Miranda).

http://dx.doi.org/10.1016/j.soildyn.2017.05.034 Received 17 November 2016; Received in revised form 25 May 2017; Accepted 29 May 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

the inherent (hysteretic) damping of the soil is neglected, having a flexible base will add an additional source of energy dissipation in the form of waves travelling away from the system. This type of energy dissipation, referred to as radiation damping, arises due to the geometry of the new system – which extends to lengths many times the largest dimension of the structure. The exact solution to the soil-structure interaction problem is complex even when using the simplest possible soil model such as a homogeneous linear elastic half-space. Obtaining the analytical response of an infinite elastic body being solicited by loads on (or within) it is a problem that puzzled mathematicians and engineers at the middle of the 19th century, and that was solved for a number of situations more than 100 years later, after the work of dozens of researchers (summarized in [10]). In the early 1970s, Veletsos and Wei [11], and Luco and Westmann [12,13] provided numerical solutions for the stiffness of rigid circular plates subjected to harmonic loads, sitting on an elastic half-space. Moreover, they showed that the stiffness and energy dissipation due to radiation damping of the soil-foundation interface could be modeled by springs and dashpots with frequency-dependent properties, respectively. This extraordinary

Soil Dynamics and Earthquake Engineering 100 (2017) 183–195

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result shows that radiation damping – a geometric energy dissipation source – can be modeled exactly using viscous dampers. Since then, there have been numerous studies solving different variations of the problem, e.g. considering embedded, non-circular, non-rigid, or piled foundations. More details can be found in several excellent literature reviews on the subject (e.g., [10,14,15]). Incorporating the flexibility of the soil in the numerical model of a building poses an additional challenge because the damping matrix of the soil-foundation-structure system is usually not classical and therefore cannot be uncoupled using normal modes. However, an approximation to the solution can be obtained using a normal modes and specifying appropriate damping values that take into account soilstructure interaction. Tsai [16] proposed a method to compute the normal-mode damping ratios by matching the amplitude transfer functions of the exact solution with that of the normal-mode approximation for a certain location of the building, evaluated at the different resonant frequencies of the system. Roesset et al. [17], Novak [18,19] and Rainer [20] computed normal-mode damping ratios using an equivalent viscous damping approach, i.e., by equating the ratio of dissipated work to the total work occurring during a period of vibration, to the theoretical work ratio that an SDOF system with a linear viscous damper would have under the same load. Novak and El Hifnawy [21] compared the damping ratios obtained using energy consideration approach of Novak [18,19] with the results of a rigorous complex eigenvalue analysis using non-classical modes. Their results show that these methods give similar results for the first mode, but can differ significantly for higher modes. Other solutions to the problem do not involve a full modal approach. Several authors have provided direct solutions to the problem (e.g., Tajimi [22], Bielak [23], Lee and Westley [24], Jennings and Bielak [7], Chopra and Gutierrez [8], Luco [25], and others summarized in [26]), some of which employ modal superposition that separate the equations that govern the superstructure alone, using fixed-base modes, and incorporate these equations to those governing the rocking and swaying of the foundation (e.g. [8,22,24]). More recently, Luco and Lanzi [27], and Lanzi and Luco [28] provided approximate analytical expressions for the system frequencies, damping ratios, and participation factors based on a perturbation approach. In spite of its importance as a significant source of flexibility of the system and its energy dissipation, soil-structure interaction is rarely explicitly taken into account in the seismic design of buildings. Usually, buildings are modeled with a rigid base, and the design is based on a response spectrum analysis. Also, most of the system identification techniques employed to infer damping ratios from the floor accelerations recorded in a building assume a fixed condition at the base (e.g. [4,29,30]). Consequently, studies on damping ratios inferred from recorded accelerations in buildings do not analyze damping ratio of the superstructure alone, but the damping ratio of a fixed-base structure that also incorporates the effects of soil-structure interaction (e.g. [5,31–33]). For this reason, the objective of this paper is to evaluate how the damping ratios of a fixed-base system should be modified in order to take into account the effects of radiation damping. First, the equations of motion of a planar N-story building with a circular foundation sitting on an elastic half-space are presented with their corresponding solution in the frequency domain. An optimization procedure is then proposed to obtain the modal properties of a replacement fixedbase structure capable of reproducing the absolute accelerations of the same structure on a flexible base. A simplified model of the superstructure is then used to evaluate the effects of soil-structure interaction in the overall damping of buildings. The influence of building height in the equivalent damping ratio of the fundamental mode is then studied. Finally, the variation with frequency of the equivalent damping ratios for higher modes is examined.

2. Multistory building on an elastic half-space 2.1. Soil-Structure interaction modeling When modeling soil-structure interaction, there are two common analysis procedures: the direct method and the substructure method. In the direct method, both the structure and the soil that surrounds the foundation are explicitly modeled, usually using the finite element method. As it is impossible to model the full length of the soil, proper boundary conditions have to be determined and applied to boundary elements in order to properly model radiation damping. The substructure method, on the other hand, significantly simplifies the analysis by separating the problem into two systems: the superstructure and the soil-foundation subsystems. Frequency-dependent force-deformation relationships between the foundation and the soil, known as the impedance, or dynamic stiffness of the soil-foundation interface, are first established. These relationships are then applied to the nodes at the soil-foundation interface to take into account the dynamic interaction. Impedance functions for different types of soil-foundations have been computed through the years (e.g., [11–13,34]), limiting the analysis to the superstructure and its interaction with the soil-structure interface. For this reason, the substructure approach was used in this investigation. Consider the soil as an elastic, homogeneous and semi-infinite medium with density ρ , shear modulus G , Poisson ratio ν , and where shear waves propagate at a velocity Vs . The foundation of the building is assumed to be a massless rigid circular disk of radius R 0 , with 2 degrees of freedom that correspond to the horizontal direction x and the rocking rotation θ (Fig. 1). The interaction forces and moments acting on the foundation for a steady-state harmonic load at frequency ω can be expressed as V0 (t ) = Veiωt and M0 (t ) = Meiωt , respectively. The corresponding horizontal displacements and rotations will be u 0 (t ) = Hu0 (ω) eiωt and θ (t ) = Hθ (ω) eiωt . The horizontal forces and displacements are related by:

{ }

⎧ V ⎫ = ⎡ KVV (a0 ) KVM (a0 ) ⎤ Hu0 ⎢ ⎥ M ⎨ ⎭ ⎣ KMV (a0) KMM (a0 ) ⎦ Hθ ⎩ ⎬

(1)

where Hu0 and Hθ are the frequency response functions associated with the swaying and rocking motions, respectively. The impedance matrix of the soil-foundation interface is defined by the terms KVV , KVM , KMV , and KMM , which are given by:

KVV = Kx⋅[k11 (a0) + ia0 c11 (a0)]

(2a)

KVM = KMV = Kx R 0⋅[k12 (a0) + ia0 c12 (a0 )]

(2b)

KMM = K θ⋅[k22 (a0) + ia0 c22 (a0 )]

(2c)

where kij and cij are dimensionless functions of the soil's Poisson ratio ν and the nondimensional frequency a0 = ωR 0 / Vs . Exact values for these parameters can be found in [11], and are shown in Fig. 2 for ν = 0.45.

