Earthquake induced stresses in a poro-elastic foundation supporting a rigid structure

Myr,Err, A. E. & ME!.C. C. (1983). Geotechnique 33, No. 3,

293-3Technische Hogeschool

Dell'

Earthquake-induced stresses in a poro-elastic foundation

supporting a rigid structure

A. E. MYNETT* and C. C. MEIt

A fluid-filled poro-elastic foundation supporting a long and rigid structure is subjected to a normally incident Rayleigh wave. For sufficiently low permeability and high frequency, the pore fluid and the solid matrix move together as a single phase, except in a thin boundary layer near the ground surface. Hence the dynamics of most of the porous medium is governed by the usual laws of elastodynamics with effective moduli which depend on both constituent phases. In this Paper, the dynamic stresses and pore pressure beneath the structure are studied. When the structure is small compared to the wave length, the neighbourhood of the structure is

approximately quasi-static with time being a parameter only. The analytical method of Muskhelishvili in elastostatics is then applied to obtain the effective stresses near the structure. By mass conservation, the corresponding pore pressure is found immediately. Boundary layer corrections are added to complete the solution. The so-called outer solution presented herein apply also to the pure elastodynamic case without pores, and the method can be easily modified for any large-scale stresses in the background. Scattered waves, being small, are not considered.

Une fondation poroelastique remplie de fluide et supportant une structure longue et rigide a ete soumise une onde de Rayleigh d'incidence normale. Pour une permeabilite suffisamment basse et une frequence suffisamment elevee le fluide interstitiel et la matrice solide se deplacent ensemble comme une seule phase, a l'exception d'une couche mince limite pres de la surface du sol. La dynamique de la plupart des milieux poreux est regie par les lois usuelles de l'elastodynamique avec des modules effectifs qui dependent des deux phases constitutives. Cet article etudie les contraintes dynamiques et la pression interstitielle au-dessous de la structure. Lorsque la structure est de grandeur reduite en comparaison de la longueur d'onde le voisinage de la structure est approximativement quasi-statique, le temps n'etant qu'un parametre. La methode analytique de Mushkelishvili dans l'elastostatique est alors appliquee pour obtenir les contraintes effectives pres de la structure. La pression interstitielle correspondante est trouvee tout de suite par la conservation des masses. Des corrections des couches limites sont ajoutees pour completer la solution. La solution dite exterieure presentee dans

Discussion on this Paper closes on 1 December 1983. For further details see inside back cover.

*Delft Hydraulics Laboratory.

t Massachusetts Institute of Technology.

Lab. v.

Scheepsbouw\kur:de'

293

l'article s'applique aussi aux cas purement elasto-dynamiques sans des pores, et la methode peut etre facilement modifiee pour des contraintes considerables a l'arriere-plan. Les ondes disperses sont negligees, puisqu'elles sont de valeur reduite.

INTRODUCTION

In this Paper the application of a recent boundary layer technique for obtaining approximate solutions of poro-elastic problems in geotechnical engineering is explored. The specific task is

concerned with a comparatively small rigid structure resting on the surface of a wetted porous ground during the passage of a seismic Rayleigh wave. Being confined to the vicinity of the ground

surface, the Rayleigh wave is attenuated weakly from the epicentre in contrast to P or SV waves,

and can therefore be the most destructive to

buildings. Knowledge of the stresses beneath a building when attacked by Rayleigh waves is

important to the estimation of foundation stability.

Mei & Foda (1981) showed that for frequencies

common to

seismic waves (5 10 rad/s), the

permeability of most of the ground materials (sand, rock, clay) is so low that relative motion between

the pore fluid and solid matrix is insignificant except near the unsealed ground surface. This

permits a boundary layer approximation whereby

a one-phase (solid-fluid composite) description

suffices away from the ground surface (i.e. in the

outer region) while a two-phase description is

necessary only within a thin boundary layer near

the unsealed ground surface. If Hooke's law is

assumed between the effective stresses and

displacements of the solid matrix, then the outer

composite medium moves according to the laws of

single-phase elasticity with appropriate material properties. In particular the following effective

density, Poisson ratio and Lame constant emerge = nP. + (I n) + 2mv ve= 2(1 + m) AC=(2v + 1/m) 1

