Wave induced stresses in a saturated poro-elastic sea bed beneath a rectangular caisson

(1)

Sea waves incident on a rigid caisson induce stresses in the sea bed by the direct action of wave pressure and by the vibration of the caisson. Under the assumption that the sea bed is an isotropic poro-elastic half-space saturated with water, the dynamics of the two phases are

governed by Biot's equations. To circumvent the

diffi-culties of solving the coupled equations an approximate

scheme is applied to achieve an analytic solution. The problem can first be reduced to a series of classical elastostatic problems which can be solved by complex

variables, and then corrected for seepage near the mud-line. Numerical results are presented.

Des vagues marines qui frappent un caisson rigide

engendrent des contraintes dans le fond marin en raison

de Faction directe de la pression des ondes et de la vibration du caisson. Si l'on considere le fond marin comme un semi-espace poro-elastique isotrope sature Teau, la dynamique des deux phases est regie par les

equations de Blot. Afin de contourner les difficultes que presente la resolution des equations couplees, un systeme approche est applique de maniere a obtenir une solution analytique. La question peut etre ramenee d'abord a une serie de problemes elastostatiques classiques qui peuvent

etre resolus al'aide de variables complexes et corriges

ensuite pour tenir compte de l'infiltration a proximite du fond de la mer. Des resultats numeriques sont presentes.

INTRODUCTION

In dynamical problems of a deformable porous medium, the motions of fluid and solid phases are coupled by Biot's equations. Even under the assumptions of linearity, it is difficult to solve analytically boundary value problems of geotech-nical interest because they are often of mixed type where stresses are prescribed over part of the boundary and displacements over the other. Most workers who use Biot's equations for studying structure-foundation interactions rely on essen-tially discrete numerical techniques.

Mei & Foda (1981a) point out that the usually

Discussion on this Paper closes 1 December 1982. For further details see inside back cover.

*Formerly Massachusetts Institute of Technology, now Delft Hydraulics Laboratory.

t Massachusetts Institute of Technology.

235

NOTATION

wave amplitude

Z. dimensionless coefficient of cohesion shear modulus of solid matrix wave number, 27r/wavelength

at stiffness parameter, nGj It( 1 2s)

n static void ratio

n' perturbation of void ratio

p dynamic pore pressure in the fluid

Po amplitude of hydrodynamic pressure S decree of saturation

velocity of fluid velocity of solid matrix

fi bulk modulus of fluid with air bubbles boundary layer thick ness

). Lame constant of solid matrix, 2vG/( I

r Poisson ratio of solid matrix p, density of solid

density of fluid

dynamic effective stress in the solid matrix

dynamic total stress in the composite medium

0, wave frequency

q, r dimensionless independent variables in the outer region

dimensionless unknowns in the outer region

dimensionless corrections in the boun-dary layer

( )5 dimensional boundary layer correction ( )" dimensional outer approximation low permeability and high wave frequency in problems of aeotechnical interest imply the existence of a thin boundary layer near the mud-line. Outside the boundary layer the pore fluid and solid matrix move in phase but relative motion (seepage) is important within it. Moreover they find that in the boundary layer, seepage does not affect the total shear stress and the total normal stress component in the vertical direction. This makes it possible in many cases to solve first an ordinary clastostatic or elastodynamics problem, and then to make a boundary layer correction.

Lab.

v.

ScheepstouvvlumL

Mvm,.-r-r. A. E. & Mhi. C. C. (1982).Georechnique 32. No. 3,235-247

Technische Hogeschool

Wave-induced stresses in a saturated poro-elastic

seQektd

beneath

a rectangular caisson

ARCH1EF

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

(2)

236 MYNETT AND MEI

Additional applications are given by Mei & Foda

(1981b, 1982). The detailed arguments of the

boundary layer approximation are given by Mei & Foda (1981a).

This recent technique is applied here to the

problem of sea waves attacking a caisson which is

in adhesive contact with a poro-elastic sea bed.

