Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

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

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils Armando Lanzi a,n, J. Enrique Luco b a b

Sapienza Università di Roma, Rome, Italy and Department of Structural Engineering, University of California, San Diego, USA Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 19 May 2013 Received in revised form 12 December 2013 Accepted 8 March 2014

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Soil–structure interaction Dynamics Perturbation Modal analysis Seismic

1. Introduction In a companion paper [1], the authors utilized a perturbation approach to study the linear dynamic soil–structure interaction problem for an elastic structure supported on a relatively stiff soil. The analysis resulted in new approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes which generalized those presented earlier by Bielak [2–3], Jennings and Bielak [4] and Veletsos and Meek [5] for the fundamental mode of a soil–structure system. The purpose of the present paper is to consider the other extreme limiting case corresponding to a structure supported on a relatively soft soil. Again, the objective is to present new approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes for this case. These expressions would generalize those presented by Beredugo and Novak [6] for the case of a rigid structure on a flexible soil. The use of the perturbation approach is selected as an alternative to a purely numerical, approximate modal analysis [7–13] of the linear dynamic soil–structure interaction problem, or to the use of the more accurate Foss' method [2,4,14], or a solution in the frequency domain. The approach has the advantage of leading to analytical expressions which offer considerable physical insight into the nature of the soil–structure interaction effects. However,

n

Corresponding author. E-mail addresses: [email protected], [email protected] (A. Lanzi).

http://dx.doi.org/10.1016/j.soildyn.2014.08.001 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

the perturbation approach does not cure the basic limitations of the approximate modal analysis for soil—structure interaction problems described, among others, by Thomson et al. [15], Clough and Mojtahedi [16], Warburton and Soni [17], and Vaidya et al. [18].

2. Statement of the problem and classical modal approach Consider the problem of forced vibrations of a linear elastic structure resting on a rigid foundation supported on a viscoelastic soil. The system is excited by elastic waves propagating through the soil and/or by external forces. The superstructure is discretized into a set of L nodal masses interconnected by massless elastic members; the rigid foundation may be partially embedded in the soil; and the soil is represented by a continuous,three-dimensional elastic or viscoelastic half-space. The fundamental equations of motion of the soil–structure system as well as those pertaining to the approximate classical modal approach have been presented in detail in the companion paper for stiff soils [1]. Here, we briefly recall the principal equations in order to facilitate the derivations that will follow. The deformed configuration of the superstructure is described in terms of a N
1 ðN ¼ 6LÞ vector fU b g ¼ ðfu1 gT ; fu2 gT ; :::; fuL gT ÞT of generalized relative displacements of the nodes fui g ¼ ðΔix ; Δiy ; Δiz ; θix ; θiy ; θiz ÞT with respect to a frame of reference attached to the moving rigid foundation. The generalized total displacement

416

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

vector for the superstructure fU bt g, which describes the motion of the nodes with respect to a fixed frame of reference, is given by fU bt g ¼ ½αfU o g þ fU b g

ð1Þ

where fU o g ¼ ðΔox ; Δoy ; Δoz ; θox ; θoy ; θoz Þ is the total foundation motion at a point of reference in the foundation and ½α is a N
6 rigid-displacement influence matrix. The total foundation motion fU o g is given by T

fU o g ¼ fU no g þfU s g

ð2Þ

where fU no g is the foundation input motion, and fU s g ¼ ðΔsx ; Δsy ; Δsz ; θsx ; θsy ; θsz ÞT is the relative motion of the foundation with respect to the input motion. The foundation input motion includes the effect of scattering of the seismic waves by the foundation [19]. For harmonic excitation, the motion of the superstructure and of the foundation is governed by the following system of equations (e.g., Lee and Wesley [20]; Luco [21]) " #( ) " #( ) " #( ) Mb Mb α Cb 0 Kb 0 Ub U_ b U€ b þ þ αT Mb Moo Us 0 Cs 0 Ks U_ s U€ s ( ) " # Fb Mb α n ð3Þ ¼  fU€ o g Fo M oo where ½M b ; ½C b ; ½K b  are the mass, damping and stiffness matrix for the superstructure on a fixed base, ½M o  is the mass matrix of the foundation, ½M oo  ¼ ½M o  þ ½αT ½M b ½α, ½K s ðωÞ þ iω½C s ðωÞ represents the foundation impedance matrix, and fF b g, fF o g are the generalized external forces acting on the superstructure and foundation, respectively. In Eq. (3), the harmonic time depending factor eiωt is omitted for brevity, and the displacement vectors fU b g and fU s g are frequency-dependent. ~ g be the jth mode and ω ~ j the corresponding natural Let fϕ j frequency of the un-damped building-foundation system. These quantities satisfy the eigenvalue problem "

Kb 0

" # 0 Mb 2 ~ g¼ω ~ fϕ j j αT M ~ 1Þ K s ðω b

Mb α M oo

# ~ g fϕ j

ð4Þ

in which the frequency-dependent stiffness matrix is approximated by a constant value corresponding to the fundamental ~1 frequency of the system. Iterations are required to evaluate ω and obtain the corresponding constant stiffness matrix. Assuming as an approximation that the system admits decomposition into classical normal modes, the ð6L þ 6Þ vector of generalized relative displacements fUg ¼ ðfU b gT ; fU s gT ÞT is written as N þ6

~ gη~ ~ fη~ g ¼ ∑ fϕ fUg ¼ ½Φ j j

damping ratios, the damping matrix of the system is evaluated at each system natural frequency, in an effort to consider in an approximate fashion the frequency dependence of the imaginary part of the impedance coefficients.

3. Perturbation approach for stiff structures on soft soils The perturbation approach for a flexible structure supported on a soft soil starts by considering that the foundation stiffness matrix 2 ½K s  is proportional to a characteristic soil shear modulus G ¼ ρβ , where ρ is a characteristic soil density and β is a characteristic shear wave velocity in the soil. Following the same approach adopted in [1], it is possible to define a dimensionless quantity ðβ =ω1 aÞ, where a is a characteristic dimension of the foundation and ω1 is the fundamental fixed-base frequency of the superstructure, which quantifies the relative stiffness between the structure and the foundation soil. In the case of a relatively stiff structure supported by a flexible soil, the dimensionless parameter ε ¼ ðβ=ω1 aÞ2 is small and the full solution can be seen as the sum of the solution for a perfectly-flexible soil and small perturbing terms, which can be expressed in a power series with respect to the small parameter ε. To start, the stiffness matrix of the foundation is written as ½K s  ¼ ε½K s 

where ½K s  represents the stiffness matrix of the foundation normalized by a term proportional to the square of the soil shear wave velocity. ~ 2 and the displacements of interest can The eigenvalues λ~ ¼ ω be expanded in terms of series of ε

λ~ ¼ λ~ 0 þ ελ~ 1 þ ε2 λ~ 2 þOðε3 Þ fU b g ¼ fU b0 g þ εfU b1 g þ ε2 fU b2 g þ Oðε3 Þ fU s g ¼ fU s0 g þ εfU s1 g þ ε2 fU s2 g þ Oðε3 Þ



ð5Þ

The modal damping ratios and participation factors for seismic excitation are defined, as usual, by ~ gT ½C fϕ ~ g þ fϕ ~ gT ½C fϕ ~ g ~ i ¼ fϕ ~ iM 2ξ~ i ω s b bi bi si si fβ~ i gT ¼

1 ~ T ~ gT ½M Þ ðfϕ bi g ½M b ½α þ fϕ oo si ~i M

ð7aÞ ð7bÞ

where ½C b  is the damping matrix of the superstructure (assumed to admit classical damping) and ½C s  is the damping matrix of the foundation, i.e. the imaginary part of the impedance matrix divided by ω. It is worth noting that, when computing the modal

ð9Þ

To obtain expressions for the coefficients of the series λ~ 0 ; λ~ 1 ; λ~ 2 ; U b0 ; U b1 ; U b2 and U s0 ; U s1 ; U s2 it is necessary to substitute Eqs. (8) and (9) into Eq. (4), and collect and set to zero the terms multiplying ε0 ; ε1 and ε2 . The approach leads to the following equations for the zero-, first- and second-order terms " #!( )     Mb Mb α U b0 Kb 0 0 ~ ¼  λ0 ð10Þ αT Mb Moo U s0 0 0 0 Kb

0

0

0

j¼1

~  is the modal matrix and fη~ g is the vector of modal where ½Φ amplitudes. Following the standard procedure, and neglecting the off-diagonal terms of the reduced modal damping matrix ~ T ½C½Φ ~ , Eqs. (3) and (5) lead to the system of N þ 6 uncoupled ½Φ equations   n on o n F ð6Þ η€~ j þ 2ω~ j ξ~ j η_~ j þ ω~ 2j η~ j ¼ fϕ~ j gT b  β~ j U€ o ; j ¼ 1; N þ6 Fo