where the parameter

ARCHla

a

=

MYNETT AND MEI NOTATION

amplitude of vertical ground acceleration

speeds of p and sv waves permeability

wave number = 27r/wave length

tiG 1

= stiffness parameter

13 1 - 2v

static void ratio

dynamic pore pressure in the

fluid amplitude of hydrodynamic pressure degree of saturation time velocity of fluid velocity of solid matrix spatial co-ordinates

dynamic effective stress in the

solid matrix

dynamic total stress in the composite medium

shear modulus of solid matrix half width of the structure 1 + m(3 - 4) = effective elasticity 1 +m parameter 2vG/(1 -2v) = Lame constant of solid matrix n G = 1 - 2v /3 (4)

essentially represents the stiffness ratio between the solid matrix and the pore fluid, G being the shear

modulus of the solid matrix and 13

the bulk modulus of the water-air mixture in the pores,

while n is the porosity and v the Poisson ratio of the matrix.

If the total stress tensor Tr, is defined to be the sum of the effective stress cif and pore pressure p,

i.e.

where 1,1 = 1.2, then the boundary layer approach

yields the important result that the total normal component r and shear component T12 near the mud line y = 0 are essentially unaffected by the

boundary layer corrections in a and p, there

being negligible boundary layer correction for a12. Hence it follows that the traction conditions on the

mud line can be applied directly to the outer

problem, which can be solved first by existing methods of elastodynamics. Afterwards,

con-servation of mass implies that the outer pore

pressure is related to the normal stress components

(2v

+ 1 Ini)G

= effective Lame 1 - 2v

constant of composite medium

InK,

14 = effective elasticity

para-27z

meter

v Poisson ratio of solid matrix ve (1 + 2mv)/2(1 +m) = effective

Poisson ratio of composite medium

# bulk modulus of fluid with air

bubbles

6 boundary layer thickness

Si) Kronecker delta Ps}

P.

density solid ty of

(

fluid) p, np, + (1 - n)p, = effective

density of composite medium dimensionless independent vari-ables in the outer region

co wave frequency

;pi, dimensionless unknowns in the

outer region

ji dimensionless corrections in the boundary layer

( )° dimensional boundary layer cor-rection

()O _{dimensional outer} approxi-mation

( )R Rayleigh wave component according to Po = -12m(ail° + 0-22°) for plane boundary 1 i104-T22°) (6a, b) 2(1 +m)

strain. Near the mud line y = 0 the

layer correction is of the form

at =

Ph =

I -i

exp (7)

where the boundary layer thickness is given by

H-

(Gkr2

rtIG I - 2v 1-1/2

w fi 2(1 - v)] (8)

and .nt is chosen such that the outer pore pressure

and its boundary layer correction combine to

balance the externally applied pressure on the mud line.

As a

consequence of Hooke's law, the

,a

_

0 Tij = (5) a22b a Cp, ,n S x, y

-L

-

_2m

EARTHQUAKE-INDUCED STRESSES IN A PORO-ELAS1 IC FOUNDATION 295

boundary layer correction for the velocity or

displacement is negligible in contrast to the outer solution.

Although numerical schemes via integral equations or finite elements are available for many

geotechnical problems, it is preferable for both

economy and insight to apply analytical methods whenever possible. This is the case with small structures whose dimensions are much less than the elastic wave length. The region everywhere

outside the boundary layer shall be called the outer

region which can be further divided into the far and near fields from the structure. In the far field the wave length is the characteristic length. In the near field, i.e. the neighbourhood of the struc-ture, the characteristic length is the structural

dimension. Inertia effects are relatively small in this

near field so that both the local stresses of the primary wave and the induced stresses by the

presence of the structure are quasi-static. For plane

strain, the technique of complex functions can be applied as in Mei & Mynett (1981) and Mynett &

Mei (1982).

In this Paper we concentrate on the near field problem for a poro-elastic ground, the boundary layer corrections near the mud line being readily added afterwards. The results for the total outer stresses are directly applicable to a non-porous elastic ground for which the pore pressure is no longer significant, if the proper effective elastic constants are replaced by the actual constants for

the single phase medium.

The properties of the incident Rayleigh wave are needed first, in particular the approximate stresses

in the near field.

