Approximate formulas for dynamic stiffnesses of rigid foundations

Artur Pais and Eduardo Kausel

Dept. of Civil Eirgirreerirlg, Roont 1-271, MIT, Caiilbririge, M A 02139, USA

Approximate formulas are proposed to describe the variation with frequency of the dynamic stiffnesses of rigid embedded foundations. These formulas are obtained by fitting mathematical expressions to accurate numerical solutions. Because of the restricted data available at the present time, only cylindrical and rectangular embcdded foundations are analysed herein; this is not a serious restriction, since these are the more common shapes used in practice. The imaginary part of the stiffnesses are approximated, for high frequencies, by their asymptotic values, which give excellent results in that range. These asymptotic values are computed assuming simple one- dimensional wave propagation theory. The approximate formulas provide a good approximation of the foundation stiffnesses and their use is very simple. Although the soil is assumed to have no internal damping, it can be incorporated by using the Corrcspondcnce Principle, if so desired.

INTRODUCTION

The first step in the study of a soil-structure interaction problem is the evaluation of the dynamic st ihess matrix of the foundation. Of special interest is the case in which the soil is much softer than the foundation; it can be assumed then that the foundation keeps its shape while vibrating, so that six components (three displacements and three rotations) are suficient to describe its motion. The dynamic stiffness matrix has then only six columns and rows.

T o find the dynamic stiffness functions, a mixed boundary-value problem must be solved, in which displacements are prescribed at the contact area between the foundation and the soil, and tractions vanish at the free surface of the soil. Since this problem is rather diflicult, it is not surprising that analytical solutions are available for only very special cases. Luco et al."." give the compliance functions for a disk foundation on an elastic halfspace, assuming frictionless contact, and for a strip foundation bonded to an elastic halfspace. Actual foundations, on the other hand, are usually embedded in the soil and have variegated shapes. To find the dynamic stiffness functions in these cases, one must use numerical procedures such as the finite element or the boundary integral methods.

While vibrating, the foundation generates waves that radiate through the soil a certain amount of energy. This introduces some damping in the motion of the foundation, which is usually referred to as radiation (or geometric) damping. To take into account this phenomenon in numerical solutions with finite elements, the soil model must include a vast region beyond the foundation. Such a large soil island however, is not needed when the model includes transmitting boundaries8.14 that reproduce the physical behaviour of

Accepted Scptcmbcr 1987. Discussion closcs Decen~brr 1988.

Q 1988 Computational Mechanics Publications

the infinite system, and which can be applied directly at the edge of the foundation. However, these boundaries are usually based on idealizations of the soil as finite strata supported by rigid rock so that any radiation into bedrock, which may be prcscnt in an elastic halfspacc (or in a very deep alluvia) is neglected in such models.

To avoid this problem, Day3 performed transient finite element analyscs for impulsive motions of an embedded cylindrical foundation, obtaining afterwards the dynamic stiffness as functions of the frequency by performing a Fourier transform of the truncated impluse response function, i.e., eliminating the reflections from the boundary. This procedure cannot be applied to layered soils, howevcr, since it is not possible to distinguish between real reflections at the interfaces of the layers and the spurious reflection at the boundary. Apse12, on the other hand, used an integral equation formulation for the

213 Soil Dyrlnrt~ics a r d Enrthq~rnke Eryirlccrirly, 1988, Vol. 7 , N o . 4

Approritmte foiwrrtlas for- ~lytlarnic stiJtlesses of rigid folrrrdatio~~s: A. Pais a d E. Knlrsel

4 P- >< A+9scl and 01, results for m t k l n g

8 b s c l and Cay results f o r hor(rmtl1 met

- aDPTOllrdiC fOI741aS 3 5 .

zo -

I5 -

Fig. 2. Variotiort of the static stifltess with the et~tbedntest (rocking arid horizottrnl modes)

static

1

Concerning rectangular foundations, Wong et d . l 5

presented compliance functions for flat foundations for several length-to-width ratios, which were obtained by dividing the contact area between the foundation and the soil into sub-regions in which a constant stress was assumed. Using the same method, Wong and Luco16 presented tables of impedance functions for flat rectangular foundations. Dominguez4 on the other hand, applied the boundary integral formulation to compute the stiffnesses of rectangular embedded foundations, examining a large number of aspect ratios in the low frequency range. Abascall, using a similar approach, presented the stiffnesses of a square embedded foundation.

