Wave-induced seabed response in a cross-anisotropic seabed in front of a breakwater: An analytical solution

Ocean Engng, Vol. 25, No. 1, pp. 49-67, 1998 © 1997 Elsevier Science Ltd. All fights reserved Printed in Great Britain 0029~018/97 $17.00 + 0.00

Pergamon PII: S0029--8018(97)00054-1

W A V E - I N D U C E D S E A B E D R E S P O N S E IN A C R O S S A N I S O T R O P I C S E A B E D IN F R O N T O F A B R E A K W A T E R : ANALYTICAL SOLUTION

AN

Dong-Sheng Jeng Centre for Offshore Foundation Systems, The University of Western Australia, Nedlands, WA 6907, Australia

(Received 10 October 1996; accepted in final form 6 November 1996) Abstract--To simplify the complicated mathematical process, most previous investigations for the water waves-seabed interaction problem have assumed a porous seabed with isotropic soil behavior, even though strong evidence of anisotropic soil behavior has been reported in soilmechanics literature. This paper proposes an analytical solution of the short-crested wave-induced soil response in a cross-anisotropic seabed. As shown in the numerical results presented, the waveinduced seabed response, including pore pressure, effective stresses and soil displacements, is affected significantly by the cross-anisotropic elastic constants. A parametric study is performed to clarify the relative differences in pore pressure between isotropic and cross-anisotropic solutions. © 1997 Elsevier Science Ltd.

1. I N T R O D U C T I O N

Water waves propagating on the ocean surface create dynamic pressure fluctuations on the sea floor. These fluctuations might induce an excessive pore pressure, accompanied by a decrease in the effective stresses, and then cause an instability in the seabed. Owing to this, large armour units have been placed on the seabed fronting marine structures to prevent scouring of the seabed. The wave-induced soil response has been recognized by marine geotechnical engineers as a key factor in analyzing the seabed instability. An evaluation of wave-induced seabed response is therefore particularly important for geotechnical and coastal engineers involved in the design of foundations for offshore structures. Conventional investigations on the wave-seabed interaction problem have only considered isotropic soil behavior for a porous seabed (Yamamoto et al., 1978; Mei and Foda, 1981; Okusa, 1985). In fact, most natural soils display some degree of anisotropy. That is, they have different elastic properties in different directions, owing to their deposition, particle shape and stress history. However, many materials show more limited forms of anistropy. For example, a cross-anisotropic (or so-called transversely isotropic) material possesses an axis of symmetry, in the sense that its properties are independent of rotation of a sample about the axis of symmetry. This implies that a cross-anisotropic material has the same properties in any horizontal direction, but not in the vertical direction. When soils are deposited vertically and then subjected to equal horizontal stresses, they are expected to exhibit a vertical axis of symmetry and be transversely isotropic (Barden, 1963; Pickering, 1970; Gazetas, 1983). When a progressive wave arrives at a marine structure, the incident wave may be reflected obliquely from the structure, resulting in a short-crested wave system (Fig. 1). The short-crested wave system not only fluctuates periodically in the direction of propa49

50

D.-S. Jeng

Z

y

~

j PLAN

SWLfree sudace

O

Soil

porous rigidimpem~ bosom seabed

ELEVATION

Fig. 1. Definitionsketch of short-crested wave system produced by oblique retlection from a breakwater, showing Cartesian coordinates used for the analysis of wave-induced soil response. gation, but also in the direction normal to it. This results in a larger wave force acting on marine structures, as well as more complex patterns of the wave-induced water-particle motions and mass-transport, which enhances the seabed scouring (Hsu et al., 1980). Recently, this wave phenomenon has attracted much attention from coastal engineers. Most previous investigations on the short-crested wave system only concerned its kinematic and dynamic properties, such as wave pressure and force on a marine structure (Fenton, 1985; Tsai and Jeng, 1992), rather than on the seabed response. However, much evidence in the literature suggests that some recent failures of breakwaters could be attributed to foundation failure rather than failure of the construction itself (Smith and Gordon, 1983; Silvester and Hsu, 1989) Thus, the analysis of the seabed response fronting a marine structure deserves critical examination. Recently, based on the assumption of a compressible pore fluid and a soil skeleton, a series of analytical solutions for the wave-induced soil response in an isotropic elastic seabed in front of a breakwater has been developed (Hsu et al., 1993, 1995; Hsu and Jeng, 1994; Jeng and Hsu, 1996). Based on these analytical solutions, Jeng (1997a) further examined the wave-induced seabed instability. Even more recently, the influences of the variable permeability and the shear modulus on the wave-induced seabed response have been investigated analytically and numerically (Seymour et al., 1996; Jeng and Seymour, 1996; Jeng and Lin, 1996, 1997; Lin and Jeng, 1997). All of these studies were only concerned with the soil response in an isotropic seabed, not in a cross-anisotropic seabed. The influence of cross-anisotropy on the wave-seabed interaction problem was first analytically studied by Jeng (1996) for a two-dimensional progressive wave condition. Later, Jeng (1997b) extended the solution to a two-dimensional standing wave condition. This paper further develops an analytical solution for the three-dimensional short-crested wave-induced soil response in a cross-anisotropic seabed. Based on the general solution, the effects of elastic cross-anisotropic constants on the soil response (including pore pressure, effective stresses and soil displacements) are detailed. 2. CROSS-ANISOTROPIC SOILS In general, an isotropic material displays same elastic properties irrespective of the orientation of the samples. Such a material can be described by two elastic constants,

Wave-induced seabed response in a cross-anisotropic seabed

51

Young's modulus, E, and Poisson's ratio, /x. However, most marine sediments display some degree of anisotropy, possessing different properties in different directions. The elastic behavior of a cross-anisotropic material is described by five independent elastic parameters (Picketing, 1970). Owing to the requirement of an advanced experimental technique for anisotropic constants, engineers tend to use the two-parameter isotropic model in practical applications, neglecting the effects of anisotropy. However, the effects of anisotropy cannot always be ignored without substantial errors. Based on three rather than five parameters (and two additional assumptions), Graham and Houlsby (1983) proposed a simplified anisotropic model for lightly overconsolidated postglacial soils. Their results indicate that the three-parameter anisotropic model can reduce the error in the predication of strains by 30-40%, compared to the predictions of the isotropic model. Five elastic parameters are required for describing such an anisotropic material (Picketing, 1970), they are: • Ev, Young's modulus in the vertical direction; • Eh, Young's modulus in the horizontal direction; • /Xhh, Poisson's ratio as the corresponding operators of lateral expansion in a horizontal direction owing to horizontal direct stresses normal to the horizon; • /Xvh, Poisson's ratio as the corresponding operators of lateral expansion in a vertical direction owing to horizontal direct stress; • Gv, modulus of shear deformation in a vertical plane. Two dependent elastic constants /Xhv and Gh can be inter-related by /~h~ Eh =~ /~vh -- Ev

