Effects of Wave Non-Linearity on Residual Pore Pressures in Marine Sediments

Transcription

1 Th Opn Civil Enginring Journal, 8,, Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Sdimnts Dong-Shng Jng* Opn Accss Division of Civil Enginring, School of Enginring, Physics and Mathmatics, Univrsity of Dund, Dund, DD1 4HN, Scotland, UK Abstract: A bttr undrstanding of th wav-inducd por prssur accumulations (i.., rsidual por prssur is a ky factor in th analysis of th wav-inducd liqufaction in marin sdimnts. In this papr, th rsidual mchanism of nonlinar wav-inducd por watr prssur accumulation in marin sdimnts is xamind. Unlik prvious invstigations, th scond-ordr Stoks wav thory is considrd in this study. Th nw modl is vrifid with xprimntal data and provids a bttr prdiction of por prssur accumulation than th prvious solution with linar wav thory. Th paramtric study concluds that th influncs of wav non-linarity incras as th following paramtrs incras (i wav stpnss (H/L and (ii rsidual paramtr (. Howvr, an opposit trnd is found for (i wav priod (T, (ii rlativ watr dpth (d/l, (iii sabd thicknss (h/l and (iv anothr rsidual paramtr (. Furthrmor, th ffct of wav non-linarity bcoms mor significant in soft sabd (Soil C. INTRODUCTION Th valuation of th wav-inducd soil rspons in marin sdimnts is particularly important for coastal nginrs involvd in th dsign of foundation of many marin installations,.g., offshor mono-pils, brakwatrs, piplins and platforms tc. Th prdiction of th wav-inducd xcss por prssur is a ky procdur in th analysis of sabd instability such as liqufaction and scour. Thrfor, it is ncssary to hav a bttr undrstanding of th mchanism of th wav-induc por prssur in marin sdimnts. Two mchanisms for wav-inducd por prssur hav bn obsrvd in th prvious fild masurmnts and laboratory xprimnts [1], as shown in Fig. (1, Th first mchanism is rsultd from th transint or oscillatory xcss por prssur and is accompanid by attnuation of th amplitud and phas lag in th por prssur changs [, 3]. Th scond mchanism is trmd th rsidual por prssur, which is th build-up of xcss por prssur causd by contraction of th soil undr th action of cyclic loading [4]. Numrous studis for wav-inducd momntary liqufaction, causd by oscillatory por prssur, hav bn carrid out sinc th 197s. Among ths, Yamamoto t al. [4] proposd a closd-form analytical solution for an infinit sabd. Mi and Foda [5] proposd a boundary-layr approximation to driv a rathr simplifid formulation for th wav-inducd transint por prssur, which is only valid for coars sand [6, 7]. Jng [3, 6-9] drivd a sris of analytical solutions for th oscillatory por prssur within marin sdimnts. His modls considrd various soil bhaviors, such as cross-anisotropic soil bhaviors and variabl prmability. Latr, Kianoto and Mas [1] and Yuhi and Ishida *Addrss corrspondnc to this author at th Division of Civil Enginring, School of Enginring, Physics and Mathmatics, Univrsity of Dund, Dund, DD1 4HN, Scotland, UK; d.jng@dund.ac.uk [11] furthr suggstd a nw and simplifid formulation for th wav-inducd por prssur in a cross-anisotropic sabd. An intnsiv rviw of th prvious rsarch for th wav-inducd oscillatory por prssur can b found in [1]. Fig. (1. Mchanisms of oscillatory and rsidual por prssurs in marin sdimnts. Th mchanism of por prssur build-up du to ocan wavs has bn studid by many rsarchrs sinc th 197s. Sd and Rahman [4] stablishd a simpl on-dimnsional finit lmnt modl by taking into account th distribution of cyclic shar strsss in th soil profil, and por-prssur dissipation. Skiguchi t al. [13] proposd an lasto-plastic modl for th standing wav-inducd liqufaction using a Laplac transformation. Latr, som numrical modls for post-liqufaction and progrssiv liqufaction and dnsification in marin sdimnts wr dvlopd [14]. In addition to numrical modling, McDougal t al. [15] proposd a st of analytical solutions for wav-inducd por prssur build-up in a uniform layr of soil, basd on th assumption of an incomprssibl soil. In thir approach, th sourc trm in th modifid Biot's consolidation quation is drivd using a linar rlationship btwn por prssur /8 8 Bntham Opn

