Approximate formulas for dynamic stiffnesses of rigid foundations

Artur Pals and Eduardo Kausel

Dept. of Civil Engineering, Room 1-271, MIT, Cambridge, MA 02139, USA

Approximate formulas are proposed to describe the variation with frequency of tile 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 embedded 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 Correspondence Principle, if so desired.

INTRODUCTION

The first step in the study of a soil-structure interaction problem is the evaluation of the dynamic stiffness 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 sufficient to describe its motion. The dynamic stiffness matrix has then only six columns and rows.

To 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 difficult, it is not surprising that analytical solutions are available for only very special cases. Luco et al. t~ i 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 t h e 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 boundaries 8'~4 that reproduce the physical behaviour of

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 present in an elastic halfspace (or in a very deep alluvia) is neglected in such models.

To avoid this problem, Day 3 performed transient finite element analyses 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, however, since it is not possible to distinguish between real reflections at the interfaces of the layers and the spurious reflection at the boundary. Apsel 2, on the other hand, used an integral equation formulation for the

I E i

Accepted September 1987. Discussion closes December 1988. Fig. 1. Cylindrical embedded foundation

1988 Computational Mechanics Publications

213 Soil D),tlamics and Earthquake Engineering, 1988, Vol. 7, No. 4

Approximate formidas for dynamic stiffinesses of rigid foundations: A. Pals and E. Kausel

6G,

35

25

20

15

10

; t a t t c

~ Aosel and )ly results for rocking

Az~sel and gay results for horiIontll mde =ocking

- - a o p n x i r ~ t e fo,rr~..=las

Eoriz;B~l

9

Fi9. 2. Variation of the static stiffness with embedment (rockin9 and horizontal modes)

the

static K

35-

25

20

Apsel and Cay results for torsion Apsel and Day results for verClcal mode

-- aOpr~xl;ate fomulas

torSlcn

verticll

Fig. 3. Variation of the static stiffness embedment (torsion and vertical modes)

':Z E/R

with the

same problem and found a very good agreement with the results of Day (see Figs 4-13). It must be added that Aspel assumed a small amount of internal damping in the soil, while Day assumed an elastic medium.

Concerning rectangular foundations, Wong et a l ) s presented compliance functions for fiat 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 L u c o 16 presented tables of impedance functions for fiat rectangular foundations. Dominguez 4 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. Abascal ~, using a similar approach, presented the stiffnesses of a square embedded foundation.

Cylindrical embedded foundations

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

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

Re(K~),

20

15

f i n i t e eler~nt solution .......

boundary integral solution . . . . . .

approximate fo~la

E/R �9 2 . 0

, E / R �9 l . O

ii o 'l 'z ; t ; ; '7

rJ~

Fig. 4. Variation with the frequency of the actual stiffiless (real part)

Im(K~) ;i o

2 0

15'

10,

finite element SolutiOn .......

boundary integral solu ........ tion

approximate formula E/R - Z .O

. . . . . . . . . . . . . . . =J . . . . . . . .

EZR - 1,0

~'IR - 0.5

E/R - O

'l '2 'B '4 ~ '7 f i r

Fig. 5. Variation with the frequency of the vertical stiffizess (imaginal 3, part)

Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4 214

Approximate formulas for dynamic stiffiwsses of rigid foumlations: A. Pals arul E. Kausel

25

20

15

I0

finite er~r, ent solution ......

boundary integral solution ......

approximate for~ula

EIR �9 2 .0

. . [ / ~ I 1 . 0

. . . . . . . . E/R - o

i '2 '3 4 ~5 '6 J7 "r,,R

Fig. 6. .Variation with the frequency of tire horizontal stiffiwss (real part)

a 0

20

15.

10

finite el~ent solution .....

boundary integral solution .......

app~xi:4te formula E/R = Z.O

"- E/R �9 1.0

~,..,,..,,,=.,..~.,,.~,.,.~.,..= ~=~. . . . . . . . . . . . . . . . . . . . E/R �9 0 .5

) i 2'

E/R �9 0

Z ~ 6 aO �9 f i r

Fig. 7. Variation with tire frequency of tire horizontal stiffiwss (imaginary part)

formulas for the stiffnesses were found:

4 G R - - (Boussinesq) (la) Vertical K~ = 1 - v

8GR Horizontal Ko = 2 - v (Mindlin, 1949) (Ib)

K o 8GR3 Rocking g=~(--i~--~__V) (Borowicka, 1943) (lc)

16GR s Torsion K ~ 3 (Reissner, 1944) (ld)

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

35

20.