(1)

Eh Gh -- 2(1 + /~hh)

(2)

and

For an isotropic soil, the nondimensional parameter 1~ is equal to 1. The vertical shear modulus in a vertical plane, Gv, can be expressed in term of the Young's modulus, Ev, as Gv = AEv

(3)

where A is the so called anisotropic constant (Gazetas, 1982): it is equal to E/2(I + ~) for an isotropic soil. Based on the generalized Hooke's law, the relationships between elastic incremental effective stresses and soil displacements are given by (Picketing, 1970):

~gu 3v ~w off = Cll ~xx + C,2 3y + C,3 ~zz c,2

bu

+ c,,

3v

+ Cl3

bw

bu bv bw off = C,3 3x + C,3 3y + C33 bz

(4) (5)

(6)

52

D.-S. Jeng

and ,

~u

by

,

T~y = Gh( ~yy + ~x ) = z~'~

(7)

Ou ~w ~'x:' = Gv( ~z + ~xx ) = ~~'

(8)

~v ~w • p zyz = Gv( fizz+ ~yy ) = ~'~y

(9)

where u, v and w are the soil displacements, while Ox', o-y' and m' are the effective normal stresses in the x-, y- and z-directions, respectively. Six effective shear stresses %f denote the shear stress in the s-direction on a plane perpendicular to the r-direction. It is worth noting that the positive sign is used for tension stresses in the present paper. In Equations (4)-(9), the coefficients Cii are given as C,, = Eh(l -- /Xhv/Xvh)/A

(10)

C12 --Eh(bl,he/.teh "t- /.thh)/A

(11)

C13 = Eh/~he(1 + jLl~hh)/m

(12)

C33 = Ev(1 - /Xhh)/A

(1 3)

A = (1 + /Xhh)(1 -- P~hh- 2b~hvbl'eh)

(14)

and

Based on the thermodynamic considerations, the strain energy of an elastic material should always be positive (Pickering, 1970). This condition requires that the following inequalities should be satisfied. E~, Ev, Gv --> 0

(15)

/X~h --> -- 1

(16)

1

~,~ (l -- ]"£hh) -- 2~LL~v ~ 0

(17)

It should be noted that the second term of Equation (17) is always negative and so the first term must always be positive. Since Eh and E~. are both positive, this requires (1 /Xhh) to be positive and, hence, /Xhh --< 1. 3. GOVERNING EQUATIONS Herein, we consider a soil column of infinite thickness in front of a vertical reflectingwall, as depicted in Fig. 1. A short-crested wave is produced by full oblique reflection of the incident waves, with obliquity angle 0 measured between a wave orthogonal and the normal direction to the wall (Hsu et al., 1979) and also between the wave crests and the wall itself. The resultant wave crests propagate in the positive x-direction parallel to the wall, while the z-direction is measured as positive upwards from the seabed surface for the waves-seabed interaction problem.

Wave-induced seabed response in a cross-anisotropic seabed

53

Generally speaking, the mechanism of wave-induced seabed response may be classified into two categories, depending upon how the excess pore pressure is generated (Zen and Yamazaki, 1990; Nago et al., 1993). One mechanism is caused by the 'residual' or 'progressive' nature of the excess pore pressure, which accumulates gradually after a certain number of wave cycles. This type of soil response is similar to that induced by earthquakes, caused by the build-up of the excess pore pressure. The other mechanism, generated by the 'transient' or 'oscillatory' excess pore pressure, appears periodically in response to each wave. In general, the transient soil response occurs after the residual soil response. In this study, only the transient mechanism will be considered. Some basic assumptions are necessary for deriving the analytical solution for the phenomenon described, such as: • the breakwater is considered as an infinitely wide and deeply embedded plane rigid structure; • the seabed is horizontal, unsaturated, hydraulically anisotropic and of infinite thickness; • the soil skeleton and the pore fluid are compressible; • the soil skeleton generally obeys Hooke's law, implying linear, reversible and nonretarded mechanical properties; • the flow in the porous seabed is governed by Darcy's law. The equations for overall equilibrium of the pore-elastic medium, in which the total stresses are resolved into the effective stresses and the pore pressure, are given by

30-/ 3~xv" a'¢z' 3p 3x + ~-y

(18)

+ 3z - 3 x

3Txy t 30"y" OTyzt Op 3x + ~ y y + 0Z - 3 y

(19)

3~xz" 3"¢z" 3o-/ 3p G x + ~ y y + az - az

(20)

where p is the wave-induced pore pressure in excess of the hydrostatic condition. Substituting Equations (4)-(9), into Equations (18), (19) and (20), the equations of force equilibrium are rendered as

32U 32U 32/,~ 32V 32W 3p CI~ 3~5 + Gh 3~y2 + Gv 3zRz 2 + (C12 + Gh) 3x3y + ( G 3 + GO 3x3z - 3x 0211

32V

32U

32W

3p

(21 ) (22)

Gh 3X 2 + Cll 3y 2- + Gv32v3z 2 + (C12 + G.) 3 ~ y + (C13 + Gv) 3y3z - 3y

and

aiw a2w 32w a2u 3% ap Gv 3x2 + Gv ~ + C33 ~ + (G3 + Gv) ~ + (Ct3 + Gv) 3y3z-- - 3z

(23)

in the x-, y- and z-directions, respectively. The consolidation equation (Biot, 1941) or storage equation (Verruijt, 1969) is generally accepted as the governing equation for flow of a compressible pore fluid in a compressible

54

D.-S. Jeng

porous medium. For a three-dimensional problem, assuming the porous bed as hydraulically anisotropic with permeabilities Kx, K~, and K~ in the x-, y- and z-directions, respectively, this equation can be expressed as Kx ~2p

g,.