(a) Static sriJhesses The relationships between forces and displacements for

a rigid disk bonded to a homogeneous elastic halfspace were investigated long ago, and the following explicit

Fig. 4. Va~iation with the freqrtetzcy of tlre octrral st@ress (real part)

20.

15.

1m(+ I GR

f i n i t e element soluticn-.-.-'-

boundary integral sotu - -------

finite element s o l u t i o n ---.--- boundary integral so lu t i on -..... approximte f o m l a -

appmxiaate formula

Fig. 3. Variatiorl of the stofic stiJtzess with the ernbedtirent (torsi011 and certical nlodes)

0

same problem and found a very good agreement with the a f l R 0'4

results of Day (see Figs 4-13). It must be added that Aspel assumed a small amount of internal damping in the soil, Fig. 5. Vnrintioa with tlrc frfq~rerrcy of the ~er t icn f while Day assumed an elastic medium. stiff,less (btnginory part)

Soil Dptmr?tics ard Earthqlrake Et~girleerirrg, 1988, Vol. 7, No. 4 214

Approxir~~nte @r~nrrlas for dyirorrric stiSfr~esses of rigid fo~rr~tlntiur~s: A. Pais arid E. Korrsel

finite clement solut4on boundary integral s o l u t i a n ------ approximate formula -

Fig. 6. .Varintiori with the frequeocy of the horizoi~tol sti,@~e.ss (red part)

Fig. 7. hriotiorl with the j k p e i ~ c y oJ the llorizor~tal stiffrless (ir~roginary port)

formulas for the stiffnesses were found:

Vertical KO--- v - 4GR (8oussinesq) 1-v

8CR Horizontal Ki=- (Mindlin, 1949)

2- 1.

8CR3 Rocking Kg=------ (Borowicka, 1943)

3(1 - 11)

Torsion KP=- 16GR3 (Reismer, 1944) 3

These formulas represent the static stiffness of a rigid circular foundation of radius R, with G and Y being the shear modulus and ~oisson's ratio of the homogeneous halfspace.

Fig. 8. Vcrriatiorl with the frequerlcy of the rocking stiflress (real part)

f i n i t e r l m c n t salution -.-.-.- bundary lntcpral solution ------- a:Pmxinate f cmula -

Fig. 9. Variatioi~ with the freqlrency of 1/11? rockirlg stiJ%ess (irtlaginory port)

215 Soil Dyrtomics orld Eortllqrrakc Erlgirleerii~g, 1988, Vol. 7, No. 4

Approxiniote for~lnrlas for dplantic stifllesses of rigid fi~rnclatiorn: A. Pais and E. Kalrsel

program was used, and the result were corrected for discretization errors. Fornlulas were developed for a maximum embedment of one and a half times the radius of the foundation.

Fig. 12. Variotiorl with the fieqrrer~cy of the colrplirlg s@ms (real part)

1 5 .

Fig. 10. firiotion with the freqrrerlcy of the torsiorlnl stif~tess ( red part)

f i n t t e e l a r n t solution ---.-. boundary integral so lut ion a p p m x i m t c formula -

f i n i t e elwent solutton -.-.-.- bounaary Inte;ral solution------

a:amxlmte f o m l a -

I LO .. CID . 2.n

, --*- I .\.wsc e-z2:z-z~----..

.-..

Fig. 11. Pi3riatior1 with the freqlretlcy of the torsior~al st$m-s (irnagitlarg part)

Concerning embedded cylindrical foundations, closed form solutions are not available, but approximate formulas have been developed from numerical solutions. Approximations for horizontal, rocking and coupling stiffnesses were developed by Elsabees, while the vertical and torsional modes were analysed by Kausel and Ushijima9. In both studies, a finite element computer

. . - - - - -. EIR -0.5 0 >

0 1 2 3 4 5 6 7 na

1 m ( 6 ) , I G R ~

Fig. 13. hriat ios with the freq~rettcj~ of the co~rplirlg stiffiless (iltlngirzury part)

- - 20 -

.