2 64 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng ratio ( ug / and cyclic ratio ( / [16, 17]. To provid a convnint practical rsult for nginrs, McDougal t al. [15] proposd thr solutions for th cass of shallow, finit and dp soil dpths, rspctivly. Ths analytical solutions ar usful for both nginrs and rsarchrs, as thy can b usd for ithr th invstigation of qualitativ bhaviors of complicatd nginring problms or th validation of numrical mthods. Rcntly, using a similar approach, Chng t al. [18] r-xamind th analytical solution of McDougal t al. [15] and proposd a numrical modl to invstigat th sam problm. As pointd out by Chng t al. [18], th analytical solution proposd by McDougal t al. [15] rvald som rrors in th formulations. Howvr, aftr a clos xamination of both th McDougal t al. [15] and Chng t al. [18] solutions, th author found numrous rrors in both publications, as rportd in [19-1]. Rcntly, a sris of analytical approximations for th wav-inducd accumulatd por prssur in marin sdimnts hav bn proposd by th author [19-1]. Both cass of infinit and finit soil layrs wr considrd in th modls. A simplifid univrsal formula was proposd for th cas of infinit sabd [19]. Howvr, all ths approximations wr limitd to linar rgular wav loadings, although it should b non-linar wavs in natural nvironmnts. In this papr, th modls dvlopd by th author [19-1] ar furthr xtndd to non-linar wav loadings. Th nw non-linar wav modl will b vrifid by comparing th prvious xprimntal data, togthr with th prvious linar wav modl. Basd on th prsnt modl, w will xamin th ffcts of wav stpnss and rsidual paramtrs on th wav-inducd accumulatd por prssur in a porous sabd. THEORETICAL FORMULATIONS Non-Linar Wav Thory Hrin, a soil matrix subjct to a two-dimnsional progrssiv wavs systm is considrd, as dpictd in Fig. (. Th wav crst propagats in th positiv x-dirction, whil th z-axis is positiv upward from th sabd surfac, as shown in Fig. (. z gh cosh kz = sin( kx t cosh kd (1 3 H coshkd sin ( kx t 3 cosh kd whr H is th wav hight, d is watr dpth, g is th gravitational acclration and t is th tim. Th wav numbr (k and wav frquncy ( can b dtrmind with th following wav disprsion rlation: = gk tanh kd ( Th watr surfac displacmnt to th scond-ordr ( can b xprssd as H = cos( kx t (3 kh cosh kd + ( + cosh kdcos ( kx t 3 16 sinh kd and th dynamic wav prssur ( P b at th sabd surfac (z= can b writtn as, P ( x, t = P + P cos( kx t + P cos( kx t (4 b w w1 w P w wkh wh =, Pw 1 = 8sinhkd cosh kd 3 wkh 1 1 w = P 8sinhkd sinh kd 3 Sinc this study focuss on th dynamic wav prssurs, th first trm on th lft hand sid of (4 is xcludd hr. Boundary Valu Problm for Sabd Th Biot consolidation quations [3] hav bn gnrally adoptd to modl th dynamic rspons of marin sdimnts for various applications. In gnral, th wav-inducd por prssur within marin sdimnts consists of two componnts: oscillatory ( p% and rsidual (u mchanism, which can b xprssd as (5 pxzt (, ; = pxzt %(, ; + pzt (, = pxzt %(, ; + uzt (, (6 fr surfac SWL whr p is th por watr prssur, p% rprsnts th oscillatory por prssur, which lads to momntary liqufaction, whil u = p rprsnts th priod-avragd por prssur, which lads to rsidual liqufaction, and dfind by d h O Watr y sabd surfac x Soil porous sabd rigid imprmabl bottom Fig. (. Sktch of wav-sabd intraction. Rfrring to a wav thory to th scond-ordr [], th vlocity potntial ( is givn as 1 t+ T u = pdt T (7 t whr T is th wav priod and t is th tim. A sris of analytical solutions for th linar wavinducd oscillatory por prssur within marin sdimnts hav bn dvlopd sinc th 197's []. In this study, rfrring to (4, th scond-ordr dynamic wav prssur can b xprssd in thr trms with diffrnt wav frquncis. Thus, th wav-inducd sabd rspons varigats can b xprssd in thr trms:

3 Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Th Opn Civil Enginring Journal, 8, Volum 65 ( m (,, = ( cos( m m m= 1 SP x z t SP z k x t (8 whr km = mk and m = m and SP dnots soil rspons variabls such as por prssur and strsss tc. ( m Th amplitud of th oscillatory por prssur ( p% and ( m shar strss ( % for ach componnt in a saturatd sabd ar givn by [3]: { ( m Pwm p% = (1 μ C C4 (1 μ % (1 μ( m m kmz kmz ( mz mz ( 5 6 } + k C C kmz {( ( = P C C k z C C k z ( m wm 1 m 3 4 m m m mz mz ( 5 6 } + k C C kmz (9 (1 ( m whr p% is th amplitud of dynamic wav prssur in ach frquncy mod; μ is th Poisson's ratio, and th Ci ( i = 1, L,6 cofficints can b found in [7]. Th rsidual por prssur (u in a homognous, isotropic soil can b drivd from th on-dimnsional Biot's consolidation quation [3], u u = cv + f t z (11 in which f is th man accumulation por prssur sourc trm associat with th surfac watr wavs [15] and c v is th cofficint of consolidation, which is givn by Gkz (1 μ cv = (1 μ w (1 To solv, th following boundary and initial conditions ar rquird: u(, t = u( z, =, uzt (, z z=h whr h is th thicknss of a sabd. Sourc Trm = (13 In this sction, w now discuss th "sourc trm" of th por prssur gnration (f. Th laboratory rsults of D Alba t al. [17] rlat th dvlopmnt of por watr prssur to th numbr of load cycls in simpl shar tsts. Thir non-linar rlationship is givn by ug 1/ 1 N 1 1 = + sin 1 Nl (14 whr u g is th por prssur gnration du to cyclic loading, is th ffctiv ovr burdn, N is th numbr of cyclic loading, N l is th numbr of cycls to liqufaction, and is th shap factor varying from.5 to. [4]. Hrin, th ffcts of th shap factor on th rlationship of por prssur gnration will b discussd. Fig. (3 illustrats th rlationships btwn ug / and th numbr of load cycls ( N / N l with various valus of. As shown in th figur, th shap factor, =.7, is likly to prsnt a linar rlation, as rportd in [4]. Th por prssur sourc trm in (11 is givn by Sd t al. [16] f u g = (15 t To simplify th problm, a linar mchanism of por prssur gnration was proposd [16, 17] ug N = N l (16 from which th sourc trm of por prssur gnration can b xprssd as u g / σ N f = = t N TN (17 l l.3 θ=.5. θ=.75 θ=1..1 θ=1.5 θ= N/N L Fig. (3. Distributions of th dvlopmnt of por prssur ( ug / vrsus th numbr of load cycls ( N / N l for various valus of th shap factor (. In (17, N l is th numbr of cycls to liqufaction, which is a function of th cyclic shar strss ratio [4], N 1/ % l = (18 whr % is th amplitud of wav-inducd shar strss, and and ar th functions of th soil typ and rlativ dnsity. Substituting (18 into (17, w hav f / % = T (19