RAYLEIGH WAVES IN THE BACKGROUND

The space co-ordinates in the outer region are scaled by K-1 where K =2n11 is the Rayleigh wave number, and time by co-1, i.e.

X, Y = K(x, y), T= rat (9)

The dependent variables are scaled as follows: ?id = (u,°, vnl(alco)

(Air' "Fii)= (1)°, aij°1 T1j°)/P0 (10)

with

GKa

P0= ₍₀₂ (11)

where a is the maximum vertical acceleration of

the ground surface and equation (13) is implied by

Hooke's law. With reference to Appendix I where

the outer problem is reduced to classical

elastodynamics as in Mei & Foda (1981), the

Rayleigh wave solution is (Graff, 1975)

(/) = Re A, egY E

çfr = Re 11,e'v E (12)

where A, and A2 are constants, and

E = ex p[iK(x CRr)] =- exp [i(X T)] = 1 CR2 IC p2 , s2 = I CR2/Cs2 (13)

and CR = (o/K denotes the phase speed of the

Rayleigh wave. The outer velocity components are ao

D,R = Re (iA egY + sA 2 er)E

ax aY 00

= = Re (qA egY iA2esY)E (14a-b)

ay ax

where the superscript or subscript R denotes

Rayleigh wave. The corresponding effective

stress components, related to the displacement of the solid matrix by Hooke's law, are given by

2v ₂

7 71)

R = Re

[(

1-2v 1_2v) x iA, 2sA2e2 E

[(2

2vq 2 2v 2v 1

2v)

x iAi eqr + 2s A

elE

a 12R =

Re[-2q,

, +(1 + s2) i A 2

elE

(15a-c)

while the outer pore pressure follows immediately from equation (6)

=

Re[1/3

(q2 1)A

eglE

(16)

nG

From equa ion (5), the corresponding total stress

components are 2v

2-2v

R = Re

f[

q2

l 2v

1-2v

+(q2

iA,eqY2SA2esY}E nG .122R= Re12 2v 2 21

1-2v

1-2v

+(q2

_nG 1)1 +2sA,

ea}

E

2R = Re [ 2qA, egr - (I + s2)iii,esr]E (17a-c)

By virtue of the boundary conditions on the ground surface i22R TI2R = 0 (18) "6-2!= Re q2 egr 2

where Y= 0; Al and A, are related by

2g

A, = iA1

1 s2

and the following expression for the wave velocity CR is obtained

_CR_{= (087+ 1.12ve)/(1 +v) (20)}

C,

Once CR is found, q and s are known and the outer solution for the Rayleigh wave is complete. From equations (14) and (19) the horizontal and vertical accelerations of the ground surface are

a,R(x,o,T) = Re A , 1 1+ s2 2qs a2R(X,O,T) = Re i A ,

q(

s22)E 1 + s (21)

Since the vertical ground acceleration a is used for normalization, we must have

A =

1+52 I (22)

1 s2 q

A1 is defined to be real so that the outer vertical acceleration of the ground surface is

(a2o)I2

(19)

where Y = 0.

The boundary layer correction PR is obtained by specifiying that PR + = o on Y = 0 (Mei & Foda, 1981; Mynett & Mei, 1982). From equation (16) we then obtain i 13 22R = R = Re (q2 1)A, nG [ex YAE

P V2 6

(24)

which must be added to

and PR for the

solution. Some numerical results have been given by Mei & Foda (1981).

Our aim is concerned with a rigid rectangular structure of width 2L and height H in welded

contact with the ground surface. We now add the

assumption that the typical structure dimension

(here the base width) L is much less than the

Rayleigh wave length, yet much greater than the

boundary layer thickness 6, i.e. IC"' »L»6. The

Rayleigh wave solution can then be approximated

for Kx,Ky«1

in

the neighbourhood of the

structure.