15

Fig. 14. Rectatzgdar ernheddrd fn~rrrdotiotz ( L 2 B )

f l n i t e e l c e n t so lut ion ' ---- boundary integral solution---- appmxirate f o m ~ l a -

.+---sy2.s7=: : :~~ .~~ .~- - - - - - - -.-.-,-,-, ------ -.-.-._._._ E l R - 2.0 -

Soil Dynmtics arul Earthqlrake Engineering, 1988, Vol. 7, No. 4 216

I . . . , . 1 2 3 4 5 10 20

LIB

Fig. 15. ' Variatior~ of the certical static st@less with the shape of the folrildatio~l

Numerical solutions for the dynamic stiffnesses of cylindrical embedded foundations were also prepared by AspelZ and by Day3, as described earlier. It should be emphasized once more that the former was computed with a small amount of internal damping, whereas the latter is fully elastic. These solutions are also used in this paper to derive the approximate solutions.

Extrapolating tl~ecurves in Figs 4 through 13, thestatic value of the impedances can be extracted (see Figs 2,3) by taking the average of the finite element and boundary element results. Figs 2 and 3 show these static stiffnesses as a function of the embedment ratio; the range considered extends from a surface foundation (E/R = 0.0) to a foundation with an embedment equal to the diameter (EJR = 2.0). It is reasonable to assume next that the effect of Poisson's ratio on the static stiffnesses is the same for a surface foundation as it is for an embedded foundation; this equivalent to the assumption that the ratio of the two is independent of v. In such case, one can seek polynomial approxi~nations to account for embedment that depend on-the embedment parameters only.

For the torsional, vertical and horizontal modes, a linear approximation is sufficient, giving less that 10% crror when compared to the numerical results. A power law with theexponent less than 1 would fit the data better, but for simplicity, the linear formula was preferred. For the rocking mode, on the other hand, a third degree polynomial was chosen because a straight line would give too much error. This has some physical justification if one remembers that the area moment of inertia of a cylinder about a horizontal axis is also a third degree polynomial in EJR. For the coupling term, the formula proposed by Elsabec was adopted. The resulting approximations are then as follows:

Vertical K;. = KF (1 + 0.54EIR) (2a)

Horizontal KZ = K; (1 -i- EJR) (2b)

Rocking K i =Kg (1 + 2 . 3 ~ / ~ + 0 . 5 8 ( ~ / ~ ) ~ )

( 2 ~ )

Torsion KT= K; (I +2.67E/R) ( 2 4

Coupling K',,, = (0.4EIR - 0.03)K;r (24

with K ? . . . etc. being given by equations (la) through (Id).

The coefficient in the approximation for the torsional stiffness is the same as in thc work by Kausel et aL9, while the vertical stiffness has been changed somewhat so as to extend its range of validity.

(b) Dyilai~lic stifJi~esses Some simple formulas are also proposed to describe the

variation of the stiffnesses with frequency of vibration. The dependence on frequency is given by a complex number that multiplies the static stiffness as follows:

Kd = ~ ' ( k + ioac) (3)

where Kqesignates the appropriate static stiffness; no is the dimensionless frequency a , = wR/C, (w = angular frequency of the motion, R=radius of the foundation, C, = shear wave velocity in the soil); k and c are functions of no, v=Poisson's ratio, and EJR=the degree of embedment,

Although k in the vertical and rocking modes depends strongly on the value of v, for simplicity its influence is not taken into account herein. This implies that for values of higher than about 0.4 the approximate formulas must be used with care, especially for high frequencies.

1 2 3 4 5 6 7 8 9 1 0 L I B

Fig. 16. Voriatioa of the l~orizontal stntic stiflitess wit11 the shape of the folrildotioi~

30-

217 Soil Dyiiainics and Enrt/iq~rake Ei~gi i~ecri i~g, 196'8, Vol. 7 , N o . 4

a Resul ts fro2 Vong and Luco

@ Resul ts f r o m Ooninguer

* Resul ts f m n Gorbunov-Posanav

Approxir11nte forrlllrlas for ~ ~ I I N I I I ~ C stiflr~esses of rigid jbrr~rlatioru: A. Pnis nrul E. Krrlrsel

30-

20-

@ Reru l ts frcm Dorninguez

a Resu1:s f r x Gor:unov-Posanov

Resu l ts f r o a W n g and L u c ~ 10. - A p r ~ x i - a t e f c w u l a

Fig. 17. kriatiorz of the rockirlg static sti~'J1ess with the shape of the fo~rrztlnriort

Nevertheless, in the high frcqucncy range, the imaginary part of the stiffness is much more important than the real part; hence, the approximations may be appropriate for engineering purposes.