4 66 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng which is a gnralisd dfinition of th sourc trm. It is notd that th linar mchanism of por prssur gnration was first applid to th wav-inducd por prssur build-up in marin sdimnt by Sd and Rahman [4]. Solutions Th shar strss in th sourc trm dpnds on both wav and soil charactristics. Th complt solution of (19 is rquird for th sourc trm. An analytical approximation for th linar mchanism of por prssur gnration was drivd in [19-1] by using a Fourir sris xpansion. Th rsidual por prssur can b xprssd as ( m cvnt/ h n 1 sin( n ( m= 1 n= 1 u = a z h a = f ( rsin( r dr (1 ( m h ( m n cv n whr = (n 1 / and n f n ( m th m-th componnt of th sourc trm, givn in (19. It is notd that th prvious modls [19-1] ar limitd to linar wav loadings, whil th prsnt study focuss on non-linar wav loadings. Comparisons In this sction, th xprimntal data of Cluky t al. [4] is compard with th prsnt solutions. Svral tsts wr conductd in a small wav tank [4]. A comparison of th calculatd and masur por prssur accumulation is shown in Fig. (4. Th input data of th xprimnts [4] ar tabulatd in Tabl (1. Th rsults of th prvious analytical solutions for linar wav loading [1], th prsnt non-linar wav loading ar includd in th comparison. Th rlativ soil dpth for th data is in th rang of.<h/l<.3, hnc th soil dpth is intrmdiat. As shown in th figur, th prsnt modl for non-linar wav loading is approaching to th xprimntal data and provids a bttr prdiction than that of Jng t al. [1] Non linar wavs Linar wavs Exprimntal data z (m Fig. (4b. Comparison of th prvious modl for linar wavs [1], th prsnt modl for non-linar wavs and xprimntal data [4]- Cas B. RESULTS AND DISCUSSIONS Th objctiv of this papr is to invstigat th ffcts of wav non-linarity on th wav-inducd por watr prssur accumulation in marin sdimnts. In this sction, A paramtric study for xamining th influncs of wav and soil charactristics and rsidual paramtrs on th quilibrium por prssur. Th input data for numrical xampls ar listd in Tabl (. In th numrical xampls, thr diffrnt sandy bds ar considrd. Tabl 1. Input Data for Exprimntal Data [4] Wav Charactristics Cas A Cas B wav priod (T [sc] watr dpth (d [m].5.5 wavlngth (L [m] wav hight (H [m]..1 Soil charactristics Non linar wavs Linar wavs Exprimntal data u (kn/m sabd thicknss (h [m].84 Poisson s ratio ( μ.49 z (m u (kn/m soil porosity ( n.46 shar modulus (G [n/m ] Soil prmability ( k [m/sc] z unit wight of soil grain ( [N/m 3 ] 18,36 s unit wight of por fluid ( [N/m 3 ] 986 w cofficint of arth prssur ( K.4 cofficint of consolidation ( c v.1165 o Fig. (4a. Comparison of th prvious modl for linar wavs [1], th prsnt modl for non-linar wavs and xprimntal data [4]- Cas A. rsidual paramtr (.46 rsidual paramtr (.165

5 Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Th Opn Civil Enginring Journal, 8, Volum 67 Tabl. Input Data for Paramtric Study Wav Charactristics 1 Non linar wavs Linar wavs Soil charactristics wav priod (T [sc] watr dpth (d [m] wav hight (H [m] sabd thicknss (h [m] 1 or various or various 3 or various or various Poisson s ratio ( μ.35 (Soil A.45 (Soil B.49 (Soil C soil porosity ( n.3 (Soil A.4 (Soil B.46 (Soil C shar modulus (G [n/m 7 ] 1 (Soil A (Soil B Soil prmability ( k [m/sc] z (Soil C 1 (Soil A 4 1 (Soil B 6 1 (Soil C unit wight of soil grain ( s [N/m 3 ].65 w cofficint of arth prssur ( K.5 Rsidual paramtr o z (m u (N/m Fig. (5a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for T=15 sc. z (m Non linar wavs Linar wavs rsidual paramtr ( rsidual paramtr (.5 or various.16 or various 9 Effcts of Wav Charactristics In this sction, thr wav charactristics will b xamind to s thir influncs on th non-linar wav-inducd quilibrium por prssur ( u. Numrical rsults ar plottd in Figs (5-(7. Wav priod is on of important wav charactristics, which is rlatd to th wav lngth and apparing in all wav charactristics such as wav nrgy, prssur and forcs, tc in trm of wav angular frquncy ( = /T. In th paramtric study, w vary th wav priod from.5 sconds to 15 sconds, which covrs all possibl rangs of ocan wavs. Fig. (5 illustrats th vrtical distribution of th quilibrium ( u vrsus soil dpth for both non-linar wav [th prsnt modl] and linar wav [1] modls. As shown in th figur, th valu of u incrass as wav priod dcrass. This is bcaus th por prssur will b accumulatd mor in a shortd wav priod, as th por prssur is mor unlikly to b discharg from soil matrix during a shortr duration. It is also obsrvd from th figur that th rlativ diffrncs btwn linar and non-linar wav modls incrass as wav priod dcrass with a fixd wav hight. This is bcaus that th wav stpnss will incras whn wav priod dcrass with a sam wav hight. That is, th wav prssur and forcs ar largr u (N/m Fig. (5b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for T=1 sc. z (m Non linar wavs Linar wavs u (kn/m Fig. (5c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for T=7.5 sc.