15,

" f in i te element solution " . . . . . . boundary intelral solu- . ......

tion

l l x a poro xi,.-4te fo n':~u I a

E/R �9 Z.O

" ' ' ' ' " ' ' - - . . . . . . _ , . . . . .

~ . E/R �9 1.0

~ - . _ ~/~ - 9

o i ~ ~ z 's 6 '7 k fir

Fig. 8. Variation with tlre frequency of tire rocking stiffiness (real part)

1~(~) a~

f i n i t e el c~.ent solution . . . . . . .

boundary integral solution . . . . . . .

approximate formula

30 . . . . . . . . . . ~ EIR �9 Z.O

25.

20.

15. ./

/

10.

E/R - 1.0

5 "E/R - O.S

o - - - - 3 4 s 6

f ir 'o'r,

Fig. 9. Variation with the frequency of the rocking stiffiress (imaginary part)

215 Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4

Approximate formulas for dynamic stiffinesses of rigid foundations: A. Pals and E. Kausel

re(K~)

25,

20.

15,

finite el~ent solution .....

boundary integral solution

~ . ~ a ~ r ~ x i ~ te for~.ula

Z.O

. . . . . . . . . . . . . . . ~I~ �9 1,0

qm

E/II - 0

a01 ~

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

20

1 5

tO,

finite el e~nent solution . . . . . . .

boundary integral solution . . . . .

epproxln~te formula

F/R - 2.n

1 .

" ~ " - , ,~ , . . . , . . . . _ , . . . , , .~ . . ~.. . ._. : .___._ ._ ... . . . .

E/R - 1 . 0

o . . . . . k . . . . . . ~" . . . . C ' " ";. "s ~, '7

Fig. 12. Variation with the frequency of tire coupling stiffiless (real part)

Fig. 10, Variation with tire frequency of the torsional stiffiness (real part)

I=(K~)

i 0

finite client solutlon .......

boundary integral solution ......

approximate for~:ula 2O

15 EIR - 2 .0

[ /R - 1.0

/ " . . . . - ~ - . - - - [1~ - OmS

0 . . . . . . . . . ~ -- E/~ - 0 , �9 m

0 1 2 3 4 5 6 nR

ao J ~'~z

Fig. 11. Variation with the frequency of the torsional stiffiless (imaginary 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 Elsabee s, while the vertical and torsional modes were analysed by Kausel and Ushijima 9. In both studies, a finite element computer

Im( K(~ r )

20

15

10

finite el~ent solu t ion ' - . . . . . boundary integral solution . . . . . . apP~xi .r.lte for~.al a

�9 ~...~jrx.~'.- "-'-''-'~''''~-'~;r~ . . . . . --.~.:'2.'~?_'.."I E/R �9 Z.O

~ [ / r * 1 . 0

E/R -O.S

o 1 2 3 4 ~ ; a O = -~S

Fig. 13. Variation with the frequency of tile coupling stiffitess (imaginary part)

2L i - (

z8 !

, ' i : , x t �9 I I

. . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . .

~z l

, t /'y ,

4, Fig. 14. Rectangular embedded foumiation (L ~> B)

Soil Dynanlics and Earthquake Engineering, 1988, Vol. 7, No. 4 216

Approximate formulas for dynamic stiffinesses of rigid formdations: A. Pais and E. Kausel

50

45

4 0

35-

3 0

25-

20-

1 5

10,

5.

-- ,~.~rexL~,~l;e formula

~ ~ ;, ; . . . . . . . . . . . . . . z0 ~o' L/B

Fig. 15. " Variation of tire vertical static stiffiress with tire shape of tire foundation

Numerical solutions for the dynamic stiffnesses of cylindrical embedded foundations were also prepared by Aspel z and by Day 3, 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 the curves in Figs 4 through 13, the static value of the impedances can be extracted (see Figs 2,3) by r~ taking the average of the finite element and boundary element results. Figs 2 and 3 show these static stiffnesses

6 0

as a fnnction 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 (E/R = 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; 3c this equivalent to the assumption that the ratio of the two is independent of v. In such case, one can seek polynomial approximations to account for embedment that depend on the embedment parameters only.

