Dynamic response and sliding distance of composite breakwaters under breaking and non-breaking wave attack

Transcription

1 Dynamic esponse and sliding disance of composie beakwaes unde beaking and non-beaking wave aack Giovanni Cuomo, Giogio Lupoi, Ken-ihio 3 Shimosako 3 & Shigeo Takahashi 4 JSPS Pos-Doc ellow a Maiime Sucues Division, Maine Envionmen and Engineeing Depamen, Po and Aipo Reseach Insiue, 3--, Nagase, Yokosuka, Japan Pincipal Enginee, Coasal Sucues Goup, HR Wallingfod, Howbey Pak, Wallingfod, OX 8BA, UK Pincipal Coasal Enginee, Hydaulics Applied Reseach and Engineeing Consuling (HAREC) S..l. Coso Tiese 4, 98, Roma, Ialy Coesponding auho. : +44 () g.cuomo@hwallingfod.co.uk Sudio SPERI S..l. s.d.i., Lungoevee delle Navi 9, 96;Roma Ialy Pincipal Sucual Enginee, Hydaulics Applied Reseach and Engineeing Consuling (HAREC) S..l. Coso Tiese 4, 98, Roma, Ialy : glupoi@sudiospei.i 3 Head of Maiime Sucues Division, Maine Envionmen and Engineeing Depamen, Po and Aipo Reseach Insiue, 3--, Nagase, Yokosuka, Japan akahashi_s@pai.go.jp 4 Execuive Reseache and Dieco of Tsunami Reseach Cene, Po and Aipo Reseach Insiue, 3--, Nagase, Yokosuka, Japan addess: shimosako@pai.go.jp Published in: Coasal Engineeing, Volume 58, Issue, Ocobe, DOI:.6/j.coasaleng..3.8 Absac Ove he las 5 yeas impoved awaeness of wave impac induced failues has focused aenion on he need o accoun fo he dynamic esponse of maiime sucues o wave impac load. In his wok a nonlinea model is inoduced ha allows evaluaing he effecive design load and he poenial sliding of caisson beakwae subjec o boh pulsaing and impulsive wave loads. The caisson dynamics is modelled using a ime-sep numeical mehod o solve numeically he equaions of moion fo a igid body founded on muliple non-linea spings having boh hoizonal and veical siffness. The model is fis shown o coecly descibe he dynamics of caisson beakwaes subjec o wave aack, including nonlinea feaues of wave-sucuesoil ineacion. Pedicions of sliding disances by he new mehod ae hen compaed wih measuemens fom physical model ess, showing vey good ageemen wih obsevaions. The model succeeds in descibing he physics ha sands behind he pocess and is fas, accuae and flexible enough o be suiable fo pefomance design of caisson beakwaes. This is he auho's vesion of a wok ha was acceped fo publicaion in Coasal Engineeing. Changes esuling fom he publishing pocess, such as pee eview, ediing, coecions, sucual fomaing, and ohe qualiy conol mechanisms may no be efleced in his documen. Change may have been made o his wok since i was submied fo publicaion. A definiive vesion was subsequenly published in Coasal Engineeing, Volume 58, Issue, Ocobe, DOI:.6/j.coasaleng Keywods Wave impacs, caisson beakwaes, wave loads, sliding, non-linea dynamics of coasal sucues, pefomance design HRPP486