The asymptotic value of the coefficient c can be found easily by assuming that the vibrating foundation generates unidirectional waves propagating perpendi- cularly to the contact surface with the soil (see Gazetas and Dobry7 for further details). Denoting this value by c, it is given by the formula

where T is a transformation matrix that is used to relate the displacements along the soil-foundation interfaces to the global motion of the foundation; and v is the celerity (velocity) of the waves generated.

This equation is similar to the formula that one would use for the calculation of the total area or the moments of inertia of the contact area, except that a weight (v/C,) is used to account for the type of waves generated.

The dominant type of waves will be longitudinal waves (L-waves) if the motion is normal to the contact surface, or shear waves (S-waves), if the motion is tangential to the surface. O n account of this fact, one can split equation (4) into two parts, the first of which takes account ofS-waves generated, while the second reflects thecontribution of L- waves:

regarding appropriate values for C,, since planc-strain conditions do not hold in the vicinity of the foundation. As a result, lateral dilation takes place which causes the value of C, to be lower than the theoretical value for P- - waves (C,= C , J2(1- v)/(l- 21.)). The discrepancy between C, and C , is particularly important for incompressible solids (r1=0.5), for which the P-wave velocity is infinity, while theeffcctive radiation velocity C , is finitk. To circumvent this difficulty, one can ehher approximate C , z C , and limit the value of r used in deriving C,, (for example, u < 2.5), or following Gazetas and Dobry7, one defines C, to be a fictitious velocity for longitudinal waves. With these approximations, the dynamic stiffness coeficients can be computed for all modes of vibration. The results are summarized in Table 1.

These formulas are similar to the ones proposed by Veletsos et al." for the case of a surface foundation, and Kausel er d9 for an embedded foundation. Figs 4-13 show a comparison of the proposed formulas with the results of Day3 and Aspelz, (taking ~ 5 0 . 2 5 ) . Because Apse1 assumed a small amount of material damping in the soil, the imaginary term of the stiffness (Ii11(K~(a~))) reported by him tends to infinity as the frequency approaches zero; nevertheless, for higher frequencies the influence of the damping can be neglected. As can be seen, the proposed formulas match well the numerical results, especially the imaginary part. For the vertical, horizontal and coupling modes, the real part of the stiffnesses evidences discrepancies in the high frequency range. However, due to the lack of reliable data, improved approximations are not warranted. The value of c for the

K= RY L60(

140C

120C

1000

800

600

LOO

200

Gi Resu l ts frm Uong and Luco

Q Resu l ts from Doainguez

with u=CL/Cs. An important question arises at this point Fig. 18. Ihriatioiz of tlle rockirlg stotic stiffi~ess with tile slznpe of tlle folrndntior~

Soil Dynnnzics m r l E~rthqtrakc Er~girlcerir~g, 1988, 161. 7 , N o . 4 218

Approsillrate forrirtrlas for dpi~oriric stiffiiesses of rigid folrrlriritiorls: A. Prris a i d E . Knirsel

rocking stiffness is different from zero in the static case, in part because the centre of coordinates was chosen to be at the base instead of at the ccntrc of stiffness. Hence, a certain degree of translation results from the rotation.

Eritberldetl rectarrgirlar fo~rrttiatior~s

( a ) Static stiffi~esses of srrrface folrrrdatiorts In the case of rectangular foundations, the lack of

cylindrical symmetry increases substantially the difiiculty of the problem so that rigorous analytical results are not

Table 1A.

Vertical Torsion

available, not even for surface foundations. In addition, the ratio of length to width, LIB (see Fig. 14), which defines the gcometry of the foundation, is another parameter that must be taken into consideration. When the foundation is very long, its stiffnesses in the short direction approaches the stiffnesses of a strip foundation (2-D problem).