For mathematical simplicity, the caisson is taken

to be infinitely long and the waves are incident

normally, so the situation is two-dimensional. The

sea bed is assumed to be an isotropic half-space with homogeneous material properties for both

fluid and solid phases. The hydrodynamic pressure

of the standing wave on one side of the caisson

directly affects the sea bed on the same side. The induced motion of the caisson further modifies the response in the surrounding sea bed. The object is to calculate the entire dynamic response. The most

involved part of the mathematical task is the solution of a mixed boundary value problem in

elastostatics for the outer approximation. BOUNDARY LAYER APPROXIMATION

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

phases are av, On, Op n2

p,(1n)

=

(1 n) +

v.i) (1) at exi ax, k au, op Op n2

pwn = n--n---(tt,v;)

(2) at axi ax; k

Darcy's resistance is incorporated in the last terms. Hooke's law is assumed between the effective stress

and strain in the solid skeleton. Subtracting the geostatic part, the dynamic Hooke's law may be

written as

o

io.

ay.)

ev,

Or

cr;

= G +

ax, aXi axK - (3)

The equations for conservation of mass in the

two phases can be combined to give

av, n Op

n(11iv)+ =

(4) ex,

ex,flOt

where ,3 is the bulk modulus of the pore fluid which

depends on the degree of saturation S. the bulk modulus of the fully saturated water /30 and the

total pore pressure pc, (Verruijt, 1969)

1 1

1S

(5)

fi

=

fib Po

The boundary layer is defined as a horizontal

slab of thickness 0(c5) below the mud-line and the

remaining part is called the outer region.

Everywhere in the porous medium a physical quantity f is written as

f

fo +.fb ₍₆₎

where f ° is the outer part and r is the boundary

layer correction which is important only within the layer.

Outer approximation

In the outer region the typical length scale

L (»<S) is either the width of the caisson or the

length of the sea wave. If Po is the characteristic

dynamic pressure due to waves it is natural to expect that (p°,6,1')= 0(P0). Hooke's law then

implies (u10, v1°) = 0(P0 toL/G), where co is the

typical frequency. Relative to the terms

and ap/ax,, the inertia terms are of the order

p, to' L/G = w2 L/G),

where Darcy's resistance is of the order tuoL2/Gk.

Typical sea wave frequencies are in the range

= (0.5 1) rad/s, the caisson width is

L = 10 50 m and the sea wave length is L =

0(100 m). For a sandy sea bed, representative

values are n =

G = l0 108 N/m2,

k= 10- 6

10'

s/kg. Hence tuol2/Gk = (102-104) and pot' L'/G = l0_6_ 10-4

It is therefore assumed that pw2 L/G « 1

no)12/Gk » 1 (8)

Equation (6) implies that inertia is negligible in equations (1) and (2). Adding assumptions (7) and (8) gives

acipaxi = Op'/0x1 (9)

This suggests the introduction of the total outer

stress tensor

= ou°

(10)

Equation (9) then becomes

= 0

which is the well-known equilibrium equation of classical elastostatics. An important consequence

is that radiation of elastic waves through the sea

bed due to caisson vibration is negligible.

With inertia omitted, assumption (7) implies

that

vi° (12)

so that the two material phases move as one. In = etaii/exi. 0-3, m3 (7)

(3)

-addition, it follows from equation (4) that

?if

/301;k°

= (13)

et n axk

where /3 is the pore fluid compressibility which depends on the degree of saturation. Combining equations (10) and (13) the stress-strain relations in equation (3) for the outer field can be rewritten in terms of total stresses, i.e.

where the effective Lame constant is =

ex j ex,

+/

-e( X5 Cl

eVi° et).°) evk°

2vG

=

2v +n

v being the Poisson ratio of the solid skeleton. Thus far the results are valid for three dimensions. For plane strain, tensor contraction of equation (14) using equation (13) relates the outer pore pressure to the total stress components according to o (Tll'H-T22')

P=

_{2( I + m)} where G n m = 11 1-2v

is essentially the stiffness ratio between the solid matrix and the pore fluid. Hence if the outer problem for the total stress components can be solved, the corresponding pore pressure is obtained immediately from equation (14). However, in general the outer pore pressure will not satisfy the appropriate boundary conditions along the free surface, and so a correction is required.