ð8Þ

"

¼ 

"  λ~ 0

Mb

αT M b

λ~ 1 Mb

Mb α

#!(

M oo #(

λ~ 1 M b α

λ~ 1 αT Mb λ~ 1 Moo  K s

Kb

0

0

0

¼



"



"  λ~ 0

Mb

αT M b

Mb α

)

U s1 ) ð11Þ

U s0 #!(

M oo #(

λ~ 1 Mb λ~ 1 M b α ~λ αT M ~λ M  K s 1 1 oo b

U b0

U b1

U b1 U s1

U b2

)

U s2 ) þ λ~ 2

"

Mb

αT M b

Mb α M oo

#(

U b0

)

U s0 ð12Þ

from which λ~ 0 ; U b0 ; U s0 ; λ~ 1 ; U b1 ; U s1 ; λ~ 2 ; U b2 ; U s2 can be obtained in sequence. In addition, to determine some of these quantities, the ~ i and the system modal stiffness K~ i given by system modal mass M " #( )   Mb Mb α Ub T T ~ M i ¼ Ub ; Us ð13aÞ αT M b Moo Us K~ i ¼ U Tb K b U b þ U Ts K s U s need to be considered.

ð13bÞ

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

Before deriving the perturbed solution, we note that the zero-order approximation, given by Eq. (10), corresponds to the classical eigenvalue problem for an elastic structure, attached to the rigid foundation but uncoupled from the soil. As it can be seen from the lower part of Eq. (10), the first six eigenvalues of this system are zero λ~ 0 ¼ 0 ði ¼ 1; 6Þ, whereas the remaining N eigenvalues have finite values λ~ 0 a0 ði ¼ 7; N þ 6Þ. Two different sets of modes will be obtained from these two groups of eigenvalues. From the case λ~ 0 ¼ 0, we will obtain a set of six system modes that involve mainly displacements of the foundation, and first- and higher order relative displacements of the superstructure; these modes will result (as seen from the bottom part of Eq. (11) with λ~ 0 ¼ 0 and fU b0 g ¼ f0g) from a perturbation of the solution of a rigid system on a flexible soil, and will be called hereafter Foundation Modes. From the case λ~ 0 a0, we will obtain a set of N system modes that involve structural displacements as well as foundation displacements; these modes will result from a perturbation of the solution of an unconstrained elastic structure, and will be called hereafter Structural Modes. It is important to note that the modes designated as Foundation Modes in the soft soil case involve significant foundation motions and small relative displacement (deformation) of the superstructure. In the stiff soil case, the modes designated as Foundation Modes in the companion paper [1] involve significant foundation motions and small total displacements of the superstructure. The frequencies of these modes for soft soils tend to zero as the soil stiffness decreases, while for stiff soils the frequencies of these modes tend to infinity in the limiting case of a rigid soil.

For k ¼ i, we obtain the second-order coefficient for the eigenvalues

ε2 λ~ 2 ¼  fψ Ri gT ½K b   1 fψ Ri g

While for k a i we obtain the first-order coefficients of the modal displacements

γ 1j ¼

ð14Þ

The six modes satisfying Eq. (14) correspond to the modes of vibration of a rigid system supported on the elastic soil. The system is composed of a rigid foundation, described by its mass matrix ½M o , and a rigid superstructure, whose inertial contribution is accounted for through the product ½αT ½M b ½α. Introducing the notation fU s0 g ¼ fϕRi g

i ¼ 1; 6

ð20Þ

i

This completes the derivation of a perturbation solution with second-order approximations for the natural frequencies and firstorder approximations for the mode shapes. 4.1. Approximate expressions for modal quantities of interest – foundation modes A simple approximate formula for the system natural frequencies is obtained by use of Eqs. (9), (15) and (19)

ω~ 2i ¼ ω2Ri  fψ Ri gT ½K b   1 fψ Ri g

ω~ 2i

¼

1

þ

1

ω2Ri ω4Ri

ð22aÞ

 fψ Ri gT ½K b   1 ψ Ri

where the term fψ Ri g ¼ ω2Ri ½M b ½αfϕRi g is a vector of modal loads. The first-order term of the foundation displacements fU s1 g and the second-order term of the eigenvalues λ~ 2 can be obtained from the bottom part of Eq. (12)   λ~ 1 ½αT ½Mb fU b1 g þ λ~ 1 ½Moo   ½K s  fU s1 g þ λ~ 2 ½Moo fU s0 g ð17Þ and expanding the foundation displacements in terms of the rigid-structure modes fU s1 g ¼ ∑6j ¼ 1 γ 1j fϕRj g. Substitution of this expansion and of the results in Eqs. (15) and (16) into Eq. (17), pre-multiplication by a generic rigid-structure mode shape fϕRk gT and use of the orthogonality condition leads to ð18Þ

ð22bÞ

Since ωRi ¼ f ð½K s Þ the implicit Eqs. (22a, b) need to be solved iteratively, calculating the foundation stiffness coefficients at the fundamental system frequency. Using also the results in Eqs. (16), (20) and (21), the system ~ i are mode shapes normalized by M ~ gT ; fϕ ~ gT Þ ~ gT ¼ ðfϕ fϕ i bi si

ð23Þ

where n

o





6

ϕ~ si ¼ ðω~ i =ωRi Þ2 ϕRi þ ω~ 2i ∑

j¼1 jai

ð15Þ

in which fϕRi g is normalized so that M Ri ¼ fϕRi gT ½M oo fϕRi g ¼ 1, the first-order term of the structural displacements fU b1 g is obtained from the upper part of Eq. (11)  εfU b1 g ¼ ½K b   1 ω2Ri ½Mb ½αfϕRi g ¼ ½K b   1 fψ Ri g ð16Þ

γ 1j ðλRi  λRj Þδkj ¼ ε2 λ~ 2 δki þ fψ Rk gT ½K b   1 fψ Ri g

ω2Ri fψ Rj gT ½K b   1 fψ Ri g ω2Rj ðω2Rj  ω2Ri Þ

To determine the remaining unknown coefficient γ 1i , the secondorder expanded versions of the system modal stiffness K~ i ¼ ω2Ri þ ~ i ¼ K~ i =λ~ i need to be 2εγ 1i ω2Ri  ε2 λ~ 2i and of the modal mass M considered. Setting γ 1i ¼ 0 leads to the convenient result !4 2 ~ i ¼ ελ1i  ε λ2i  ωRi þ Oðε2 Þ ð21Þ M ελ1i þ ε2 λ2i λ~

1 First, to obtain the foundation modes, we consider the case λ~ 0 ¼ 0. From the upper part of Eq. (10) it follows that fU b0 g ¼ f0g and that the modes are rigid-body motions of the superstructure. To determine fU s0 g we use the bottom part of Eq. (11), leading to

ελ~ 1 ¼ λRi ¼ ω2Ri ;

ð19Þ

which, to the same order of accuracy, can also be written in the form

4. Perturbation about foundation modes

ð½K s   λ~ 1 ½M 00 ÞfU s0 g ¼ f0g

417

~ g ¼ ðω ~ i =ωRi Þ2 ½K b   1 fψ Ri g fϕ bi

fψ Rj gT ½K b   1 fψ Ri gn

ω ω 2 ð Rj

2  Rj

ω

2Þ Ri

ϕRj

o

ð24Þ

ð25Þ

Once the system mode shapes are obtained, the damping ratios and the participation factors for seismic excitation can be computed in the classical way using Eqs. (7a, b), (24) and (25). An approximate, closed-form expression for the system damping ratios is

 

 ω~ i 3 1 ξ~ i ¼ ξRi þ fψ Ri gT ½K b   1 ½C b ½K b   1 ψ Ri ð26Þ 2ωRi ωRi where ξRi ¼ fϕRi gT ½C s fϕRi g=ð2ωRi Þ are the damping ratios of the rigid system. The first term in Eq. (26) represents the contribution of the radiation and material soil damping, the second term is related to structural damping. It is important to point out that the approximation given by Eq. (26), which clearly shows the role of the different sources of damping in the system damping ratio, is obtained considering only the zero-order terms of the modal displacements of the foundation. A better estimate of the modal damping ratios for the foundation modes can be obtained by the more general Eq. (7a).

