Simple lumped‐parameter models of foundation using mass‐spring‐dashpot oscillators

Journal of the Chinese Institute of Engineers, Vol. 24, No. 6, pp. 681-697 (2001) 681

SIMPLE LUMPED-PARAMETER MODELS OF FOUNDATION

USING MASS-SPRING-DASHPOT OSCILLATORS

Wen-Hwa Wu* and Cheng-Yin ChenDepartment of Construction Engineering

National Yunlin University of Science and TechnologyYunlin, Taiwan 640, R.O.C.

Key Words: soil-structure interaction, lumped-parameter model,foundation impedance function.

ABSTRACT

Simple lumped-parameter models are developed in this study forgenerally representing the horizontal, vertical, rocking and torsionalvibrations of rigid foundations. A spring and a dashpot are first at-tached in parallel to a mass to constitute a single-degree-of-freedomoscillator. Depending on the requirement of accuracy, the lumped-parameter model is then designed to consist of one or several oscilla-tors connected in series. In addition, a new expression for the dynamiccompliance function, instead of the conventional expression for thedynamic impedance function, is adopted in this research to improveaccuracy. The effectiveness of these models is demonstrated for vari-ous cases including surface square foundations, surface circular foun-dations and embedded square foundations. It is found that the perfor-mance of the new models is better in approximating translational foun-dation vibrations than in cases of rotational foundation vibration. Com-pared with the other existing lumped-parameter models, the modelsproposed in this paper have advantages in requiring fewer parametersand featuring a more systematic expansion.

*Correspondence addressee

I. INTRODUCTION

The dynamic analysis of soil-structure interac-tion (SSI) systems has been well studied in the pastfew decades. To consider SSI effects, the analysisshould be extended from the structure to include thetotal structure-foundation-soil system. The substruc-ture method, in which the discrete superstructure andthe unbounded continuous soil are separatelymodelled, is commonly adopted in the SSI analysisto take advantage of the appropriate formulationsfor the respective subsystems. It consists of twomajor steps in the substructure method: (1) solvingthe dynamic force-displacement relationship of a

vibrating foundation sitting on soil, i.e., the dynamicimpedance function of the foundation; (2) assembl-ing the foundation impedance function with theproperty matrices of the superstructure to formulatethe governing equations for the whole system.Therefore, a successful SSI analysis strongly de-pends on the application of accurate foundation im-pedance functions and their efficient incorporationwith the structural parameters to form a total systemmatrix.

Since most of the analysis complexity resultsfrom the frequency dependence of the foundationimpedance function of the foundation, many attemptshave been made to s implify SSI analysis by

682 Journal of the Chinese Institute of Engineers, Vol. 24, No. 6 (2001)

representing the continuous soil with frequency-in-dependent models. Early studies usually employedrepresentative constant values for the foundation stiff-ness and damping to approximate the soil system(Perelman et al., 1968; Parmelee et al., 1969; Jenningsand Bielak, 1973). While satisfactory for the analy-sis of some typical structures, this approximation maynot be adequate for other cases, such as concrete grav-ity dams (Chopra and Gutierrez, 1974). Several stud-ies (Veletsos and Nair, 1974a; Meek and Wolf, 1992;Meek and Wolf, 1993) were also devoted to develop-ing a truncated semi-infinite cone model for generalapplications in foundation vibration. Even thoughthose cone models can provide valuable insights andyield good accuracy, they are not popularly appliedin engineering practice due to their inability to rep-resent the influence of surface waves and to modelthe portion of soil outside the cone (Wolf, 1994). Itwas further shown that equivalence could be estab-lished between the cone models and certain discretephysical models (Veletsos and Nair, 1974b; Wolf,1994), which naturally lead to simple lumped-param-eter models.

The advantages of easy incorporation with con-ventional dynamic analysis and direct applicabilityto nonlinear structural analysis have continued tomotivate the recent development of several improvedlumped-parameter models (Wolf and Somaini, 1986;Nogami and Konagai, 1986; de Barros and Luco,1990; Jean et al., 1990; Wolf, 1991a; Wolf, 1991b;Paronesso and Wolf, 1995). These models basicallyare chosen by arranging a few sets of connectedsprings, dashpots and masses with unknownparameters, which are determined by minimizing thetotal square errors between the dynamic impedanceof the lumped-parameter model and that for the ac-tual system. While these discrete models have beendemonstrated to have good accuracy, the intricatearrangement of vibration units usually prevents easyand systematic extension of these models with highaccuracy. In addition, difficulties may also be en-countered in obtaining the property matrices associ-ated with these models.

Aimed to congregate the vibration units in amore systematic way for easy expansion, a set ofconsistent lumped-parameter models is developed inthis study to approximate vibrating foundationssitting on or embedded in an elastic half-space. Simi-lar configurations using mass-spring-dashpotoscillators are constructed for all four different vi-brating modes; horizontal, vertical, rocking and tor-sional directions. Moreover, a new expression forthe dynamic compliance function, instead of the con-ventional expression for the dynamic impedancefunction, is adopted in this research to further im-prove accuracy and reduce the parameters required

for the model. The effectiveness of these lumped-parameter models is evaluated by comparing the com-pliance functions for various cases of circular andsquare foundations.