For the torsional, vertical and horizontal modes, a 2~ linear approximation is sufficient, giving less that 1 0 ~ error when compared to the numerical results. A power law with the exponent 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 ~0 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 5. about a horizontal axis is also a third degree polynomial in E/R. For the coupling term, the formula proposed by Elsabee was adopted. The resulting approximations are then as follows:

Vertical K~, = K ~ (1 + 0.54E/R) (2a)

Horizontal KnS -- Kn~ (1 + E/R) (2b)

Rocking KR--KR ~ -- o (1 + 2.3E/R +O.58(E/R) 3)

(2c)

Torsion K,'--K,~ (1 +2.67E/R) (2d)

Coupling K~, = (0.4E/R - 0.03)K~1 (2e)

with K ~ etc. being given by equations (la) through (ld).

The coefficient in the approximation for the torsional stiffness is the same as in the work by Kausel et al. 9, while the vertical stiffness has been changed somewhat so as to extend its range of validity.

(b) Dynamic stiffnesses 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:

K,n = K~(k + ioac ) (3)

where K s designates the appropriate static stiffness; ao is the dimensionless frequency ao=coR/Cs (~o=angular frequency of the motion, R = radius of the foundation, C, = shear wave velocity in the soil); k and c are functions of a0, v=Poisson 's ratio, and E/R=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 ofv higher than about 0.4 the approximate formulas must be used with care, especially for high frequencies.

-- Approximate formula

ReSults frc~ Wong And Luco

| Results from Dominguez

x Results from Gorbunov-Posanov

H Y

s .

, LIE

Fig. 16. Variation of the horizontal static stiffness with the shape of the foundation

217 Soil Dynamics aml Earthquake Engineering, 1988, Vol. 7, No. 4

Approximate formulas for dynamic stiffnesses of rigid foundations: A. Pais and E. Kausel

$ KRx

40

q.

/ / / ~ x Resu] ~,~r, ,Wunov-Posancv

/ ~ Res~lr.s frc,~, .Wong. an{:l Luco

~ '3 '4 s '~ '7 '8 '9 '~o k/8

Fig. 17. Variation of the rockin9 static stiffi~ess with the shape of the foumtation

Nevertheless, in the high frequency 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 Dobry ~ for further details). Denoting this value by E', it is given by the formula

(4)

$ KRr 1600

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 (viCe) 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. On account of this fact, one can split equation (4) into two parts, the first of which takes account of S-waves generated, while the second reflects the contribution of L- waves:

~- R~ (fs TrT, dS + c~ fs TrT2 dS ) (5)

with ct = Cc/Cs. An important question arises at this point

regarding appropriate values for C L, since plane-strain conditions do not hold in the vicinity of the foundation. As a result, lateral dilation takes place which causes the value of CL to be lower than the theoretical value for P- waves (Cv=Cs x / 2 ( l - v ) / ( l - 2 v ) ) . The discrepancy between CL and Cv is particularly important for incompressible solids (v=0.5), for which the P-wave velocity is infinity, while the effective radiation velocity CL is finite. To circumvent this difficulty, one can either approximate CL~ C v and limit the value of v used in deriving C~, (for example, a ~< 2.5), or following Gazetas and Dobry 7, one defines Cc to be a ficticious velocity for longitudinal waves. With these approximations, the dynamic stiffness coefficients 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.~3 for the case of a surface foundation, and Kausel et al. 9 for an embedded foundation. Figs 4-13 show a comparison of the proposed formulas with the results of Day 3 and Aspel 2, (taking v=0.25). Because Apsel assumed a small amount of material damping in the soil, the imaginary term of. the stiffness (lm(Ka(ao))) 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 ofc for the

1400 Results from ~ong and Luco

0 Results fro~ Oo~inguez

Results from Gorbunov-Posanov 1200

1000

800

600

400

200

Approxir,~e for~ula

10 L/8

Fig. 18. Variation of tile rocking static stiffiless with the shape of the foundation

Soil Dynamics and Earthquake Engineering, 1988, Vol. 7., No. 4 218

Approximate formulas for dynamic stiffnesses o f rigid foundations: A. Pals and E. Kausel

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

Embedded rectangular fomulations

(a) Static stiffiwsses o f surface foundations In the case of rectangular foundations, the lack of

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

Table IA.