418

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

The corresponding result for the system participation factors for seismic excitation is ~ i =ωRi Þ2 ðfβ Ri gT þ fψ Ri gT ½K b   1 ½M b ½αÞ fβ~ i gT ¼ ðω fψ Rj gT ½K b   1 fψ Ri g

6

~ 2i ∑ þω

ω2Rj ðω2Rj  ω2Ri Þ

j¼1

fβ Rj gT

ð27Þ

j ai

where fβ Ri gT ¼ fϕRi gT ½M oo  is the vector of participation factors for seismic excitation of the rigid system. The previous Eqs. (22)–(27) can be recast in an alternative form by making use of the modal properties of the superstructure on a fixed-base. This form might be

more  convenient for practical implementation. Let λ i ¼ ω2i and ϕi ði ¼ 1; NÞ be the ith fixedbase eigenvalue and eigenvector satisfying the characteristic equation ð½K b   λi ½M b Þfϕi g ¼ f0g

ω~ i ωRi

3

ð29Þ

N

ξRi þ ∑ ν k¼1

2 ik



ω~ i ωk

3

ξk

ð30Þ

j¼1 j ai

νik fβ k gT 2 ω k¼1 k ! ω~ 2i ω2Ri N νjk νik T f β g ∑ Rj ðω2Rj  ω2Ri Þ k ¼ 1 ω2k

N

~ g¼ω ~ 2i ∑ ðνik =ω2k Þfϕk g fϕ bi o



ϕ~ si ¼ ðω~ i =ωRi Þ2 ϕRi 6

þ ∑

j¼1 j ai

2ðK H K M  K 2HM Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v" # " #ffi u u ðK I þ K m  2K s Þ 2 2 Þ ðm I  s oo oo H oo M oo HM oo oo 7t  2ðK H K M K 2HM Þ ðK H K M K 2HM Þ ð36Þ

and pffiffiffiffiffiffiffiffi fϕRi gT ¼ ðρRi ; 1Þ= M Ri

ð37Þ

where ω2 s  K K  ω2Ri I oo u ρRi ¼ R ¼ Ri oo 2 HM ¼ MM 2 θR i K HH  ωRi moo ωRi soo  K HM

ð38Þ

M Ri ¼ I oo þ 2ρRi soo þ ρ2Ri moo

ð39Þ

1

βR1i ¼ pffiffiffiffiffiffiffiffiðρRi moo þ soo Þ;

ð31Þ

ð32Þ



νjk νik ω~ 2i ω2Ri n o ∑ ϕRj 2 2 2 k ¼ 1 ωk ðωRj  ωRi Þ N

1

βR2i ¼ pffiffiffiffiffiffiffiffiðρRi soo þ Ioo Þ M Ri

ð40Þ

Finally, the modal damping ratios can be calculated as

k¼1

n

ðK H I oo þ K M moo  2K HM soo Þ

M Ri

N

þ ∑

¼

The participation factors for seismic excitation are

~ i =ωRi Þ2 fβRi gT þ ω ~ 2i ∑ fβ~ i gT ¼ ðω 6

1

ω2R1;2

and where the modal mass is

N ν2 1 1 ¼ 2 þ ∑ ik2 2 ω~ i ωRi k ¼ 1 ωk



from which ωRi and fϕRi gT ¼ ðuRi ; θRi Þ (i¼1,2) can be easily obtained. The resulting frequencies and normalized mode shapes are given by

ð28Þ

in which the eigenvectors are normalized so that M i ¼ fϕi gT ½M b fϕi g ¼ 1. Also, let fβi g ¼ ½αT ½M b fϕi g and ξi be, respectively, the vectors of participation factors for seismic excitation and the modal damping ratios of the superstructure on a fixed base. Making use of the standard results ½α ¼ ½Φ½β T and ½Φ½Ω  2 ¼ ½K b   1 ½M b ½Φ, where ½Ω ¼ diagðωi Þ, ½Φ ¼ ½fϕ1 g; …; fϕN g and ½β ¼ ½fβ 1 g; :::; fβ N g, and introducing the coefficients νik ¼ fβk gT fϕRi g ðk ¼ 1; N; i ¼ 1; 6Þ, it follows that

ξ~ i ¼

the structure with respect to their center of mass; and H g is the height of the center of mass of the entire system. The basic eigenvalue problem required to start the perturbation process corresponds, in this case, to " #   K HH  ω2Ri moo K HM  ω2Ri soo 0 fϕRi g ¼ ð35Þ 2 2 K MH  ωRi soo K MM  ωRi I oo 0

! ð33Þ

Some typical values for the coefficients νik are presented in Section 6.

ξRi ¼

ρ2Ri C HH þ2ρRi C HM þ C MM 2ωRi M Ri

ð41Þ

It should be noted that expressions (36) through (37) are consistent with those proposed by Beredugo and Novak [6] for the analysis of a rigid embedded footing. For a more precise comparison with the results in [6], it is necessary to recast the equations of motion of the rigid system by using the center of mass as point of reference. This is done in Appendix A. Once the basic eigenvalue problem of the rigid system has been solved, the natural frequencies, mode shapes, damping ratios and participation factors for the foundation modes of the actual interacting system can be obtained by use of Eqs. (29)–(33). 4.3. Foundation system modes for 1-DOF structure and comparison with earlier results

4.2. The case of plane vibrations To further analyze the foundation modes, we specialize our results to the case of a structure undergoing coupled horizontalrocking vibrations in the yz plane. In this case, the mass matrix for the rigid system is " # moo soo ½M oo  ¼ ð34Þ soo I oo N where moo ¼ mo þ ∑N i ¼ 1 mi , soo ¼ moo H g ¼ mo hg þ ∑i ¼ 1 mi hi and 2 2 N N I oo ¼ I g þmo hg þ ∑i ¼ 1 I i þ ∑i ¼ 1 mi hi are, respectively, the total mass, moment of masses and moment of inertia of the rigid system (foundation and structure) with respect to the reference point O; mo and mi are the masses of the foundation and of the floors; hg is the height of the center of mass of the foundation; I g and I i are the mass moments of inertia of the foundation and of

We further specialize the results for plane vibrations to the case of a 1-DOF elastic superstructure, with mass m1 , centroidal mass moment of inertia I 1 , height h1 , natural frequency ω21 ¼ k1 =m1 and damping ratio ξ1 ¼ c1 =ð2m1 ω1 Þ, supported by a rigid foundation with mass m0 and mass moment of inertia I 0 resting on an elastic soft soil with shear wave velocity β . In this case, there are two foundation system modes and one structural system mode. pffiffiffiffiffiffiffi In the case of a 1-DOF structure, fβ1 gT ¼ m1 ð1; h1 Þ, fϕRi gT ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðρRi ; 1Þ= M Ri and νi1 ¼ ðρRi þ h1 Þ m1 =M Ri in which ρ1i and M Ri are given by Eqs. (38) and (39), respectively. Using the notation defined in the previous section, the modal quantities of interest for the two foundation system modes are 1

ω~ 21;2

¼

1

þ

1

m1

ω2R1;2 ω21 M R1;2

ðρR1;2 þ h1 Þ2

ð42Þ

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

ξ~ 1;2 ¼ n



ω~ 1;2 ωR1;2

o

β~ 1;2 ¼



3

ξR1;2 þ



ω~ 1;2 ω1

3

m1 ðρ þ h1 Þ2 ξ1 M R1;2 R1;2

ð43Þ

 ffi   2 qffiffiffiffiffiffiffiffi  ω~ 1;2 2

m1 βR1;2 þ ω~ω1;21 β1 þ ::: ωR1;2 M R1;2 ρR1;2 þ h1

ω ω þ 2 ω1 ð ω ω

ρ

ρ

β

ω~ 21;2  1 ρR1;2 þh1 M R1;2 ω21

ϕ~ b1;2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi U n

o

1 M R1;2

ϕ~ s1;2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi U

"

ω~ 1;2

2 

ωR1;2

ρR1;2

 ~ 21;2 2R1;2 2 2 R2;1  R1;2 Þ

ω m1 ðρ þ h1 ÞðρR2 þ h1 Þ 2 þ M R2;1 R1 ω1 ðω

ω

ω

ρR2;1

M R2  m1 h1 ðm0 þ I 0 Þ½1 2 ðωM0 =ωR2 Þ2 

ð53bÞ

#

1

2

m0 {m1 and I 0 þ I 1 {m1 h1 . Under these assumptions, M R1  m1 : pffiffiffiffiffiffiffi ðρR1 þ h1 Þ2 , ν11  1, fβR1 gT  m1 ð1; h1 Þ, ρR1  ðω2M0 =ω2H0 Þh1 and 1