II. EXPRESSIONS FOR FOUNDATIONIMPEDANCE FUNCTION

The dynamic impedance function of a rigid foun-dation supported by unbounded soil is defined as theratio between the exerted harmonic excitation force(or moment) and the resulting steady-state displace-ment (or rotation). The foundation mass is usuallyassumed to be zero and only its rigidity is consideredin this analysis for more general applicability of theimpedance function. The dynamic force and displace-ment are generally out of phase and consequently, thefoundation impedance is represented by a complex-valued function depending on the excitation frequencyω. The real part of the impedance function is deter-mined from the in-phase component of the force-dis-placement relationship and defines the elastic restrain-ing action of the soil medium. On the other hand, theimaginary part of the impedance function is decidedfrom the 90-degree-out-of-phase component of theforce-displacement relationship and describes the ef-fects of radiational energy dissipation.

1. Conventional Expression

For a concise presentation, the dynamic imped-ance function Kd(ω) is conventionally normalizedwith respect to its corresponding static impedance Ks

and expressed as

Kd(ω)=Ks[α(a0)+ia0β(a0)] (1)

where α (a0) and β(a0) represent the dimensionlessstiffness and damping coefficients, respectively. InEq. (1), a0 is a dimensionless frequency parameterdefined as

a 0 = ωdvs

(2)

to incorporate the excitation frequency ω, the shearvelocity of soil vs and a characteristic length of foun-dation d (e.g., the radius of a circular foundation orthe half side-length of a square foundation). FromEq. (1),α(a0) and β(a0) are commonly adopted in mostof the studies to express the real and imaginary partsof the impedance function, respectively.

2. Euler Expression

From previous works on the impedance

W.H. Wu and C.Y. Chen: Simple Lumped-Parameter Models of Foundation Using Mass-Spring-Dashpot Oscillators 683

functions of various foundations (e.g., Gazetas, 1983;Gazetas, 1993), it is evident that the dimensionlessstiffness and damping coefficients of the impedancefunction usually fluctuate with increasing values ofthe frequency parameter. This feature is the majorreason why numerous artificial degrees of freedomand parameters are required in most of the lumped-parameter models already developed. To overcomethis difficulty, an alternative formulation usingEuler’s formula to denote complex quantities wassuggested (Wu, 1998; Wu et al., 2000) for express-ing the dynamic impedance function:

Kd(a0)=Ks(ρeiθ) (3)

where i= – 1 and

ρ(a 0) = (α)2 + (a 0β)2 and θ(a 0) = tan– 1(a 0βα ) (4)

In other words, the normalized dynamic impedancefunction is expressed in terms of its amplitude andphase angle, instead of the real and imaginary parts.Therefore, ρ and θ are referred to as the amplitudeand the phase coefficients, respectively. The advan-tages of using ρ and θ over α and β to express theimpedance function primarily reside in the fact thatρ and θ are generally smoother functions of a0.Fig. 1 illustrates this tendency for the case of a circu-lar foundation resting on incompressible soils(Poisson’s ratio ν=0.5). Besides, it is also shown inFig. 1 that the stiffness coefficient α may becomenegative in this special case, which has caused prob-lems for a number of previously suggested models.

Fig. 1 Comparison of expressions for dynamic impedance functions (surface circular foundations, ν=0.5)


The amplitude and phase coefficients, on the otherhand, still hold their positive and monotonically in-creasing characteristics. With these properties, theerrors for the lumped-parameter model to fit the ac-tual impedance function of foundation can be con-siderably reduced.

3. Further Adapted Expression for the Definitionof Error Function

In addition to a smoother expression for thefoundation impedance, the definition of error func-tion in determining the optimal model parameters isalso a crucial constituent for further improving thelumped-parameter model. As long as the complex-valued impedance function is divided into two sepa-rate components (real and imaginary parts, or ampli-tude and phase angle), the errors respectively corre-sponding to these two components have to be com-bined in defining the total error function. Most ofthe previous studies directly added these two inde-pendent errors together without exploring the possi-bility of a more reasonable aggregation. In general,the two components of the complex-valued founda-tion impedance may not be of similar magnitude,which can also be observed from Fig. 1. Accordingly,the optimization process would intend to representthe component of larger magnitude with betteraccuracy.

In this paper, the expression for the impedancefunction is further modified to tackle this problem.It is suggested to synthesize the two individual com-ponents of dynamic impedance through parametricrepresentation in the complex plane using the dimen-sionless frequency a0 as the tracing parameter. More-over, since the amplitude of the foundation imped-ance function for unbounded soil generally increaseswith the frequency parameter, it is difficult to assurethe high-frequency convergence of the lumped-pa-rameter model to the actual soil system. Therefore,the foundation compliance function Fd (the recipro-cal function of foundation impedance), which ap-proaches zero in the high frequency limit, is adoptedin this research for defining the error function. Simi-lar to Eq. (3), Fd can also be normalized with respectto its corresponding static compliance Fs as

Fd(a 0) = 1K d(a 0)

= 1K s

[ 1ρ(a 0)e

iθ(a 0)] = Fs[

1ρ(a 0)

e – iθ(a 0)]

(5)

Eq. (5) shows that the amplitude of the normalizedcompliance function is the reciprocal of its corre-sponding normalized impedance function and thephase angle holds a different sign. This expression

is also illustrated in Fig. 1 for the impedance func-tions of a circular foundation.

III. SYSTEMATIC LUMPED-PARAMETERMODELS

The lumped-parameter model proposed in thisstudy to approximate the dynamic impedance of arigid foundation is illustrated in Fig. 2 for all fourvibrating directions. The coupled impedance betweenhorizontal and rocking vibrations, usually insignifi-cant, is neglected so that the lumped-parameter mod-els can be independently established correspondingto different modes of foundation vibration. A basicunit of this model is a single-degree-of-freedom(SDOF) oscillator, which is aggregated by attachinga spring and a dashpot in parallel to a mass. The massof this SDOF oscillator is used for modeling the in-ertial effect of soil, while the spring and the dashpotare employed to simulate the elastic behavior andenergy radiation of the soil, respectively. Depend-ing on the requirement of accuracy, a set of lumped-parameter models can be systematically constructedby linking one or several SDOF oscillators in seriesand then connecting to a rigid end. Each oscillator isspecified by three parameters (mass, damping andspring coefficient). Consequently, if the lumped-pa-rameter model consists of N SDOF oscillators, it willgenerally provide 3N parameters for minimizing thediscrepancy between the compliance function for themodel and that for the actual system.