NEAR FIELD APPROXIMATION OF THE BACKGROUND RAYLEIGH WAVE The near field dimensionless variables are

( (x, y)/L = (X, Y)/KL, T = tot (25)

v

= Re (q2 1) A ,

1v nG

.612K = 0

while keeping the other normalizations of

equation (10) unchanged. The Rayleigh wave solution (equation (15)) are then expanded for

small KL = E« 1. Keeping only the linear terms in the expansion and taking the real parts we get

tf(l) +f(3)

5-22R = E./(4) +1)6) & 12R =

(7) EPs) ./(9)

where the expansion coefficients are

= A,

( 2v

q2 2 2v

4sq )

1 2v

1 2v

1 +52 COS T

fa)

2v _{q 2}

2-2v

4s2

2v

1-2v

1 + s2 _To) _A

i( 2v

1 _21,q

1-2v

+1+52 sin r

2-2v

4sq )

=

Ai(2-2v

.72 2v 4sg

1-2v

1-2v

1+52 sin r f(6) f(7) (26)

Similarly, the normalized pore pressure and the

solid displacements can be expanded for small K L to give

pit Ep(I) Ep12) p(3)

CR a_ piR _oift Eu(I) +Eu(2)1+ ui3)

pit

7R_

-2R_{a = EV 0/(2) + 1/(3)(28ac)}

where the coefficients for the pore pressure are po) A1 (g- I) cos t m ( I 2v (2) A

(q2 1)

.

sint

P

-

_{(I 2v)}

p13)= A (q2 l)sin r (29ac) m (l 2v)

The coefficients for the horizontal displacement

components are i)

2sg )

A

,(

I

+s, sin r

U2 = A I ± S2 1(q 2s2 q)COS T f (4) =A

,(2

2va 2 2v

4sq )

I 2v'

1-2v

1+52

cost

f (5) A 2 2v 2 5 +2.52) =

i(

q2

1-2v

I12.v 14 sin r1

=0

= A , 2g(s q) cos t

=0

(27ai) = a sin K(x t) (23)

)

-= &1R =

f(4

(

sin

-= +

-q

U3=

-

AI (1 2sci

1+s2) cos

T (30a-c)

and the coefficients for the vertical displacement components are 1/(11= - A

i(q

2)

I" 1+S2

v(2=

A1( 2s , sin 1

+)

s-2q

V(3) = A

i(q (31a-c)

sin r I +s2

All coefficients have a simple harmonic time

dependence. Also, since both fiR and 6R are linear functions of ( ,ri) the same holds for o If the wave field in the half space is that of P-incidence or SV-incidence, as discussed in, e.g. Graff (1975), similar approximations result

in linear stress fields too. In other words, our

subsequent analysis can be readily modified for

any non-Rayleigh waves by changing the expansion coefficients f('), TO, etc.

These results give the background stress field.

We now examine the additional effect of the

structure.

GROUND RESPONSE DUE TO THE MOTION OF THE SMALL STRUCTURE

In general the structure not only scatters waves

but also radiates waves by its induced motion. Around the structure whose dimension is L the variation of stresses must be comparable to that of

the Rayleigh waves, and is 0(KaKLGIco2). By

Hooke's law the associated skeleton displacement relative to the Rayleigh wave must be 0(K2aLlw2). Hence the associated skeleton inertia per unit

volume is 0(i) K2aL2). Now the stress force on a

unit volume of the skeleton is 0(K2aG1w2). The ratio of the skeleton inertia per unit volume to the stress force acting on it is then

0(p w2 L2/G,_{) 0 Psw2 2m}

K (KL)2

G

= 0(KL)2 «1 which is very small since

0(p,co2IK2G)= 0(1)

(cf. equation (65), Appendix I). Consequently the induced stress field is also quasi-static. Below the

boundary layer near the unsealed mud line, the

outer boundary value problem for the near field is

to seek a biharmonic function f subject to the

boundary conditions

-.122 = il2 =0 (32) where I I > I, =

= u

0 (33)

r= vo+Ewc, (34)

where I <1, = 0; Uo, Vo and Wo are the

horizontal, vertical and the angular displacements, respectively, of the rigid base, which are undetermined functions of T. The dimensionless

rotation Wo is 0(r,) times smaller than the dimensionless vertical displacement, owning to the

long wave length. The background stresses

function, due to

the free Rayleigh wave FR , depends on and rl linearly, so that the difference F PR must not have a linear growth as

+112)1,2 _{CO. As in Mynett & Mei (1982), the}

EARTHQUAKE-INDUCED STRESSES IN A PORO-ELASTIC FOUNDATION 297

problem can be separated into two parts

=

+ P

(35)

where the superscript S designates the disturbance

due to the structure. Now Fs is also biharmonic

and must satisfy the following boundary conditions on r! = 0 1.22s = 12s (36) where I I >I Cs=

CR = v0u13)Euo)

(37) = vo +Ewo OR =

V0

v(3) +E(wo v"1) (38) while should vanish away from the structure, i.e.