2 . Inoducion The capabiliy of wave impac loads o cause he sliding of composie-ype beakwaes had been poved in he ealy sixies by Nagai (966) who saed I was poven by / and / scale model expeimens ha, a he insan when he esulan of he maximum simulaneous shock pessues jus exceeds he esising foce, he veical wall slides. Analyses of failues caied ou in Euope and Japan ove he pas 5 yeas confim impac loads induced sliding o be he mos impoan cause of failue fo caisson beakwaes. Neveheless, despie he impoance of impulsive loadings and hei effecs on he dynamic of caisson beakwae have been widely ecognised, a simple and compehensive mehodology fo he assessmen of cumulaive sliding disance is sill missing. This pape pesens a simple bu consisen mehod fo modelling he dynamic esponse and sliding disance of composie beakwaes subjec o wave aack. In he following, documened cases of sliding-induced failues of caisson beakwaes ae biefly summaised (.) and findings fom pevious eseaches on dynamics of caisson beakwaes eviewed (.-.3). A non-linea dynamic model fo he esponse of caisson beakwaes subjec o wave loading is hen pesened ( 3), ogehe wih a pocedue fo he geneaion of wave foce ime-hisoies fo use in dynamic analysis ( 4). The effeciveness of he model is hen veified using simplified foce ime-hisoies ( 5) and finally compaed o measuemens fom physical mode ess on sliding of caissons subjec o boh pulsaing and beaking wave aack ( 6) showing vey good ageemen wih boh analyical soluions and expeimenal obsevaions.. Lieaue eview Reseach on he dynamics of caisson beakwaes subjec o wave loading has mainly concenaed on suveying damaged and failed sucues, undesanding he physics ha sands behind he dynamics of caissons and defining wave loads fo use in dynamic analysis. Accodingly, in he following we summaise documened failue of caisson beakwaes and mos significan effos owads he undesanding of caisson dynamics... Documened ailues Oumeaci (994) gave a eview of analysed failue cases fo boh veical and composie beakwaes: 7 failue cases wee epoed fo veical beakwaes and 5 fo composie o amoued veical beakwaes. The auho idenified wave beaking and beaking clapois as he mos fequen damage souce of he disases expeienced by veical beakwaes, by means of (in ode of impoance): sliding, shea failue of he foundaion and oveuning. anco (99 and 994) and anco and Passoni (99) summaised he Ialian expeience in design and consucion of veical beakwaes giving a hisoical eview of he sucual evoluion in he las cenuy and ciically descibing he majo documened failues (Caania, 933; Genova, 955; Venoene, 966; Bai, 974; Palemo, 983; Bagnaa, 985; Naples, 987 and Gela, 99). In all cases he collapse was found o be due o unexpeced high wave impac loading, esuling fom he undeesimaion of he design condiions and he wave beaking on he limied deph a he oe of he sucue. Knowledge on failue mode of veical beakwaes has been widened by he lage expeience inheied in ecen yeas fom obsevaions made all hough las decades in Japan. Among he ohes, Goda (974) HRPP486