Vertical Torsion

l ( f = K~.(k + iaoC ) K~ = K~(k + iaoC )

k= 1.0 0.35% 2

k= 1 . 0 - - - 1.0+ao 2

2

rc(2 + 2.0 E/R) ~ (1 + 4 E/R) b-~a 2

K~/GR KI/GR 3

1 ~t = CL/C, b 0.37 + 0.87(E/R) 213

available, not even for surface foundations. In addition, the ratio of length to width, LIB (see Fig. 14), which defines the geometry 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 dependence 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 L/B, and for Poisson's ratio v = 1/3. These figures are based on

Table lB.

Horizontal Rocking

l(all = K~lt(k + iaoc)

k= 1.0

n[1.O + (1.O + ~)E]R] C ~

K~tt/GR K}ffGR 3

l(aR = K~(k + iaoc)

k=l.O 0"35a~ 1.0+% 2

[~ l+~t 2 a z b - + E/R + - - - c = 4 ( 2 ) (E/R)3]~+O84(I+ct)(E/R)2"'

3 JO+a o b+aZo

2 with b = - -

1 + E/R

/ ( ~ l t = (0 .4 E/R --0.03)/(~t

Table 2.

Dominguez "~ Wong and LUCO 16

Abascal (1) (a) {b) v = 1 ]3 v = 0.45 Value taken

K~?/(2- v) 9.41 9.47 9.35 9.22 9.16 9.2

GB

K?,O -~') 4.75 4.88 4.75 4.66 4.57 4.7

GB

K~ 4.38 3.85 3.79 4.17 4.04 4.0 GB 3

K o 8.71 7.53 7.48 8.31 8.42 8.31

GB 3

K~ - - - 0.508 0.302 0 GB 2

(a) Relaxed boundary conditions (b) Nonrelaxed boundary conditions

219 Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4

$ K t

160,

140

120.

100,

80.

60,

40

| Results from 0ominsuez

Results fro~ 7ong and Luco

Approximate formula

20.

Approxinlate fornndas for dynamic stiffnesses of rigid foundations: A. Pais and E. Kausel

i.

LIB

Fig. 19. Variation of the torsional siatic stiffness with the shape of the foundation

the data presented by Wong et alJ 6, Dominguez 4, and Gorbunov-Posanov (from Ref. 6). Use of these figures led to the following approximations (with L/B>>. 1):

Vertical K~ _ 3.1(L/B)O.75 + 1.6 (17) GB

Kttx(2_ ~ , Horizontal =6.8(L/B) ~ +2.4 (18) GB

K%(2- v) K~ v) + 0.8(L/B- I)

GB GB

(19)

K~ -v) Rocking GB 3 - 3.2(L/B) + 0.8 (20)

K~ - v ) 3.73(L/B)Z'4 + 0.27 (21)

GB 3

Torsion K~ =4.25(L/B) T M + 4 . 0 6 (22) GB 3

The exponent of (L/B) is less than 1 for the vertical and horizontal modes, equal to 1 for rocking around the longitudinal axis, and greater than I 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 alJ 6 (v= 1/3), Dominguez% and the formulas proposed, for 1 <~L/B<~4. As can be seen, the agreement is very good, the largest diffferences being less than a few percent.

(b) Dynamic stiffnesses of surface foundations To describe the variation of the stiffnesses with

frequency, the results by Wong and L u c o 16 w e r e 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

6

5

. . . . . Approximate form.ula 2

_ _ Wong and kuco

!

.w._~9 ao C$

Fig. 20. Variation of the stiffness with frequency; surface foundation L/B = 1 (vertical and horizontal modes)

9 ~ (b)

6,

5. . . . . . . . Approx imate fo r'=,,ul a

~,'ong and Luco 4.

(a) relK~x) 3.

(b) Im(K~xlla ~ 2-

(C) Re(Kdy) I

(d) Im(Kdy)/ao

a =~- . o C S

Fig. 21. Variation of the stiffiness with frequency: surface fomldation L/B = 2 (horizontal mode)

Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4 220

Approx ima te formulas f o r dynamic stiffnesses o f rigid foumlat ions: A . Pais amt E. Kause l

Table 3.