For the lumped-parameter model illustrated inFig. 2, let zi denote the absolute displacement of thei-th SDOF oscillator and mi, ci and ki signify the cor-responding mass, damping and spring coefficient,respectively. When subjected to a single force ap-plied on mi, the relationship between displacement(z1) of the first oscillator and the applied force (p)should be able to accurately represent the correspond-ing relationship (u−p) for the actual system with amassless foundation sitting on or embedded in an elas-tic half-space. In this case, the equations of motionfor the lumped-parameter model including N SDOFoscillators are given by

m 1z 1 + c 1(z 1 – z 2) + k 1(z 1 – z 2) = p

m 2z 2 + c 2(z 2 – z 3) + k 2(z 2 – z 3) – c 1(z 1 – z 2)

– k 1(z 1 – z 2) = 0

m N – 1z N – 1 + cN – 1(z N – 1 – z N) + k N – 1(z N – 1 – z N)

– cN – 2(z N – 2 – z N – 1) – k N – 2(z N – 2 – z N – 1) = 0

m Nz N + cNz N + k Nz N – cN – 1(z N – 1 – z N)

– k N – 1(z N – 1 – z N) = 0

(6)


To insure that the compliance function of the foun-dation is exactly modelled in the limiting static case(ω=0), the spring coefficients of the SDOF oscilla-tors have to satisfy

1k 1

+ 1k 2

+ + 1k N

= 1K s

(7)

For convenience, all the spring coefficients of themodel are taken in this study as

k1=k2=...=kN=NKs (8)

Restrained by Eq. (8), Eq. (6) can be further

normalized and listed in matrix form as

Mz + Cz + Kz = 1(p

NK s) (9)

where

M = 1ω2

a 02γ1 0 0

0 a 02γ2

a 02γN – 1 0

0 0 a 02γN

;

Fig. 2 Lumped-parameter models for vibrating foundations on an elastic half-space


C = 1ω

a 0δ1 – a 0δ1 0 0

– a 0δ1 a 0(δ1 – a 0δ2

+ δ2)

0 – a 0δ2 0

a 0(δN – 2 – a 0δN – 1

+ δN – 1)

0 0 – a 0δN – 1 a 0(δN – 1

+ δN)

;

(10)

K =

1 – 1 0 0– 1 2 – 10 – 1 0

2 – 10 0 – 1 2

; z =

z 1z 2

z N – 1z N

;

1 =

10000

and

γi =m ivs

2

NK sd2

and δi =c iv s

NKsd(11)

are the dimensionless mass and damping parameters

of the i-th SDOF oscillator, respectively. It is im-plied in Eq. (10) that there remain totally 2N param-eters to be selected for the lumped-parameter modelin matching the frequency dependency of the actualfoundation compliance function.

Taking the Fourier transform of Eq. (9), thetransfer function from the displacement z1 of the firstoscillator to the applied force p (i.e., the compliancefunction of the lumped-parameter model) can besolved as

Fd(ω) = 1Kd(ω)

=Z 1(ω)P(ω)

= 1NKs

1T( – ω2M + iωC + K)– 11 (12)

where Kd (ω) denotes the dynamic impedance func-tion of the model and P(ω) and Z1(ω) represent theFourier transforms of p(t) and z1(t), respectively.Equation (12) can be further simplified as

Fd(ω)Fs

=Ks

Kd(ω)= 1

N1T( – ω2M + iωC + K)– 11

= 1N

1TH(ω) 1 =H 11(ω)

N(13)

where

H(ω) = ( – ω2M + iωC + K)– 1

=

1 – a 02γ1 + ia 0δ1 – 1 – a 0δ1 0 0

– 1 – ia 0δ1 2 – a 02γ2 + ia 0(δ1 + δ2)

0 0

– 1 – ia 0δN – 1

0 0 – 1 – ia 0δN – 1 2 – a 02γN + ia 0(δN – 1 + δN)

– 1

(14)

and H11(ω) denotes the element of the first row andthe first column in the matrix H(ω). The specificformulae for the normalized compliance function interms of the dimensionless parameters are listed inthe appendix for lumped-parameter models consist-ing of one, two and three SDOF oscillators. FromEq. (13) and the expression for H(ω) in Eq. (14), it iseasy to show that the normalized compliance

function for the model approaches zero in the limit-ing high-frequency case (a0→∞), which resembles thehigh-frequency behavior of the actual soil system.Combination of this high-frequency convergence andthe static convergence guaranteed by Eq. (8) auto-matically leads to doubly asymptotic convergence forthese lumped-parameter models. For correspondingto the normalized foundation compliance of the


actual soil system as expressed in Eq. (5), it isappropriate to reformulate Eq. (13) into

Fd(a 0)Fs

= 1ρ(a 0)

e – iθ(a 0) (15)

where

1ρ(a 0)

= Amp{H 11(a 0)

N} and – θ(a 0) = Ang{

H 11(a 0)N

}

(16)

and Amp{.} and Ang{.} signify the amplitude andthe phase angle of a complex-valued quantity,respectively.