Table 2 shows the static stiffnesses of a square foundation found by several authors, but scaled by the factor giving the dependence on Poisson's ratio for circular footings. The underlying assumption is that the dcpendcnce on Poisson's ratio is the same for rectangular and circular foundations. If this is true, then the results in the table are independent of Poisson's ratio. Judging by the numbers in columns four and five in this table, this assumption appears to be reasonable.

The values in this table match each other reasonably well, except for the rocking and torsional modes, where Dominguez's results seem too low. The values chosen for the static stiffnesses are displayed in the last column (based mainly on Wong and Luco's results). The coupling stiffness has been neglected because its value is small for a surface foundation.

Figs 15 through 19 show the static stiffnesses of rectangular foundations in terms of the aspect ration LIB, and for Poisson's ratio 1- = 113. These figures are based on

7irble IB.

Horizontal Rocking

2 with b=- 1 + EIR

Tnble 2.

Dominguezi Wong and Luco16 Abascal (1) (4 (b) v = 113 v = 0.45 Value taken

@,(I -r)

GB

K i ( 1 - v )

G B 3

KP GB'

K h - GB2

(a) Relaxcd boundary conditions (b) Nonrelaxed houndary conditions

219 Soil L ) J ' I I N ~ ~ I ~ C S mid Eorthqtrnke Elrgiiwerir~g, 1988, Vol. 7, NO. 4

Approxiimte forrrtlrlas jbr rlyrirrr~~ic stifltesses of rigid foirirtlntiorts: A. Pois arid E. Kamel

O R e s u l t s f rom Coninpuez

a Resu1:s f r o 3 Gong and Luco

- -p;rcxica:e f o n u l a

Fig. 19. Vnrintioi~ of the torsiorid stoticst@iess with the slinpe of the fo~rizdutioit

the data presented by Wong et d l 6 , Dominguez4, and Gorbunov-Posanov (from Ref. 6). Use of these figures led to the following approximations (with L I B 2 1):

Vertical

Horizontal K'x(2-v) = 6.8(L/B)'," + 2.4 (18) G B

Rocking

Torsion -- KP - 4.25(L/B)2.45 + 4.06 (22) GB3

The exponent of (LIB) is less than 1 for the vertical and horizontal modes, equal to 1 for rocking around the longitudinal axis, and greater than 1 for torsion and for rocking around a transverse axis. These values approach

the stiffnesses of a strip foundation as the length/width ratio increases.

Table 3 shows a numerical comparison of the stiffnesses found by Wong et nl.16 (v= 1/3), ~ o m i n g u e z ~ , and the formulas proposed, for 1 $LIB d 4. As can be seen, the agreement is very good, the largest diffferences being less than a few percent.

(b) Dyrmrlic stiflresses of strrfoce folrr~datior~s To describe the variation of the stiffnesses with

frequency, the results by Wong and Luco16 were used as reference, since they are available for a reasonably extended range of frequencies; their plots are shown in Figs 20-33 (solid line). The shape of these plots is quite

-.--. Acprox imate f o r n u l a

- Yong and L u : ~

Fig. 20. Voriatiorl of the stifliess wit11 frequeitcy; strrfnce fotrndation L/B= 1 (ccrticol artd 11orii011t~l nlodes)

4 1

- Kong and LUCO

Fig. 2 hrirrtioii oJ the st~fiiess with freqrre~rcy; sirrfi7ce folrr~dntiort L/B= 2 (horizoirtnl rnode)

3 .

2 -

1 -

Soil Dyrmilics oittl Enrthqirnkc Ertgirteerirtg, 1988, 161. 7, No. 4 220

( a 1 Re(KgZ) / GB

(b) 1 n ( 4 ~ ) / a , / GB

(c ) R ~ ( K & ) / GB

( d l ~ m ( ~ : ~ ) / a ~ / GB . 0 I 2 3 4 5 6

Vertical

K?,(l -1,)

GB

Horizontal-s

Rocking-x

KgJI - r)

GB'

Torsion

KP GB3

Wong (r = 113)

Dominguez

Formula

Wong (r = 1/3)

Dominguez

Formula

Wong (s = 113)

Domingu-z

Formula

Wong (r = 113)

Domingucz

Formula

Wong (v = 113)

Dominguez Formula

Wong (r = 113) 8.31

Domingucz 7.53 Formula 8.3 I

Vertical Torsion

k:: = KP(k + ia,c) (la;

k = 1.0-- c= 4[+ (L/Bj3 +f (LIB)] a:

b+ai KP I G B ~ f+d

Trrble 4b.