Boundary layer correction

Within the boundary layer, the characteristic length scales in the horizontal and the vertical directions are Land 6 respectively withLli5»1 so

that e

0(L)--/°

b j, = 6 ex (18) (14) (15)

WAVE-INDUCED STRESSES BENEATH A CAISSON 237

The reasoning leading to equation (18) is described by Mei & Foda (1981a). For simple harmonic motion the solution of equation (18) is given by

pb ₌_(Re)F0(x)exp

[I,/2

6 not]

(21)

where the boundary layer thickness is

(Gkr2 ( 1

2v nG)-1-1

)

2( I v)

)

which depends on the stiffness parameter tn. F0(x) is the spatial variation along the free surface, which has yet to be determined. The corresponding corrections for the effective stress components are

b

allb =

_1vP

a2 2b Pb

(22)

From this it can be found that the pore pressure correction should satisfy the one-dimensional Terzaghi equation y= H rik y= ti Caisson a epb 02 pb Pc Pw (19) a y2

where C is the consolidation coefficient Gk

C-

(20) Sea bed/

(1 2v)[m+1/2(1 v)] Fig. I. Problem analysed

b_0

₍₂₅₎

Equations (24) and (25) in conjunction with equation (10) yield the important result that no boundary layer corrections are required for the total stress components r,, and r22 because

,226= 0 ₍₂₆₎

Hence, if the boundary conditions are expressed in terms of the total stress, the outer problem (equation ( I I )) can be solved a priori as an ordinary elastostatic problem, and the outer pore pressure is obtained from equation (16). A variety of applica-tions has been explored by Mei & Foda (1981a,

1981b).

The problem analysed in this Paper (Fig. 1) involves mixed boundary conditions: displace-ments along one part and stresses elsewhere. However, from Hooke's law it can be shown that the boundary layer corrections for the solid displacements are negligible compared with the outer displacements, specifically

t,ib _0(5/L)2vio ₍₂₇₎

V2b co/L),20 (28)

Hence the displacement conditions along the G

at

X23) (24)

(4)

238 MYNETT AND MEI

structure base can be applied directly to the outer

problem. which is again one of elastostatics,

although still of the mixed type. After the outer solution is obtained, boundary layer corrections are easily made according to equations (21) and

(23)-(25). Thus the primary mathematical task is to solve the outer problem.

FORMULATION AND DECOMPOSITION OF THE OUTER PROBLEM

A rigid caisson of height H and width 2L rests on the horizontal sea bottom in water of depth h. Monochromatic gravity waves of frequency oi and

amplitude are normally incident from the left

and then reflected. As the movement of the caisson and of the sea bottom is expected to be far less than the sea wave amplitude ao, the flow field above the

sea bed and to the left of the caisson (x<

LI is

essentially that of a standing wave of which the hydrodynamic pressure is, for x< -L

cosh K

p = (Re) Po cos K(x +1) exp(-iwi)

cosh Kli

(29)

where Po = 2p To the right of the caisson the

fluid is undisturbed so that for x > L

p = 0 (30)

Therefore, the boundary conditions on the

mud-line for the complete stress field (i.e. for the sum of outer and boundary layer stresses) arc

7,2= 0

(31)

= -p = -(Re)P0

sech Kli cos K(.v + L)exp(- hot) (32)

where x< -L, y = 0, on the up-wave side, and

722 = 0 (33)

where x> L, y = 0, on the down-wave side. It is assumed that the caisson base is in welded

contact with the sea bed,

i.e. there is infinite

friction. Let 1/) denote the solid displacement,

related to the velocity by v1=

'/it. Then the

conditions beneath the caisson for I x I <L, y = 0, are

V, = (Re)U0exp(- hot) (34)

V, = (Re)(Vo+ flo x)exp ( - hot) (35)

where UO, Vo and

are the amplitudes of

horizontal and vertical translations and rotation of the base about the origin. These amplitudes are all unknown at first.