Fig. 1. Simplified model of the soil-foundation system.

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Fig. 2. Dynamic stiffness for a rigid circular disk on an elastic half-space with ν = 0.45.

radius R 0 and mass m 0 , sitting on an elastic half space with no inherent damping. Consider that the building is subjected to a ground motion ug (t ) . Due to the flexibility of the soil, there will be a relative horizontal deformation u 0 (t ) between the foundation and the rigid-body motion of the ground. Also, the base moment of the structure will induce a rotation θ in the foundation. The equations of motion of the system under this loading are given by:

It can be seen that the dimensionless dynamic damping coefficient associated with pure horizontal translation, c11 is much larger than the one associated with rocking/rotation of the foundation, c22 . Furthermore, c11 remains approximately constant with changes in a 0 while c22 increases with increasing a 0 , particularly for values of a 0 smaller than 2. Kx and K θ are the static translational and rotational stiffness coefficients of the foundation, respectively. They represent the force (or moment) required to generate a unit horizontal displacement (or rotation) in the disk under static loading. For a circular disk, these static stiffnesses Kx and K θ are given by:

8GR 0 Kx = 2− ν Kθ =

(3)

8GR 03 3(1 − ν )

[Ms]{u}̈ + [Cs]{u}̇ + [Ks]{u} + [Ms]{1} ü0 + [Ms]{h} θ ̈ = −[Ms]{1} ug̈

(5)

{1}T [Ms]{u}̈ + mt ü0 + L0r θ ̈ + V0 (t ) = − mt üg

(6)

{h}T [Ms]{u}̈ + L0r ü0 + Ib θ ̈ + M0 (t ) = − L0r üg

(7)

where Eq. (5) represents the dynamic equilibrium of lateral forces at each floor, Eq. (6) corresponds to the dynamic equilibrium of shear at the base, and Eq. (7) represents the dynamic equilibrium of moments at the base; [Ms], [Cs], and [Ks] are the mass, damping, and stiffness matrices of the superstructure, respectively; {u} , {u}̇ , and {u}̈ are vectors containing the horizontal displacements, velocities, and accelerations of the different floors with relative to the foundation; Ib is the mass moment of inertia of the building with respect to its base; {h} is a vector N containing each level's height hj ; L0r = ∑ j = 1 mj hj is an auxiliary variable; V0 (t ) and M0 (t ) are the base shear and moment that arise due to soil-structure interaction, respectively. If the rotational inertia of each N floor (including that of the foundation) is negligible then Ib ≈ ∑ j = 1 mj hj2 Given that the superstructure is assumed to have classical modes, its lateral response can be computed via modal superposition, i.e. {u (t ) } = [ϕ] {q (t )} . If only Nm are modes participating in the structural response, then substituting in Eq. (5), pre-multiplying by [ϕ]T and dividing by the modal mass leads to the following system of differential equations:

(4)

2.2. Structural model and equations of motion for earthquake loading Consider the planar N-story building shown in Fig. 3. The building's mass has been lumped at floor levels, with each story having mass mj . Each level is assumed to act as a rigid diaphragm, the rotational inertia of each story with respect to a horizontal axis is assumed to be negligible, as well as the axial deformations of the columns, so there is only one dynamic degree of freedom per floor. It is also assumed that the superstructure has classical damping, which allows decomposition into classical normal modes. The building has a rigid, circular foundation of

qn̈ + 2ζn ωn qṅ + ωn2 qn + Γnh ü0 + Γnr θ ̈ = −Γnh ug̈

(8)

where ωn and ζn are the modal frequencies and damping ratios of the superstructure, respectively. Γnh and Γrn are defined as the horizontal and rotational modal participation factors of the n-th mode, which are given by:

Γnh =

{ϕn}T [Ms]{1} Mn

(9a)

Γnr =

{ϕn}T [Ms] {h} Mn

(9b)

{ϕn} corresponds to the n-th mode shape, and Mn = {ϕn}T [Ms]{ϕn} to the modal mass of the n-th mode. Defining Dn = qn /Γnh and substituting in Eq. (8) leads to:

Fig. 3. Idealized building-foundation-soil system.

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Γr Dn̈ + 2ζn ωn Dṅ + ωn2 Dn + ü0 + nh θ ̈ = −üg Γn

(10)

+ KVM Hθ

Note that if the soil is assumed to be rigid, then ü0 = θ ̈ = 0 , leading to the well-known modal equation for fixed-base systems with classical damping. Substituting Eqs. (9)–(10) into Eqs. (6) and (7), the equilibrium of horizontal forces and moments at the base of the structure can be written as: Nm

∑

2 (Γnh) Mn Dn̈

+ mt u ̈ 0 +

L0r θ ̈

+ V0 (t ) = − mt ug̈

(18b) N

− ω2 ∑n =m1 Γnr Γnh Mn HDn − ω2L0r Hu0 − ω2Ib Hθ + KMV Hu0 = − L0r + KMM Hθ (18c) which corresponds to a linear system of Nm+2 equations on the unknowns Hθ , Hu0 , and HDn . The solution of the system is: ∼∼ ∼∼ FA − DC Hθ = ∼∼ ∼∼ (19a) EA − DB

(11)

n=1

Nm

∑ Γnr Γnh Mn Dn̈ + L0r ü0 + Ib θ ̈ + M0 (t ) = −

L0r üg

(12)

n=1

∼ ∼ C B Hu0 = ∼ − ∼ Hθ A A

2.3. Solution in the frequency domain

(ω) eiωt

(13b)

u 0 (t ) = Hu0 (ω) eiωt

(13c)