Horizontal Rocking

Kb,= K&,(k + ia,c) K$,= Kg# + ia,c)

irregular, so that their approximation by simple formulas excreised in thcir use; since the real parts are not very is not easy. Use of regression analyses on this data led to reliable, particularly at high frequencies. The imagninary the formulas in Table 4. These formulas are also plotted parts, on the other hand, are quite good, and valid even as dashed lines in Figs 20-33. Some care must be for very high frcqucncies. As in the case of cylindrical

221 Soil Dyrraritics m r l Eortliqrrokc Eilgiileeririg, 1988, Vol. 7, N o . 4

Approsbrote forrlllrlas for ilyr~nrllic stifllessfs of rigid fo~rr~dntior~s: A. Pois clnd E . Kolrsel

t Rocking

---.-- Approx inla t e fu1-111u1a

h'ong and Luco

( a ) = Re(aix) 1 GB

Fig. 22. T4rrirrtior1 of tlre st$~ess with freqlrer~cy; surface folrrltlatioll LIB = 3 (horizorltnl mode)

foundations, the influence of Poisson's ratio on the variation of the stiffnesses with frequency has been ignored.

(c ) Stntic stifllesses of enlberlded rectanglrlnr folrr~datior~s

Only scarce data are available for the stiffnesses of rectangular embedded foundations. Dominguez4 presents result for square and rectangular (L/B=2) embedded foundations, while Abascal' studied the case of a square foundation. I n both works, the maximum depth of embedment considered was an cxcavation equal to the width of the foundation (E /B= 2).

Figs 33-36 present the effect on stiffness caused by the embedment. For the torsional, vertical and horizontal modes of a square foundation, the results by Dominguez are too high when compared to the ones by Abascal. This can be due to the fact that Dominguez did not account for discretization errors, whereas Abascal did, so the latter's results seem more accurate. On the basis of these data, we propose the following formulas for the stiffnesses as a function of the degree of embedment, and which are represented, in Figs 34-36, r~sing dashed lines:

Vertical

Horizontal

1.6 K;(, = Kg, [ 1 .O + E/B + (0.3 5 + (LIB)') ("B)~] (3 2,

Torsion

These formulas agree well with Abascal's results; ascan be seen, their dependence on the degree of embedment is less than linear (exponent of Eli3 less than I ) , except for rocking, where a second degree parabola gives good agreement. For the influence of the shape of the foundation, the only data available are Dominguez's; thus, some intuitive choices had to be made. Thc asymptotic values for a strip foundation were matched for both rocking about x and for swaying along y. Since a strip foundation has only two sides instcad of four, the effect of the embedment was thought to be split evenly between each side. The decay with the ratio (LIB) is such that the error relative to Dominguez's rcsults is rnorc or less constant.

i ------- Approximate formula

i - Uong and Luco

Fig. 23. kriot ion of the stiffi~ess with freqlrerlcy; s~rrfoce Jolr~ldntioil L/B=4 (horizor~tnl mode)

Soil Dyirmnics arrd Eortilqlrrrkc Errgirreerirrg, 1988, t6l. 7, No . 4 222

Approximnte forr~rlrlas for dylrantic stiffizesses oJ rigid folrndations: A. Pois orul E . Kalrsel

As shown by Dominguez, the height of the centre of ,m(e, stiffness is approximately 113 of the height of embedment.

aoJ I GB Because of its lesser importance, the coupling stiffness can IS be taken simply as

4

3 .

2

1 .

t 1 2 3 4 5 6 -a .wB

Cs

Fig. 24. Thriarion oJ rlre st@ress with fieqlrertcy; srrrjke Jolri~dntior~ LIB = 2 (vertical r ~ ~ o d e )

- Wong and Luco

Fig. 25. I'nriatiorr of the stiflress wit11 fi-eqlrerrcy; sw$~ce Fig. 27. Vnricztiorl oJ the stifizess with fierl~rerrcy; slrrfnce fo~rizciatior~ LIB = 3 (certicnl 1~1odc) folrrzdatio~l LIB= I (rockilzg old torsior!)