0 as r (c2 ,12)2. (39)

The above boundary value problem can be

solved by the function-theoretic method of Muskhelishvili (1977). Specifically, we search in the complex r, plane

= +M) for

an analytic

function 0g)

in terms of which the stress

components are

ills+i225=

2[00+(C)]

(40)

i22S ii12S = 46V;) (41)

while the displacements satisfy

0

-

_ 2 [Us+ i c c¢' (0 (42) The parameter,

k` -

1 +in 1+ m(3- 4v) (43)

varies monotonically from 1 to 3-4v as in varies

from 0 (incompressible fluid) to cc, (dry soil); see Mynett & Mel, 1982 for a plot. On the mud line, vanishing of surface traction z12 = i22 = 0 over

I I > 1 leads (from equation (41)) to the boundary

condition .0

P

= 0 + + I

298

+ =0 (44)

where I I >1, = 0, (k+M (1)(+i0) and

(1)(xi0). Since the displacement is formally prescribed over I I <1 we have from (equation

(42))

a

()+ K (;) = (us + vs) 2g5'() (45)

a

where I I <1, = 0, II s'(*) = WO fitt'() and 0 gR'

= (//g

+il7R)

'

2sq = { 1 1 + 52 sin .1- -1; 2q

i q

)

/

1 +s2 COS T

The corresponding solution is

X(Of=2rci _ X +

C)

+ ( W, + iW2)X(C)

The near

field

stress components may be

obtained from equation (40) and (41), and the

amplitude of rotation Wo and the complex function of time (W, + iW2) are found. These coefficients

may be

obtained by invoking the dynamic

equilibrium condition of the structure. DYNAMIC EQUILIBRIUM OF THE STRUCTURE

While the inertia of the ground matrix

is

negligible in the near field, inertia of the structure is

of the order 0(psal2) and is comparable to the

force acting on the structure by the ground

0(K2 aG121w2), since the ratio of the two forces is 0(p,, w2IGK 2) =

A small structure in long Rayleigh waves is

expected to follow

the motion of the wave

displacement which is 0(aco- 2). In contrast, the

additional displacement due to the induced motion of the structure is only

(K2 LaG)L

(02 G

(KL)2

_(.72«072.

a a

Thus the acceleration of the structure can be

approximated by the acceleration of the Rayleigh wave at the centre of the base,

= = 0). Using

this, the structural inertia in dimensionless form

becomes Ma,° 2sq horizontal

=,(1

cos r Po L 1 + s2 (50) and M a vertical 2q sin r (51).

P L

cl A

,(q

s2

where M is

the mass of the

structure and

M = Ma/P0 L. There are two contributions to the inertial moment about the origin. The first is due to the height of the centre of gravity 1112 above the ground surface. This inertial moment is of 0(p, 1.,3 a) for H 0(L). The other is due to the

angular acceleration of the structure 1w20. Now / = 0(p, L4), and 0 = 0fK(alca2), hence

1w20 = 0(p,L4K(a w2) co2) = 0((KL) ps a 12) which is much less than the first. Hence the inertial moment is dominated by IC1(H12L) A,(1 2sq

cost

+s2

+s2-2sy

= 11-4-(HI2L) 1

q(1 s)

cos r (52)

These inertial forces must be balanced by the total forces acted on the base of the structure.

From equation (43)

(22

12) = (/)-()+ (PIO

(53) where I <I, since the Rayleigh wave part does

not have any surface traction. Using equation (47) and Plemelj's formula we can show that

1i22()

al 2(0) + 1 =

gs'()+

X + ()(WI +1W2) K, K 1 X +() 11 (54) 'C 2ni

J-1 X+(0)(0)

where the last integral is the Cauchy principal

value. By integrating equation (54) over the entire base, we obtain the resulting forces which are then

used to

write down the linear and angular

momentum equations for the structure.

Ci22()+iii2(0](14.