þ

1

ð47Þ

ω2M0 ω2H0

in which ω2H0 ¼ K HH =m00 and ω2M0 ¼ K MM =I 00 . The resulting approximations to the system modal quantities of interest are 1 1 1 1 ¼ þ þ ω~ 21 ω2M0 ω2H0 ω21

ξ~ 1 ¼



ω~ 1 ωH0

2

ξH0 þ



ω~ 1 ωM0

and

ð45Þ

4.3.1. First foundation system mode – comparison with earlier results At this point, to compare with the results for the fundamental mode of a 1-DOF structure supported on a relatively stiff soil, we consider the case of a flat foundation (hG ¼ 0) for which the coupling terms of the impedance matrix can be neglected with sufficient accuracy (K HM ¼ K MH  0). In addition it is assumed that



2

fϕR2 gT ¼ ðρR2 ; 1Þ=

pffiffiffiffiffiffiffiffiffi M R2

ð53cÞ

sffiffiffiffiffiffiffiffiffiffiffiffi 2 m1 h1 ν21   M R2

ðm0 ω2M0  I 0 ω2H0 Þ ðω2M0 þ ω2H0 Þ

ð53dÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi m0 þ I 0 fβR2 gT   m1 2 ðω2 ;  h1 ð1 þ I 0 Þω2M0 Þ ðωM0 þ ω2H0 Þ H0

ξR2 

ξM0 þ



ω~ 1 ω1

3

ξ1

ð49Þ

pffiffiffiffiffiffiffi fβ~ 1 gT  m1 ð1; h1 Þ

ð50Þ

pffiffiffiffiffiffiffi ~ gT ¼ ððω ~ 1 =ω1 Þ2 ; ðω ~ 1 =ωH0 Þ2 ; h1 1 ðω ~ 1 =ωM0 Þ2 Þ= m1 fϕ 1

ð51Þ

~ 1 C HH =2K HH and ξM0 ¼ ω ~ 1 C MM =2K MM . where ξH0 ¼ ω The approximate results in Eqs. (48)–(51) for a 1-DOF structure supported on a flat foundation resting on a relatively soft soil have the same form as those obtained in a companion paper [1] by a perturbation approach for the case of relatively stiff soils. The results also agree with those presented earlier by Bielak [2], Jennings and Bielak [4] and Veletsos and Meek [5], for the fundamental system mode of a single-story structure supported on an elastic or viscoelastic soil. These results as well as the numerical results that will be presented in Section 6 and those reported in [1] and in other studies [3,9] indicate that under certain conditions the approximate expressions for the fundamental system mode for the stiff-soil case are also sufficiently accurate for very soft soil conditions. 4.3.2. Approximate second foundation mode To gain a better understanding of the second foundation system mode for a 1-DOF superstructure, we introduce the same

ð53eÞ

ðξH0 ω2H0 þ ξM0 ω2M0 Þ ðω2M0 þ ω2H0 Þ n

n

ð53fÞ

where now ξH0 ¼ ωR2 C HH =2K HH and ξM0 ¼ ωR2 C MM =2K MM . Approximations to the modal quantities of interest for the second foundation system mode are: n

n

ðm0 ω2  I 0 ω2 Þ2 1 ðm0 þI 0 Þ þ 2 2 M0 2 2 H0  2 2 2 ω~ 2 ðωM0 þ ωH0 Þ ω1 ðωM0 þ ωH0 Þ ðm0 þ I0 Þ

ξ~ 2 



ω~ 2 ωR2

3

ð54aÞ

3 n n ðω2M0 ξM0 þ ω2H0 ξH0 Þ ðm0 ω2M0  I 0 ω2H0 Þ2 ω~ 2 þ ξ ω1 ðω2M0 þ ω2H0 Þ2 ðm0 þ I0 Þ 1 ðω2M0 þ ω2H0 Þ ð55aÞ



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ω~ 2 2 m1 ðm0 þ I 0 Þ β~ 21  

ωR2

þ

ð48Þ

2

ð52Þ

ð53aÞ

Note that the damping ratios given by Eq. (43) are very approximate, as only zero-order terms of the foundation displacements are considered.

1

ðm0 þI 0 Þ ðω2M0 þ ω2H0 Þ



ρR2   h1 ð1 þ I 0 Þ½1  ðωM0 =ωR2 Þ2 

ð46Þ

ω2R1

ω2R2

ð44Þ



1

assumptions of the previous section, including the assumptions 2 that m0 ¼ m0 =m1 {1 and I 0 ¼ ðI 0 þI 1 Þ=m1 h1 {1. This leads to the following approximate results for the second rigid-structure mode 1

~ 21;2 2R1;2  

 m1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R1 þ h1 R2 þ h1 R2;1 2 2 M R1 M R2 R2;1  R1;2 Þ

419

β~ 22 

ðm0 ω

ω ω21 ðm0 þ I0 Þ



ω~ 2 ωR2



2 M0  I 0

2 h1

ðm0 ω

2 H0 Þ 2 ð

ðω

ω ω2R2  ω2R1 Þ 2 R1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" m1 ðm0 þ I 0 Þ

ω ω21 ðm0 þ I0 Þ 2 M0  I 0

2 H0 Þ 2 ð

ω

2 H0 2 þ 2 Þ M0 H0

ðω

ω

#

ω

2 M0 2 þ 2 Þ M0 H0

ω ω2R2  ω2R1 Þ 2 R1

ω

#

ð56aÞ



1 ω~ 2 2 ðω2H0 I0  ω2M0 m0 Þ ~ gT   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; fϕ 2 ωR2 ½ω21 ðm0 þ I 0 Þ m1 ðm0 þ I 0 Þ !

ωM0 2 1 ;  1 þI 0  h1 ωR2

ð57aÞ

in which only the dominant terms in the expressions for the system damping ratio and mode shape have been kept. In the case of a very soft soil and when ωM0 {ωH0 , the results above reduce to "

ωH0



ω ω~ 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ H0 ω1 ðm þ I Þ 0

"

ξ~ 2  ξnH0 þ

0

2

2

#  1=2

2

I0

ω3H0 I0 ξ ω31 ðm0 þ I0 Þ5=2 1

#"

1þ

ð54bÞ

2

ðm0 þ I 0 Þ

ωH0 ω1

2

#  3=2

2

I0 ðm0 þ I 0 Þ

2

ð55bÞ

420

β~

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ω2 m1 ðm0 þI 0 Þ 1  M0 21   2

#

I0

While for k a i we obtain the first order coefficients of the displacements

ω1 ðm0 þ I0 Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" 2 ω ω2 ~ β 22  h1 m1 ðm0 þ I0 Þ M0 þ M0 2 2

ωH0

I0

# ð56bÞ

ω1 ðm0 þ I0 Þ

"



2 1 ωH0 2 I0 ~ gT   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1þ fϕ 2 ω1 ðm0 þ I0 Þ2 m1 ðm0 þ I 0 Þ ! ω2H0 I0 ; 1;  1=h 1 ðm0 þ I 0 Þ ω21

εα1j ¼

fϕf j;s gT ½K s fϕf i;s g ðω2f i  ω2f j Þ

ð64Þ

Finally, it is necessary to derive the first-order coefficient of the modal displacements α1i ðk ¼ iÞ, which is not defined by Eq. (64). This term can be obtained analyzing the perturbed expression for ~ i ¼ 1 þ 2εα1i . Setting α1i ¼ 0, the system the system modal mass M modal mass is approximately equal to one.

#1 

ð57bÞ

which imply that for extremely soft soils: (i) the system frequency increases linearly with the shear wave velocity β in the soil, (ii) the system damping ratio is essentially equal to the typically large n radiation damping ratio ξH0 . The system damping decreases as the soil becomes stiffer. The additional contribution of the structural damping is very small, particularly when ωH0 {ω1 , (iii) if ωM0 {ωH0 {ω1 the participation factor β~ 22 for rocking seismic excitation is very small, and (iv) for ωH0 {ω1 the deformation of the superstructure is very small and the system mode shape for total displacement has an approximate node at the location of the mass m1 .