3 epoed ad e-analysed a lage numbe of hisoical sliding-induced failues of veical caisson beakwaes in Japan, Hiachi (994) descibed he damage of Musu Ogawaa Po (99), Takahashi e al. (994) discussed he failues occued a Sakaa ( ) and Hacinohe (99) Pos. Takahashi e al. (998) discussed esuls fom an exensive field suvey of Japanese beakwaes and summaised caisson wall failues in he peiod Among ohe findings, he auhos confimed impulsive beaking wave pessue o be he main cause of damage fo caisson beakwaes, ogehe wih he collision of concee blocks agains he caisson walls. Moe ecenly, Takahashi e al. () analysed 33 majo failues occued beween 983 and 99 and epoed ypical failues of composie beakwaes; he auhos idenified sliding of caissons and sucual failues due o impulsive wave pessue as he mos impoan failue modes fo caisson beakwaes insalled on a seep foeshoe and subjec o beaking wave aack... Exising Models o Dynamics Of Caisson Beakwaes Mainski and Oumeaci (99) gave a eview of he CIS (fomely Sovie Union) design expeience on dynamic esponse of veical sucues subjec o beaking wave foces. Mos of he mehods developed in he CIS assumed he dynamics of veical beakwae o be well descibed by ha of a igid body on a homogenous, elasic and isoopic half space wih he soil paamees adoped in he model diving he oveall esponse of he sysem. Reviewing he available lieaue (almos always in Russian), he auhos idenified hee schools of houghs, based especively on heoeical woks by Peashen (956), Sminov and Mooz (983) and Loginov (96 and 969). The mehod suggesed by Loginov is he only one o have been included in he Russian guidelines fo he evaluaion of he loadings and hei effecs on maiime sucues; The model combines he swaying and oaing moions of he caisson in wo ocking moions aound wo sepaae cenes (locaed especively above and below he cene of gaviy of he caisson) and neglecs he effec of damping. De Goo e al. (996) exensively eview (a ime) sae of he a mehods fo design of caisson beakwae foundaion, including exising appoaches o dynamics. On his gound, simplified models fo he dynamic behaviou of caisson beakwaes have been developed wihin he famewok of he PROVERBS (PRObabilisic design ools fo VERical BeakwaeS) eseach pojec (see, among ohes, Oumeaci and Koenhaus, 994; Oumeaci e al., 99; Klamme e al., 994). Despie is elaive simpliciy, he model poposed by Oumeaci and Koenhaus (994) epesens an efficien ool fo he exploaion of he dynamic esponse of caisson beakwaes o wave impac loads and a emakable aemp o quanify he elaive impoance of he applied dynamic load and he dynamics (mass, siffness and damping) of he beakwae (including he supesucue, is foundaion soil and he suounding wae) on he oveall dynamic esponse of he sysem as a whole. o hese easons, his model is biefly descibed in he following. The igid body in he idealised D lumped sysem skeched in igue has wo degees of feedom, especively he hoizonal anslaion and he oaion aound A. o such a sysem, he equaion of moion can be e-wien in maix fom as follows: M u + C u + K u = () ( ) ( ) ( ) ( ) whee: m M = x () m θ HRPP486 3

4 cx cx ( H c y A ) ( ) ( ) x H c y A cθ + cx H c y A k x k x ( H c y A ) ( ) ( ) x H c y A kθ + k x H c y A x ( ) ( ) ( ) ( ) ( ) x yl y A + y x A xl ( ) ( ) θ C = (3) c K = (4) k = (5) u u = x u (6) Tems m x, m θ, k x, k θ, c x, c θ in Equaions - 4 epesen he oal mass ( m ), he siffness ( k ) and he damping ( c ) of he sysem agains sliding ( x ) and ocking ( θ ) and x L and y L ae especively he leve am of he veical ( y ) and hoizonal ( x ) foces, x A and y A ae especively he coodinaes of he cene of oaion of he caisson. Accoding o he auhos, he siffness ems can be deemined accoding o Mainski and Oumeaci (99) while he oal mass of he sysem is given by he summaion of he mass of he caisson, he hydodynamic mass and he geodynamic mass. The damping coefficiens wee obained expeimenally by means of pendulum ess on he caisson beakwae model iself and fo diffeen degees of immesion (Oumeaci e al. 99). x kx kθ cθ θ cx y igue Dynamic model of caisson beakwae (afe Oumeaci and Koenhaus, 994) Moving fom ealie obsevaions duing small-scale model ess (Klamme e al. 994) a simple model fo he evaluaion of he pemanen displacemen of caisson beakwaes unde impac loads has been suggesed by Oumeaci e al. (995) de Goo e al. (996) and Koenhaus and Oumeaci (996). Accoding HRPP486 4