Mode L/B = 1 L/B = 2 LIB = 3 L/B = 4

Vertical

K~,(I--v)

GB

Horizontal-x

GB

Horizontal-)'

K~?~,y(2- v)

GB

Rocking-x

K~( I --v)

GB 3

Rocking-):

K~ - v)

GB ~

Torsion

K, o

GB ~

Wong (v = 1/3)

Dommguez

Formula

Wong (v = 113)

Dominguez

Formula

Wong (v = 1/3)

Dommguez

Formula

Woug (v= 1/3)

Dommguez

Formula

Wong (v = 1/3)

Dommguez

Formula

Wong (v = 1/3)

Dommguez

Formula

4.66 6.73 8.56 10.22

4.88 7.0 8.9 10.7

4.70 6.81 8.66 10.36

9.22 12.95 16.19 19.15

9.47 13.1 16.3 19.3

9.20 13.07 16.29 19.14

9.22 13.75 17.79 21.48

9.47 14.0 18.1 21.8

9.20 13.87 17.89 21.54

4.17 7.18 10.30 13.18

3.85 6.8 9.75 12.8

4.00 7.20 10.40 13.60

4.17 20.21 52.26 104.21

3.85 19s 50.6 105.3

4.0 19.96 52.37 104.18

8.31 28.32 67.41 131.03

7.53 26.8 65.7 129.2

8.31 27.28 66.77 130.95

Table 4a.

Vertical Torsion

- d _ _ 0 . " K~ - K~. (k + laoc) l?,a~ = K ~ (k + iaoc )

k = 1.0 da~ 4~L/B da2o 4[-~ (L/B) 3 +~ (/.fiB)] " ao 2 - - - - c = - - k= 1 . 0 - - - c=

b+a~ K ~ b+a~ K ~ f + a z

0.2 1 0 1.4 d = 0.33 - 0.03 x / L / B - 1 f = 1 + 3(L/B- 1 )0.7 d = O . a + L ~ b I+3(L /B- I )

0.8 b=

1 +0.33(L/B- 1)

Table 4b.

Horizontal Rocking

gall:, = K~t~(k + iaoc)

4L/B k= 1.0 c=

gJIr = K~y(k + iaoc )

4L/B k= 1.0 C=

K~,.

g~.= KL(k +iaoc)

4= da2o -~ L/B a2 ~

k= 1 . 0 - - - c = - - b+.~ KL f + . o ~

d=0.55 +0.01 v/L/B-- 1

0.4 0.4 b = 2 . 4 - - - f = 2 . 2 - - -

(L/B) 3 (L/B)'

R~r = K~r(k + iaoc)

4x 3 k = 1.0 -- 0.55 a__.....~ S (L/B) a2 ~

b+a~ K~ f + a 2

1.4 1.8 b = 0 . 6 + - - f =

(L/B) 3 1.0 + 1.75(L/B - 1)

i r regu la r , so tha t the i r a p p r o x i m a t i o n by s imple f o r m u l a s is n o t easy. U s e o f r eg ress ion ana lyses on this d a t a led to t he f o r m u l a s in T a b l e 4. These f o r m u l a s a re a lso p lo t t ed as d a s h e d l ines in F igs 20-33 . S o m e c a r e m u s t be

exerc i sed in the i r use, ' s ince the real par t s a re n o t ve ry re l iable , pa r t i cu l a r ly at h igh f requenc ies . T h e i m a g n i n a r y par ts , on the o t h e r h a n d , a re qu i t e g o o d , a n d va l id even for ve ry h igh f requenc ies . As in the case o f cy l indr ica l

2 2 1 Soil Dynamics and Ear thquake Engineering, 1988, Vol. 7, N o . 4

Approximate formulas for" dynamic stiffnesses of rigid foundations: A. Pais amt E. Kausel

14'

13,

1 2

11.

10,

9 i Bj

6.

5

4

3

2

1

Approx imate formula

Rong and Luco

{a) = Re(r~x)

(b) = Im(r~x)/a 0

(c) = Re(K~y)

(dl = hn(K~ylla ~ = ~ d O C

S

o i 2 ; 4 5

Fi9. 22. Variation of the stiffiwss with frequency; surface foundation L/B=3 (horizontal mode)

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

(c) Static stiffnesses of embedded rectangular foundations

Only scarce data are available for the stiffnesses of rectangular embedded foundations. Dominguez 4 presents result for square and rectangular (L/B=2) embedded foundations, while Abascal I studied the case of a square foundation. In both works, the maximum depth ofembedment considered was an excavation 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, using dashed lines:

Vertical

K~,=K~ / ) ' j 0 " 2 5 ' E B ~ s-I (29)

Horizontal

K~z=KO[1.O+(O.33q I~_~/B)(E/B). ' 08-] (30)

Rocking

K~Rx=KOx[1.O+E/B [ 1.6 ' z-] +~0.35+ L/B) (E/B) J / (30

K~,=K~ B [ 1.6 ] +~O.35~-(L/B)4)(E/B) z3 (32)

Torsion

K : = K ~ 1-3 +L--~)( / ) " J 1"32' E B 09-] (33)

These formulas agree well with Abascal's results; as can be seen, their dependence on the degree of embedment is less than linear (exponent of E/B less than 1), 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. The 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 instead of four, the effect of the embedment was thought to be split evenly between each side. The decay with the ratio (L/B) is such that the error relative to Dominguez's results is more or less constant.