-=

/i3A,(q

2g )sin

r 1 +s2 2sq ia A (1 COS I ±S2

f

= Re f 22 -1 (55a) where X(C) = (t.", -1- 0- 1/2 +ig.( 1)- 1/2 with In lc Pc = " )7r

-4)

I

06)

= (47) (48) (49) 0(1). _1 = 2q,l(G)

---= ICI(H/2L)'A (1

1 + s2

2sq )

CQS T (55b) Equations (55a and b),constitute three real

equa-tions to determine the three unknown coefficients Wo, W, and W2. In the Boussinesq theory for a,line

/load, the linear displacements U0 and 1/0 are

indeterminate in the two-dimensional context. As gs( *) is constant with respectto all the integrals in equation (55) can be evaluated by the techniques

explained in Muskhelishvili (1977). Details are

available in Mynett (1980). From equation (55a) we have 2 . 1-1.1 ±ii12):g .cl. Kc 1 ,

1

i

1 ' -+ (.11/,+1W2)i .V(0c1 1 K, + 1 +Kc+ 1 2_ ,f

o

X

f

Kc

"

f 2nt 's _1

+(o),I(o 0

'I X+ I

go

2q5 'OW, + iW2)27r1 e _2gS4 Ice ' Kc -_--7: 2ni(W1 + iW2) = A71 . 1 1,(q -

)sint

1 +s2 2q Ct A 1(11

)

cost

i +s2 2sq

Taking the reall and imaginary parts we obtain (1 =-2ir 1.+s2 COST 2sq Wv = _A - i - A , 1

2q )

2n I + .92 sin T

Also from equation; (55) we get

lC.

111 (--i22+7'El2gd gs'

f

t (W! .i14(2)1 4".(0)rd

-K+1

fa+gi

e 27ti j X + (o) +iw2)2ikc+h2.n+igs1:(1' + 4/42) (58)

Equating the real part of the preceding ?result( to equation (52) yields 2n wo W2 zg,

(1 +.4p)

c,-F 1 (56). (57a, b) 2so (H/20 A I ' I +S2)cos T (59) which gives W0.

With the outer total stresses thus determined

throughout the near

field,

the pore pressure

follows from equation (6). Afterwards a boundary layer correction similar to equation (24) for pore pressure and effective stresses can be added. All of

Fig. 11. Dynamic outer pore pressure f) = e/Pc, near a 'rigid'

structure subjected to a seismic Rayleigh wave. Dimen-sionless time r = cut is marked on the right of each figure EARTHQUAKE-INDUCED STRESSES IN A PORO-ELASTIC _FOUNDATION 299

K, 1 Ai) = K, -0

= (W,

-2 -3 -5 03 2 01 00 0-1 0 = =

this should be superimposed to the background

Rayleigh wave solution (outer part plus boundary

layer correction) in order to obtain the complete

stress field.

NUMERICAL RESULTS

To demonstrate the essential features of the

complete solution, we present some sample results of computed dimensionless pore fluid pressure and effective solid stress components for the following inputs H/2L = 1; pjpw = p structure/p

water = 2-5; m= G/13 = 1 (with n = v =

a = vertical ground acceleration = 0-1g and

KL = E = 04.

Thus the structure is a solid of square

cross-section. The value of the

stiffness parameter m = 0/13 = 1 corresponds to a compressible pore fluid (for fully saturated pore fluid on =G/13 = 0-0044; see Mei & Foda, 11984

First we present the dynamic outer solution in the neighbourhood of the structure, by including

the near field approximation for the background

Rayleigh wave field, but ignoring the correction in

the boundary layer which

is

too thin to be

incorporated in the contour plots. Quantitative

correction due to the boundary layer presence will

be considered later. The static response to the

weight of the structure is also excluded. In Figs* 1

4 contours of p and are plotted for different

;instants in a half period cot = r = 0, n/4, n/2,

3n/4, n. These results show the modification of the 'background pressures and stresses in the vicinity of the structure. After referring to equations (26)

(29) for the behaviour of the Rayleigh wave

solution, we may observe for the particular instant

- 11

of time cot = = 0 that the outer solutions io

and all are antisymmetric

in while the corresponding stress components of the Rayleigh

'wave in the background are also antisymmetric.