The absolute acceleration at the j-th story ütj at time t is given by:

∑ Γnh ϕnj Dn̈ (t ) + ü0 (t ) + hj θ ̈ (t )

∼ 2 N C = ω2 ∑n =m1 (Γnh) Mn Gn − mt

(20c)

∼ N D = ω4 ∑n =m1 Γnr Γnh Mn Gn − ω2L0r + KMV

(20d)

∼ N E = ω4 ∑n =m1 (Γnr )2Mn Gn − ω2Ib + KMM

(20e)

∼ N F = ω2 ∑n =m1 Γnr Γnh Mn Gn − L0r

(20f)

If the ground acceleration (i.e.ug̈ (t ) = eiωt ), then:

is

assumed

to

be

harmonic

Nm

H Aj (ω) = 1 − ω2

−1 ωn2 − ω2 +2iζn ωn ω

(20g)

Most buildings are designed using a linear elastic response modal spectrum analysis, in which the base structure is typically assumed as fixed. The same assumption is made in several system identification techniques employed to infer damping ratios from the accelerations recorded in a building. Therefore, it is of interest to evaluate what properties should a “replacement” fixed-base structure have in order to replicate as best as possible the absolute acceleration transfer function of a building with a flexible base. The equations of motion of a structure with a fixed base are obtained by imposing that the base of the building cannot rock or sway, i.e. θ (t ) = u 0 (t ) = 0 in Eq. (10), resulting in the following well-known system of equations:

(15)

n=1

∑ Γnh ϕnj HDn (ω) − ω2Hu0 (ω) − ω2hj Hθ (ω) n=1

(16)

where H Aj (ω) is the frequency response function (FRF), or transfer function, of the total acceleration at the j-th story. Given that the FRF of a system does not depend on the input, then for any ground motion acceleration üg the problem can be solved by:

1 2π

(20b)

3. Effective damping ratios

Nm

uẗj (t ) =

(20a)

∼ N B = ω4 ∑n =m1 Γnr Γnh Mn Gn − ω2L0r + KVM

(14)

Using modal decomposition with Nm modes:

uẗj (t ) = üg (t ) +

(19c)

∼ 2 N A = ω4 ∑n =m1 (Γnh) Mn Gn − ω2mt + KVV

Gn =

utj̈ (t ) = üg (t ) + uj̈ (t ) + ü0 (t ) + hj θ ̈ (t )

Γr ω2 nh Gn Hθ Γn

where

(13a)

θ (t ) = Hθ (ω) eiωt

(19b)

HDn = Gn − ω2Gn Hu0 −

Given that the force-deformation relationships at the soil-foundation interface are frequency dependent, it is convenient to obtain the solution of the system in the frequency domain. The solution to this problem was first presented by Jennings and Bielak [7] using Laplace transforms. Here, we used Fourier transforms as done in other investigations (e.g., [8]) and explicit solutions to the transfer functions of absolute accelerations are developed. Consider a linear and time invariant (LTI) system and suppose that we are interested in a specific response quantity y (t ) , characterized by its frequency response function Hy (ω) . If the input x (t ) of the system is harmonic (i.e. x (t ) = eiωt ), then the response quantity can be computed as y (t ) = Hy (ω) eiωt , where i = −1 . In particular, for the soil-structure interaction problem:

Dn (t ) = HDn

2

N

− ω2 ∑n =m1 (Γnh) Mn HDn − ω2mt Hu0 − ω2L0r Hθ + KVV Hu0 = − mt

Dn̈ + 2ζn ωn Dṅ + ωn2 Dn = −üg

If only Nm modes are assumed to participate in the structural response, then the absolute horizontal acceleration at the j -th floor is given by:

∞

∫−∞ HA (ω) Ug̈ (ω) eiωtdω j

(17)

Nm

utj̈ (t ) = üg +

where Ug̈ is the Fourier transform of the ground acceleration. From Eqs. (16) and (17) it can be seen that obtaining the absolute accelerations in the structure only requires knowing the transfer function H Aj (ω) or the frequency response functions HDn , Hu0 , and Hθ ; and the ground acceleration ug̈ (t ) . To obtain the transfer functions, the equations of motion are computed assuming that the ground motion acceleration is a harmonic load. Substitute Eq. (1) and (13) into Eqs. (10)–(12), we obtain:

(− ω2 + 2iζn ωn ω + ωn2) HDn − ω2Hu0 − ω2

Γnr

Hθ Γnh

= −1

(21)

∑ Γnh ϕnj Dn̈ n=1

(22)

Similarly, the transfer function of the absolute acceleration for a fixed-base structure can be obtained by: Nm

H AFj (ω) = 1 − ω2

∑ Γnh ϕnj Gn (ω) n=1

(23)

where the superscript F denotes that the structure has a fixed base, and Gn is given in Eq. (20g). The main objective of this study is to determine the effective modal damping ratios of the “replacement” fixed-base

(18a) 186

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Fig. 4. Contribution of the different modes to the fixed-base transfer function of the horizontal relative acceleration, evaluated at the roof.

structure that, together with effective modal periods and fixed-base modes, provide good approximations of the absolute accelerations of the same structure on flexible soil. It should be noted that if, in addition to absolute accelerations, relative displacement need to be computed, then the excitation, or equivalently the modal participation factors, need to be modified [26]. Fig. 4 illustrates the contribution of the different modes to the fixedbase transfer function of horizontal acceleration evaluated at the roof, for the 25-story example building described in Section 3.2. The figure plots the sum of the second term of Eq. (23) and each of its different modal components. Note how the curves of each mode have a peak located approximately at their corresponding modal frequency, decaying to zero at low frequencies, but not vanishing completely at higher frequencies. This causes the curves to overlap. As a consequence, the sum of the modes contributing to the response, depicted in segmented lines, will have different peak locations, amplitudes, and widths than its individual modal components. The difference is larger at frequencies between higher modes, as they are affected by the overlapping with all previous modes. For this reason, trying to infer equivalent damping ratios based on the bandwidth of the filtered individual modal components does not yield accurate results, especially for higher modes. To avoid such issues, the effective damping ratios were computed employing a transfer function matching approach, via optimization. The idea of matching the transfer function of two different structural models was first employed by Tsai [16], who performed the match between the exact transfer function and the transfer function obtained employing a normal-mode approximation to the problem, which still considers the stiffness and damping properties of the soil. In his work, he only performed the match at the resonant frequencies of the soil-foundation-structure system at one structural location. In this investigation, the fit is performed between the exact solution and that of a fixed-base model that cannot rotate or sway at the base, over a vast range of frequencies, and at several structural locations. An optimization routine was developed to find the equivalent fixedbase parameters that would best reproduce the transfer function of the building with a flexible base. That is, to find the equivalent modal ∼ h∼ ∼ periods Tn , damping ratios ζn , and effective mode shapes Γ ñ ϕnj , that would make the curve defined by Eq. (23) as close as possible to that defined by Eq. (16). To this end, the objective function considered for this process is defined as the difference squared between the transfer function of the flexible base HA (ω) and that of the replacement fixedbase structure HAF (ω) , summed over all the frequencies in the range from 0.001 rad/s to 1.3ω Nm , over all the floors at which the match is desired, and normalized by the modulus of the flexible base transfer function summed over the range of frequencies where the two functions are being matched:

2 Nf

J (Θ) =

Ω

H Aj (iΔω) − H AFj (iΔω)

∑∑

Ω

∑k = 1 H Aj (kΔω)

j=1 i=1

(24)

where Nf is the number of stories being considered for the fit, Δω and Ω are the frequency increment and the number of points in the function, respectively. This objective function provides an overall measure of fit between the two transfer functions being matched. The purpose of the normalization shown in Eq. (24) is to give equal weight to each floor, otherwise the optimization would converge towards parameters that provide a better fit in floors experiencing larger accelerations, which in general occur at higher floors of the building. The effective Nm (2+Nf ) fixed-base parameters are found as those that minimize the objective function

min J (Θ)

(25)

where

∼

h∼

⎛ ⎧ T∼1 ⎫ ⎧ ζ1 ⎫ ⎡ Γ1̃ ϕ11 Θ = ⎜ ⋮ ; ⋮ ;⎢ ⋮ ∼ ∼ ∼ ⎜⎨ TNm ⎬ ⎨ ζ ⎬ ⎢ Γ1h̃ ϕ ⎝ ⎩ ⎭ ⎩ Nm ⎭ ⎣ 1Nf

h ∼ ⋯ Γ Ñ m ϕ Nm 1 ⎤ ⎞

⋱ ⋮ ⎥⎟ h ∼ ⋯ Γ Ñ m ϕ Nm Nf ⎥ ⎟

⎦⎠

(26)

3.1. Simplified model of the superstructure To evaluate the effects of soil-structure interaction in the overall damping of the system, the superstructure was modeled using a simplified model. The model considers the building as a continuous system consisting of a cantilever flexural beam laterally attached to a shear beam such that both undergo the same lateral deformation along the height. Here, the lateral rigidity in the shear and flexural beams is assumed to remain constant along the height of the building, such that the dynamic properties are fully defined by a small number of parameters, namely the fundamental period T1, the modal damping ratios ζn , and the lateral stiffness ratio α , which measures the relative contribution of the flexural rigidity and shear rigidity in the flexural and shear beams, respectively. Based on these parameters, mode shapes, period ratios, and modal participation factors are easily computed (e.g., [35]). This model has previously been used to obtain approximate floor acceleration demands, interstory drift ratio demands, and to identify the dynamic properties of buildings [35–37]. The mathematical description of the model, as well as recommended values for α depending on the structural system of the building, can be found in [38] while closed form equations of modes shapes, period ratios and modal participation factors can be found in [36] and [35]. For this investigation, the model was adapted to have the mass lumped at the different floor levels (such as in Fig. 3) while the lateral stiffness was provided by the continuous 187

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Fig. 5. Comparison of the absolute acceleration transfer function of the building at the roof considering fixed and flexible bases.

flexural and shear beams.

for higher modes. Regarding the amplitude of the different peaks, it can be seen that there is an increase in the amplitude of the transfer function of the building with flexible base for first mode with respect to that of the fixed-base structure, but that the peak amplitude decreases for the second and higher modes. Furthermore, the reduction in amplitude of the second mode is almost negligible but it is significant for the third and higher modes, with the reduction increasing with increasing frequency.

3.2. Example of building with soil-structure interaction A mathematical model of a 25-story building was constructed using the aforementioned simplified method. The building has a height of 75 m, with a uniform story height of 3 m. The superstructure has a fundamental (fixed-base) period T1= 3.0 s. The building has a dual lateral resistant system, consisting of moment frames and shear walls i.e. α = 6 [38]. The damping ratio of the superstructure was considered to be 2% for all modes. Each story was assumed to be circular with a radius R 0 = 15 m. The seismic weight is 10.5 kN/m2 (220 psf). The foundation has the same radius R 0 , but only half the mass of a typical story. The building was assumed to be sitting on a soil with shear wave velocity Vs = 250 m/s (corresponding to NEHRP site class D), mass density ρ = 2 kN s2/m4, and Poisson ratio ν = 0.45. It was assumed that 5 modes contributed significantly to the structural response. Fig. 5 shows the absolute acceleration transfer function of the system – computed with Eq. (16) – and compares it to the fixed-base case of Eq. (23). In the figure, the location of the peaks of the transfer functions correspond to the modal frequencies, while the height of the peaks is related to the corresponding modal damping ratios. Note that in both cases, the transfer functions have 5 clearly distinguishable peaks each one corresponding to each of the 5 modes included in the modeling, after which it decays to zero. It can be seen that soil-structure interaction shortens the system’s frequencies, thus elongating the modal periods. As previously noted by Jennings and Bielak [7], this frequency shift, which arises due to the increased flexibility of the base, affects primarily the fundamental mode being very small or almost negligible

3.3. Effective damping ratios obtained for the fixed-base building

∼ ∼ The modal periods Tn , damping ratios ζn , and effective mode shapes

h∼ Γ ñ ϕnj

of a replacement fixed-base structure capable of replicating the response of the 25-story flexible-base building were then investigated. These periods and damping ratios in the replacement MDOF fixed-base structure are referred here as effective periods and damping ratios, similarly to how they were referred to for the replacement fixed-base SDOF in ATC3-06 [39]. For this purpose, the absolute acceleration transfer function of the flexible-base building was evaluated at the roof, 19th, 12th, and 5th floors. The optimization routine was employed to perform the curve-fitting to minimize the difference between the transfer function of the building on flexible base and its equivalent fixed-base counterpart. Fig. 6 shows the transfer function of the replacement or “equivalent” fixed-base building computed with the resulting effective parameters, and compares it to the original flexiblebase one at the roof, while Fig. 7 shows these two transfer functions evaluated at the rest of the considered floors. It can be seen that, for all practical purposes, the transfer functions of the two systems are Fig. 6. Comparison of the absolute horizontal acceleration transfer function at the roof of the building with a flexible base and the replacement building with a fixed base.