K z Oxz + ~

O2p

7w~On'/3Op

32p3y2 + Oze

K~

3t

7wW De

(24)

K~ Ot

where Yw is the unit weight of the pore water, n' is the soil porosity,/3 is the compressibility of the pore fluid, t is the time and ~ is the volume strain defined by bu

3v

bw (25)

• = Ox + 03; + Oz

The compressibility of the pore fluid,/3, is related to the true bulk modulus of elasticity of water, Kw (may be taken as 2 × 109 N/m2), and the degree of saturation S~, such that 1

1 - S~

/3 = Kw + Pw0

(26)

where Pwo is the absolute water pressure. If the soil skeleton is absolutely air-free, i.e. fully saturated, then/3 = 1/Kw, since S, = 1. For a homogeneous seabed of infinite thickness, as shown in Fig. 1, the evaluation of seabed response requires the solution of Equations (21)-(24) with the appropriate boundary conditions. Firstly, zero displacements and no vertical flow occur at the horizontal bottom. Thus, the bottom boundary condition is given as u=v=w=p=0

as

z~-~

(27)

Secondly, the vertical effective normal stress and shear stress vanish at the seabed surface, i.e. o-.' = r,_' = r,.' = 0 as z -- 0

(28)

and the pore pressure on the upper soil boundary is consistent with the wave pressure at the seabed surface, Pb P = Pb

7wH~ 2 cosh kd cos nky cos(mkx

wt) = Po cos nky cos(mkx - wt)

(29)

= _Po Re{ei~,,~ + ,,k~..... ~ + ei~,,k~ ,,k,. ~,,} 2 where cos nky cos(mkx - cot) denotes the spatial and temporal variation within the wave field described above. The amplitude factor Po is given by the first-order short-crested wave theory (Yamamoto et al., 1978), H~ denotes the wave height of short-crested wave. Parameters m and n are the obliquity components relating to the wave number k in the x- and y-directions, respectively, kx = 2rr/L, = m k = k sin 0 and ky = 27r[Ly = nk = k cos 0

(30)

where Lx and Ly are the crested length of the short-crested waves, as seen in Fig. 1. From Equation (30), the relationship m 2 + n 2 = 1 is applicable to the short-crested wave system.

Wave-induced seabed response in a cross-anisotropicseabed

55

4. GENERAL SOLUTIONS FOR THE SEABED RESPONSE

4.1. A general solution for a cross-anisotropic seabed Since all equations are linear, it is expedient to employ complex variables in the analysis. The analytical solution for the pore pressure and soil displacements are assumed as

=

ei~,,k~+

nky -

~ot)+

P1

ei(mkx- nky- o,,)

(3 1)

P2

where i( = ~ / - 1) stands for the complex variable. In Equation (31), the subscript 1 denotes the incident wave component, while 2 represents the reflected wave component. Firstly, we consider the reflected wave component, with the subscript 1. Introducing Equation (31) into the governing Equations (21)-(24), after some algebraic manipulation, renders the final form as

nU~ - mV1

=

al erlkz + a2e -r, kz

(32)

{GvD 3 + (Gv + C~3 - C~)k2D}U~ + {(Gv + C~3 - C33)D2

(33)

+ GvkE}(imkW1) = nkari{Gv + C13 - Gh - C12}(ale krlz - a2e- kriz) and

{GvD4 - (s°Gv + CllkZ)D2 + k2[ s°C~ - ~i'Ywt° -z ]} UI+

(G~+Cla)D 3+

-so(G~+C13)+~

= nk 2 (so - ~kZ)(Gh + C12) -

D (imkWl)

(34)

(ale r'kz + a2e- r,kz)

where rt =

(35)

and

2{Kx Ky ~ So = k ~ m 2 + Kzz n2}

i31wtOn'[3 Kz

(36)

Combining Equations (32).(34), it renders to (SlD6 + s2D4 + s302 + S4) imkW1

tn~2(al e~lkz - a2e- qkz)J

(37)

in which D denotes the differential operator ( = d/dz), and the coefficients s~ (i = 1, 2, 3, 4) and ~ i (J = 1, 2) are given as

D.-S. Jeng

56

(38)

o~1 = C33Gv

or2 = k2(Cl3 + Gv)z - k2(G {

+ C33C11 ) -

(39)

GvC3361

0/3 = kZ{ - (C~3 + Gv)262 + C33C,~83 + k2G2v&}

(40)

Ot4 = - k4GvC1163

(41)

and @, = k4{~(Gv + C13)2(r~1 k2 - 65) -I- r'~lbC33(Gh + C12)(61 - 66)

(42)

- Gv(Gh + Cl2)(OZo - 66)} ~'~2 =

(43)

k S r l ( G ~ - Gh)(G~ + C13)(65 - ~k 2)

In Equations (38)-(43), the coefficients 5~ (i = 1-6) are given by Ky

elK:,

2/'Kv

62

2[K~

63

)

iyw~O(

l)

(44)

m 2 + K ) n2 - ~

n'[3+C,3+G--;

(45)

m2+~

rl'~+Cll

(46)

n2 -- ' U Kv

CI,)

i ~/w(/) (

Cll t

(47)

m=+~ ~+ O,.J-X~- ~'~+ O~vJ 65

\< m z + K ~ n e - K :

(48)

n'I3+GV+cI3

and

6 6 = k2r~l +

i'y~to

(49)

K~(Gh + Ci2)

The general solution of the sixth-order ordinary differential Equation (37) can be expressed as: Ul = {n67(al e''k: + a2e ,,k~) + (a3er2z + a4e- r2~) + (ase r3~ + a6e-r~z) + (a7e r4z + ase ~4:)}

(50)

where the coefficients r2, r3 and r4 are given by

Oe2

21/3(3OqOt3-- 0/2)

-- - 30/, - -

3 0 / , % ~ '3

0/2

0/2 - - 30/1 + (1 - i q S ) _

_

_

-

0///3

(51)

+ 5a,,3~,

21/3(30/10/3 -- 0/2) • '-'~' '-'l"s

Otl/3 532m°fi 21/3(3OqOt3 - OZ~)_ (1 + i'~3) O/113 1081,3,~,a~, 3

5321,3a,

(52)