5 o he auhos, he ineacion of he supe-sucue wih he foundaion soil is diven by adhesion while he hoizonal foce does no exceed he ciical value xƒ : x,c [ ] ( ) = µ W ( ) s y Whee µ s is he saic ficion coefficien, W is he weigh of he caisson in wae and y () is he waveinduced uplif foce. When he caisson sas o move, a consan ficion coefficien µ d is applied unil he hoizonal foce educes again o he ciical value x,c () in Equaion 7, when he adhesion foce sas dominaing again. Moe ecenly, he dynamic analysis model poposed by Oumeaci and Koenhaus (994) has been exended by Wang () o accoun fo educion in effecive impac foces due o sliding and ocking moion of caisson beakwaes. Among he possible moions of caisson beakwaes unde dynamic load, he auho disinguished a pue vibaing moion, a vibaing-sliding moion and a vibaing-ocking moion; fo each of hem, a slighly diffeen vesion of Equaion is poposed which accouns fo he elaive moion of he caisson and is foundaion. Alenaive models have been suggesed by eseaches in Japan. Goda (994) suggesed modelling he dynamics of a composie beakwae as ha of a igid body suspended on a sysem of mass and dual spings fo oaional and hoizonal moions, and using he momenum heoy of impulsive beaking wave foces o esimae he sliding disance of he supesucue. Takahashi e al. (994), invesigaed he dynamic esponse and he sliding of beakwae caissons agains wave impac loads by mean of a EM model. A moe sophisicaed, non-linea model was successively adaped by Takahashi e al. (998) o evaluae cumulaive sliding of caisson beakwae unde beaking wave aack. The model was shown o be able o descibe he chaaceisics feaues of he ineacion of he caisson wih is soil foundaion including sliding; neveheless, he model was found o be elaively complex and is applicaion ime-consuming. Shimosako e al. (994) and Shimosako and Takahashi (999) pesened a simplified model fo he esimaion of disance of caisson sliding ha was found o compae saisfacoily well wih daa fom small-scale physical model ess. Accoding o he auhos, he pemanen displacemen can be evaluaed as: 3 d ( x,max y,max µ d W ) ( x,max y,max + µ d W ) S = (8) µ m W ( ) 8 d c x,max y, max Whee d epesens he duaion of iangula wave hus, µ d is he ficion coefficien, W is he caisson weigh in wae, x,max is he peak value of wave hus upon he caisson and y x,max is he uplif exeed upon he boom of caisson. The mehod has been ecenly adoped in pefomance-based design (Shimosako and Takahashi, 999) and eliabiliy design (Goda and Takagi, and Kim and Takayama, 3) mehods fo caisson beakwaes. An alenaive simplified mehod fo he evaluaion of he pemanen displacemen of composie beakwae subjec o wave impacs has been pesened by Ling e al. (999). Neveheless, he mehod only consides a sinusoidal, long lasing, load and hus suffes fom no accouning fo he impulsive naue of beaking wave loads. A summay of he mos significan models fo dynamics of caisson beakwae poposed in lieaue is given in Table. (7) HRPP486 5

6 Table Summay of exising models fo he dynamics of caisson beakwaes Loinov (96) Swaying and oaing moions ae combined in wo ocking moions aound wo sepaae cenes. Goda (994) Rigid body suspended on a sysem of mass and dual spings fo oaional and hoizonal moions, includes conibuion of he geodynamic mass. Oumeaci and Koenhaus (994) Modified by Koenhaus and Oumeaci (996) o accoun fo sliding of he supesucue when he hoizonal foce exceeds he adhesion foce. Takahashi e al. (994) EM - Modified by Takahashi e al. (998) o accoun fo non-lineaiy of he soil foundaion..3. Exising models fo ime-hisoy wave loads on caisson beakwaes Example ime-hisoy load exeed by a beaking wave on he seawad face of a veical beakwae is ploed in igue, showing a shap pulse supeimposed o a slowly vaying, pulsaing load. In ode o pefom numeical simulaion of he dynamic esponse of caisson beakwae o wave loading, a paameeised ime-hisoy load needs o be assumed fo use in he compuaions. An example idealised load-hisoy is supeimposed on an oiginal signal in igue ; he iangula spike is chaaceized by he maximum eached by he signal duing loading (P max ) and he ime ( ) aken o ge o P max fom. The shaded aea in igue epesens momenum ansfe o he sucue duing he impac: he impulse. HRPP486 6