223 Soil D,wnrnics nr~d Earthq~rnke Er~girrecrir~g, 1988, Vol. 7 , NO. 4

Approxiiniite forrwrlas for dytrnr~lic stiffiresses of rigid Joirrtdotiorls: A. Pnis and E. Knrcsel

Fig. 28. Vtrrirrtic~rl of the stiflress wid1 fieq~tency; srrrfice jorrrdntiorl L/B= 2 (rocking)

----- Approxlt~~ate fontula

- Uong and Luco

0 1 2 3 4 5 *

Fig. 29. Vnriafiot~ of the sti@less with frequerlcy; strrfnce fourldntiorl L/B=3 (rocki~rg)

More data would bc ncccssary to improve the reliability of these approximate formulas.

the stiffness in the low frequency range (a, < 1.5 and a,< 2.0 respectively). Because of this lack of data, it will be assumed hcre that the variation of the stiffnesses with frequency is the same for surface as lor embedded

Fig. 30. Vnriatiow of the stijliiess with frequency; srrrfnce for~rldotion L/B = 4 (rocking)

((1) Dynamic st$ress of erlrbedderi forr~rdntiot~s Fig. 31. Vnrintion o f t h e stifiless wit11 freqrreircy; sirrjince Dominguez4 and Abascall present only the variation of forrrrdntiorl L/B = 2 (torsiolt)

Soil Dynnntics N I I ~ En~dtqu~lke E~lgirreerirlg, 1988, Vd. 7, No. 4 224

Approsinrnte for~?lrrlns for. dy~rmlic st~firesses of rigid folrrrrlatiorn: A. Pais ar~d E. Karrsel

--.--- Approximate formulas

Uong and Luco

60

5 0

Fig. 32. firintion of the st@less with freqlrerlcy; srrrface fo~rrrdatiorl L/B = 3 (torsiorr)

lso-l Approximate formulas ----- I Won~ and Luco -

Fig. 33. Krriatiorl of the stifl~ess with fieqlrerrcy; slrrjiace jblrrrdotiorl L/B=4 (torsion)

foundations. The formulas describing this variation is given in Table 5.

The asymptoticvalues of the coefficient c were obtained by computing the geometrical inertias and areas, as was also donc for thc cylindrical foundations. The rocking modes exhibit a nonzero value of c in the static case, which agrees with Abascal's results.

The dynamic coupling stiffnesses were obtained

multiplying the horizontal stiffnesses by 1/3 of the height of embedment, as was done for the static case.

Applicatiorls to viscoelastic mnteriols The previous results can be extended to the case of a

Kembl Kno emb

1 stat stat

----- A p p r o r i ~ a t e f o m l a

R e s u l t s by Coningez

Results by Abascal

Fig. 34. Vorintiorl of the static stiJkesses with the elnberherlt (lrorizot~tol omi uertical)

emb 0

t "'" ' Approximate formula

- Results by Daminguez

O Resul ts by Abascal

Fig. 35. Vorintiorl of the static stfilesses with the er,rDetl,ner~t (rocking)

225 Soil Dy11nmics arrd Enrthq~rake E~rgirleer-ir~g, 1988, Vol. 7, No. 4

A p p r o s b ~ n f e for~ntifos for cfyr~olttic stiffi~esses o j rigid jozrl~tlnrioits: A. Pois nrtd E . Korrsef

Table 5a

Vertical Torsion

= ~ : . ( k + ia,c) E;d= ~ : ( k +ia,c)

Table 5b.

Horirontal Coupling

Table 5c.