Comparing the results from Eqs. (5) and (15),the optimization process is then applied to determinethe dimensionless parameters of the model throughminimizing the discrepancy between the normalizedcompliance function for the model and that for theactual system. With these dimensionless parametersγi’s and δi’s available, the correlative equation of Eq.(11)

m i =

NKsd2

vs2

γi and c i =NKsd

vsδi (17)

can be utilized to eventually decide the mass anddamping coefficient for each SDOF oscillator of thelumped-parameter model. It should be reiterated thatthe models derived in this section are applicable tohorizontal, vertical, rocking and torsional vibrationsof foundation. Consequently, force and displacementare generalized to indicate the corresponding momentand rotational angle in rocking and torsional founda-tion vibrations, whereas mass implies the correspond-ing rocking or torsional moment of inertia in thesetwo cases.

1. Definition of Error Function in DeterminingOptimal Parameters

For determining the dimensionless parametersof the model, a total error function has to be definedas the objective function used in the optimizationprocess. As deliberated in the previous section, thedirect addition of the errors from the two indepen-dent components of dynamic impedance may lead toa biased optimization process emphasizing the accu-racy of the component with larger magnitude. Adopt-ing the relative error for each component, instead ofthe absolute error, to compose the total error func-tion can lessen this problem. However, questions ofrationality in engineering applications may also beraised with this definition where equal weightings areenforced for the two separate components.

Using the modified expression proposed in thisstudy to entirely describe the compliance function inthe complex plane with diminishing high-frequencyamplitude, the deviation between the model and theactual soil system can be appropriately taken as cer-tain measures on the relative errors of their corre-sponding complex compliance values. Directly tak-ing the amplitude of the relative error in the complexplane, the total error function in this study is conse-quently defined as

Φ =

Fd(a 0j)

Fd(a 0j)– 1

2

Σj = 1

L=

K d(a 0j)

K d(a 0j)– 1

2

Σj = 1

L

=

ρ(a 0j)

ρ(a 0j)e i[θ(a 0j) – θ(a 0j)] – 1

2

Σj = 1

L

= L + {[

ρ(a 0j)

ρ(a 0j)]2Σ

j = 1

L– 2[

ρ(a 0j)

ρ(a 0j)]cos[θ(a 0j) – θ(a 0j)]}

(18)

where L symbolizes the selected number of differentfrequencies and a0j denotes the value of the dimen-sionless frequency parameter a0 corresponding to thej-th selected frequency. Since L is constant, the ob-jective function subject to minimization can be takenas

Φ = {[ρ(a 0j)

ρ(a 0j)]2 – 2[

ρ(a 0j)

ρ(a 0j)]cos[θ(a 0j) – θ(a 0j)]}Σ

j = 1

L(19)

Eq. (19) essentially provides an appropriate solutionto combine the two relative errors from the ampli-tude and phase coefficients with geometric rationality.Substituting Eq. (16) into Eq. (19), the objective func-tion Φ can be expressed as a nonlinear function of2N dimensionless parameters. Nonlinear optimiza-tion analysis is then applied to minimize this objec-tive function for the determination of these 2Nparameters.

IV. APPLICATIONS TO DIFFERENT TYPESOF FOUNDATION SYSTEMS

The effectiveness of these new consistentlumped-parameter models is first demonstrated in thissection by applying them to various cases of squareand circular foundations, all resting on or embeddedin an elastic half-space. By slightly modifying theelements in each SDOF oscillator, these models canalso be generalized for application in cases where thematerial damping of soil is considered.


1. Surface Square Foundations

Based on the values for the impedance functionsof a surface square foundation recently calculated byWong and Luco (2001) in the range of a0 from 0 to 5,the optimal parameters of various lumped-parametermodels are determined and listed in Table 1 for thesoil with Poisson’s ratio ν=1/3 and ν=0.45. For trans-lational (horizontal and vertical) vibrations, two mod-els consisting of one set of SDOF oscillators (N=1)and two sets of SDOF osci l la tors (N=2) areconstructed; while three different models including

one to three sets of SDOF oscillators (N=1~3) are es-tablished for the rotational (rocking and torsional)vibrations. In discussing the following two types offoundation systems, the same group of lumped-pa-rameter models will also be chosen for evaluation.Adopting the determined optimal parameters, the nor-malized compliance functions of the lumped-param-eter models are plotted in Figs. 3 and 4 to comparewith the corresponding values of the actual soilsystem. It is clearly observed that the lumped-pa-rameter model using two sets of SDOF oscillatorsalmost perfectly duplicates the actual foundation

Table 1 Optimal parameters for lumped-parameter model of surface square foundations

Poisson’s Horizontal Vertical Rocking TorsionalParameter

ratio N=1 N=2 N=1 N=2 N=1 N=2 N=3 N=1 N=2 N=3

γ1 .0038 .0026 .0233 -.0054 .0334 .0174 .0625 .0199 .0184 .0467δ1 .7014 .3385 1.0830 .5894 .3579 .1020 .2269 .2917 .0412 .1804γ2 . .2714 . .4653 . .1081 .0201 . .0892 .0230

v=1/3 δ2 . 1.0028 . 1.1623 . .3493 -.3228 . .3445 -.3038γ3 . . . . . . .1510 . . .1284δ3 . . . . . . .5743 . . .5055

γ1 -.0023 .0030 .1160 -.0134 .0537 .0259 .0689 .0198 .0182 .0471δ1 .6744 .3086 1.2153 1.0044 .3472 .1074 .2172 .2938 .0429 .1817γ2 . .2838 . .3253 . .1011 .0217 . .0899 .0232

v=0.45 δ2 . .9933 . .8887 . .3582 -.3340 . .3449 -.3047γ2 . . . . . . .1551 . . .1295δ3 . . . . . . .5890 . . .5077

Fig. 3 Comparison of compliance functions in the complex plane for a surface square foundation (ν=1/3)


compliance functions in the cases of translationalvibration. Moreover, even the degraded model usingonly one SDOF oscillator can practically representthe actual soil system in good accuracy. As for thecases of rotational vibration, the proposed models arenot as effective because of the more intricate behav-ior inherited in the corresponding foundation com-pliance functions. However, acceptable accuracy isstill illustrated for the model employing three sets ofSDOF oscillators in these cases.