6 68 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng 1 Non linar wavs Linar wavs Non linar wavs Linar wavs 3 z (m 5 z (m u (kn/m Fig. (5d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for T=5 sc u (kn/m Fig. (6a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for d/l=.3. 1 Non linar wavs Linar wavs Non linar wavs Linar wavs 3 z (m 5 z (m u (kn/m Fig. (5. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for T=.5 sc. Watr dpth is anothr important wav paramtr, which is also involvd in th dtrmination of wavlngth and th maximum wav hight bfor wav braking. In this study, w vary th rlativ watr dpth from.5 (shallow watr to.3 9narly dp watr. Fig. (6 prsnts th distribution of por prssur ( u vrsus soil dpth for diffrnt rlativ watr dpth (d/l. As illustratd in th figur, por prssur ( u incrass as rlativ watr dpth (d/l dcrass. That implis that th rsidual por prssur will accumulat mor in shallow watr, compard with dp watr. It is also obsrvd from th figur that th rlativ diffrncs btwn linar and non-linar wav modls incrass as rlativ watr dpth (d/l dcrass with a fixd wav hight. Wav hight is an important factor in th dtrmination of wav forcs and wav nrgy acting on marin structurs. In this study, w considr non-braking wavs, which has th maximum wav stpnss with th following critrion: H d =.14 tanh L L max ( u (kn/m Fig. (6b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for d/l=.. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (6c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for d/l=.15.

7 Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Th Opn Civil Enginring Journal, 8, Volum 69 Non linar wavs Linar wavs Non linar wavs Linar wavs z (m 1 z (m u (kn/m Fig. (6d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for d/l= u (N/m Fig. (7a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for H/L=.5. Non linar wavs Linar wavs Non linar wavs Linar wavs z (m 1 z (m u (kn/m u (N/m Fig. (6. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for d/l=.5. On of th aims of this study is to invstigat th ffcts of wav non-linarity on th wav-inducd accumulatd por prssur in marin sdimnts. Fig. (7 illustrats th ffcts of wav stpnss on th quilibrium accumulatd por prssur u. Gnrally spaking, th quilibrium accumulatd por prssurs ( u incras as wav stpnss (H/L incrass, comparing Fig. (7a-(7. Furthrmor, th diffrncs of por prssur ( u btwn linar and non-linar wav componnts incras as th wav stpnss (H/L incrass. All ths diffrncs majorly com from th scondordr wav componnts. Soil Charactristics As rportd in [5], svral important soil paramtrs will significant affcts th non-linar wav-inducd oscillatory sabd rspons. Thy ar: sabd thicknss and soil typ (in a combination of shar modulus, soil prmability, Poisson s ratio and porosity. In this sction, w will focus on th ffcts of ths two soil paramtrs. Fig. (7b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for H/L=.1. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (7c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for H/L=..

8 7 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng Non linar wavs Linar wavs In addition to th wav and soil charactristics, th influncs of two rsidual paramtrs, and, will b xamind in this sction. Basically, ths two rsidual paramtrs, and, wr proposd by Sd and Rahman [4]. Ths two paramtrs ar normally dtrmind by laboratory xprimnts with crtain rangs of valus. For xampl, varis from. to.5, whil varis btwn.16 and.4. In this paramtric study, w vary and within ths rangs. z (m 1.5 Non linar wavs Linar wavs u (kn/m Fig. (7d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for H/L= Non linar wavs Linar wavs u (N/m x 1 3 Fig. (8a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for Soil A. z (m 1.1 Non linar wavs Linar wavs u (kn/m.5.6 Fig. (7. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for H/L=.6. Soil typ is on of important soil charactristics in most gotchnical problms. Th proprtis of thr diffrnt soil typs ar tabulatd in Tabl (. Fig. (8 illustrats th influnc of soil typs on th por prssur ( u. As shown in th figur, Th ffcts of wav non-linarity on th quilibrium por prssur ( u bcom significant in softr sabd (Soil C. Sabd thicknss somhow will caus 4% rlativ rrors of oscillatory por prssur, if w us th solution of an infinit sabd is usd [7]. Fig. (9 illustrats th diffrnc of linar and non-linar wav modls on th wav-inducd rsidual por prssur ( u. As shown in th figur, th influncs of sabd thicknss incras as h/l incras whn h/ L.. Howvr, an opposit trnd is obsrvd for th cas of h/ L >.. Rsidual Paramtrs u (N/m Fig. (8b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for Soil B. Figs. (1 and (11 illustrat th influnc of rsidual paramtrs on th accumulatd por prssur. As shown in th figurs, th rsidual por prssur incrass as dcrass, and th diffrnc btwn th prvious linar wav modl and th prsnt non-linar wav modl incrass as incras. This is bcaus a small will incras th sourc trm, which will nhanc th por prssur build up. Howvr, th influnc of is opposit to that of, comparing Figs. (1 and (11. It is also notd that a minor chang in and will caus a rmarkabl chang in th quilibrium por prssur ( u. Thrfor, dtrmination of ths rsidual paramtrs rquirs xtrmly attntions, bcaus thy ar vry snsitiv to u.