(53)

Wave-induced seabed response in a cross-anisotropic seabed % = 4"~4

2 ~ 2 ~ 23 + -- ~t,tlt,~Et,t

4%% + 4%%%

57

18%1%%% + 27%10t24

- 2 % + 9 % % z % - 27%2%

(54)

From the bottom boundary condition, Equation (27), the coefficients a> a4, a6, and a8 are required to be zero. Thus, Equation (50) can be simply rewritten as U, = {n87ale rlkz + a3e r2z + ase r3z + ave r4z}

(55)

where 87 is given by ~'

8 7 • %k6r61 -1-- %k4r~l -{- %k2r~l + %

(56)

From the final form of the governing equations, Equations (32)-(34), the other soil displacements and pore pressure can be expressed in a similar form, and related to the coefficients al (i = 1, 3, 5, 7). The remaining four unknowns, aj (j = 1, 3, 5, 7) can be determined by employing the boundary conditions Equations (28) and (29). Employing the same procedure as that for subscript 1 above, a solution for the incident-wave components, i.e. the second terms with subscript 2 in Equation (31), can also be found. After some algebraic manipulation, the wave-induced soil displacements can be presented as U = mpo{C2e ~2z + C3 er3z + C4 er4z} cos nkye i(mk~ or)

(57)

v = inpo{C2er2 z + C3e~3z + Cner4~} sin nkye i~mkx- too

(58)

ipo {88C2er2z +

W=T

89C3er3z + 8'°C4er4z} c o s n k y e i(mkx- ,or)

(59)

p = po{81 l C2e ~2z + 812C3er3z + 8,3C4 er4z} c o s nkye i(ml~r- tot)

(60)

and the pore pressure is

where the adding 8~ coefficients (i = 8-13) are given as ra2Gv + r2k2(Gv + C13 - C l l ) 68 = ~(Gv + C13 - C33) + k2Gv

(61)

r33Gv + r3k2(Gv + C13 - C l l ) 69 = ~(Gv + C13 - C33) + k2Gv

(62)

r344Gv + rnk2(Gv + Cl3 810 =

8,,= 8,2 =

8,3 =

CIL)

r~4(Gv + C13 _ C33) + k2Gv

(63)

ywto(k 2 + r238) kKz(~ - % )

(64)

3"~t°(k2 + r289)

(65)

kKz(~

-

%)

ywto(k 2 + r281o) kKz(r]4 _ % )

(66)

58

D.-S. Jeng

It is noted that the coefficient a~ in Equation (55) is equal to zero, after some algebraic manipulation. The coefficients Ci (i = 2--4) in Equations (57)-(60) are given by C2 = 63 =

1-

81364

-

81263

(67)

811 811(88 -- 81o)(C~3 k2 + 6337"21"4) "F (F 4 -- r2)(Cl3 k2 + C338881o )

(68)

6o 811[(~9 - 88)(C13 k2 + 633F2F3) -- (/"2 -- r3)(Ct2k z + 6338889)]

C4 = Co---

Co [811(r3 - 89) - 812(r2 - 88)][Ci3k2(811

(69) - 813)

q- 633(810811/'4 - 81388F2)] -- [811(F4 -- 810) -- 813(F2 -- 88)][613k2(811

- 612)

(70)

+ 633(81169/3 - 81268r2)]

Substituting Equations (57)-(60) into the relationship of stresses and displacements, the wave-induced effective normal stresses can be obtained as

ipo

0"~' = ~--{(G lk 2 - 2n2k2Gh + 88r2Ci3)Cze ~2~ + ( C 11k2 - 2nZkZG h + 69r3613)63 er3z + ( C 1lk 2 - 2nZk2Gh

(71)

+ 81or.~Cl3)Cae"4~} COS nkye i~mk". . . . .

o -,= ipo T {(Cllk 2 - 2rn2k2Gh + 68r26j3)Cee "2~ + ( C l l k 2

2m2k2Gh + 89r3613)

C3e"3z + (61 lk 2 - 2m2kZGh + 81or4Cl3)C4e r4z} cos nkye i~'~ ,o,,

,

=

lpo{

(72)

(k2613 + ¢~8r2633)62 er2z + (k2C1369r3C33)C3 er3:

+ (k2C13810r3C33)C4 er4z} c o s n k y e i~m*':- ''~

(73)

and the effective shear stresses are "(~," = - 2mnGhpo{C2er2 ~ + C3er3: + C4 er4z} sin n k y e i~m~ ,oo

(74)

"(~./ = mG,cpo{(r 2 - 88)62e~2: + (r369)C3e"3~ + (r 4

-

-

81o)64 er4z} c o s nkye i~mk. . . . . )

(75)

%.f = inGvPo{(r2 - 88)62 e~z~ + ( r - 389)63 e~3~

+ (r4 - 8 t o ) 6 4 e r4z} sin g/kye ifmkx o~t)

(76)

Since the short-crested waves produced by oblique reflection from a marine structure are doubly periodic in two perpendicular directions (see Fig. 1), their influences are seen in wave obliquity m and n.

W a v e - i n d u c e d seabed r e s p o n s e in a cross-anisotropic seabed

59

4.2. A general solution f o r an isotropic s e a b e d To simplify the c o m p l e x mathematical process, conventional solutions for the shortcrested wave-induced seabed response have assumed that the seabed is an isotropic medium. The analytical solution of a short-crested wave-induced seabed response in a cross-anisotropic seabed is n o w available, the solution of its special case, an isotropic seabed, can be obtained from the reduction of the solution here. For an isotropic seabed, the soil characteristics are only dominated by two elastic parameters: Y o u n g ' s modulus E( = Eh = Ev) and Poisson's ratio/z( = P~hh = ~-Lhv= ~vh)" Based on these two elastic parameters, the shear modulus can be obtained from Equation (2) as E/2(1 + / x ) . The coefficient Cij in Equations (10), (11) and (12) can then be interpreted for an isotropic seabed as Clt = C33 -

2G(1 - / x ) 1 -

2tx

2G~ , C12 = C13 - - 1 -

(77)

2~

Substituting Equation (77) into Equation (37), the final form of the equation governing soil displacement can be rendered as 2G2(1 - p,) {D 6 - (2k2824)D4 + (k 4 + 2k2t~4)D 2 -1 - 2/x --