2. Surface Circular Foundations

The case of circular foundations sitting on anelastic half-space is also evaluated. Numericalvalues for the vertical impedance functions presentedby Veletsos and Tang (1987) in the range of a0 from0 to 6 are used as the continuum values. For the cor-responding horizontal and rocking impedancefunctions, results listed by Veletsos and Wei (1971)in the range of a0 from 0 to 8 are adopted. In addition,

Fig. 4 Comparison of compliance functions in the complex plane for a surface square foundation (ν=0.45)

Table 2 Optimal parameters for lumped-parameter model of surface circular foundations



γ1 .0047 .0001 .0237 -.0019 .0189 .0078 .0427 .0115 .0069 .0260δ1 .6310 .3248 1.0063 .5380 .3552 .1446 -.2121 .2743 .0910 -.1584γ2 . .1865 . .3933 . .0914 .0058 . .0606 .0064

v=1/3 δ2 . .8474 . 1.0068 . .2952 .2074 . .2755 .1587γ3 . . . . . . .1101 . . .0814δ3 . . . . . . .4612 . . .3948

γ1 -.0002 .0006 .1605 .0826 .0425 .0238 .0445 .0115 .0069 .0260δ1 .5998 .2872 .8813 .4163 .2888 .0910 -.1904 .2743 .0910 -.1584γ2 . .2071 . .4546 . .0794 .057 . .0606 .0064

v=0.5 δ2 . .8364 . 1.1865 . .3239 .1908 . .2755 .1587γ2 . . . . . . .1106 . . .0814δ3 . . . . . . .4501 . . .3948


the comparison for the torsional impedance functionsis based on the values from Luco and Westmann(1971) in the range of a0 from 0 to 8. The optimalparameters of different lumped-parameter modelsare determined and listed in Table 2 for the soil

Fig. 5 Comparison of compliance fucntions in the complex plane for a surface circular foundation (ν=1/3)

Fig. 6 Comparison of compliance function sin the complex plane for a surface circular foundation (ν=0.5)

with Poisson’s ratio ν=1/3 and ν=0.5. Since thetorsional vibration of a circular foundation is anaxisymmetric problem, its corresponding impedancefunction is independent of the soil’s Poisson’s ratio.Applying these determined optimal parameters,


the normalized compliance functions of the lumped-parameter models are depicted in Figs. 5 and 6 tocompare with the values of the correspondingcontinuum system. Similar accuracy is observed forvarious lumped-parameter models as in the case ofsquare foundations. For translational foundationvibrations, the continuum values using the modifiedcompliance expression virtually follow smooth half-circle curves, which can be easily produced by thefour-parameter or even two-parameter model. Moreelaborate continuum values for rotational vibrations,on the other hand, requires a model with more pa-rameters to attain acceptable accuracy. This trend isparticularly evident in the case of incompressible soil(ν=0.5).

3. Embedded Square Foundations

The effectiveness of the lumped-parameter mo-del is further investigated for the case of square foun-dations embedded in an elastic half-space. The im-pedance functions reported by Mita and Luco (1989),which cover the values for a0 from 0.1 to 3, areadopted as the fitting basis. The ratio between theembedded depth h and the half side-width of founda-tion d is usually used to characterize the embedmenteffects for this type of foundation. The optimal pa-rameters of different lumped-parameter models aredetermined and listed in Tables 3 and 4 for the foun-dation with the embedment ratio h/d=0.5 and h/d=1.5 and for the soil with Poisson’s ratio ν=0.25 and

Table 4 Optimal parameters for lumped-parameter model of embedded square foundations (embeddedratio h/d=1.5)



γ1 .0360 -.0029 .0337 -.0096 .0624 .0370 .1802 .0551 .0474 .1237δ1 1.5296 .8265 1.2716 .9288 .6388 .2181 -.4527 .5319 .0521 .2889γ2 . .9380 . 1.2705 . .3274 .0257 . .2017 .0657

v=0.25 δ2 . 2.1130 . 2.1838 . .7544 .4532 . .5694 -.5058γ3 . . . . . . .5454 . . .3380δ3 . . . . . . 1.2113 . . .9033

γ1 .0936 -.0147 .0612 -.0033 .0742 .0358 .1725 .0569 .0493 .1229δ1 1.5488 1.0498 1.2817 .9261 .6300 .2288 -.4419 .4299 .0495 .2806γ2 . .5880 . 1.1170 . .3047 .0266 . .2013 .0681

v=0.4 δ2 . 1.8123 . 2.0284 . .7482 .4326 . .5714 -.5096γ2 . . . . . . .5116 . . .3358δ3 . . . . . . 1.1824 . . .8890

Table 3 Optimal parameters for lumped-parameter model of embedded square foundations (embeddedratio h/d=0.5)