5.1. Approximate expressions for modal quantities of interest – structural modes The N additional frequencies and mode shapes related to the structural modes, obtained with a first order perturbation analysis are

ω~ 2i ¼ ω2f i þ fϕf i;s gT ½K s fϕf i;s g

ð65Þ

and n

o

n

o

Nþ6 f

ϕ~ i ¼ ϕf i þ ∑

j¼1

ϕf j;s gT ½K s fϕf i;s gn o ϕf j ω2f i  ω2f j

ð66Þ

jai

5. Perturbation about structural modes Returning to the general case, approximations for the remaining N (i¼ 7, N þ6) system modes, designated here as structural modes, require solution of the eigenvalue problem for a perfectlyflexible soil given by Eq. (10). The corresponding N þ 6 perfectlyflexible soil eigenvalues and eigenvectors are represented by

λ~ 0 ¼ λf i ¼ ω2f i ;

ðU Tb0 ; U Ts0 ÞT ¼ ðϕf i;b ; ϕf i;s ÞT ¼ fϕf i g T

T

i ¼ 1; N þ6

in which the eigenvectors are normalized so that " # Mb Mb α M f i ¼ fϕ f i gT fϕf i g ¼ 1 T α M b Moo

ð58Þ

ð59Þ

The first six eigenvalues are such that λf i ¼ ω2f i ¼ λ~ 0 ¼ 0 ði ¼ 1; 6Þ and, consequently, the corresponding modes are not uniquely defined. Since any set of six orthogonal rigid-body motions could be used for this purpose, we select the six rigid-system modes given by Eq. (15), so that

ω2f i ¼ λ~ 0 ¼ 0;

fϕf i gT ¼ ðU Tb0 ; U Ts0 ÞT ¼ ðf0gT ; fϕRi gT ÞT ;

i ¼ 1; 6

ð60Þ

Next, the first-order terms of the frequencies and mode shapes for the structural modes can be obtained from Eq. (11). First, we expand the displacement vector in terms of the perfectly-flexible soil modes ( ) ( ) ϕf j;b Nþ6 U b1 ¼ ∑ α1j ð61Þ ϕf j;s U s1 j¼1 Substituting the expansion (61) in Eq. (11), pre-multiplying by  ϕTfk;b ϕTfk;s and making use of the orthogonality conditions, it results that Nþ6

∑ α1j ðω

j¼1

2 fj 

ω δ ¼ λ~ 1 δki  fϕf k;s gT ½K s fϕf i;s g 2 f i Þ kj

ð62Þ

For k ¼ i, we obtain the first order coefficients for the eigenvalues

ελ~ 1 ¼ fϕf i;s gT ½K s fϕf i;s g

ð63Þ

from which the modal damping ratios and participation factors for seismic excitation can be obtained by use of Eqs. (7a) and (7b). A first approximation for the modal damping ratios can be obtained by considering only the first term in Eq. (66). The resulting expression is ~ gT ½C fϕ ~ g þ fϕ ~ gT ½C fϕ ~ g ~ i  fϕ ~ iM 2ξ~ i ω s b f i;b f i;b f i;s f i;s

ð67Þ

~ g is independent of the soil properties and ½C  increases Since fϕ s f i;s approximately linearly with the soil shear wave velocity, Eq. (67) implies that modal damping ratios for the structural modes increase as the soil becomes stiffer. Similarly, a first approximation to the modal participation factors for seismic excitation for the structural modes, obtained by considering only the first term in Eq. (66), corresponds to fβ~ i gT 

1 ~ ~ gT ½M Þ ¼ f0gT ðfϕ f i;b gT ½M b ½α þ fϕ oo f i;s ~i M

ð68Þ

which is zero based on the lower portion of Eq. (10). Thus, the contribution to the system participation factors fβ~ i g for the structural modes arises only from first- and higher-order terms of the perturbed mode shapes. As it will be confirmed by the numerical results presented below, it is reasonable to expect that, for very soft soils, the contribution of the structural modes to the response of the system to seismic excitation is very small. The derivations presented in Sections 4 and 5 allow to approximate the response of a soil–structure interacting system ~ gη~ of N þ6 replacement by combining the modal responses fϕ i i ~ i and system oscillators, characterized by system frequencies ω n damping ratios ξ~ i , subjected to the base accelerations fβ~ i gT fU€ 0 g. 5.2. Structural mode for a 1-DOF structure on a soft soil In the case of a 1-DOF superstructure there is only one structural system mode. To obtain the modal characteristics of this mode we need to solve first the eigenvalue problem of the structure uncoupled from the soil. In this case, we have that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m m h ωf 1 ¼ ωf 2 ¼ 0; ωf 3 ¼ ω1 1 þ 1 þ 1 1 ð69Þ m0 I 0 þ I 1

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

fϕf 1;2 gT ¼ ð0; fϕR1;2 gT Þ;



ωf 3 m1 ω1 m1 h1 ω1 1 fϕf 3 gT ¼ pffiffiffiffiffiffiffi ; ; m0 ω f 3 I 0 þI 1 ωf 3 m1 ω 1 ð70Þ

ξf 1 ¼ ξf 2 ¼ 0;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m m h ξf 3 ¼ ξ1 1 þ 1 þ 1 1 m0 I 0 þ I 1

fβf 1;2 gT ¼ fβR1;2 gT ;

ð71Þ

fβ f 3 gT ¼ ð0; 0Þ

ð72Þ

At this point we introduce again the additional assumption of a flat foundation (hG ¼ 0) for which the coupling terms of the impedance matrix can be neglected (K HM ¼ K MH ¼ 0). In this case, the system frequency for the structural mode is given by 2 3 ! 2 2 ω21 4 m1 2 K H m1 h1 KM 5 2 2 ð73Þ ω~ 3 ¼ ωf 3 þ 2 þ I 0 þI 1 m1 h2 ω f 3 m0 m1

421

Table 1 Fixed-base dynamic properties of the test structure ðM tot ¼ 12:2
106 kg; Htot ¼ 45:7 mÞ. Mode #

Natural frequencies

Damping ratios

j

ωj [rad/s]

fj [Hz]

ξj [–]

1 2 3 4 5 6 7 8 9

13.57 40.34 66.02 89.89 111.31 129.69 144.54 155.44 162.11

2.16 6.42 10.51 14.31 17.72 20.64 23.00 24.74 25.80

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Normalized participation factors

Modal amplitude

pffiffiffiffiffiffiffiffiffiffi βj = M tot 0.8683  0.2841 0.1640  0.1101 0.0782  0.0560 0.0389  0.0247 0.0120

pffiffiffiffiffiffiffiffiffiffi ϕj;9 M tot 1.4580 1.4182 1.3398 1.2248 1.0764 0.8986 0.6963 0.4750 0.2408

pffiffiffiffiffiffiffiffiffiffi β2j =Htot M tot 0.6277 0.0109 0.0443  0.0074 0.0174  0.0052 0.0080  0.0026 0.0024

1

ð1Þ

ð2Þ

ð3Þ

~ gT ¼ ð ϕ ~ ;ϕ ~ ;ϕ ~ Þ and the corresponding system mode shape fϕ 3 3 3 3 is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1Þ 1 m m h ϕ~ 3 ¼ pffiffiffiffiffiffiffi 1 þ 1 þ 1 1 ð74aÞ m0 I 0 þ I 1 m1 "

#

ð2Þ K H ρ2R1 ρ2R2 ρR1 ρR2 ð2Þ ð3Þ K M ~ ϕ 3 ¼ ϕf 3 1 þ 2 þ þ þ ϕf 3 2 ωf 3 MR1 MR2 ωf 3 MR1 MR2

ð74bÞ

"

#

ð3Þ KM 1 1 ρR1 ρR2 ð3Þ ð2Þ K H ~ þ ϕf 3 2 ϕ 3 ¼ ϕf 3 1 þ 2 þ þ ωf 3 MR1 MR2 ωf 3 MR1 MR2

ð74cÞ

where ð2Þ f3 ;

ðϕ

ϕ

ð3Þ f3 Þ ¼



1 ω1 m1 m1 h1  pffiffiffiffiffiffiffi ; m1 ωf 3 m0 I 0 þ I 1

Table 2 Modal properties of the rigid-structure case. β [m/s]

i

fRi [Hz]

ρRi/Htot

ξRi

pffiffiffiffiffiffiffiffiffiffi β1;Ri = M tot

pffiffiffiffiffiffiffiffiffiffi β2;Ri =H tot M tot

120

1 2

0.75 4.27

0.055  0.741

0.045 1.011

0.878  0.478

0.651 0.026

180

1 2

1.13 6.41

0.056  0.741

0.046 1.011

0.878  0.478

0.651 0.027

components of the foundation input motion are given by ð75Þ

The modal damping ratio and participation factor for seismic excitation can be obtained by use of Eqs. (74a,b,c) and (7a,b). A first approximation for the modal damping ratio can be obtained by considering only the dominant terms in Eq. (74a,b,c). The resulting expression is 2 3 !