The outer problem is quasi-static, and so the

stress and displacement components at all points are in phase; hence the harmonic time dependence

can be factored out and attention given to the

spatial variation only. Dimensionless outer

vari-ables, distinguished by (-) are now adopted.

y) = LG1. = q) (36)

(p",c 7,j", r,°) = (Re) Po(ii, exp ( -ion) (37) L

j", = (Re) i'dexp( -ion) (38)

P0 L,

= (Re) - 17,exp( -not) (39)

Ii = Lii (40)

K = L (41)

The governing equations for the spatial part of the outer problem are now

= (42)

= 0 (43)

ill +1'22 =

-27(1- (44)

where in is given by equation (16). By using

equation (26) the boundary conditions (31) (33)

can be rewritten as '112 = 0 (45)

= --P =

A ocos + 1) (46) for ,;=< -1, q = 0 and il2 = 22 = 0 (47) for ,;> I. q = 0 where jo = sech (48)

Beneath the caisson the normalized conditions (34)

and (35) also apply to the outer displacements

directly, so that

= Co (49)

= 17o + ,1 (50)

for IJ.,I < I,i = 0.

The outer boundary value problem governed by equations (44), (45)-(47), (49) and (50) is reduced to

two well-known problems of elastostatics as is

done by Mei & Mynett (1982) for the strictly static

problem of seepage through the deformable

ground beneath a dam.

In the first problem (A) the shear and vertical

stress are prescribed for the enitre mud line, i.e.

, = ,,10 cos KG.:+ I) (51)

= 0 (52)

for

- I.

= 0 and

172A 7e,,A 0 (53)

for

> - I,

= 0.

In

the second problem (B) the shear and

(

(5)

vertical stress vanish for I ;1> 1 but the displace-ments are prescribed to be the base displacedisplace-ments (equations (49) and (50)) minus the induced

displacements of the first problem. i.e.

122B = ii2" = 0 (54)

for )1 = 0 and

for I I, = 0.

For half-plane problems use is made of the fact that only the horizontal gradient of the diplace-ment affects the stress field. The superposition of both these problems is the original problem.

These elastostatic half-plane problems are called the first and the mixed fundamental problems by Muskhelishvili (1977) and have been formally solved in general terms by means of complex variables. In particular, the stresses and displace-ments can be related to a complex function (I)(;)

(55)

and a bar designates the complex conjugate. Two kinds of complex conjugations are distinguished here. If

(K) = + ,11 (60)

where P and Q are the real and imaginary parts of (I), then

(1)(:) = PG, )11 (61)

(1)(0 = PG", (62)

As the effective Lame constant is introduced in equations (14) and (15) the effective elasticity parameter K.

1+ m(3 40

= (63)

I +III

appears in equation (58). In the limit of in = 0, corresponding to a fully saturated incompressible fluid, h. becomes I. which implies that no shear is transmitted. However, for in = x the classical value of K= 3-4v for a pure solid holds, and no effect of the pore fluid is felt. For Poisson's ratio

2

0

0 2 4 6 8 10

Fig. 2. Elasticity parameters of the solid matrix K and of the composite medium K., as functions of stiffness para-meter in; Poisson ratio of solid matrix r = 1/3, void ratio

= 1/3 so that in =

v = 1/3, both lc and K are plotted in Fig. 2. The boundary conditions on the real axis ; = may be expressed in terms of 11)(). By letting approach the real axis from below and hence from above, 4)1( - () and (1)(:).(1)+(;'). Then from equations (57) and (58)

(I)+G')

4)-0 =

i220+

(64)

= 2[()+i)] (65)

_rc.

FIRST FUNDAMENTAL PROBLEM FOR THE OUTER REGION

The first outer stress boundary value problem is to seek (1)(;) which is analytic in the entire ; plane subject to

(1),+ G.:") =/locos k(+ 1) (66)

for! and

+G")(1),),-(,D = 0 (67)

for ; > I. where KL., and /To is given by equation (48).