γ1 .0071 .0056 .0337 -.0153 .0544 .0473 .1124 .0563 .0579 .0927δ1 1.0283 .5078 1.2716 .7306 .4092 .0336 .2619 .3411 -.0362 .2030γ2 . .5424 . .6116 . .1869 .0723 . .1513 .0996

v=0.25 δ2 . 1.5615 . 1.5839 . .5425 -.5076 . .4993 -.5249γ3 . . . . . . .3023 . . .2354δ3 . . . . . . .8572 . . .7891

γ1 .0069 .0071 .0612 -.0531 .0517 .0425 .1061 .0575 .0583 .0937δ1 .9931 .4796 1.2817 .9743 .3963 .0311 .2483 .3400 -.0382 .2068γ2 . .5325 . .4638 . .1758 .0766 . .1512 .0994

v=0.4 δ2 . 1.4958 . 1.2075 . .5195 -.5157 . .5013 -.5204γ2 . . . . . . .2820 . . .2372δ3 . . . . . . .8433 . . .7964


ν=0.4. Applying these determined optimal para-meters, the normalized compliance functions of thelumped-parameter models are displayed in Figs. 7 to10 to compare with the values of the correspondingcontinuum system. Similar accuracy is obtained forvarious lumped-parameter models as in the case of

surface foundations.

4. Nonnegative Parameters and Comparison withOther Models

It is noteworthy from Tables 1 to 4 that

Fig. 7 Comparison of compliance functions in the complex plane for an embedded square foundation (ν=0.25, h/d=0.5)



negative optimal values of the dimensionless param-eters may result and no physical realizations can beimplemented using these values. Similar circum-stances were found in some of the previously devel-oped lumped-parameter models (Jean et al., 1990;

Paronesso and Wolf, 1995). This is not surprisingconsidering that all the parameters of the lumped-parameter model are selected for minimizing theobjective function in a purely mathematical sense.Nevertheless, investigations also show that the

Fig. 9 Comparison of compliance in the complex plane for an embedded square foundation (ν=0.25, h/d=1.5)



accuracy of this model is barely affected even if allthe parameters are further restricted to nonnegative.Taking the case of a surface circular foundation sit-ting on the soil with ν=1/3 as an example, the nonne-gative optimal parameters can be determined to beγ1=0, δ1=0.5299, γ2=0.4063, and δ2=1.0154 for thevertical model using two SDOF oscillators and γ1=0.0079, δ1=0.0364, γ2=0.0331, δ2=0.0995, γ3=0.0934and δ3=0.3589 for the rocking model using threeSDOF oscillators. For demonstration, the lumped-parameter model adopting the original optimal param-eters as listed in Table 2 is compared in Fig. 11 withthe model using the nonnegative optimal parametersfor the vertical and rocking vibrations.

In addition, comparison with the other existinglumped-parameter models is also made in Fig. 12 forthe new model to illustrate its advantages. The modeldeveloped by de Barros and Luco (1990) using fiveparameters is taken to compare with the new modelemploying two SDOF osci l la tors ( i .e . , fourparameters) for the vertical compliance function of asurface circular foundation. It is clearly shown inpart (a) of Fig. 12 that the new model achieves betteraccuracy even with fewer parameters. As for the rock-ing compliance function of a surface circular

foundation, the model proposed by Jean et al. (1990)requiring ten parameters is adopted in part (b) of Fig.12 to compare with the new model employing fourSDOF oscillators (i.e., eight parameters). The newmodel performs equally well or even better in themedium to high frequency range while minor ineffi-ciency can be observed in the low frequency range.

5. Consideration of Material Damping for Soil

In the above applications, the soil is assumed tobe a perfectly elastic medium where the energy ofvibration is dissipated only through radiation of wavestoward infinity. To be more realistic, the materialdamping commonly existing in soil should be alsoincluded in the foundation vibration analysis. By ap-plication of the correspondence principle, the dynamicimpedance functions for the cases considering soildamping can be computed from their correspondingfunctions for the elastic foundation. Rather than de-termining independent lumped-parameter models torepresent the impedance functions of viscoelasticfoundations, however, a more convenient way to con-struct the models for the viscoelastic foundation hasbeen developed by Meek and Wolf (1994) by directly

Fig. 12 Comparison of different lumped-parameter models for a surface circular foundation (ν=1/3)

Fig. 11 Comparison of the lumped-parameter models using the original and nonnegative optimal parameters for a surface circular founda-tion (ν=1/3)


applying the correspondence principle to the discreteelements of the lumped-parameter models corre-sponding to the elastic foundation. It was shown thatthe Voigt viscoelasticity can be introduced throughaugmenting each original spring in the lumped-pa-rameter model by a dashpot and each original dash-pot by a mass, attached in a special way. Besides,more realistic nonlinear-hysteretic damping can alsobe characterized by replacing the augmenting dash-pots and masses by frictional elements. Since the dis-crete elements of the new lumped-parameter modelproposed in this study are no more than springs,dashpots, and masses, similar augmentation canbe made to generalize the new model for its applica-tion in the foundation system incorporating soildamping.

V. CONCLUSIONS

Simple lumped-parameter models are developedin this study to generally represent the horizontal,vertical, rocking and torsional vibrations of rigidfoundations. It is demonstrated for various surfaceand embedded foundations that these models can ef-fectively and efficiently replace the frequency-depen-dent foundation-soil system. Compared with the otherlumped-parameter models to achieve the same degreeof accuracy, the models proposed in this paper havethe advantages in requiring fewer parameters andfeaturing a more systematic expansion. Furthermore,it is also shown that the effectiveness of these mod-els barely deteriorates for nearly incompressible soilwith Poisson’s ratio close to 0.5, which has causeddifficulties for a number of previously suggestedmodels. As for their applications in different vibra-tion modes, the performance of the new models isbetter in approximating the translational foundationvibrations than in the cases of rotational foundationvibration.