2 2 2 2 CH m1 h1 CM 5 ~ξ  ωf 3 ωf 3 ξ þ 1 ω1 4 m1 ð76Þ þ 3 ω1 ω~ 3 1 2 ω2f 3 m0 ω~ 3 m1 I0 þ I1 ω ~ 3 m1 h21 which implies that the modal damping ratio for the structural mode increases as the soil becomes stiffer. Also, since ωf 3 c ω1 ~ 3  ωf 3 , then the system modal damping ratio for the and ω structural mode is much larger than the fixed-base damping ratio of the superstructure (ξ~ 3 c ξ1 ).

6. Numerical results To evaluate the accuracy of the modal quantities obtained by the perturbation approach for soft soils, we performed numerical analysis considering the plane, coupled, horizontal-rocking response of a multi-story building subjected to a seismic excitation represented by vertically incident SH-waves. The particular ninestory building model used in the study is the same as that considered for the stiff-soil perturbation analysis [1]. The characteristics of the model are described in detail in the Appendix of the aforementioned paper. For completeness, Table 1 lists some important dynamic properties of the structure on a fixed base. The elements of the impedance matrix ½K^ s  ¼ ½K s  þ iω½C s  for the embedded foundation were obtained from the solution by Apsel and Luco [22]. The seismic excitation was represented by vertically incident SH waves. The translational and rotational

U n0y ðωÞ ¼ Syy ðωÞU gy ðωÞ;

aθ0x ðωÞ ¼ Rxy ðωÞU gy ðωÞ n

ð77a; bÞ

where U gy ðωÞ is the free-field motion on the ground surface, and Syy ðωÞ and Rxy ðωÞ are the scattering coefficients for the embedded foundation (e.g., Luco and Wong [19]).

6.1. Numerical values of the modal quantities for the foundation modes Before presenting a thorough study about the accuracy of the perturbation solution, it is possible to gain a better understanding of the key parameters that govern the response of the foundation modes, by analyzing the numerical values for the 9-story test structure. Table 2 lists the modal properties (natural frequencies, mode shapes, damping ratios and normalized participation factors for two soil stiffnesses considered) for the rigid-structure case, which represents the basic problem required to start the perturbation approach for the foundation modes. The rigid-system mode shapes are represented in terms of the ratio between the coefficient ρRi and the total height of the structure and foundation H tot . Analysis of Table 2 reveals that only the system natural frequencies vary with the soil stiffness, while the mode shapes, damping ratios and participation factors remain practically unchanged. In particular, except for the dependence of ½K s  on the normalized frequency a0 ¼ ωa=β , the variation of the system natural frequencies of the rigid system is linear with respect to the soil shear wave velocity, while the other modal quantities are independent from it. As noted by Barkan [23], Richart et al. [24] and Beredugo and Novak [6], the ratio ρRi ¼ ðuR =θR Þi represents the distance between the reference point on the foundation O (bottom of the foundation in our case) and the center of rotation of the rigid system. The first rigid system mode represents a rotation about a point lying below the reference point O, while in the second mode the center of rotation is above the point O.

422

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

Table 3 lists the coefficients νik and the terms ν2ik ðω1 =ωk Þ2 and ξk ν2ik ðω~ i =ωk Þ3 appearing in Eqs. (29) (through 33), needed to obtain the system natural frequencies and damping ratios for the

6.2. Comparison with classical modal analysis

Table 3 Numerical values of the coefficients in Eqs. (29) (through 33). k

νik

ν2ik ðω1 =ωk Þ2

ξk ν2ik ðω~ 1 =ωk Þ3

½
100

β¼ 120 m/s

1 2 3 4 5 6 7 8 9

foundation modes. Analysis of the coefficients νij indicates that the contribution of the fixed-base modes j4 2 to the fundamental mode of the system is practically negligible.

β ¼ 180 m/s

i¼ 1

i ¼2

i¼ 1

i¼ 2

i¼ 1

i ¼2

i¼ 1

i¼ 2

0.97  0.01 0.08  0.02 0.03  0.01 0.01  0.01 0.00

 0.04 0.58  0.20 0.19  0.11 0.10  0.05 0.04  0.02

0.9332 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0017 0.0383 0.0017 0.0009 0.0002 0.0001 0.0000 0.0000 0.0000

0.0336 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0107 0.0788 0.0022 0.0008 0.0001 0.0001 0.0000 0.0000 0.0000

0.0953 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0281 0.2077 0.0058 0.0021 0.0003 0.0002 0.0000 0.0000 0.0000

Σ

0.9334

0.0429

0.0336

0.0927

0.0953

0.2443

The accuracy of the perturbation results is first assessed by a set of detailed comparisons with the solution of the classical eigenvalue problem for the complete soil–structure system. Fig. 1 shows a comparison of the system natural frequencies for the test structure supported on different soils, obtained by the perturbation approach for soft soils (circles), the perturbation approach for stiff soils (stars) and by direct numerical solution of the classical eigenvalue problem (continuous lines). The system frequencies normalized by the fundamental fixed-base frequency of ~ j =ω1 Þ are shown versus the normalized soil the superstructure ðω shear wave velocity parameter ðβ=ω1 aÞ. Values of β=ω1 a ¼ 0:8 and 1.21 correspond to shear wave velocities of 120 and 180 m/s, respectively. For a better visualization of the results relative to softsoil conditions, the maximum value of relative stiffness considered is β=ω1 a ¼ 4, which corresponds, for the nine-story building model, to

Fig. 1. System natural frequencies. Comparison between classical modal analysis and perturbation approaches.

Table 4 System natural frequencies in Hertz. Mode #

β¼ 120 m/s

j

Modal analysis

β ¼180 m/s Perturbation soft-soil

Perturbation stiff-soil

Modal analysis

Perturbation soft-soil

Perturbation stiff-soil

Foundation modes 1 2

0.713 3.883

0.713 3.952

0.732 4.830

1.008 5.080

1.010 5.459

8.585 9.139 12.791 15.716 18.769 21.240 23.364 24.885 25.839

8.443 8.921 12.751 15.619 18.710 21.203 23.345 24.877 25.837

1.029 5.545

Structural modes 1 2 3 4 5 6 7 8 9

8.027 8.501 12.484 15.442 18.603 21.142 23.313 24.864 25.834

7.960 8.479 12.473 15.423 18.592 21.136 23.310 24.863 25.834

– – – – – – – – –

– – – – – – – – –

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

a soil shear-wave velocity β ¼ 600 m=s. Except for the two foundation modes, the results of the perturbation approach for soft-soils are plotted only for β=ω1 a o 2:1, while the results of the perturbation approach for stiff-soils are plotted for β=ω1 a 4 1:4. For very soft soils (i.e. β=ω1 a o 1), the results in Fig. 1 indicate that, as expected, the frequencies obtained by the perturbation analysis are very accurate for all modes. As the relative soil-tostructure stiffness increases, the perturbation frequencies show increasing deviations from those obtained through classical modal analysis. As noted in Luco and Lanzi [1], the frequencies obtained by the perturbation approach for stiff soils are accurate for all modes when β=ω1 a 4 3:5. Finally, it appears that both the perturbation approach for soft- and stiff-soils are effective in predicting the fundamental frequency of the system in the entire range of soils considered. It is worth recalling that, for the fundamental mode, the

423

results of the perturbation approach for stiff and soft soils are consistent with those given by the well-known formulae presented in Refs. [2–5]. A comparison of the numerical values for the system natural frequencies obtained by direct numerical solution of the classical eigenvalue problem and by the perturbation approach for softand stiff-soils for the two cases β ¼120 and 180 m/s is presented in Table 4. For a better comparison, the frequencies of the structural and foundation modes are listed separately. The perturbation approach is very accurate for the softer soil, with maximum deviations from the modal solution of less than 2%. For the stiffer soil, the perturbation method is very accurate in predicting the frequencies of the first foundation mode (fundamental mode), and of the structural modes (from the third structural mode), while the error increases to 7.5% for the second foundation mode. The results

Fig. 2. Total displacement mode shapes for β ¼120 and 180 m/s.