20, L

19.

18.

17 (d ' x - - - . ) (b) ,6, ::<. ~ .

14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I I . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

9.

8 . . . . . . . . Approx imate formula

7 . ' - Won 9 and Luco

6- (a) = Re(K~x ) 5

(b) = Im(K~x)/a ~ 4

3 (c) = Re(K~y)

2 (d) = lm(K~y)/a ~

I

| I i t 0 1 2 3 4

a =~dL o C s

5

Fig. 23. Variation of tire stiffness with frequency; smfitce foundation L/B = 4 (horizontal mode)

Soil Dynanzics and Earthquake Engineering, 1988, Vol. 7, No. 4 222

Approximate formulas for dynamic stiffinesses of rigid foundations: A. Pals and E. Kausel

16. . . . .

15.

16.

13.

. . . . . . . Approximate formula

12. Wong a~d Luco

It.

9

8

5

4

3

2

!

t ! i i ~ 3 4 ;

lm(L~)

a o

I I wB

6 aO=~- s

Fig. 24. Variation of the stiffiless with fi'equency; surface foundation L/B = 2 (vertical mode)

2 5 .

2 0 .

1 5 .

1 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nm(K~)

.... ~ a~

. . . . . . . Approximate fon~tula Wong and Luco

. . . . --.--'~.,~.~.. Re(g~)

�9 I i I

w8

Fig. 25. Variation of the stiffiness with frequency: surface foundation L/B = 3 (vertical mode)

As shown by Dominguez, the height of the centre of stiffness is approximately 1/3 of the height of embedment. Because of its lesser importance, the coupling stiffness can be taken simply as

K~tR~ = ~ (E/B)K~I~ (34)

KnRr = ~ (E/B)Knr (35)

31~

25.

20

15.

lO,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Im(K~)

. . . . . . A p p r o x i m a t e f o r m u l a

- - Nong and l u c o

. . . . . . . . . . . . Re(K~)

a 0 C s

I I . - - & t - -~ 1 2 3 4 5

Fig. 26. Variation of the stiffiness with fi'eqztelzc),; surface foundation L/B = 4 (vertical mode)

! ,~ t . . . . . , o = 1 . . 9( Wong and Luco

(it) - Im(Kdp,)/a ~ 8 t "~".'%. 7~ --'.~.. (b} " Ira(r~)/a 0

";"" (a) ~S zl / ~ ' ~ ( ' ) ao " E" �9 , . . . . . . . $

1 2 3 J* 5 6 7 8 9 i

Fig. 27. Variation of the stiffness with frequency; surface fotmdation L/B = l (rocking and torsion)

223 Soil D),namics and Earthquake Engineering, 1988, P'ol. 7, No. 4

Approximate formulas for dynamic stiffi~esses of righl foundations: A. Pals aml E. Kat~sel

. . . . . . A p p r a x 1 = ~ : e for'~ul a

- - h'ong a~a Luco

the stiffness in the low frequency range (a o < 1.5 and ao < 2,0 respectively), Because of this lack of data, it will be assumed here that the variation of the stiffnesses with frequency is the same for surface as for embedded

20

15.

10 ,

Tm(K~)

~e(~:yl

/

a o

. . . . . . dp f l rox in~ te fo rmula

I~ong and Luco

�9 ao

0 1 2 3 t, 5 6

C s

Fig. 28. Variation of the stiffiness with fi'equency; smface foundation L/B= 2 (rocking)

Im(K~x) (a) =

a o

(b) ~ eelK~x) i

8 J . . . . . l~pprox imate f o r ~ u l a " 1

,0r , - - _

Z/

0 1 2 3 4 5

$

Fig. 29. Variation of the stiffi~ess with frequency; st~rface foundation L/B=3 (rocking)

More data would be necessary to improve the reliability of these approximate formulas.