ACKNOWLEDGEMENT

This study was supported by the National Sci-ence Council, Republic of China, under Grant NSC89-2211-E-224-003. This support is great lyappreciated.

NOMENCLATURE

a0 dimensionless frequency parameter (=ωd/vs) C normalized damping matrix of the lumped-pa-

rameter model, [T]ci damping coeff ic ient of the i - th SDOF

oscillator, [M/T or ML2/T]d characteristic length of foundation [L]Fd dynamic compliance function of foundation,

[T2/M or T2/ML2] Fd dynamic compliance function of the lumped-

parameter model, [T2/M or T2/ML2]Fs static compliance function of foundation,

[T2/M or T2/ML2]H transfer matrix of the lumped-parameter model

(=(−ω2 M +iω C + K )−1))H11 element of the first row and the first column in

Hh embedded depth of foundation, [L]

K normalized stiffness matrix of the lumped-pa-rameter model

Kd dynamic impedance function of foundation,[M/T2 or ML2/T2]

Kd dynamic impedance function of the lumped-pa-rameter model, [M/T2 or ML2/T2]

Ks static impedance of foundation, [M/T2 or ML2/T2]

ki spring coefficient of the i-th SDOF oscillator,[M/T2 or ML2/T2]

M normalized mass matrix of the lumped-param-eter model, [T2]

mi mass or area moment of inertia of the i-thSDOF oscillator, [M or ML2]

P Fourier transform of p, [M/T2 or ML2/T]p applied force or moment, [ML/T2 or ML2/T2]vs shear velocity of soil, [L/T]ω excitation frequency, [1/T]Zi Fourier transform of zi, [LT]zi absolute displacement of the i-th SDOF

oscillator, [L]z absolute displacement vector of the SDOF

oscillators, [L]α dimensionless stiffness coefficientβ dimensionless damping coefficientγi dimensionless mass parameter of the i-th SDOF

oscillator (= m iv s

2

NKsd2

)

δi dimensionless damping parameter of the i-th

SDOF oscillator (= c iv s

NKsd)

θ dimensionless phase coefficient of foundationθ dimensionless phase coefficient of the lumped-

parameter modelν Poisson’s ratio of soilρ dimensionless amplitude coefficientρ dimensionless amplitude coefficient of the

lumped-parameter modelΦ total error function

REFERENCES

1. Chopra, A. K., and Gutierrez, J. A., 1974, “Earth-quake Response Analysis of Multistorey Build-ings Including Foundation Interaction,” Earth-quake Engineering and Structural Dynamics, Vol.


3, No.1, pp. 65-77. 2. de Barros, F. C. P., and Luco, J. E., 1990, “Dis-

crete Models for Vertical Vibrations of Surfaceand Embedded Foundations,” Earthquake Engi-neering and Structural Dynamics, Vol. 19, No.2, pp. 289-303.

3. Gazetas, G., 1983, “Analysis of Machine Foun-dation Vibrations: State of the Art,” Soil Dynam-ics and Earthquake Engineering, Vol. 2, No. 1,pp. 2-42.

4. Gazetas, G., 1991, “Formulas and Charts forImpedances of Surface and Embedded Founda-tions,” Journal of Geotechnical Engineering,ASCE, Vol. 117, No. 9, pp. 1363-1381.

5. Jean, W. Y., Lin, T. W., and Penzien, J., 1990,“System Parameters of Soil Foundations for TimeDomain Dynamic Analysis,” Earthquake Engi-neering and Structural Dynamics, Vol. 19, No.2, pp. 541-553.

6. Jennings, P. C., and Bielak, J., 1973, “Dynamicsof Building Soil Interaction,” Bulletin of the Seis-mological Society of America, Vol. 63, No. 1, pp.9-48.

7. Luco, J. E., and Westmann, R. A., 1971, “Dy-namic Response of Circular Footings,” Journalof Engineering Mechanics, ASCE, Vol. 97, No.EM5, pp. 1381-1395.

8. Meek, J. W., and Wolf, J. P., 1992, “Cone Mod-e l s fo r Homogeneous So i l , ” Journa l o fGeotechnical Engineering, ASCE, Vol. 118, No.5, pp. 667-685.

9. Meek, J. W., and Wolf, J. P., 1993, “Cone Mod-els for Nearly Incompressible Soil,” EarthquakeEngineering and Structural Dynamics, Vol. 22,No. 5, pp. 649-663.

10. Meek, J. W., and Wolf, J. P., 1994, “MaterialDamping for Lumped-parameter Models ofFoundations,” Earthquake Engineering andStructural Dynamics, Vol. 23, No. 4, pp. 349-362.

11. Mita, A., and Luco, J. E., 1989, “Impedance Func-tions and Input Motions for Embedded SquareFoundations,” Journal of Geotechnical Engineer-ing, ASCE, Vol. 115, No. 4, pp. 491-503.

12. Nogami, T., and Konagai, K., 1986, “Time Do-main Axial Response of Dynamically LoadedSingle Piles,” Journal of Engineering Mechanics,ASCE, Vol. 112, No. 11, pp. 1241-1249.

13. Parmelee, R. A., Perelman, D. S., and Lee, S. L.,1969, “Seismic Response of Multiple-story Struc-tures on Flexible Foundations,” Bulletin of theSeismological Society of America, Vol. 59, No.7, pp. 1061-1070.