424

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

The plots for the two foundation modes also show a displaced and rotated coordinate axis moving with the rigid foundation (thin gray lines). The first foundation mode involves significant rotation of the foundation and a small amount of structural deformation in the shape of the fundamental fixed-base mode. The second foundation mode involves translation and rotation of the foundation together with some structural deformation following the first and second fixed-base mode shapes. The mode shapes of the 3rd– 9th structural modes are essentially independent of the soil stiffness and reflect the mode shapes of the structure and foundation when detached from the soil. Table 5 presents a comparison of the modal damping ratios. For the classical modal analysis results, also the contribution of the different sources of damping (structural, foundation translation, rotation and coupling terms) are listed separately. For the softer soil ðβ ¼ 120 m=sÞ, the damping ratios obtained by the perturbation approach for soft-soils are accurate for all the modes, with maximum deviations from the modal analysis solution of about 15% for the 1st structural mode. For the case β ¼ 180 m=s, large differences can be observed for the second foundation mode and for the first two structural modes. For the higher structural modes, the perturbation approach leads to errors of less than 10%. The perturbation approach for stiff-soils is accurate in predicting the damping ratio of the fundamental mode in the soft soil case.

obtained with the perturbation approach for stiff-soils are only listed for the first two modes, as for the higher modes they would involve large errors. As expected, the error is still very small for the fundamental mode (2.7% and 2.1% respectively for the two soils considered), whereas it is higher for the second foundation mode (24% and 9%); also, the error decreases with increasing soil stiffness. The 11 mode shapes (for total displacements) obtained by numerical solution of the eigenvalue problem and by the perturbation approach for soft- and stiff-soils, for the two soil stiffnesses considered, are compared in Fig. 2. As for the case of the natural frequencies, the perturbation approach is very accurate for the softer soil. For β ¼180 m/s, small differences are observed for the 2nd foundation mode and more significant ones for the 1st and 2nd structural modes. It is worth to note that the first and second structural modes are characterized by very close natural frequencies, which lie in a region where a curve veering phenomenon occurs [1]. The perturbation approach used, which assumes wellseparated frequencies, may not work well in these regions of strong interaction between modes. The mode shapes obtained with the perturbation approach for stiff-soil are represented only for the two foundation modes; they are very accurate for the fundamental mode, while differences are noticeable for the second foundation mode. Table 5 System damping ratios. Mode # β ¼120 m/s j

β¼ 180 m/s

Modal analysis

Perturb. soft-soil

Perturb. stiff-soil

Structural Translation Rotation Coupling Total

Modal analysis

Perturb. soft-soil

Perturb. stiff-soil

Structural Translation Rotation Coupling Total

Foundation modes 1 0.000 0.009 2 0.001 0.771

0.016 0.036

0.012  0.095

0.038 0.038 0.714 0.764

0.038 0.368

0.001 0.004

0.007 0.430

0.012 0.021

0.010  0.056

0.031 0.399

0.031 0.543

0.031 0.207

Structural modes 1 0.022 2 0.016 3 0.017 4 0.013 5 0.013 6 0.011 7 0.011 8 0.010 9 0.010

0.094 0.004 0.010 0.000 0.001 0.000 0.000 0.000 0.000

 0.046 0.014 0.014 0.003 0.004 0.001 0.001 0.000 0.000

0.169 0.245 0.146 0.121 0.077 0.048 0.030 0.018 0.012

– – – – – – – – –

0.025 0.008 0.016 0.013 0.013 0.011 0.011 0.010 0.010

0.011 0.456 0.160 0.193 0.116 0.072 0.037 0.015 0.004

0.123 0.024 0.012 0.001 0.002 0.000 0.000 0.000 0.000

 0.019  0.053 0.020 0.004 0.006 0.002 0.002 0.000 0.000

0.141 0.435 0.208 0.210 0.136 0.086 0.050 0.026 0.014

0.322 0.346 0.230 0.211 0.130 0.079 0.046 0.025 0.014

– – – – – – – – –

0.100 0.211 0.104 0.105 0.060 0.036 0.018 0.008 0.002

0.192 0.227 0.148 0.121 0.076 0.048 0.030 0.018 0.012

Table 6 Normalized participation factors. Mode #

β¼ 120 m/s

j

pffiffiffiffiffiffiffiffiffiffi β~ 1j = M tot M.A.

β¼ 180 m/s pffiffiffiffiffiffiffiffiffiffi β~ 1j = M tot

pffiffiffiffiffiffiffiffiffiffi β~ 2j =H tot M tot P. Soft

P. Stiff

Foundation modes 1 0.880 2  0.464

0.880  0.461

0.857  0.346

Structural modes 1 0.050 2 0.078 3  0.031 4 0.022 5  0.013 6 0.008 7  0.005 8 0.003 9  0.001

0.056 0.069  0.030 0.020  0.012 0.007  0.004 0.003  0.001

– – – – – – – – –

M.A.

P. Soft

0.651 0.029

0.650 0.029

 0.018 0.005  0.004 0.001  0.001 0.000 0.000 0.000 0.000

 0.016 0.011  0.004 0.001  0.001 0.000 0.000 0.000 0.000

pffiffiffiffiffiffiffiffiffiffi β~ 2j =H tot M tot

P. Stiff

M.A.

P. Soft

P. Stiff

0.633 0.037

0.881  0.425

0.880  0.425

0.864  0.354

– – – – – – – – –

 0.018 0.181  0.068 0.053  0.032 0.021  0.013 0.008  0.004

0.118 0.149  0.067 0.045  0.026 0.016  0.010 0.006  0.003

– – – – – – – – –

M.A.

P. Soft

0.650 0.028

0.649 0.029

0.037  0.011  0.007 0.002  0.002 0.001  0.001 0.000 0.000

 0.035 0.023  0.009 0.002  0.002 0.001  0.001 0.000 0.000

P. Stiff

0.636 0.031

– – – – – – – – –

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The system damping ratios for the second foundation mode and for the first few structural modes are quite large as a result of radiation damping associated with (mostly) translation of the foundation. The damping ratios for the structural modes increase with the stiffness of the soil in full agreement with Eq. (67) and for the reasons stated in Section 5.1. Finally, estimates of the damping ratios for the foundation modes through the simplified Eq. (26), which are not listed in Table 5, appear to be very accurate for the fundamental mode (maximum errors of about 7%), while for the second foundation mode their accuracy decreases rapidly with an increase of the relative soil stiffness. Comparisons of the modal participation factors are presented in Table 6. The participation factors for translational ðβ~ 1j Þ and rotational ðβ~ 2j Þ excitation are normalized with respect to the total mass and the total height of the structure and foundation. The deviations of the perturbation approaches from the modal analysis solution show a trend similar to that for the mode shapes and for the damping ratios. The participation factors for the structural modes are quite small but tend to increase with the stiffness of the soil. As noted in the discussion of Eq. (68), these modes will not be excited significantly by seismic excitation in the case of soft soils. 6.3. Comparison with exact frequency response The accuracy of the perturbation method in predicting the structural response is finally evaluated by analyzing the frequency response at different locations on the structure and foundation. Transfer functions of normalized relative displacements are computed for a unit free-field acceleration input, using four analysis methods: (1) exact solution in the frequency domain, considering the frequency-dependence of the impedance functions, (2) classical modal analysis, (3) perturbation approach for soft soils and (4) perturbation approach for stiff soils. The analysis is performed in the range of frequencies from 0.02 to 42 Hz, with a frequency step of 0.01 Hz. The amplitudes of the transfer functions for the relative

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displacement of the first floor, relative displacement of the top floor, and the relative translation and rotation of the base are shown in Figs. 3 and 4 for soils with shear wave velocities of 120 and 180 m/s, respectively. The plots of the transfer functions indicate a very good agreement of the solutions obtained with classical modal analysis and with the perturbation approach for soft soils in the entire frequency range considered. As expected, the agreement is excellent for the softer soil (β ¼ 120 m=s), while small deviations are noticeable for the case β ¼ 180 m=s. Both approaches based on a classical normal mode decomposition (classical modal analysis and perturbation) appear to be accurate only in the vicinity of the fundamental system frequency, and lead to significant deviations from the exact solution for higher frequencies. However, as indicated by the analytical results presented above, it is evident that the contribution to the response of the higher modes becomes progressively smaller as the soil stiffness decreases. Finally, in agreement with the analytical results presented in Section 4.3 and with the numerical results about the modal quantities, the solution obtained using the perturbation approach for stiff soils provides an accurate estimate of the fundamental mode response even though, for the soil stiffness considered in this study, the basic assumption of the method is violated. To obtain an overall quantitative measure of the error of the classical modal analysis and of the perturbation approaches with respect to the exact frequency response, the two quantitative measures of error defined in [1] are considered

μ1 ¼

    ∑i jH iex j2  jH i j2 Δωi ∑i jH iex j2 Δωi

;

μ2 ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u∑i ðH i  H i ÞðH i  H i ÞΔωi ex ex t i

∑i H iex H ex Δωi

ð78a; bÞ where H ex represents the values of the exact complex transfer functions, H represents the transfer functions obtained by classical

Fig. 3. Transfer functions for β¼ 120 m/s (relative displacement, unit acceleration input).

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Fig. 4. Transfer functions for β ¼ 180 m/s (relative displacement, unit acceleration input).