The shear stress 5.12, however, is symmetric in

both in the background Rayleigh wave in the

resulting field.

However at cat=t= n/2,

the solutions for 15, and a22 become symmetric, while Er,2 is antisymmetric in In general the

solutions behave quite differently after one quarter

wave period, but return to their original form

except for a sign reversal after one half wave

period.

The outer pore pressure does not vanish at the

ground surface as the boundary condition requires, therefore a boundary layer correction is necessary. In Fig. 5 the value of j'7(, 0, "C) is plotted for t = 0, n/4, 7r/2, 3n/4, it. The required boundary layer correction is simply the real part of

*Throughout Figs 1-4, the abscissa is = x/L, while the, ordinate is = y/L..

77

77

77

Fig. 2. Dynamic outer effective normal stress component = (7110/P0 near a rigid structure in Rayleigh wave 1/3); = = 2 3 4 -77 2 3 4 -5 0 0.3 0-0 0-1 0.2 0.3 2 = -5 -3 3 5

-2 3 0 1 2 3 1 2 3 -4 2 0-0 -0-1

EARTHQUAKE-INDUCED STRESSES IN A PORO-ELASTIC FOUNDATION 301

0.3 0-2 1 0.1 - 0-1 - 0-2 - 0-2 -03 -0-3

404

5 "7 - 0.1 - 0.2

\

-03 0.0 -0-1 0-0

0.0-

-4--5

Fig. 3. Dynamic outer effective normal stress component

= o-22 Po near a rigid structure in Rayleigh wave

0.2_---N-0!'t0 3r -4-4 0 '7 o -3 - 4 -5 0 -1 -2 -3 -4 -5 0 -1 -2 -3 -4 -5 -5 -3 -1 1 3 5 I Qz_CV I 0-1 --N?L° 076

-_---0 .1 0 1 ---°. 2 0 -3 0-5 -

- 11---(07.0

- 0 .2 -- 0.6 0-1 - 0.2 3r -4- '7 o

-

1 - 3 -4 -5 -1 -2 -3 -4 -5 - - '01 -0-01 -0-02 -0-05 0.01 CY05 0.02 oa

Fig. 4. Dynamic outer effective shear stress component a-12 = 120/P near a rigid structure in Rayleigh wave

0.1 _0-2 0-01 0-10 0-02 0-2 0-1 -0-10 - 0-01 - 0-02 -5 -3 -2

302 1 -4 -2 0 2 4 08 0.4 -04 -08 0 8 of= 0 (of = IT 0.4

u

---

0 0 -0.4 -0.8 ^ 0.8 _cot= IT 4 3rr (Dt = -4--0.4 p 00 -0.4 -0.8

Fig. 5. Dynamic outer pore pressure ;5 = elP along the I

free surface (ti = 0) near a rigid structure in Rayleigh wave= -4 -2 0 2 4

EARTHQUAKE-INDUCED STRESSES IN A PORO-ELASTIC FOUNDATION 303

--):5(,0,r)exp (1

)E

The boundary layer feature for the same instants is shown in Fig. 6 where the cross-sectional profiles for the pore pressure are plotted for

= 2 and 2.

The solid line represents the complete solution including the boundary layer correction, which

vanishes at the free surface; and differs from the

outer solution (represented by the dotted curve)

only in the thin region near the free surface. In Fig. 7 we further plot-fr.,i22 and i 2 along the

base I

_<I,

= o-. in is nearly the same as j) all the time so that the effective stress all is relatively small and the dynamic load is largely borne by the pore fluid. This sharing of load is expected to vary with the degree of saturation which influences the rigidity m of the composite medium.

In conclusion, we have shown that the dynamic interaction of a rigid footing with a fluid-saturated

foundation can be solved by an approximate

analytical method if the porosity is low and the

footing is much smaller than the wave length but

much larger than the thickness of the boundary layer. The same idea can of course be extended

to three dimensions. ACKNOWLEDGEMENTS

We are grateful for the support of US National

Science Foundation through Grant (CME

792-1993). Computational support was also received

from Delft Hydraulics Laboratory during the

preparation of this Paper.