14. Paronesso, A., and Wolf, J. P., 1995, “GlobalLumped-parameter Model with Physical Repre-sentation for Unbounded Medium,” EarthquakeEngineering and Structural Dynamics, Vol. 24,

No. 5, pp. 637-654.15. Perelman, D. S., Parmelee, R. A., and Lee, S. L.,

1968, “Seismic Response of Single-story Inter-action Systems,” Journal of Structural Division,ASCE, Vol. 94, No. 12, pp. 2597-2608.

16. Veletsos, A. S., and Nair, D. V. V., 1974a, “Tor-sional Vibration of Viscoelastic Foundations,”Journal of Geotechnical Engineering, ASCE, Vol.100, No. GT3, pp. 225-246.

17. Veletsos, A. S., and Nair, D. V. V., 1974b, “Re-sponse of Torsionally Excited Foundations,”Journal of Geotechnical Engineering, ASCE, Vol.100, No. GT4, pp. 476-482.

18. Veletsos, A. S., and Tang, Y., 1987, “VerticalVibration of Ring Foundations,” Earthquake En-gineering and Structural Dynamics, Vol. 15, No.1, pp. 1-21.

19. Veletsos, A. S., and Wei, Y. T., 1971, “Lateraland Rocking Vibration of Footings,” Journal ofSoil Mechanics and Foundation Division, ASCE,Vol. 97, No. SM9, pp. 1227-1248.

20. Wolf, J. P., 1991a, “Consistent Lumped-param-eter Models for Unbounded Soil: Physical Re-presentation,” Earthquake Engineering and Struc-tural Dynamics, Vol. 20, No. 1, pp. 11-32.

21. Wolf, J. P., 1991b, “Consistent Lumped-param-eter Models for Unbounded Soil: Frequency-independent Stiffness, Damping and MassMatrices,” Earthquake Engineering and Struc-tural Dynamics, Vol. 20, No. 1, pp. 33-41.

22. Wolf, J. P., 1994, Foundation Vibration AnalysisUsing Simple Physical Models, Prentice-Hall,Englewood Cliffs, NJ.

23. Wolf, J. P., and Somaini, D. R., 1986, “Approxi-mate Dynamic Model of Embedded Foundationin Time Domain,” Earthquake Engineering andStructural Dynamics, Vol. 14, No. 5, pp. 683-703.

24. Wong, H. L., and Luco, J. E., “Impedance Func-tions for Rectangular Foundations on a Viscoelas-tic Half-space,” submitted to Journal of Geote-chnical and Environmental Engineering, ASCE.

25. Wu, W. H., 1998, “A Uniform Representation forDynamic Impedance Functions of RectangularFoundations Incorporation Different AspectRatios,” Journal of Science and Technology, Vol.7, No. 3, pp. 259-270.

26. Wu, W. H., Ke, T. C., and Kao, H. Y., 2000,“Effects of Foundation Flexibility on DynamicImpedance Funct ions - Two-dimensionalAnalysis,” Journal of the Chinese Institute ofCivil and Hydraulic Engineering, Vol. 12, No. 3,pp. 475-485.

APPENDIX

For the lumped-parameter models consisting of


different number of SDOF oscillators, the correspond-ing normalized compliance functions can be ex-pressed in terms of the dimensionless parameters asthe following:(a) the model including one SDOF oscillator

�� !"#$%&'()*+,-./012�� !

�� !"#$

�� !"#$%&'(

��

�� !"#$%&'()*+,-./01+23456789:8;

�� !"#$%&'()*+,-./0123456789:;*<=

�� !"#$%& '()*+,-./0�123456789:8(;

�� !"#$%&'()*+,-./0123456789:;2<=>

�� !"#$%&'()*+,'-./01234567894:*+,

�� !"#$%&� '()*+,-./012345�6789#:;

�� !"#$%&'()*+

�� !"#$%&'()*+,-.(/0123,4

Fd(a 0)Fs

=1 – a 0

2γ1 – i(a 0δ1)

1 + a 02(δ1

2 – 2γ1) + a 04γ1

2

(b) the model including two SDOF oscillators

Fd(a 0)Fs

=( – 2 + a 0

2γ2) – ia 0(δ1 + δ2)

– 2{[1 – a 02(2γ1 + γ2 + δ1δ2) + a 0

4γ1γ2] + i[a 0(δ1 + δ2) – a 03(γ1δ1 + γ1δ2 + γ2δ1)]}

(c) the model including three SDOF oscillators

Fd(a 0)Fs

= ND

where

N = [3 – a 02(2γ3 + 2γ2 + δ1δ2 + δ1δ3 + δ2δ3) + a 0

4γ2γ3]

+ i{2a 0(δ1 + δ2 + δ3) – a 03[γ2(δ2 + δ3) + γ3(δ1 + δ2)]}

and

D = – 3{ – 1 + a 02[3γ1 + 2γ2 + γ3 + δ1(δ2 + δ3) + δ2δ3]

– a 04[δ1δ2(γ1 + γ2 + γ3) + δ1δ3(γ1 + γ2) + δ2δ3γ1

+ 2γ1(γ2 + γ3) + γ2γ3] + a 06γ1γ2γ3} – 3i{ – a 0(δ1 + δ2 + δ3)

+ a 03[2γ1(δ1 + δ2 + δ3) + γ2(2δ1 + δ2 + δ3)

+ γ3(δ1 + δ2) + δ1δ2δ3] – a 05[γ1γ2(δ2 + δ3)

+ γ1γ3(δ1 + δ2) + γ2γ3δ1]}

Manuscript Received: Nov. 27, 2000Revision Received: Jan. 09, 2001

and Accepted: May 11, 2001

Simple lumped‐parameter models of foundation using mass‐spring‐dashpot oscillators

Documents