2G2(1 - / x ) 1 -

( 9 2 -- k Z ) Z ( D 2 -

k4~214 }

t~24) = 0

2/x

(78)

Following the same procedure presented above for a cross-anisotropic seabed, the waveinduced soil displacements and pore pressure can be expressed as 2Gku* - - m{(C~ + C~3z)ekz + C]4e~14z}cos nkye i~mk~- o~,) Po

2Gkv*

- in{(C~ + C3z)e kz + Cae6,4z} sin nkye i(mkx- ,or)

Po 2Gkw* - = i{[kC2 - (1 + 2615)C3 + C3kz]e kz + 6C4e&4Z} cos nkye i<'k~ - o,t) Po

(79)

(80) (81)

and p*

24 -- k 2

1

Po - 1 - 2p~

{(1 - 2/x - ~15)C~3ekz + - -

k

(1 - i~)C4e&4Z} cos nkye i("k~- o,t) (82)

where -- it~lS[/d,(~14 -- k) z

-

i(t~14 -- 614/./~ alC~3 = (614 -

2k)] k/x + k~15)

~14(~14 -

C 2 = k(t~14 _ k ) ( ~ 1 4 _ ~14/.L +

k/x) k615)

614/J~ -1- kj[z -I-

(83) (84)

60

D.-S. Jeng

ikrls ~rK~ ~:K" ] 824 = k - [ ~ m2+ n2

(l - 2

){k

[

iWTw[ 1-2l, ] n'/3 + - ~ 2G(t -/x)J

Kxm2

1 - K_

-

K;

]

+

iWyw

8,5 = ~ ~K~ ' ; - K , ? _ ] iW}w 1-, k z 1 - ' m - ~ n e + ...... n/3

(86)

n'/3}

i - 2/*]

(87)

It is worth noting that Equations (79)-(82) are identical with the previous analytical solution for an isotropic seabed (Hsu et al., 1993). 5. NUMERICAL RESULTS The general solution for the wave-induced pore pressure (p), soil displacements (u, v, w) and effective stresses (o-,', ~r,.', m', r~', %f and Z~y') has been derived previously. This analytical solution contains three groups of parameters:

1. wave characteristics--these are wave height (H~), wave period (T) and the relative water depth (d/L); 2. basic soil parameters--these include hydraulic permeabilities (K,, Kv and K.), porosity (n') and degree of saturation (&) in term of compressibility (/3);

3. cross-anisotropic constants--these are the Poisson's ratios (/*hh, /Xh~and P.~h), Young's moduli (Eh and Ev) and shear moduli (Gh and G,,). By considering several examples, it was found that the wave-induced soil response is insensitive to variations in hydraulic anisotropy (KJK~I and KJK.4:1) over a range of (KJK~) and (KJK~) from 1 to 100; hence, only hydraulically isotropic conditions (K~ = K,. = K~) are considered here.

5.1. Effect of wave obliquit3, The vertical distributions of the maximum values of Lpt/po and 1o'_']/po vs z/L can be calculated from Equations (60) and (73). They are found to be independent of the wave obliquity 0 (Fig. 2). The figure clearly shows that the pore pressure Lpl/po decreases as z/L increases. However, the vertical effective normal stress ]ofI/Po increases as z/L increases near the seabed surface, and then decreases as z/L increases further. Figures 3 and 4 illustrate the vertical distribution of the other soil response (lcrx'l/po, Io-,,'llpo, ]r,~'llpo and [r,..-'l/po) vs zlL for various wave obliquities, 0. It was found that the maximum ]~x' ]/Po increases as the incident angle 0 increases and the general patterns are similar (Fig. 3). The curves for 1o-,,']/Po are also presented in Fig. 3. They are compensated with the value of IO'x'l/po. For example, the curve of 0 = 30 ° for ]o~']/po is the same as for 0 = 60 ° to I~r,.'l/po. Similar trends are observed in Fig. 4 for ]r,~'l/po and [r,..,.']lpo. 5.2. Effect of cross-anisotropic constants Poisson's ratio, /*hh, describing the strain in one horizontal direction, is caused by the strain on another horizontal direction normal to the first direction. The vertical distributions

Wave-induced seabed response in a cross-anisotropic seabed 0.0

''1

. . . .

i

. . . .

)

. . . .

-0.2-0"I" /

/

. . . .

i

. . . .

i

. . . .

i

. . . .

i . '

I

/

...)..z.,

tt ,h 0.45 A=0.6 ~=0.6 e,=~o'M/. ~ n'=0.4 K~ K) Kz 104m/sec =

=

=

=

s;=o.~5 '

....

0.1

~ .... 0.3

0.2

1

L = 205.09,,, "" = 0.4.5

/

/

t

d=#om

//

/

12.Ssec

T=

/

/

-0.4

0.0

i

/

/

-0.5

. . . .

.

/ -0.3

i

61

, .... 0.4

, .... 0.5

, .... 0.6

Ip IlPo ;Io

, .... 0.7

~ .... 0.8

, .... 0.9

l.O

t" I l P o

Fig. 2. Vertical distribution of the maximum ~P[/Po and [(rz'[/po for various wave obliquities 0.

-0.1

"""~"'.. "~

-0,2

~ i i

i

i

/

!

i /

i

t'4

.,.

i

s

i

,

/ i -0.5

" 0.0

i

/ /

i / "

/

/

i

•

.' .

. .' / '/

/ ..

/J /"

/

Jo~l/po(la~i/p°; - -

// /

/ -11.4

/

"

I

/.

/' /

./

././"

"

// -0.3

\

0°(9o o) 300(6O* )

.......

/

............. 45°(45°)

..

.

..." / ./"

.

.

.

.

.

.

.

.

.

.

.

.