9 Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Th Opn Civil Enginring Journal, 8, Volum 71.1 Non linar wavs Linar wavs. Non linar wavs Linar wavs u (kn/m Fig. (8c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for Soil C Non linar wavs Linar wavs Fig. (9c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for h/l= u (kn/m Non linar wavs Linar wavs u (kn/m u (kn/m Fig. (9a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for h/l=.5. Fig. (9d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for h/l=...1 Non linar wavs Linar wavs ( h/l=.3 Non linar wavs Linar wavs u (kn/m Fig. (9b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for h/l= u (kn/m Fig. (9. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for h/l=.3.

10 7 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng Non linar wavs Linar wavs Non linar wavs Linar wavs z (m 1 z (m u (kn/m Fig. (1a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for = u (N/m Fig. (1d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.4. Non linar wavs Linar wavs Non linar wavs Linar wavs z (m 1 z (m u (kn/m Fig. (1b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.5. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (1c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for = u (N/m Fig. (1. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.5. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (11a. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.16.

11 Effcts of Wav Non-Linarity on Rsidual Por Prssurs in Marin Th Opn Civil Enginring Journal, 8, Volum 73 Non linar wavs Linar wavs Non linar wavs Linar wavs z (m 1 z (m u (kn/m Fig. (11b. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.18. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (11c. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.. z (m 1 Non linar wavs Linar wavs u (kn/m Fig. (11d. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for = u (kn/m Fig. (11. Vrtical distribution of quilibrium por prssur ( u vrsus soil dpth (z for =.4. CONCLUSIONS In this papr, ffcts of wav non-linarity on th wavinducd por prssur accumulations in marin sdimnts ar xamind, and th following conclusions can b drawn. 1. A comparison against th prvious wav tank xprimnts show that th prsnt modl with non-linar wav thory providd a bttr prdiction of th wav-inducd quilibrium por prssur ( u.. Numrical xampls indicatd that th ffcts of wav non-linarity on por prssur ( u incrass as wav stpnss (H/L and th rsidual paramtr (. Howvr, an opposit trnd is found for wav priod (T, rlativ watr dpth (d/l, sabd thicknss (h/l and anothr rsidual paramtr (. 3. Th ffct of wav non-linarity bcoms mor significant in soft sabd (Soil C. REFERENCES [1] K. Zn, and H. Yamazaki, Mchanism of wav-inducd liqufaction and dnsification in sabd, Soils and Foundations, vol. 3, pp. 9-14, 199. [] T. Yamamoto, H. Koning, H. Sllmjjr, and E. van Hijum, On th rspons of a porolastic bd to watr wavs, Journal of Fluid Mchanics, vol. 87, pp , [3] D.-S. Jng, Wav-inducd sabd rspons in front of a brakwatr, PhD Thsis, Th Univrsity of Wstrn Australia, [4] H. B. Sd, and M. S. Rahman, Wav-inducd por prssur in rlationto ocan floor stability of cohsionlss soil, Marin Gotchnology, vol. 3, pp , [5] C. C. Mi, and M. A. Foda, Wav-inducd rspons in a fluidfilld porolastic solid with a fr surfac--boundary layr thory, Gophysical Journal of th Royal Astrological Socity, vol. 66, pp , [6] J. R. C. Hsu, and D.-S. Jng, Wav-inducd soil rspons in an unsaturatd anisotropic sabd of finit thicknss, Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, vol. 18, pp , [7] D.-S. Jng, and J.R.C. Hsu, Wav-inducd soil rspons in a narly saturatd sabd of finit thicknss, Gotchniqu, vol. 46, pp , [8] D.-S. Jng, Soil rspons in cross-anisotropic sabd du to standing wavs, Journal of Gotchnical and Gonvironmntal Enginring, ASCE, vol. 13, pp. 9-19, 1997.