The solution for this problem is given by the Cauchy integral

Ao'

_cos :2c_+ 1)

(DAG) = (68)

-(Muskhelishvili, 1977, p. 476). On making the change of variables Y= - R(," + 1) (69)

X=

+1) (70) equation (68) becomes

jo

cos X' AV) = dX' (71) 27ri Jo X' Z

with Z = X +iY. The cosine integral may alter-natively be expressed in terms of exponential integrals, which are tabulated by Abramowitz & Stegun (1972). Also, from their series expansions, they are easily evaluated numerically. Assuming that Z is on neither the real nor the imaginary axis. equation (71) can be manipulated in the complex defined in the entire ; plane by analytic

continuation

li

+

= 4(1)()+4)0]

(56)

= 01;1 (1)(:)+ (57)

24-[t1[Pi+i1721 =K.:OG-1+010f, 0,1)(c.) (58)

where

;

= (59)

g

+

(6)

240

plane to give

for

(first

Z in second and third quadrant

fourth

The boundary values along the positive real axis, taken from above or below, are given by

27ri OA +(X) =. [ +nt Ci4X)] cos X A0 [Si(X)+ sin X (73) 2 A0 27r1

[

(X)] cos X

[SifX)+1

sin X [74) 2'

-for X > 0, where Ci and Si are the cosine and sine integrals. Along the negative real axis

27tidWX) = Ci (X) cos X

_[Si

(X)i-7-11sin X

(75) for X <O.

From this solution, the horizontal gradient of

the induced displacements along the base can be found by using equation (65)

0

2-0 2[17( +

=

1)-2ni

x{Ci(X0) cosX0 +

[Si (,(0)_,]

sinxo

(/6)

where

x0= K(0+1)

(77)

The outer total stress components can readily be obtained from equations (56), and (57), and the

corresponding pore pressure follows from

equation (44).

Some computed results for g = hi = 1 are given

in Fig. 3. The periodic influence of the standing

wave is evident. Moreover, it may be seen that the

f+

in exp (iZ) 27ri

--(1)A(Z)'

A

utexp,(-1Z)

+1[expi(iZ)Ei(iZ)+.ekp(iZ)Ei( iZ)] (72),

MYNETT AND MEI

-42L0fi -0.04 - -0.02" O02 -0.06 -0-04 -0.02 0.00 722A -42A 0.00 75A

Fig. 3. Contours of outer stresses and pore pressure for first problem; R =in = Glli = I

outer pore pressure does not satisfy the boundary condition along the free surface (equations (45) and (46)). Thus a boundary layer correction is needed later.

MIXED FUNDAMENTAL PROBLEM FOR THE OUTER REGION

In the mixed boundary value problem of the

outer stress field a function 4:013(0 is sought, which

is analytic in ; over the entire ; plane and subject

to +( .1+ K OB-(0= 24)0'() (78), where' and (1)0 +( -)(1)11-10 = 0 (79) where If > In equation (78) c)-01) is given by ju'f,D =

-

1(80)

where flo is the dimensionless base rotation and

YA(c-) is given ,by equation (76). The solution is formally expressed by -(X) = = x/L -6 -4 -2 04 0 2 002 000 I = 0 -2 -4 A 000 0'02 0

(7)

11 vl t"--2 4 2 4 2 -4--11B B 722 B 7.2213 .0-00 0.05

0.10-Fig. 4. Contours of outer stresses and pore Pressure FOC second problem; J = m = GI 13 = 0-4 02 0 zr.95 1, 0, - 02 0-4 =Xg)

f

+ _2g131) ± _{X() (841} 0.4

x(ouLi

0-2 (Muskhelishvili, 1977).

The basebase rotation f/o, as well as the complex

coefficient

CH = iC2 (84)

remains to

be determined by invoking the

equilibrium conditions of the caisson. Ifpcdenotes the caisson density, then its mass and weight are of the order O(p, L2)and 0(/), gL2)irespectively: As the

ground acceleration is of the order

0(co2p,ciao LIG), the ratio of inertia force to the

wave force on the caisson is of the order

0-.4 :R= R=1)-25 = 0,0044 OD). -2 e= x/L (c), 111 III

Fig. 5. Variation along the mud-line of normalized outer pore pressure "j) =p"12P0 and normalized wave pressure 12- )(co 2 p,cps, LIG) ps(02 L2

«1

(86) p-=pwl2P0 due to standing waves for

g

= 2.0,, 0.25

,p1,c1 Lao and for in = Gill = 0-0044, II, -4; Po = 2Pga0

where

= 1(t,+ 4)- t14:4 (82)

p, = Intc,127r (83)

WAVE-PNDUCED STRESSES BENEATH A CAIISSON 241

4 0-05 0.00 x/L -2 4 -0.10 -0.05 0.00 0.00 -0-10 0-00 -0-05 I 27ri , (a) = (p, ₌ = = = -2

(8)

242 08 0-4 -04 -0-8 0-8 04 0 0 .4 (a)

MYNETT AND ME!