................ •

'

0.1

'

'

J 0.2

i

i

I 0.3

60"(300) ~°(o0) ,

,

J

i 0.4

. Fig. 3. Vertical distribution of the maximum [crx'l/po and [(ry'l/po for various wave obliquities 0. Input data is indicated in Fig. 2. o f the maximum pore pressure Lol/po vs z / L for various values of/Xhh are depicted in Fig. 5. It was found that the maximum Lol/po generally increases as the Poisson's ratio /~hh increases (Fig. 5). A similar trend is observed in Fig. 6 for the Poisson's ratio /Xvh. Compared with the influence of the Poisson's ratio /Xhh on the wave-induced pore pressure, the Poisson's ratio /~vh affects the maximum pore pressure Lollpo more significantly (Figs 5 and 6). Besides the Poisson's ratios, /Xhh and /&h, the other three cross-anisotropic parameters,

62

D.-S. Jeng 0.0

~¢.,.'

I

. . . .

I

-0. I

. . . .

"

I

. . . .

"~

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

"~

', i

-0.2

i

! i

i

/

/

/

/

~ ,. . . .

i

,

t

J

/

/

/

,

-0.4

/ /

/

j

i

t

/ ./

00(90

_

°)

- ......

30o(60

. )

...........

450(45

°)

/ 600(30

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

/"

90o(0

°) ° )

/

! /'

. ..'. ~/'.

0.0

_

./

"/"

/

i -0.5

!!:

Ix 'I/p (l't~I/p)

/' //

/

-0.3

/

~ ....

0.1

0.2

, ....

0.3

,

,

....

0.4

....

,

t ....

0.5

0.6

....

~ ....

0.7

0.8

,

....

0.9

1.0

I x:~l/p o; Ix,,zl/p o I~,:l/p,, versus z / L for various wave obliquities 0. Input data is indicated in F i g . 2.

F i g . 4. Vertical distribution o f the m a x i m u m ['r~fl/po and

0.0

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

I ' ' '

-0.1

S;" 40,.;;;;.;;

-0.2

1,4

;";~;';:::'~I' d= :C' ::'

/5.

L = 205.09m

///

-0.3

0 = 45"

//," ///

lI ~ ^^

/,/ /f:

-0.4

~'/ /t

'/ -0.5 0.0

,,,

.

.

~ ,h = 0 . 4 5 ~ 0.6

- - o : o

.=oo

""

0.15 0.30

E=IO'N/m' n J=0.4

0.45

S

--

!.

t

..... . . . .

.

~

--_ l (

" ' x - ' "

/

i ,.~

I

0.1

i

. . . . . . . .

0.2

I

0.3

. . . .

I

. . . .

0.4

I

. . . .

I

0.5

. . . .

0.6

. .

= K = I 0 .r ~ / ~ p r - " , -

= 0.~75 I

0.7

. . . .

. . . . . . . .

" I

0.8

. . . .

I

0.9

. . . .

1.0

Ipl/p o F i g . 5. Vertical distribution of the m a x i m u m

[PI/Po

vs

z/L

for various Poisson's ratio /Xh,.

Eh, Ev and Gv, are related to two non-dimensional parameters, 12 and A, as presented in Equations (1) and (3). The vertical distributions of the maximum Lol/po vs z/L for various values of lq = Eh/Ev are illustrated in Fig. 7. It is found from the figure that the maximum pore pressure Lp]/po increase as 12 increases. Figure 8 illustrates the effects of anisotropic parameter A on the wave-induced pore pressure. Unlike other anisotropic parameters, the pore pressure Lp]/po decreases as the anisotropic parameter A = Gv/Ev increases. The figure clearly shows that the pore pressure

W a v e - i n d u c e d seabed response in a cross-anisotropic seabed 0.0

. . . .

i

'i

. . . .

. . . .

i

. . . .

i

. . . .

I

. . . .

i

. . . .

i

. . . .

i

. . . .

63

i ' ' '

• ..................

"......

:

-0.1

.'."2" -'~2"~2'~

..

"- .

-0.2 . ~.~)5/

, ; ~,/" -7,:/

t'4

q).3

...

,',/ ,'///

""

"1

t i

~ -0.5 0.0

.~

.. ..... .:

'~'/

-0.4

d=40m ~ zos.og m

~ "*

.""

- -

-

-

.

0.1

0.2

i ....

0.3

0.6

¢ i ....

0.4

._~.,,s

A--

o.0 ti = 0.6_ -_-._ ~~ ==IO'NI,." ,v ,,.

-

, ,,'~ i . . . . . . . . . . . .

o = 453 ..

-- ~o',.,.< i ....

0.5

i ....

0.6

i ....

0.7

0.8

i ....

0.9

1.0

Ip I/Po Fig. 6. Vertical distribution o f the m a x i m u m ~llpo vs z/L for various P o i s s o n ' s ratio /xvh.

0.0

....

i ....

i ....

i ....

i ....

i ....

i ....

i ....

i ....

b'''

-0.2 .~.l

t.4

j

/,',,"

-o3

d = 40

o = 4s;

/,,/

A"/ I / /

- - - - - -

I~

IS,:

-o.5 0.0

l

~

0.1

m

~ 20S..Ogm

/,,.."

~

.

. .

.

~=

04

A--o.:

o.4s ......

0.6n:'=0.4

x

<

=

Jo ' ,./,,~

<

.

, 0.2

~ 0.3

.

. 0.4

, 0.5

.

. . . . . . . . . . .

0.6

0.7

0.8

0.9

1.0

I p I/Po Fig. 7. Vertical distribution of the m a x i m u m Lpllpo vs z/L for various anisotropic constant f i ( = EdEv).

~ l l p o decreases gradually as z/L increases for the case of small A, for example, A = 0.4 and 0.6. On the other hand, the [ollpo decreases as z/L increases near the seabed surface for large values of A. Once hol/po reaches a minimum value, it increases as z/L increases. Furthermore, the pore pressure is affected more significantly by anisotropic parameter A than ~ (Figs 7 and 8).

64

D.-S. Jeng

u

I-

-03~-

':"

.

/

,./

t ,. - o• ~I ~ [, 0.0

-/,

,'/

,'

i' /

0.1

,

-

o

,,,

0=45°_..

,

o.~,='o'~',, ~ .