(d) D),~mmic stiffitess of embedded folmdatio,s Dominguez 4 and Abascal~ present only the variation of

IOC

5C

, !

1 s e

I

, i

J

f

!

!

e r r

|

(hi

. w.._B_B a o Cs

Fig. 30. Variation of the stiffi~ess with frequency; surface fotmdation L/B = 4 (rocking)

3 0 .

25-

2 0

15

10

. . . . . . Approxffnate for~ula

ae(t~l

~B

. = t

Fig. 31. Variation of the stiffizess with frequency; stiff ace folmdation L/B= 2 (torsion)

Soil Dynamics and Earthquake Enuineering, 1988, Vol. 7, No. 4 224

Approximate formulas for dynamic stiffnesses of rigid foumlations: A. Pals and E. Kausel

8O

70.

60

50,

40

30

2O

10

. . . . . . Approximate formulas

~ W o n g and Luco

~ ~ Re(K~)

. . - - - , . . . . . . . .

"77 . . . . "-" s

Im(K~) To

= t~B

a o ~-- i I ~ I ' ' i 1 2 3- 4 5

Fig. 32. Variation of the stiffizess with frequency; surface fozmdation L/B=3 (torsion)

150- Approximate formulas . . . . .

Wong and Luco

10(

/ , . " /" ao

5 0 -

a = I~I.BB , , , o

0 I 2 3 4 5

Fig. 33. Variation of the stiffness with fi'equency; surface foundation L/B=4 (torsion)

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

The asymptotic values of the coefficient c were obtained by computing the geometrical inertias and areas, as was also done for the 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 I/3 of the height of embedment, as was done for the static case.

Applications to viscoelastic materials The previous results can be extended to the case of a

. . . . . Approximate formula

~ R e s u l t s by Dominguez

Results by Abascal

. ~ ' .. 1 hori-

2P3 i 4'/3 E/B

Fig. 34. Variation of the static stiffilesses with the embedment (horizontal and vertical)

B.

6.

4.

2,

I

I o

. . . . . . Approximate formula

Results by Dominguez L/B-I | /

| Results by Abascal r

/

/ I "I'I

I " i ,~ .z i

i /

I"'" / "

2'13

Fig. 35. Variation of embedment (rocking)

4't3 2 EIB

the static stiffilesses with tlle

225 Soil Dynanffcs and Earthquake Engineering, 1988, Vol. 7, No. 4

Approximate formulas for dynamic stiffi~esses o f rigid foumlations: A. Pals aml E. Kausel

Table 5a.

Vertical Torsion

- d F s . �9 K, = K~(k + moC ) K~ = K~.(k + moc ) ~a .~ . .

da~ 4[ctL/B+ E (I + L/B)] da 2 d=O'33-O'O3x/~/B-I k=l.O-b+a-~o c K'~ k=l'O-b+a~o

d = 0 . 4 + 0"2 b = 10.0 LIB 1 + 3 (LIB - 1 )

F ~ . ~z 1 1 4L (L/BI(E/B) + 7 (L/B)3(E/B) +(LIB)" (E/B) + 7 (E/B) +-j (L/B) 3 +-~ (L/B)]

C =

0.8

l + 0.33(L/B- l)

gl

1.4

f = 1 + 3 ( L / B - 1) ~

~o f+a~

Table 5b.

Horizontal Coupling

k = l . 0 ~ c:

k = 1.0 c =

F.,]t~, = Kh~(k + iao c) 4[ L/B + E/B(~ + L/e)]

KJtx - - d - - * s . " Knr - K ny(k + tao c) 4[LIB+E/B(I +~LIB)]

KJtr

- a ' (E/B)I(~. KHRx = 5

~ d I - - d K H R r = ] (E/B)Knr

Table 5c.