08 0-4 -J 0 0-4 0.8 08 0-4 0-4 -0-8 r22 -0-4 -08 -0.8 -0.4 0 0-4 08 ;=x/L 712 - '22 (0 -08 -04 0 04 08

Fig. 6 (above and facing page). Variation beneath the caisson of normalized total outer stresses ii2 = r12/2P0() and

-r°/2P0 and normalized outer pore pressure j = p°/2P; = 20. 1.0, 0.25 and tn = = 0-0044, I. x IZ-= 0 25, G/i3= 0 0044 (e) R=025, G/t3 = 10 08 r12 04 2.0, G/fi = = = = =

(9)

o 8 0 4 0 -04 -0 8 -04 -0 8 Fig. 6 contd

k-

1 0, G/i3 = 712 - 722 (g) (h)

Thus the caisson inertia is negligible because of

equation (7). The total vertical and horizontal

stress distribution along the base may be obtained from equation (64) after using equation (81)

K, 1

-"1,2(0)+iii2Gfo) = K, K,A- 1 V(:0) 27ri

J- X+ ()N--0]

+ X + ())C,, (86)

where the integral is

to be interpreted as the

Cauchy principal value and is calculated by

Gauss-Chebyshev quadrature (Mynett, 1980,

appendix C). The equilibrium equations are

expressed by the conditions

[--inG.0)+iii2(0)]C'o}

= v (87) Re{f [-'1221,>o) + 2((>0)] d,;() = (88) f Re{

fir=".

0, -2 2.'0, i7r_- 2,,()f." II,,de= = CIVV - I (89)

where Tv, Pw and Mw are the externally applied horizontal, vertical forces and torque respectively,

with being positive if downward. These

equations areformally the same as those given by

Mei & Mynett (1982) provided proper changes are

made for and M. Substituting equation (86)

using the explicit forms (80) and (76) into the

conditions (87)-(89) gives (.2 = TW (90)

Cl= Pw

(91) 2// (1 +412)no = (92) h.,+ I 77-r

where NIA accounts for the contribution due to the induced displacements of (1)A, i.e.

-I

I' = _K I._1 27ri

K+I

."'() X+(tfo) ' 2 X3-X ij d,7,) + (93)

The right-hand sides of the conditions (87)--(89) contain the external forces exerted by the standing

wave on the structure. Integrating the

hydro-dynamic pressure along the vertical face

= - I of

08 04 -.J 0 -08 0 8 04 -J 0.4 -t22 0.25..G/P = + + MA = I -0-8 x/L 0,4 ₀₈

(10)

244 MYNETT AND MEI the caisson gives

:4-of cosh R7 d9 = sinh Kh (94)

Pw =0

A0 h k

Jori cos tidq Ao

=

sinh g7-cosh

+ 1] (96)

The constants no, lc,

and C,

are thus

determined. This completes the solution for the

mixed fundamental problem. The corresponding stresses and pore pressure can then be computed.

Some computed results for K = m = I presented in Fig. 4 show the features of the solution. The

pore pressure does not satisfy

the condition

= 0 along the rigid base. Hence a boundary layer correction is required.

BOUNDARY LAYER CORRECTION

The corrections for the shear stress and displace-ment components for the boundary layer structure

are at most of order E = ó/L and can thus be

conveniently ignored. Expressed in terms of the

dimensionless outer variables, the stresses are

F(;)

To give an estimate one can take

v = n = 1/3, G_ 107iosN/m2,

k =10-6-10'9

w =0.5-1 rad/s

and

L 10-50m.

Then the dimensionless boundary layer thickness is in the range

E = 6/L-,0.01-0-1

for ni3O-0(1). The explicit form for the function

FfiD must now be found by satisfying the pore pressure boundary condition on the mud-line.