_

~ 0.2

-

~ - - -

/

, :/

L-- 2o5.o9

A

~r

/

t

'\

_

n'=0.4 •

v

. . . . . . . ~2 s;=o~75 0.3

0.4

0.5

Ip

0.6

0.7

0.8

z

0.9

1.0

IlPo

Fig. 8. Vertical distribution of the maximum ]Pl/l)~vs z/L for various anisotropic constant A( = Gv/E,.). 5.3. Relative differences between a cross-anisotropic and an isotropic solution Unlike the conventional solution (Yamamoto et al., 1978; Hsu et al., 1993), which is based on the assumption of an isotropic seabed, the wave-induced soil response in a crossanisotropic seabed is dominated by five anisotropic parameters. It is interesting to find the relative differences of the wave-induced soil response between cross-anisotropic and isotropic solution. It has been suggested that the wave-induced seabed response is affected significantly by the degree of saturation and soil permeability in an isotropic seabed (Hsu and Jeng, 1994; Jeng and Hsu, 1996). Furthermore, as the analytical solution presented previously, the vertical Young's modulus, Ev, has been included directly or indirectly in most coefficients. The influences of these three soil parameters on the relative differences between the pore pressure obtained from cross-anisotropic and isotropic solution, ([Pc]- ~])/Po, are illustrated in Figs 9, 10 and 11. In these figures, p,. and p~ denote the pore pressure obtained from cross-anisotropic and isotropic solutions, respectively. As shown in Fig. 9, the effects of the degree of saturation (St) on the relative differences of pore pressure, ([Pcl - [Pil)/P0, may be divided into two categories: saturated and unsaturated seabed. It was found that the conventional isotropic solution overestimates the pore pressure under a fully saturated condition. For an unsaturated seabed, the isotropic solution, however, underestimates the pore pressure near the seabed surface. Once the (Loci-Lo~l)/po reaches a maximum point, it decreases as z/L increases. This trend becomes more significant as the degree of saturation decreases. Furthermore, the maximum relative difference of pore pressure may reaches 0.2po at Sr = 0.90. It is well known that the seabed instability (including liquefaction and shear failure) occurs in the region near the seabed surface under an unsaturated condition. Thus, the seabed would suffer liquefaction or shear failure, if the conventional isotropic solution is used in a cross-anisotropic seabed. Figure 10 demonstrates the vertical distribution of the (Loci - ~il)lpo vs z/L for various

Wave-induced seabed response in a cross-anisotropic seabed 0.3

. . . .

,

. . . .

I

. . . .

,

' /

-

-

"

1.0

....... g~

0.975

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

. . . .

L = 2 0 5 . 0 9 ra 0=45 ° p~=0.45

0.99

~. . . . . . . . . . . . . .

,

T = 12.5 s e c d=4Ora

S 0.2

. . . .

65

P ,~ = 0 . 4 5

0.95

~ = 0.6 0.6

0.1

A=

"i

k

i

00.

"~"

.

.

.\;..

.

I

,

,

,

i

l

i

-0.1

i

.

-02'

K

=K=K=lOZm/~ec

I

i

i

'

. . . .

'., -

. . . .

0.0

'

. . . .

'

-0.1

i

.

i

i

i

l

i

*

'

. . . .

i

i

•

. . . .

-0.2

-0.3

-0.4

-0.5

z/L Fig. 9. Relative differences of pore pressure (Loci - Lo~l)/po vs z/L for various values of the degree of saturation S .

0.3

. . . .

,

. . . .

i

K

. . . .

,

=Ky=Kz(ra/sec

0.1

-

"

10 i

.......

1 0 -z

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

10 4

,

. . . .

T = 12.5 sec d=40m L = 205.09 m

)

0.2 -

. . . .

0=45*

p~=0.45 p ~, = 0 . 4 5 ~=0.6 A=0.6

I

E= lOZNhn 2 S=0.975

I | - -

0.0

-0.1

................. L ~ L . 2 . 2 . 2 . 2 .

-0.2

. . . . 0.0

i -0.1

. . . .

x~._.~ ~ ~ . . . ~ . . . - - ~ .

i

. . . .

-0.2

i -0.3

. . . .

i -0.4

. . . . -0.5

z/L Fig. 10. Relative differences of pore pressure (Loci - Lo~l)/po vs z/L for various vertical permeability K z.

values o f soil permeability. For a coarser seabed (for example, Kz = 10-', 10 -2 m/s), the relative differences (~Ocl - ~oil)/po increase as z/L increases near the seabed surface, and then decreases after reaching a peak. This phenomenon is more obvious in a porous seabed with K z = 10 -l m/s than K z = 10 -2 rn/s. However, for a finer seabed (such as K z = 10 -4 m ] s ) , the relative difference o f pore pressure (~ocl - Loil)/po decreases rapidly as z/L increases near the seabed surface, and then increases smoothly. It was found that the conventional isotropic solution underestimates the pore pressure in a coarser seabed, but overestimates it in a finer seabed.

66

D.-S. Jeng 0.3

. . . .

f

. . . .

i

f /

\

/ ~2~

\ \

]

0.0

. . . .

T = 12.5 sec d=40m L = 205.09 m 0=45"

,o,

. ~ = 045

- - - -

5xlO 7

II~, =0.45 ~.2 = 0.6 A = 0.6 n'=0.4 K = K = K = 10 2 m/sec S = 0.975

\ \ \

22--'"

~,

02

i

5x10

\

/ ~ ",

4).1

~

. . . .

.......

\

/ [ 0.1

i

E ( N / m z)

0.2

~" -..~"

. . . .

i;;

4). 1

-0.2

Fig. 11. Relative differences of pore pressure (Ip~l -

4).3

4).4

-0.5

z/L Ip~l)/p,,vs c/L for various vertical Young's modulus E,.