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Fig. 7. Comparison of the absolute horizontal acceleration transfer functions of the building on flexible base and replacement fixed-base at the rest of the floors considered for the optimization.

effective damping, reaching up to an increment of 81% for the fifth mode. It is interesting to note that the effective damping ratios increase ∼ with increasing effective modal frequency f , which is shown in Fig. 8. It can be seen that this increment is approximately linear, in complete agreement to recent observations of modal damping ratios of higher modes in tall buildings [5,33]. Fig. 8 also shows the slope β and intercept ξ0 of a linear fit of the data, obtaining a coefficient of determination R2 = 0.99. Veletsos and Meek [9] proposed a method to compute the effective damping ratio of an SDOF system. For validation purposes, this method was extended for MDOF structures by considering each mode as an SDOF system with mass and height equal to the effective modal mass and effective modal height of the mode being analyzed, respectively. Fig. 8 compares the results with those obtained using the method of Veletsos and Meek [9]. It can be seen that the effective damping ratios obtained using both methods are in close agreement. Finally, the resulting effective mode shapes of the equivah∼ lent structure Γ ñ ϕnj were analyzed. Fig. 9 compares the effective mode shapes of the superstructure with the equivalent ones, evaluated at the stories in which the curve-fitting process was performed. It can be seen that the differences between the two models are negligible, which suggests that the effects of soil structure interaction may be taken into account approximately by just modifying the periods and damping ratios in the replacement fixed-base superstructure.

Table 1 Effective periods and damping ratios for the example building. Mode

T [s]

∼ T [s]

∼ T /T

ζ [%]

∼ ζ [%]

∼ ζ /ζ

1 2 3 4 5

3.000 0.885 0.432 0.249 0.159

3.129 0.890 0.439 0.252 0.161

1.04 1.01 1.02 1.01 1.01

2.00 2.00 2.00 2.00 2.00

1.79 2.09 2.51 3.04 3.62

0.89 1.04 1.26 1.52 1.81

3.4. Effective damping ratio of the fundamental mode The variation of the effective damping ratio of the fundamental mode in the replacement fixed-base building models as a function of the building height was then analyzed. For this purpose, the height of the building model described on Section 3.2 was modified by adding or removing stories. The fundamental period of the superstructure was assumed to vary as

Fig. 8. Variation with frequency of the effective damping ratio of the 25-story building.

essentially identical – showing that it is possible to represent an MDOF building with a flexible base with a replacement, effective, or “equivalent” fixed-base MDOF structure. Table 1 lists the modal periods T and damping ratios ζ of the fixed-base superstructure, and the cor∼ responding properties of the replacement or “equivalent” structure T ∼ ∼ and ζ . As expected, the effective periods T are longer than those of the superstructure. Period elongation affects primarily the fundamental mode, which elongates 4% while the periods of higher modes elongate, on average, just 1%. Regarding damping, the effective damping ratio of the fundamental mode decreases in 11% with respect to the superstructure, but for the second and higher modes, there is an increase in

T = 0. 0905⋅H 0.8

(27)

with H in meters, which corresponds to the regression found by Goel and Chopra [40] for steel moment frame buildings. The damping ratio of the superstructure was again set to 2% for all modes. Fig. 10 shows the variation of the effective damping ratio of the fundamental mode with increasing building height. It can be seen that the effective damping decreases with the height of the building. It was found that the 189

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Fig. 9. Comparison of the mode shapes of the superstructure with the effective fixed-base mode shapes computed at the 25th, 19th, 12th, and 5th floors.

Fig. 11. Variation of the imaginary part of the rocking dynamic stiffness of an embedded foundation (adapted from [34]). Fig. 10. Variation of the effective damping ratio of the fundamental mode with building height.

functional form that best approximated this trend was a power function. The coefficients of the power regression are also shown in Fig. 10. Moreover, for buildings with heights lower than approximately 30 m (10 stories) the damping ratio is larger than that of the superstructure, increasing exponentially as the height diminishes. On the other hand, the equivalent damping is lower than that of the superstructure for buildings taller than 30 m, and the rate of decrease diminishes with increasing height. The results of Fig. 10 were obtained under the assumption of a superficial foundation sitting on an elastic half-space with no inherent damping. The influence of these assumptions on the effective damping ratio of the fundamental mode was investigated next. Embedding a foundation below ground surface will increase its dynamic stiffness [41–43]. Fig. 11 shows the damping term of the dynamic stiffness associated with rocking of the foundation for 4 embedment ratios (embedment depth E over the radius of the foundation R 0 ). It can be seen that the damping term associated with rocking of the foundation will not go to zero at low frequencies, and that the value increases with the ratio of embedment, which will increase the effective damping ratio of tall buildings. The aforementioned numerical simulations were repeated, now considering 3 different embedment ratios E /R 0 : 0.00, 0.25, and 0.50. The impedance functions used for this purpose correspond to the closed-form approximations of Pais and Kausel [34]. Fig. 12 shows the variation of the effective damping ratio of the fundamental mode with increasing building height for the different embedment ratios considered. As expected, embedment tends to attenuate the trend observed in Fig. 10 by reducing the damping effective damping ratio of short buildings and increasing it for tall

Fig. 12. Variation of the effective damping ratio of the fundamental mode as a function of building height level of foundation embedment.

buildings. Nevertheless, the effective damping ratio of the fundamental mode in all cases decreases with increasing building height even for embedment ratios of 0.5. The effects of including the inherent damping of the soil were investigated next. The two most common models to incorporate the inherent damping of the soil are to idealize the half-space as a Voigt solid or as a constant hysteretic solid. In a Voigt solid, the shear modulus of viscosity of the soil G’ is assumed to be constant, while in a constant hysteretic solid the product of ωG′is assumed to be constant. For a Voigt solid, the damping ratio is defined as ξ = Vs G′/(RG ) , whereas for the hysteretic solid the damping is characterized as tan (δ ) = ωG′/ G in which δ is the loss angle. Both models were considered in this 190