7 Seveal simplified models exis in lieaue fo ime-hisoy load fo use in dynamic analysis of caisson beakwae subjec o impulsive and quasi-saic wave loading (see Cuomo, 5 fo a eview), seleced models ae summaised in Table. Paamee qs+ in Table sands fo he quasi-saic seawad foce as pediced by Goda s mehod in is oiginal fomulaion (Goda, 974) while impac maximum ( max ), ise ime ( ) and duaion ime ( d, igue ) need o be evaluaed accoding o guidance given in he efeed documens. Model by Shimosako and Takahashi (999) is he mos o sophisicaed up o dae, and has been widely adoped in Japan as a basis fo he evaluaion of sliding of caisson beakwaes. Based on obsevaions fom small scale laboaoy expeimens, Kim e al. (4) suggesed educing he ampliude of he quasi-saic componen of abou % when assessing he pemanen displacemen due o sliding and iling of caisson beakwaes subjec o wave impacs. Among he models summaised in Table, he chuch-oof poposed by Oumeaci and Koenhaus (994) and he model by Shimosako e al (994) (and successive modificaions by Shimosako and Takahashi, 999, Kim and Takayama, 3 and Kim e al. 4) have been found o give he mos deailed epesenaion of he vaiaion in ime of he oveall loading. In his wok we used he model oiginally poposed by Shimosako e al. (994) as i includes boh impulsive and pulsaing (posiive - landwad and negaive seawad) foce. Obseving Tables and, i is eviden ha since he ealy 6 s, models fo caisson dynamics have coninuously impoved side by side wih he gain in knowledge on wave loading and hei effecs on caisson beakwaes. Alhough epesening mos of he main feaues of caisson dynamics, exising models do no neveheless allow o epesen he effec of he vaiaion in ime of he loading on he coupling beween he dynamic esponse of he sucue and he beaing capaciy of he soil. While he full-dynamic EM pesened in Takahashi e al. 994 is he only o make an excepion, i is no ye suiable fo use in pefomance design of caisson beakwaes, fo which a lage numbe of simulaions is needed. Beaing his in mind, in he following a non-linea model is poposed ha allows accouning fo wave-sucue-soil ineacion bu is fas and accuae enough o be used in pefomance design of caisson beakwae based on accepable sliding disance in he design life ime (Lupoi e al. 7). igue Example pessue ime-hisoy ecoded on he seawad face of a seawall duing lage scale physical model ess a CIEM unde VOWS eseach pojec (Cuomo e al. 7). HRPP486 7

8 HRPP486 8 Table Summay of exising wave ime-hisoy loads fo use in he dynamic analysis of caisson beakwaes Lundgeen(969) π ; cos max max Goda (994) d fo fo < qs+ max Oumeaci and Koenhaus (994) d d fo fo fo < < < d d max max Shimosako and Takahashi (994) fo fo fo < < < max max Shimosako and Takahashi (999) fo fo π < < π < π T sin T sin ; max T sin ; max i qs i qs max i qs max Oumeaci e al. () d d fo fo fo < < < qs+ d d max max

9 3. A non-linea model fo he dynamics of caisson beakwae 3.. Dynamics of caisson beakwae subjec o wave loading In he following, a non-linea model is inoduced ha allows o evaluae he sliding of caisson beakwae subjec o boh pulsaing and impulsive wave loads. The mahemaical idealisaion of he poblem is shown in igue 3, whee he caisson is epesened by a igid block siing on a maess of non-linea spings. When a wave his he supesucue, he caisson ansmis a combinaion of (hoizonal and veical) foces o he foundaion soil. The esisance o hoizonal and veical loads ae especively epesened by he Coulomb s ficional foce developed a caisson-soil ineface and by he beaing eacion of he soil. b y( ) x( ) h Suppoing suface igue 3 Mahemaical idealisaion of a caisson beakwae adoped in he pesen model, he spings a he ineface beween he supesucue and is soil foundaion epesen non-linea elemens descibed in 3. and 3. o a saic sysem, he condiions fo a es ae expessed by: x ( ) µ s [( mc mw ) g y ( ) ] [( m mw ) g y ( ) ] ( ) [ ( ) x + ( m m ) g x ] x (9) c () y L c w g () y L whee µ s is he saic ficion coefficien beween he caisson sucue and he foundaion soil, m c is he mass of he caisson sucue, m w is he mass of he volume of wae displaced by he caisson, x L and y L ae especively he leve ams of veical and hoizonal wave-induced foces, x g is he level am of he gaviaional load. Equaions 9 and descibe he equilibium in he hoizonal and veical diecion, especively, while equaion epesens he momen equilibium aound he cones. The onse of he movemen of he body is deemined when one of he pevious equaions is no longe valid oiginaing, especively, sliding (Eq. 9), uplif (Eq. ) and ocking (Eq. ) moion. HRPP486 9