Three parts of the free surface are distinguished: to the left, beneath and to the right of the structure. To the left of the caisson the pore pressure at the soil surface must equal that of the standing wave

p = - = Ao COS ± 1) (99)

(95)

when i = 0, < - 1, where j) = 'f,A+ i5B. It follows that

F() = Jo cos K(+ 1)- f)', 0)

(100)

when c,=< -1. The total

pore pressure in the

boundary layer is

p = 1)+1,-4-0 cos 0)1 exp

-N./2

(101)

for

-At the caisson base, there can be no relative

velocity between the two phases. As inertia effects are negligible, the momentum equation of the pore fluid gives

i't1

=0

(102)

for q = 0, lc:I 1 along the base of the caisson.

Differentiating equation (98) with respect to 9 gives

F()=

I + i (103)

N/2 e// Pi= 0

for I

I 1 so that the corrected solution beneath

the base is expressed by

1+i irp

P = E

1. This correction is only of order OW and important.

the right of the

should vanish reads

p = p+f) = 0 (105)

for q = 0, > 1. Hence the amplitude is simply

caisson the dynamic pore

at the free surface, and the where (pb, (7,1') = (Re)Po(P,5-ii)exp(-iwt)

F()

-. 11 (97) (98) for I I not very To pressure condition F( ') 1 - v e

p(1

\/2 E

)

, 0

and the complete solution becomes

P = exp (107)

0 E

11-i

The boundary layer corrections for the effective

stress components are readily inferred from

equation (98).

RESULTS

The total outer stress field Tit is readily obtained

by summing the stress fields A and B; the outer pore pressure follows from equation (44). The boundary layer corrections can then be

super-imposed, making the solution complete.

As the boundary layer correction is quite simple

to make but difficult to plot, only the extent to which the correction is needed is demonstrated

F()= 15

(106) exp

11- i

(104) = IZ-( , + I

(11)

-0.10

R= 025

Fig. 7. Contours of normalized outer pore pressure

p = p°/2P0 for g = 2.0, I 0, 025 and in = GIII = 0 0044

here. In particular, the outer pore pressure along the sea bed will be presented and compared with

the applied bottom pressure (equations (45) and

(46)) due to the standing wave.

For ji = h/L -= 1.5 three

wave numbers

K L = K 2.0, 1-0 and 0-25, representing short,

medium and long waves

respectivelyare

considered. It is assumed that n = v = 1/3 so that

in = GIII and in is taken as 0-0044, 1 and

corresponding to full (S = 1, cf. equation (5)) and

partial saturation, and dry soil respectively. The

results are presented in Fig. 5, where the applied wave pressure is denoted by pw which is the right-hand side of equation (99). A correction is clearly required not only beneath the standing wave to the left of the caisson, but also to the right of it. For a

small stiffness parameter in = 0-0044, the

correction is confined to the immediate

neighbourhood of the structure. Half a wavelength or so away from the caisson, the outer and applied

pore pressures differ very little. For short waves (see Fig. 5(a) for K = 2) the horizontal variation

becomes more rapid, but because the surface

amplitude decays according to sech Kb, no

significant pressure is felt directly at the bottom.

The wave forces are transmitted to the sea bed through the caisson only, which behaves as an

o7N/1i 6.

0.16 0.08 -0-16 0-00 -0.08 0.08 -2 4 = x/L -3.0 -10 1.0

0,00. sd,

-0.16 0 16 -0-08 0.08 000 7;12

Fig. 8. Contours of outer effective stresses Fri, = aiiIP for

= I and in = Glli = 0-0044

oscillating stamp. The quasi-static outer stress

field is _{obtained by solving a classical stamp}

problem as discussed by Muskhelishvili (1977, p. 486). For long or low frequency waves (Fig. 5(c))

the applied bottom pressure equals the outer

pressure almost everywhere at the surface. Close to the base the physical picture resembles the case of seepage underneath a dam due to different water levels on both sides (Mei & Mynett, 1982). Indeed the boundary layer thickness ö is proportional to

co- ". As w = K ,l(gh) for long water waves,

6-, K-1/2 and is greater for longer waves.