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investigated next. For this purpose, it is convenient to analyze the variation of the effective damping ratio of the fundamental mode with the dimensionless wave parameter σ , defined as:

σ=

Vs Tn hn*

(28)

where hn* is the effective height of the mode being analyzed, defined as: N

hn* =

∑ j = 1 hj mj ϕnj N

∑ j = 1 mj ϕnj

(29)

The effective modal height may be interpreted as the height that an equivalent SDOF system with period Tn , damping ratio ζn , and mass 2 equal to the effective modal mass Mn* = (Γnh) Mn should have in order to produce the same base shear and overturning moment as those in the nth modal contribution in the MDOF structure. Please note that, depending on the mass distribution in the building. the effective modal height may be negative for some of the higher modes, meaning that the signs of the base shear and overturning moment are opposite. In contrast, h1* is always positive. Since the sign is not relevant for the analysis, the absolute value of the effective modal height is employed in the definition of the wave parameter. Veletsos and Meek [9] found that the wave parameter σ and the aspect ratio H /R 0 are the most important parameters for determining the effects of soil-structure interaction phenomenon in SDOF systems. To examine their influence in multiple degree of freedom (MDOF) systems the building described in Section 3.2 was modified in the following manner: 3 building models were created having the same geometric properties in plan, but a different number of stories: 5, 10, and 25, corresponding to aspect ratios of 1, 2, and 5, respectively. The stiffness of each building was then varied systematically, obtaining different fundamental periods, and hence, σ values. For each σ value, an optimization was conducted to obtain the effective damping ratio of the fundamental mode. Fig. 15 shows the resulting effective damping ratios, normalized by that of the superstructure as a function of the wave parameter for the 3 different structures. It can be seen that, in all cases, the equivalent damping ratio tends to the damping of the superstructure for large values of σ . Note that the effects of soil-structure interaction increase as σ decreases. The specific effect of soil-structure interaction in the overall damping of the structure varies significantly with the aspect ratio of the building. Squatty structures with low aspect ratios experience an increase in equivalent damping as σ decreases, obtaining effective damping ratios that can be much higher than that of the superstructure. The opposite occurs for slender structures, where the effective damping ratio decreases as σ diminishes, experiencing values lower than that of the superstructure when the wave parameter is less than about 20. This trends reflect the influence of rocking in the

Fig. 13. Rotational dashpot coefficient for a rigid circular disk on a viscoelastic half-space with ν = 0.45. The solid lines correspond to the results for a Voigt solid, the dashed lines to hysteretic damping (adapted from [44]).

Fig. 14. Variation of the effective damping ratio of the fundamental mode as a function of building height and level of inherent damping in the soil.

investigation. To this end, the soil was modeled as a linear viscoelastic half-space using the closed-form approximate solutions of Veletsos and Verbic [44] for the impedance functions of a rigid, superficial, massless disk. Fig. 13 shows the radiation damping coefficient c22 associated with the rocking degree of freedom of the impedance function for 3 different damping ratios (or tan (δ ) values): 3%, 1%, and 0%. It can be seen that in all cases, c22 increases with increasing damping ratio. For values of a 0 larger than about 1.0, there are no significant differences between the two damping models. There are significant differences, however, for low frequencies. In the constant hysteretic model, c22 tends to infinity as a0 approaches 0, whereas it converges to a value close to 0 in the Voigt solid model. Consequently, the effective damping ratios for the fundamental mode computed assuming a Voigt solid will not differ much from the elastic case with no damping, but those computed assuming a constant hysteretic model will be larger. This is especially significant for the case of tall buildings with low fundamental frequencies, where a0 can reach values lower than 0.1. Fig. 14 shows the variation of the effective damping ratio of the fundamental mode with increasing building height for the different soil damping models considered. As expected, the results for the Voigt solid model do not significantly differ from the case with no damping, while the hysteretic model tends to attenuate the effects of SSI for tall buildings by increasing the effective damping due to the divergence of the impedance function at very low frequencies. Nevertheless, it can be seen that the descending trend of the effective damping ratios with increasing building height is still clearly observed. The results shown in Figs. 10, 12, and 14 are consistent with the findings of several investigations that have inferred the equivalent damping ratio of monitored buildings [5,31,32,45], clearly indicating that the decrease in damping ratios that has been observed in instrumented buildings is due to soil-structure interaction effects. The influence of the fundamental period of the superstructure was

Fig. 15. Variation of the effective damping ratio of the fundamental mode changes in the wave parameter.

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actual buildings. The tallest building was selected to have an aspect ratio AR = 8.0, which is close to the aspect ratios of some of the tallest buildings in Los Angeles, like Citicorp Plaza's 777 Tower (H = 218.1 m; AR = 8.22), or the Figueroa at Wilshire building (H = 218.2 m; AR = 8.14). In these buildings the aspect ratio was computed by dividing their height by the radius of a circle with an area equal to that of the base of the building. Fig. 16 shows the resulting variation of equivalent damping ratios with effective modal frequencies for all six cases. It can be seen that in all but one case the effective damping ratios increase monotonically with frequency, reaching values that can be 100% higher than the assumed damping ratio of the superstructure (ξ = 2%). For buildings having aspect ratios equal or greater than 4, this variation is approximately linear. As the aspect ratio goes down, the effective damping ratio of the fourth and fifth modes start saturating, and may even start to decrease. In low-rise buildings, however, higher modes do not contribute significantly to their seismic response. A linear fit to the data of those modes that contribute significantly to the structural response (filled dots), along with its corresponding slope β and coefficient of determination R2 , is also shown in Fig. 16. It can be seen that a linear variation is a good predictor of the equivalent damping ratios for the participating modes. The effects of embedment and inherent soil damping on the effective damping ratios of higher modes were examined next. To this end, one of the cases of Fig. 16 was selected (H = 96 m, H /R 0 = 6.4) and modeled with three different embedment ratios (E /R 0 = 0.00, 0.25, and 0.50) and three different soil damping ratios (0%, 1%, and 3%) using both Voigt and constant hysteretic damping models. Fig. 17 (left) shows the results obtained for the considered embedment ratios. It can be seen that, in all cases, the effective damping ratios increase with