Table 7 Comparison of the error measures for relative displacements, Response parameter

μ1

μ2

β ¼120 m/s

U1 U9 Usy ysx

β ¼ 120 and 180 m/s.

β ¼ 180 m/s

β ¼ 120 m/s

β ¼ 180 m/s

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

0.133 0.124 0.126 0.135

0.128 0.122 0.122 0.130

0.620 0.382 0.399 0.330

0.130 0.121 0.129 0.129

0.112 0.111 0.186 0.106

0.377 0.335 0.345 0.314

0.219 0.186 0.452 0.159

0.216 0.180 0.457 0.153

0.559 0.351 0.811 0.354

0.198 0.173 0.433 0.152

0.180 0.145 0.459 0.122

0.382 0.326 0.776 0.334

Table 8 Comparison of the error measures for relative displacements, fundamental mode analysis. The perturbation results are obtained considering only the fundamental mode. Response parameter

μ1

μ2

β ¼120 m/s

U1 U9 Usy ysx

β ¼ 180 m/s

β ¼ 120 m/s

β ¼ 180 m/s

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

M.A.

P. Soft

P. Stiff

0.133 0.124 0.126 0.135

0.137 0.123 0.156 0.140

0.471 0.353 0.483 0.329

0.130 0.121 0.129 0.129

0.123 0.109 0.140 0.117

0.385 0.323 0.439 0.313

0.219 0.186 0.452 0.159

0.234 0.179 0.458 0.154

0.428 0.342 0.813 0.354

0.198 0.173 0.433 0.152

0.204 0.144 0.458 0.123

0.369 0.323 0.778 0.334

normal mode or by the perturbation approach, and H denotes complex conjugate. As shown in Table 7, the errors associated with the perturbation approach for soft soils are of the same order as those obtained with the classical modal analysis. In general, the error μ1 related to any of the two approximate solutions, i.e. modal analysis or perturbation for soft soils, is always less than 22%. It is found that, except for the translation of the foundation, the perturbation analysis leads to a slight decrease of both the error measures.

Use of the perturbation approach for stiff soils leads to a generalized increase of the wide-frequency measures of error in the steady-state response. As expected, the errors increase with decreasing soil stiffness. However, the resonant response at the fundamental system frequency (which would be an important quantity in a response-spectrum analysis) is still predicted with a good accuracy. Finally, the analytical derivation presented in Section 5 showed that the participation factors for seismic excitation for the structural

A. Lanzi, J. Enrique Luco / Soil Dynamics and Earthquake Engineering 66 (2014) 415–428

modes are related only to first- and higher-order terms of the perturbation series. In addition, the damping ratio of the second foundation mode, which involves mainly translation of the foundation, is typically very high. As confirmed from the numerical results shown, it seems reasonable to expect that, for soft soils, the contribution to the response of the higher modes (j41) can be neglected. Table 8 lists the values of the two measures of error obtained considering, for both the perturbation approaches, only the contribution of the fundamental mode. The maximum increase of the error measure μ1 is about 3% for the foundation translation in the case β ¼ 120 m=s.

7. Conclusions The paper presents an approximate method of solution of the linear dynamic three-dimensional soil–structure interaction problem, valid for structures resting on relatively soft soils. The method is based on the application of a perturbation analysis to solve the eigenvalue problem of a multistory structure on an elastic or viscoelastic soft soil. The solution provided, which corresponds to an approximation of the classical modal analysis approach, complements that presented in a companion paper [1], which was valid for relatively stiff-soil conditions. The approach leads to analytical expressions which offer considerable physical insight into the nature of the soil–structure interaction effects, but do not cure the basic limitations of the approximate modal analysis for soil–structure interaction problems. The soft-soil perturbation approach is based on the assumption that the solution of the interacting system can be obtained from that of an unconstrained elastic structure, with the addition of small perturbing terms proportional to the square of the soil shear wave velocity. The approach leads to two separate sets of system modes. The first set corresponds to perturbations of the modes of a rigid structure on a flexible soil and involves mostly displacements of the foundation with small deformations of the superstructure. The second set of system modes involves structural displacements as well as foundation displacements. It has been shown that this second set of modes provides a small contribution to the system response in the case of ground motion excitation. The characteristics of the foundation modes, which provide most of the contribution to the system response for very soft-soils, have been analyzed in much detail for the particular case of plane, coupled horizontal-rocking vibrations, and comparisons have been made with the earlier results of Beredugo and Novak [6] for the case of a rigid structure. The results obtained here for the fundamental system mode of a 1-DOF structure supported on relatively soft soils have been compared with those presented in a companion paper for the case of stiff soils [1] which in turn coincide with the earlier results obtained by Bielak [2,3], Jennings and Bielak [4] and Veletsos and Meek [5] under various assumptions. These important earlier results have been adopted by several technical standards and have been widely used in engineering practice to assess the effects of soil compliance on the structural response. The analysis shows that the results for soft soils reduce to the earlier results if the mass of the superstructure is much larger than that of the foundation and if the slenderness ratio of the superstructure is sufficiently high. Simple analytical expressions for the modal quantities of interest for the second foundation mode and the single structural mode for a 1-DOF structure supported on soft soils have also been obtained. The accuracy of the perturbed solution has been assessed through numerical analysis of a nine-story test structure. In the case of plane, coupled horizontal-rocking vibrations, the modal frequencies, mode shapes, damping ratios and participation factors for the system modes obtained by the soft-soil perturbation approach are very accurate with respect to the corresponding

427

classical modal analysis results. Comparison with the transfer functions for the exact solution reveals, instead, that both the classical modal analysis and the perturbation approach, which are based on a normal-mode decomposition, are not able to accurately predict the system response for frequencies higher than those of the foundation modes. This limitation may have little practical consequences, as the contribution of the structural modes decreases progressively with decreasing soil stiffness. The perturbation approach for relatively soft-soils presented here does not substantially vary the error inherent to the classical modal analysis approach to soil–structure interaction problems. For example, a measure of the error of the classical modal approach for the structural response at the top of a 9-story test structure is 12% for a soil with a shear wave velocity of both 120 m/s and 180 m/s. The corresponding errors for the soft-soil perturbation analysis are the same, if not slightly smaller. Considering the contribution of the fundamental mode only leads to an increase of the wide-frequency error of about 3%. In the general case of multi-story structures supported on soft soils, the use of the approximate expressions for stiff soils leads to a generalized increase of the wide-frequency measures of error, but the resulting estimate of the peak response at the fundamental frequency is still reasonably accurate. Finally, it should be noted that the results presented here to study the dynamic response of a soil–structure system can also be used to obtain an approximate solution for the dynamic response of base-isolated structures on rigid soils. Approximate formulae for the modal quantities of interest for the isolated system can be derived from those presented here, by simply neglecting the rocking flexibility of the base. The resulting formulae for the natural frequencies and mode shapes of the base-isolated system are similar to those presented by Tsai and Kelly [25,26] and are consistent with the results discussed by Skinner et al. [27].

Appendix A For a better comparison with the results in Beredugo and Novak [6] for the analysis of a rigid embedded footing, it is necessary to recast the equations of motion of the rigid system, expressing the equilibrium with respect to its center of mass. Introducing the transformation ( ) ( )  ( u ) ug uo g 1  Hg ¼ ½A ¼ θg θg ; θo 0 1 ( ) ( ) " ) #( 1 0 Fg Fo Fo ¼ ½AT ¼ ðA:1Þ Mg  Hg 1 Mo Mo the system mass matrix and the soil impedance matrix become " # " # mgg 0 K^ HH K^ HM ^ ¼ ; ½K^ s  ¼ ^ ðA:2Þ ½M 0 I gg K HM K^ MM where mgg ¼ moo ; I gg ¼ I oo moo H 2g ; K^ HH ¼ K HH ; K^ HM ¼ K HM  K HH H g and K^ MM ¼ K MM þK HH H 2g  2K HM H g . With these definitions, the natural frequencies and mode shapes for the rigid system can be written in the form ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! v !2 u 2 u1 K^ ^ HH K^ MM K K^ MM K^ 1 HH ω2R1;2 ¼ þ  þ HM ðA:3Þ 8t 2 mgg 4 mgg I gg I gg mgg I gg

ρRi;g ¼



uR;g

θR;g

¼ i

K^ MM  ω2Ri I gg ð  K^ HM Þ ¼ 2 ^ K HH  ωRi mgg ð  K^ HM Þ

ðA:4Þ

which correspond to the results in Eqs. (24) and (25) in Beredugo and Novak [7]. Modal masses, participation factors and damping ratios are given by Eqs. (39)–(41), after substituting I oo with I gg and soo with sgg ¼ 0.

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