The total outer stress distribution along the base is now presented as Riven by equation (86) with equations (80) and (76). The resulting normal and

shear stress components as

well as the pore

pressure are presented in Fig. 6 for three wave

numbers (R 2-0, 1-0, 0.25) and three stiffness

parameters On = 0.0044, 1, x,).

For a

highly

incompressible fluid (in, x.) the pore pressure

coincides with the total normal stress component and hence the structure is largely supported by the fluid without the help of the skeleton.

The dynamic pore pressure distribution beneath

x/L -ao -1-0 1.0 3-0 5.0 ₃₀ ₅₀ = 0-00 -622 -0)06 =

(12)

11

246 MYNETT AND MEL

the caisson is shown in Fig. 7 for the three different

Wave numbers but only' for a

fully saturated

porous bed (al = 0-0044). The pore pressure is fell in a larger zone for longer waves and the boundary layer corrections along the free surface do not alter

this result qualitatively.

The outer effective stress components ao

corresponding to K = 1 are presented in Fig. S. The results clearly show the need for boundary layer corrections, not only to the left, hut also to the right or the structure. As the formulation is

based on Hooke's law, both compressive and

tensile stresses are present. If these results are

applied to a granular medium, at least the region

of pure tension would require inelastic analysis.

The boundary layer correction can be added

simply and is omitted here.

To obtain further indication where inelastic theory is needed, the static and dynamic stress

fields are combined and the maximum

instantaneous stress angles calculated using

MohrCoulomb analysis, again without the boundary layer correction. The static stresses are composed of the ,geostatic stresses

iL2`2b = ,111"(PsP,411'

612 =0 (1l110)i

;as well as the static response to the. buoyant weight of the caisson. The latter corresponds to the well-known stamp problem (Muskhelishvili, 1977, p.

486) with a vertical force (A, H ph)01... acting

downwards. Numerical values are given by

Mynett (1980, appendix B). For a saturated pore fluid (az« I) equation (16) gives.

+ (MO

.Fig. 9. Mohr circle combining static and dynamic. stresses; for m = Gip« I. the dynamic stresses c and a22 are out of phase by it, r = ml, (1 is internal friction angle, c is cohesion coefficient and 0 is maximum stress angle

.1108)

4109)'

-1171enee the corresponding effective stresses

agi =

024 = 22P

= n --- flrI30'

)which are of equal magnitude but opposite sign.

Thus the dynamic normal stress components are

out of phase by

it radians. The Mohr circle is

.shown in Hu.. 9. By also allowing for different

cohesion coefficients, the maximum instantaneous stress angles 47 can he calculated from

6.-22)2 2112'

'(/) _{+ (71}2

X[am .4-(3""-14COO]

where 0 is the angle of internal friction and e the

coefficient of cohesion The value of 0 ranges

from 30

for sand to 50

for sandstone, and c

ranges from 0 kN/m.' for sand to 104 kN/m1 for

sandstone. As Po = 2p.ga0 = (02 x 103) kN/m.' sample values of 0 = 30 andZ.

= 0,05 and I are

used to calculate the maximum stress angles 0.

The results are shown in Fig. 10, where the relative

caisson height and density are 11/1. = 2.0 and

= 25, the water depth, wave amplitude and

wave number are

= 1.5, (111, = 0.25 and

-ao o 1 0

80 ao 50

20

Fig. IR Contours of maximum stress angle 0 for = Po = 0, 0.5, 1-0' when I: = I and in = G/ 0-oo44 = = are

i,2)

1 -2 = = = =

(13)

= KL = 1.0 respectively. In the absence of cohesion there is a large region around the caisson where the stress angle exceeds 30" which is the typical failure angle for sand. With cohesion the predicted stress angles are reduced, danger of momentary failure is lessened. Modification for the boundary layer can be carried out if desired but changes will he confined in a region too small to be

plotted here.

ACK NOWLE DG EM ENT

The Authors are grateful to the US National Science Foundation for supporting the research reported by grant ('ME 792-1993.

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Biol. M. A. (1956). Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoust.

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Mei, C. C. & Foda, M. A. t198(a) Wave-induced responses in a fluid-filled poro-elastic solid with a free

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