response: as the aspect ratio increases, the rocking degree of freedom tends to dominate the soil-structure interaction. Since taller structures have lower fundamental frequencies, the damping coefficient of the rocking dynamic stiffness function c22 will be very small, making radiation damping negligible (see Fig. 2). The increased flexibility of the system will lead to a larger response, making the equivalent damping ratio lower than that of the superstructure. The opposite occurs for short structures: as the aspect ratio diminishes, the swaying of the foundation controls the interaction. Again, Fig. 2 shows how the dynamic stiffness coefficient of the horizontal degree of freedom c11 is relatively large for all frequencies, consequently, there will always be radiation damping associated with horizontal swaying. Moreover, lower slender structures will have larger fundamental frequencies and therefore the effect of rocking will, in this case, increase the equivalent damping even further. Similar trends were observed by Veletsos and Meek [9] for equivalent SDOF systems. 3.5. Effective damping ratio of higher modes In engineering practice modal damping ratios are either assumed to be constant (i.e., same damping ratio for all modes) or to change with frequency based on assuming Rayleigh damping and fixing the damping ratio at two modal frequencies. Therefore, of particular interest is to investigate how soil-structure interaction effects modify effective damping ratios with increasing frequency. Most damping models specify damping ratios based on the modal frequency (e.g. stiffness and/or mass proportional damping), therefore, the variation of the equivalent damping ratios with their corresponding equivalent modal frequencies was studied first. From the analyses of damping against building height (Fig. 10), 6 cases were selected to cover most of the height range of

Fig. 16. Variation of the effective damping ratio for increasing effective modal frequency.

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Fig. 17. (Left) Effect of embedment and (right) effect of soil damping on the effective damping ratios of higher modes for building H = 96 m; H/R0 = 6.4.

ratios T1/ Tn , and hence smaller modal σ values. Therefore, with all other properties being equal, the effects on higher modes will be larger in buildings with structural systems that deflect approximately like flexural beams (e.g., buildings with shear walls) than those that deflect approximately like shear beams (e.g., buildings with moment resisting frames). For the cases studied, unlike the fundamental mode, the effective damping ratio of higher modes is always larger than that of the superstructure. This occurs because the mode shape of higher modes have sign changes which cancel out a significant portion the modal overturning moments at the base, reducing the effects of rocking. The relative contribution of higher modes to the base moment is significantly lower than their relative contribution to base shear. Consequently, the effects of soil structure interaction on higher modes of vibration are governed primarily by the swaying of the foundation, resulting in increasing effective modal damping. This is consistent with previous findings by Jennings and Bielak [7], who found that soilstructure interaction had little effect on the properties of higher modes of a 10-story shear building with an aspect ratio H /R 0 of 6.

increasing effective modal frequency, but that the rate of increment decreases as the embedment ratio increases. Fig. 17 (right) shows the results of including soil inherent damping. It can be seen that, regardless of the damping model employed, the inherent damping of the soil affects primarily the effective damping ratio of the fundamental mode, but that it does not have a significant effect on the damping of higher modes. The approximately linear increment of the effective damping ratio with increasing effective modal frequency is consistent with inferred values of damping ratios of higher modes in tall buildings subjected to earthquakes [5,33], indicating that the linear increase in modal damping ratios observed in instrumented buildings is due to soilstructure interaction effects. Fig. 18 shows the variation of the effective modal damping ratios of higher modes with changes in the wave parameter σ . In this figure the effective modal damping ratios are normalized by the damping ratio in the superstructure. It can be seen that as soil-structure interaction effects become more significant with decreasing σ the effective damping ratios increase. Similarly, as σ increases the effective modal damping ratios tend to that of the superstructure. It can be seen that for smaller aspect ratios damping starts increasing at larger values of σ . These figures also show that the effects of soil-structure interaction in higher modes will be more pronounced for buildings having larger period

4. Conclusions The effects of soil-structure interaction on damping ratios of multi-

Fig. 18. Variation of the effective damping ratio of higher modes with the wave parameter.