Rocking, direction x

I(~ = K*Rx(k + iaoc )

d~g d = 0.55 + o. I , , / - s l k=l.O-b+a----~o

0.4 b = 2 . 4 - - - (L/B) 3

1 1 3 ct s r a t f c=4[-~(E/B)+-~(E/B) +~(L/B)(E/B) +(E/B)(L/B)+~(L/B)]K~x f-~a~ +Df +a~

f=2.2

4 L 0.4 D 3(Ct-B+I)(E/B)3

(L/Bp K ~

Rocking, directiony

l(ngy = Kk~.(k + iaoe)

k = 1 . 0 - - - 0.55a~ 1.4

b = 0 . 6 + - - b + ag [L/B) 3

I cg 1 3 4[3 (L/B)3(E/B)+3 (E/B)3(L/B)+3 (E/B) + (E/B)(L/B)2 +3 (L/B)3] at ~_D f_~a~

c= K~r f +a~

4 1.8 -j (LIB + r a

f = D 1.0 + 1 .75 ( L / B - - 1 ) Khr

Note: L~> B; a o = o~B/Cs; a = CtJC,

Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4 226

Approximate formulas for dynamic stiffinesses of rigid foundations: A. Pais and E. Kausel

The founda t ion stiffness becomes:

8 . . . . . . A p p r o x i r , . a t e f o , , ~ u l . K d = K~(k(a'd) + ia~c(a~))(1 + 2ifl) Results by Dominguez

o Results by Ab~scat where a~ = a o ( 1 - ifl).

/ LIB-1

o - 2~3 ~ ;I~ ~ '

Fig. 36. Variation of the static stiffiless with the embedment (torsion)

viscoelast ic halfspace by app ly ing Biot 's co r respondence principle. This pr inciple states tha t it is sufficient to subs t i tu te the real modu l i of the soil by complex modu l i to account for mater ia l damping . Usua l ly it is assumed tha t the value of Po isson ' s ra t io does not depend on the a m o u n t of mate r ia l d a m p i n g . F o r s impl ic i ty it can be assumed that bo th P-waves and S-waves have the same a m o u n t of a t t enua t ion . The complex wave celerities become then:

c_ C, ~_ Cp (12) C , - l _ i f l and C p - l _ - - ~

where fl represents the a m o u n t of mate r ia l d a m p i n g in the soil.

Because the value of fl is genera l ly small c o m p a r e d to unity, the complex shear modu lus can be wri t ten as

G'=p(C~)2=G(1-i f l ) -2~G(l+2i f l ) (13)

(14)

R E F E R E N C E S

1 Abascal, R. Estudio de Problemas Dinamicos en lnteraccion Suelo-Estructura por el Metodo de los Elementos de Contorno, Doctoral Thesis, Escuela Tecnica Superior de lngenieros Industriales de la Universidad de Sevilla, 1984

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

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

4 Dominguez, J. Dynamic Stiffness of Rectangular Foundations, Report No R78-20, MIT, 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 of the Art, International Journal of Soil Dynamics and Earthquake Eng., 1983, 2(1), 2-42

7 Gazetas, G. and Dobry, R. Simple Radiation Damping Model for Piles and Footings, Journal of the Eng. Mech. Dirision, ASCE, June 1984, 110(EM6), 937-956

8 Kausel, E. Forced Vibrations of Circular Foundations on Layered Media, Report No. R74-11, MIT Dept. of Civil Engineering, Cambridge, Massachusetts, 1974

9 Kausel, E. and Ushijima, R. Vertical and Torsional Stiffness of Cylindrical Footings, Report No. R76-6, MIT Dept. of Civil Engineering, Cambridge, Massachusetts, 1976

10 Luco, J. E. and Westmann, R. A. Dynamic Response of Circular Footings, Journal of Eng. Mech. Dirision, ASCE, 1971, 97(EM5), 1381-1395

I 1 Luco, J. E. and Westmann, R. A. Dynamic Response of a Rigid Footing Bonded to an Elastic Halfspace, Journal of Applied Mech., ASME, 1972, 39, 527-534

12 Luco, J. E., Frazier, G. A. and Day, S. M. Dynamic Response of Three-Dimensional Rigid Embedded Foundations, Cal. Univ., Report No. NSF/RA-780499, Natl. Tech. Inf. Serv., 1978

13 Veletsos, A. S. and Verbic, B. Basic Response Functions for Elastic Foundations, Journal of the Eng. Mech. Dit'ision, ASCE, 1974, 100(EM2), 189-202

14 Waas, G Analysis Method for Footing Vibrations through Layered Media, PhD Thesis, University of California, Berkeley, 1972

15 Wong, H. L. and Luco, J. E. Dynamic Response of Rigid Foundations of Arbitrary Shape, Earthquake Eng. and Struct. Dynamics, 1976, 6, 3-16

16 Wong, H. L. and Luco, J. E. Tables of Impedance Functions and Input Motions for Rectangular Foundations, Report No. CE78- 15, Univ. of Southern California, 1978

227 Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 4