Rocking, direction x

Rocking, direction1

4 - (LIB +z)(E/B)'

1.8 3 D =

f = ~ . 0 + 1 . 7 5 ( ~ / ~ - I) Kkr I GB3

Note: L a B; a,= wB/C,: a= CL/C ,

Soil Dyrrrinzics trrd Enrthqrrc~ke E~~girteeririg, 1988, Vol. 7 , No . 4 226

Approsiinnte forrillrlas for d y i ~ a i ~ i c stifllesses of rigid folrndatioiu: A. Pais m ~ d E. Kalrsel

emb , K n ~ emb "stat stat The foundation stiffness becomes:

- Results by Dominguez

Resul ts by Abescal where a: = a,(l - i j ) ,

LIB-1

REFERENCES

1 Abascal, R. Estudio de Problcmas Dinamicos en lnteraccion Suclo-Estructura por el hlctodo de 10s Elementos de Contorno, Doctoral Thesis, Escuela Tecnica Superior de Ingenieros Industriales de la Universidad de Sevilla, 1983

2 Apsel, R. J. Dynamic Green's Functions for Layered Media and Appltcations to Boundary-Value Problems, PhD Thesis, Univ. of California at San Diego, 1979

3 Day, S. M. Finite Element Analysis of Seismic Scattcring Problems, PhD Thesis, Univ. of Calif. at San Diego, 1977

4 Dominguez, J . Dynamic StifTness of Rectangular Foundations, Report No R78-20, hlIT, Cambridge, Massachusetts, 1978

5 Elsabee, F. and Moray, J. P. Dynamic Behavior of Embedded Foundations, Report No. R77-33, MIT Dept. of Civil Engineering, Cambridge, Massachusetts, 1977

6 Gazetas, T. Analysis of Machine Foundation Vibrations: State L of the Art, ~rtr~rn~tiorlnl Joirrttalof Soil Dyttarnicsarid Earrhqlroke

1 4 / 3 z Ertg., 1983, 2(1), 2-42 . . ff8 7 Gazetas. G. and Dobrv, R. Simple Radiation Damping hlodel

Fig. 36. Varintiol~ e~t~bedrilent (torsion)

for piles and ~ootings; ~ourrlai of the Eng. hfe'lr. Dicisiort, oJ the static stt$lrss wifh the ASCE, June 1981, 1 lO(EM6). 937-956 8 Kausel, E. Forced Vibrations of Circular Foundations on

Layered Media, Report No. R74-11, hlIT Dept. of Civil Eneineerine. Cambridge. Massachusetts. 1974

viscoelastic halfspacc by applying Biot's correspondence principle. This principle states that it is sufficient to substitute the real moduliof thesoil by complexmoduli to account for material damping. Usually it is assumed that the value of Poisson's ratio does not depend on the amount of material damping. For simplicity it can be assumed that both P-waves and S-waves have the same amount of attenuation. The complex wave celerities become then:

where /3 represents the amount of material damping in the soil.

Because the value of p is generally small compared to unity, the complex shear modulus can be written as

~ G s e l , ~ . & d ~shi j ima, R. Vertical and Torsional StifTness of Cylindrical Footings, Report No. R76-6, MIT Dept. of Civil Engineering, Cambridge, Massachusetts, 1976 Luco, J. E. and Westn~ann, R. A. Dynamic Response of Circular Footings, Joctrtlal of Eng. Mech. Dicisiori, ASCE, 1971, 97(Eh15), 1381-1395 Luco, J. E, and Westmann, R. A. Dynamic Rcsponse of a Rigid Footing Bonded to an Elastic Iialfspace, Jotrrr~ol of Appfitd Afech., ASAIE, 1972,39,527-534 Luco. J . E., Frazier, G. A. and Day, S. hl. Dynamic Response of Three-Dimensional Rigid Embedded Foundations, Cal. Univ., Report No. NSFIRA-780199, Natl. Tech. Inf. Scrv., 1978 Veletsos, A. S. and Verbic, B. Basic Response Functions for Elastic Foundations, Jo~rrtlal of rhe Etig. Alech. Ditisior~, ASCE, 1974,10O(EM2), 189-202 Waas, G Analysis Method for Footing Vibrations through Layered Media, PhD Thesis, Uni\ersity of California, Berkeley, 1972 .. -

Wong, H. L. and Luco, J. E. Dynamic Response of Rigid Foundations of Arbitrary Shape, Earrliqirake Eng. and Srrircr. Uyrlarnics, 1976,6, 3-16 IVong, H. L. and Luco, J. E. Tables of lmpcdance Functions and Input Motions for Rectangular Foundations, Report No. CE78- 15, Univ. of Southern California, 1978

227 Soil Dyrnii~ics aild Enrtlrqtroke Ei~gir~eerirlg, 1988, Vol. 7 , N o . 4