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1943-541X.0001628. [6] Parmelee RA, Perelman DS, Seng-Lip L. Seismic response of multiple-story structures on flexible foundations. Bull Seismol Soc Am 1969;59:1061–70. [7] Jennings PC, Bielak J. Dynamics of building-soil interaction. Bull Seismol Soc Am 1973;63:9–48. [8] Chopra AK, Gutierrez JA. Earthquake response analysis of multistorey buildings including foundation interaction. Earthq Eng Struct Dyn 1974;3:65–77. http://dx. doi.org/10.1002/eqe.4290030106. [9] Veletsos AS, Meek J. Dynamic behaviour of building-foundation systems. Earthq Eng Struct Dyn 1974;3:121–38. http://dx.doi.org/10.1002/eqe.4290030203. [10] Kausel E. Early history of soil-structure interaction. Soil Dyn Earthq Eng 2010;30:822–32. http://dx.doi.org/10.1016/j.soildyn.2009.11.001. [11] Veletsos AS, Wei YT. Lateral and rocking vibration of footings. J Soil Mech Found Div 1971;97:1227–48. [12] Luco JE, Westmann RA. Dynamic response of a rigid footing bonded to an elastic half space. J Appl Mech 1972;39:527–34. http://dx.doi.org/10.1115/1.3422711. [13] Luco JE, Westmann RA. Dynamic response of circular footings. J Eng Mech Div 1971;97:1381–95. [14] Roesset JM. A review of soil-structure interaction. In: Johnson JJ, editor. Soilstructure Interact. status Curr. Anal. methods Res., Report Nos. NUREG/CR-1780 and UCRL-53011. US Nuclear Regulatory Commision and Lawrence Livermore Laboratory; 1980. [15] Dobry R. Simplified methods in soil dynamics. Soil Dyn Earthq Eng 2014;61–62:246–68. http://dx.doi.org/10.1016/j.soildyn.2014.02.008. [16] Tsai N-C. Modal damping for soil-structure interaction. J Eng Mech 1974;100:323–41. [17] Roesset JM, Whitman RV, Dobry R. Modal analysis for structures with foundation interaction. J Struct Div 1973;99:399–416. [18] Novak M. Effect of soil on structural response to wind and earthquake. Earthq Eng Struct Dyn 1974;3:79–96. [19] Novak M. Additional note on the effect of soil structural response. Earthq Eng Struct Dyn 1975;3:312–5. [20] Rainer J. Damping in dynamic Structure-foundation interaction. Can Geotech J 1975;12:13–22. http://dx.doi.org/10.1139/t75-002. [21] Novak M, El Hifnawy L. Effect of soil-structure interaction on damping of structures. Earthq Eng Struct Dyn 1983;11:595–621. [22] Tajimi H. Discussion: building-foundation interaction effects. J Eng Mech Div 1967;93:294–8. [23] Bielak J. Earthquake response of building-foundation systems [Report No. EERL 7104;]. California Institute of Technology; 1971. [24] Lee TH, Wesley DA. Soil-structure dynamic interaction effects on the seismic response of an aribitrary three-dimensional structure [GA-10437]. Boston, MA: Gulf General Atomic Company; 1971. [25] Luco JE. Soil-structure interaction effects on the seismic response of tall chimney. Soil Dyn Earthq Eng 1986;5:170–7. [26] Luco JE. Linear soil-structure interaction [UCRL-15272, PDA. No. 7249808]. Livermore: CA: Lawrence Livermore Lab.; 1980. [27] Luco JE, Lanzi A. Approximate soil-structure interaction analysis by a perturbation approach: the case of stiff soils. Soil Dyn Earthq Eng 2013;51:97–110. http://dx.doi. org/10.1016/j.soildyn.2013.04.005. [28] Lanzi A, Enrique Luco J. Approximate soil-structure interaction analysis by a perturbation approach: the case of soft soils. Soil Dyn Earthq Eng 2014;66:415–28. http://dx.doi.org/10.1016/j.soildyn.2014.08.001. [29] Safak E. Identification of linear structures using discrete-time filters. J Struct Eng 1991;117:3064–85. [30] McVerry GH. Structural identification in the frequency domain from earthquake records. Earthq Eng Struct Dyn 1980;8:161–80. http://dx.doi.org/10.1002/eqe. 4290080206. [31] Satake N, Suda K, Arakawa T, Sasaki A, Tamura Y. Damping evaluation using fullscale data of buildings in japan. J Struct Eng 2003;129:470–7. http://dx.doi.org/10. 1061/(ASCE)0733-9445(2003)129:4(470). [32] Bernal D, Dohler M, Kojidi SM, Kwan K, Liu Y. First mode damping ratios for buildings. 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story buildings with a circular rigid foundation sitting on an elastic halfspace were investigated. For this purpose expressions of the transfer functions of the horizontal absolute accelerations were developed in the frequency domain. A procedure to obtain the effective modal properties of a replacement or “equivalent” fixed-base building that can obtain good estimates of the dynamic response of a building with flexible base was developed. The procedure uses optimization for finding the effective modal periods, effective damping ratios, and effective mode shapes that minimize the difference between the ordinates of the transfer functions of the replacement fixed-base structure to those of the structure with flexible-base. Results indicate that as the wave parameter decreases, soil structure interaction effects become more significant. Soil-structure interaction effects tend to increase the effective damping ratio of the fundamental period of vibration as their height and aspect ratio decreases. On the contrary for slender long-period structures, such as it occurs in tall buildings, soil-structure interaction effects tend to reduce the effective damping ratio of the fundamental period of vibration leading to effective damping ratios that may be smaller than those of the superstructure on fixed base. A parametric study was conducted to study the variation of effective modal damping ratios of the fundamental mode as a function of the height of the building, by keeping the dimensions in plan and setting the relationship between the height of the building and the fundamental period of vibration to agree to that of empirical data. It was shown that effective damping ratios of the first mode decrease with increasing height with a power function. This decreasing trend computed analytically is consistent with that reported from damping ratios inferred from records obtained in instrumented buildings subjected to seismic or wind loadings, indicating that the reduction in damping ratio observed in buildings as the height of the building increases is primarily due to soil-structure interaction effects. Effective damping ratios of higher modes were also investigated. Their values increases with decreasing wave numbers, that is, as soil structure interaction effects become more significant. Of particular interest is that effective modal damping ratios for modes that have significant contributions to the seismic response of buildings increase approximately linearly with increasing frequency. This linear increasing trend of damping ratios with frequency, computed analytically in this study, is consistent to the trend recently observed from damping ratios inferred from acceleration records obtained during earthquakes in tall buildings, indicating that such linear increasing trend is caused by soil-structure interaction. These results are significant as they suggest that stiffness proportional damping models are more appropriate to capture radiation damping in buildings. Acknowledgements The authors would like to acknowledge CONICYT – Becas Chile, and the Blume Earthquake Engineering Center at Stanford University for the financial support to the first author for conducting doctoral studies at Stanford University under the supervision of the second author. The authors also wish to thanks comments and suggestions by Prof. Enrique Luco. References [1] ASCE. ASCE/SEI 7-10: minimum design loads for buildings and other structures. Reston, VA: ASCE; 2010. http://dx.doi.org/10.1061/9780784412916. [2] FEMA, NEHRP. FEMA P-1050-1: Recommended seismic provisions for new buildings and other structures. vol. 1. Washington, DC: Building Seismic Safety Council; 2015. [3] Jacobsen LS. Steady force vibration as influenced by damping. Trans Am Soc Mech Eng 1930;52:169–81. [4] Beck JL, Jennings PC. Structural identification using linear models and earthquake records. Earthq Eng Struct Dyn 1980;8:145–60. [5] Cruz C, Miranda E. Evaluation of damping ratios for the seismic analysis of tall buildings. J Struct Eng 2016;143:4016144. http://dx.doi.org/10.1061/(ASCE)ST.

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[44] Veletsos AS, Verbic B. Vibration of viscoelastic foundations. Earthq Eng Struct Dyn 1973;2:87–102. http://dx.doi.org/10.1002/eqe.4290020108. [45] Fritz WP, Jones NP, Igusa T. Predictive models for the median and variability of building period and damping. J Struct Eng 2009;135:576–86. http://dx.doi.org/10. 1061/(ASCE)0733-9445(2009)135:5(576).

medium: an integral equation approach. Earthq Eng Struct Dyn 1987;15:213–31. [42] Elsabee F, Morray J. Dynamic behavior of embedded foundations [Report No. R7733]. Cambridge, MA: Massachusetts Institute of Technology; 1977. [43] Gazetas G. Formulas and charts for impedances of surface and embedded foundations. J Geotech Eng 1991;117:1363–81. http://dx.doi.org/10.1061/(ASCE)07339410(1991)117:9(1363).

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