The effects of the vertical Y o u n g ' s modulus Ev on the relative difference of pore pressure (]p~.] - [Pd)/Po are presented in Fig. 11. The patterns of the trends of (]P~.I - [Pil)/Po for various E,. are similar. The relative difference, (]p~[ - ]p~l~/po, increases as z/L increases near the seabed surface, and then decreases after a certain burial depth. This implies that the isotropic solution underestimates the pore pressure near the seabed surface. Furthermore, the m a x i m u m values of the relative differences ([p~.] - [Pd)/Po increase as E,. increases, as seen in Fig. 11. 6. CONCLUSIONS In this paper, an analytical solution for the wave-induced soil response in a cross-anisotropic seabed in front of a breakwater is proposed. From the numerical results presented the following conclusions were drawn: 1. The wave-induced pore pressure [l)]&o and vertical effective normal stress ](r='[/po are independent of the wave obliquity 0. The effective stresses ]o-~'I/Po and t'c,:" ]/Po increase as the incident angle 0 increases, and compensate with ]o-,,'/po and ]%:'/Po, respectively. 2. As in the numerical results presented, the pore pressure IPl/Po increases as the Poisson's ratios, /Xhh, /Xvh, and anisotropic constant, [L increase. This trend is more significant tot various /Xvh than the other two parameters. On the other hand, the wave-induced pore pressure decreases as the anisotropic constant A increases, and the influence of A on ~]/Po is more significant than for the other parameters. 3. The relative differences of the pore pressure between a cross-anisotropic and an isotropic solution, ( ) ~ l - [P~I)/Po, are investigated with various values of S,, K. and Ev. Generally speaking, the m a x i m u m values of (~Pcl - Ipil)/po increase as Sr and K. decrease, but as E, increases. It is worth noting that the m a x i m u m relative difference may reach 20% of Po, as shown in Figs 9-11.

Wave-induced seabed response in a cross-anisotropic seabed

67

Acknowledgement--The computing work of this paper was performed when the author worked at the Department of Environmental Engineering, The University of Western Australia. REFERENCES Barden, L. (1963) Stresses and displacements in a cross-anisotropic soils. G~otechnique 13, 198-210. Biot, M. A. (1941) General theory of three-dimensional consolidation. Journal of Applied Physics 12, 155-164. Fenton, J. D. (1985) Wave forces on vertical walls. Journal of Waterways, Port, Coastal and Ocean Engineering, A.S.C.E. 111, 693-718. Gazetas, G. (1982) Stresses and displacements in cross-anisotropic soils. Journal of the Geotechnical Engineering Division, A.S.C.E. 108, 532-553. Graham, J. and Houlsby, G. T. (1983) Anisotropic elasticity of a natural clay. G~otechnique 33, 165-180. Hsu, J. R. C. and Jeng, D. S. (1994) Wave-induced soil response in an unsaturated anisotropic seabed of finite thickness. International Journal of Numerical Analysis Methods in Geomechanics 18, 785-807. Hsu, J. R. C., Jeng, D. S. and Tsai, C. P. (1993) Short-crested wave induced soil response in a porous seabed of infinite thickness. International Journal of Numerical Analysis Methods in Geomechanics 17, 553-576. Hsu, J. R. C., Jeng, D. S. and Lee, C. P. (1995) Oscillatory soil response and liquefaction in an unsaturated layered seabed. International Journal of Numerical Analysis Methods in Geomechanics 19, 825-849. Hsu, J. R. C., Silvester, R. and Tsuchiya, Y. (1980) Boundary-layer velocities and mass transport in short-crested waves. Journal of Fluid Mechanics 99, 321-342. Hsu, J. R. C., Tsuchiya, Y. and Silvester, R. (1979) Third-order approximation to short-crested waves. Journal of Fluid Mechanics 90, 179-196. Jeng, D. S. (1996) Wave-induced liquefaction potential in a cross-anisotropic seabed. Journal of the Chinese Institute of Engineers 19, 59-70. Jeng, D. S. (1997a) Wave-induced seabed instability in front of a breakwater. Ocean Engineering 24, 887-917. Jeng, D. S. (1997b) Soil response in a cross-anisotropic seabed due to standing waves. Journal of Geotechnical and Geoenvironmental Engineering, A.S.C.E. 123, 9-19. Jeng, D. S. and Hsu, J. R. C. (1996) Wave-induced soil response in a nearly saturated seabed of finite thickness. G~otechnique 46, 427-440. Jeng, D. S. and Lin, Y. S. (1996) Finite element modelling for water waves-soil interaction. Soil Dynamics and Earthquake Engineering 15, 283-300. Jeng, D. S. and Lin, Y. S. (1997) Non-linear waves induced response of porous seabeds: a finite element analysis. International Journal of Numerical Analysis Methods in Geomechanics 21, 15-42. Jeng, D. S. and Seymour, B. R. (1996) Wave-induced pore pressure and effective stresses in a porous seabed with variable permeability. Proceedings of the 15th International Conference on Offshore Mechanics and Arctic Engineering, ASME, Florence, pp. 327-334. Lin, Y. S. and Jeng, D. S. (1997) The effect of variable permeability on the wave-induced seabed response. Ocean Engineering 24, 623-643. Mei, C. C. and Foda, M. A. (1981) Wave-induced response in a fluid filled porous elastic solid with a free surface--a boundary layer theory. Geophysics Journal of the Royal Astronomy Society 66, 597~537. Nago, H., Maeno, S., Matsumoto, T. and Hachiman, Y. (1993) Liquefaction and densification of loosely deposited sand bed under water pressure variation. Proceedings of the 3rd International Offshore and Polar Engineering Conference, Singapore, pp. 578-584. Okusa, S. (1985) Wave-induced stresses in unsaturated submarine sediments. G~otechnique 35, 517-532. Picketing, D. J. (1970) Anisotropic elastic parameters for soil. Ggotechnique 20, 271-276. Silvester, R. and Hsu, J. R. C. (1989) Sines revisited. Journal of Waterways, Port, Coastal and Ocean Engineering, A.S.C.E. 115, 327-344. Seymour, B. R., Jeng, D. S. and Hsu, J. R. C. (1996) Transient soil response in a porous seabed with variable permeability. Ocean Engineering 23, 27-46. Smith, A. W. and Gordon, A. D. (1983) Large breakwater toe failures. Journal of Waterways, Port, Coastal and Ocean Engineering, A.S.C.E. 109, 253-255. Tsai, C. P. and Jeng, D. S. (1992) A Fourier approximation for finite amplitude short-crested waves. Journal of the Chinese Institute of Engineers 15, 713-721. Verruijt, A. (1969) Elastic storage of aquifers. In: Flow Through Porous Media, ed. R. J. M. De Wiest. Academic Press, New York, pp. 331-376. Yamamoto, T., Koning, H. L., Sellmeiher, H. and Van Hijum, E. V. (1978) On the response of a poro-elastic bed to water waves. Journal of Fluid Mechanics 87, 193-206. Zen, K. and Yamazaki, H. (1990) Mechanism of wave-induced liquefaction and densification in seabed. Soils and Foundations 30, 90-104.