APPENDIX I. ELASTIC WAVE EQUATIONS FOR A TWO PHASE MEDIUM

The linearized Biot (1956) equations for the momentum conservation of the solid and fluid phases are

avi Op

n-p,(1 n) =

(1 n)+(ui vi)

(60) Or ex; Ox, k au, Op n2

pen

= - n---

(141- (61) at ex, k

Where Darcy's law of flow resistance is incorporated as the last term in each equation. Hooke's law is assumed between the effective stresses and strains in the solid skeleton. Subtracting the geostatic part, the dynamic Hooke's law may be written as

Oa, Gfar. au;)+ Aeu,

(62) et (3)e.,

3.0

ex,

The equations for conservation of mass in the two phases can be combined to yield the storage equation

a ay, nOp

n(Ui

= (63)

ex, ax, # et

where 13 is the bulk modulus of the pore fluid which depends on the degree of saturation S. the bulk modulus of thefullysaturated water /30 and the total pore

71

-

17-0 17-0 -5 -02-01 01 0.2 -0.2-0.1 0.10 2 3ff

Fig. 6. Cross-sectional profilesofpore pressure. Solid line: total (p°+ph)/ P and dotted line: outer approximation p. The scaleof the pore pressureisindicated inside the figures

5 3 -1 1 -5 --0.2-0.1 -02 0.1 02

ti

Fig. 7. Stress and pore pressure distribution along the

base of a rigid structure in Rayleigh wave. Curve 1:: curve,

2: 2 29 curve 3: t2 4s, 0 8 0 4 0 0 -0.8 08 0.4 0.0 0 4 0.8 1 -08 -04 00 0.4 08 0.8 0'4 0-0 0-4 -08

pressure po

1

I S

= + (64)

Pb Po

In terms of the dimensionless variables defined in equations (9)-(11), the dimensionless momentum equations (60) and (61) are then characterized by two ratios: ow2/G10 and tuolkGK2. The first ratio signifies the importance of inertia while the second interphase resistance. For a wide variety of soils and earthquake frequencies

EARTHQUAKEINDUCED STRESSES IN A PORO-ELASTIC FOUNDATION 305

ruolkGIC2» I (66)

From either momentum equation follows the result that

The storage equation (63) reduces to

. T

"G

13 OT

The momentum equations can be combined with equation (67) to give

p,c02 /2 a2

v2 +

V (V i;)- V (69)

G Pt 2 1-2v

where )2 denotes the effective density defined by equation (60). Hence, by eliminating 0j5/0-r

Nco2 12 02 172

(

1 /3

= V2 + + V(V (70)

G at2 1 -2v nG

The velocity vector of plane strain elastodynamics can be described by two scalar potentials 4, and 0, i.e.

It = V0+V xiiez (71)

where ez = (0,0,1).0n substituting this into equation (70) and taking the divergence and curl alternatively, the elastic wave equations are obtained

zic+ 2G 2

i1r2 p,w2 12 V (/'

02 IP G v2

(3,2 p, a,' 12

where and correspond respectively to compressional and shear waves with propagation speeds Cr and C

REFERENCES

Biot, M. A. (1956). Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoust. Soc. Am. 28, 168-191.

Graff, K. F. (1975). Wave motion in elastic solids. Ohio State University Press.

Mei, C. C. & Foda, M. A.(1981). Wave-induced responses in a fluid-filled poro-elastic solid with a free surface a boundary layer theory. Geophys. J. R. Asir. Soc. 66, 591-637.

Mei, C. C. & Mynett, A. E. (1983). Two-dimensional stresses in a saturated poro-elastic foundation beneath a rigid structure. I: A dam in a river. Numer. Analyt. Meth. Geomechan. 7, 57-74.

Muskhelishvili, V. I. (1977). Some basic problems of the mathematical theory of elasticity. Amsterdam: Nordhoff.

Mynett, A. E. (1980). Dynamic stresses and pore pressure in poro-elastic foundation beneath a rigid structure. DSc thesis, Massachusetts Institute of Technology. Mynett, A. E. & Mei, C. C. (1982). Wave-induced stresses

in a saturated poro-elastic sea bed beneath a rec-tangular caisson. Geotechnique 32, No. 3, 235-247.

pco21G10 = 0(1) (65) Cr2 = + 2G)//),; C, = Glp, (74)

and in physical dimensions.

(68)

-(721