12 74 Th Opn Civil Enginring Journal, 8, Volum Dong-Shng Jng [9] D.-S. Jng, and B.R. Symour, Rspons in sabd of finit dpth with variabl prmability, Journal of Gotchnical and Gonvironmntal Enginring, ASCE, vol. 13, pp , [1] T. Kianoto, and H. Mas, Boundary-layr thory for anisotropic sabd rspons to sa wavs,, Journal of Watrway, Port, Coastal and Ocan Enginring, ASCE, vol. 15, pp , [11] M. Yuhi and H. Ishida, Simplifid solution of wav-inducd sabd rspons in anisotropic sabd, Journal of Watrway, Port, Coastal and Ocan Enginring, ASCE, vol. 18, pp. 46-5,. [1] D.-S. Jng, Wav-inducd safloor dynamics, Applid Mchanics Rviw, ASME, vol. 56, pp , 3. [13] H. Skiguchi, K. Kita, and O. Okamoto, Rspons of porolastoplastic bds to standing wavs, Soils and Foundations, vol. 35, pp. 31-4, [14] S. Sassa, and H. Skiguchi, Wav-inducd liqufaction of bds of sand in cntrifug, Gotchniqu, vol. 49, pp , [15] W. G. McDougal, Y. T. Tsai, and P. L.-F. Liu and E. C. Cluky, Wav-inducd por watr prssur accumulation in marin soils, Journal of Offshor Mchanics and Arctic Enginring, ASME, vol. 111, pp. 1-11, [16] H. B. Sd, P. O. Martin, and J. Lysmr, Th gnration and dissipation of por watr prssur during soil liqufaction, Collg of Enginring, Univrsity of California, Brkly, CA, [17] P. D Alba, H. B. Sd, and C.K. Chan, Sand liqufaction in larg-scal simpl shar tsts, Journal of Gotchnical Division, ASCE, vol. 1, pp. 99-8, [18] L. Chng, B.M. Sumr, and J. Frdso, Solutions of por prssur build up du to progrssiv wavs, Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, vol. 5, pp , 1. [19] D.-S. Jng and B.R. Symour, A simplifid analytical approximation for por-watr prssur build-up in a porous sabd, Journal of th Watrway, Port, Coastal and Ocan Enginring, ASCE, vol. 133, pp , 7. [] D.-S. Jng, B. R. Symour, F. P. Gao, and Y.X. Wu, Ocan wavs propagating ovr a porous sabd: Rsidual and oscillatory mchanisms, Scinc in China, Sris E, vol. 5, pp , 7. [1] D.-S. Jng, B.R. Symour, and J. Li, A nw approximation for por prssur accumulation in marin sdimnt du to watr wavs, Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, vol. 31, pp , 7. [] R. G. Dan and R. A. Dalrympl, Watr Wav Mchanics for Enginrs and Scintists, World Scintific, Nw Jrsy, [3] M. A. Biot, Gnral thory of thr-dimnsional consolidation, Journal of Applid Physics, vol. 1, pp , [4] E. C. Cluky, F. H. Kulhawy, P. L.-F. Liu, Laboratory and fild invstigation of wav-sdimnt intraction, Josph H.Dfrs Hydraulics Laboratory, School of Civil and Environmntal Enginring, Cornll Univrsity, Ithaca, NY, [5] D.-S. Jng, and Y. S. Lin, Non-linar wav inducd rspons of porous sabds: A finit lmnt analysis, Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, vol. 1, pp. 15 4, Rcivd: March 31, 8 Rvisd: April 8, 8 Accptd: May 3, 8 Dong-Shng Jng; Licns Bntham Opn. This is an opn accss articl distributd undr th trms of th Crativ Commons Attribution Licns ( which prmits unrstrictiv us, distribution, and rproduction in any mdium, providd th original work is proprly citd.