10 The equaion of moion can be wien accoding o he moe geneal fomulaion (valid fo boh he saic and dynamic sysem) as: ( ) + C u ( ) + K u( ) ( ) M u =, () whee a do denoes diffeeniaion wih espec o ime, M, C and K ae, especively, he mass, he damping and he siffness maixes a he fee degees of feedom. In he simplified fomulaion adoped in his pape, he veco u is composed by he displacemen along x, he displacemen along y and he oaion, θ. The veical and he hoizonal displacemen a each sping, u xi u x, i and u y, i, ae elaed o he hee degees of feedom (u x, u y, u θ ) by he following equaions: u x,i = ux (3) u = u + uθ x, (4) y,i y i whee x i is he posiion of he sping wih espec of he cenoid of he igid block. The esising foces a he hee degees of feedom ae hen evaluaed as a funcion of eacion a each sping, by means of he following se of equaions: n sping x ( ) = mx u x ( ) + c u x ( ) + kx,i ux ( ) (5) i= n sping y ( ) = my u y ( ) mw g + cy u y ( ) + k y,i ( u y ( ) + uθ( ) xi ) (6) i= n sping θ ( ) = mθ u θ( ) + cθ u θ( ) + k y,i xi ( u y ( ) + uθ( ) ) (7) i= In he numeical soluion of diffeenial equaions (Eq.) he acceleaion, velociy ad displacemen a ime + ae expessed in em of acceleaion, velociy and displacemen a ime. The evoluion equaions ae hee solved by means of Newmak s mehod (Newmak, 959), while he Newown-Raphson algoihm scheme (Chopa, ; Chape 5) has been adoped o solve fo he non-lineaiy of he sysem ( 3.). Pecisely, a each ime sep, we fis evaluae he unknown acceleaion, velociy and displacemen a ime +, hen we check he equilibium beween he applied foces and he esising ones and ieae unil convegence is achieved. The dynamic popeies (mass, siffness and damping) of he sysem ha ae epesened in he model include conibuions by he supesucue, he wae suounding he sucue and he soil foundaion. In he pesen fomulaion, he wo conibuions ae evaluaed accoding o Equaions - 4 (Oumeaci and Koenhaus, 994), whee he ansiional and oaional hydodynamic masses ae given especively by (Pedesen, 997): m =. 543ρ d (8) m hyd,x hyd w, θ =. 8ρw d (9) whee ρ w is he wae densiy and d he wae deph in fon of he wall. A moe sophisicaed model is employed fo he descipion of he dynamics popeies of he soil foundaion, whose linea ( 3.) and nonlinea ( 3.) aspecs ae descibed in deails in he following. HRPP486

11 3.. Dynamic popeies of soil foundaion, linea aspecs o massless sip foundaion esing on he suface of an elasic undamped homogeneous half plane Wolf (988, Chape ) deived he following expessions fo he veical ( K y ) and hoizonal ( K x ) siffness (pe uni lengh): K y = + 4ν () = () K x ( ) G ( + 5ν ) G whee ν=.33 is he Poisson s aio and G is he shea modulus. o he damping em ( C ) he classical expession by Lysme and Richa (966) has been adoped: 3. 4 C = Gρs, () ν in which ρ s is he mass densiy of he soil and he chaaceisic dimension of he equivalen cicula fooing has been aken as = bl / π, b and l being he plana dimensions of he caisson. The shea modulus G in Equaions, and above is given by G = E [ ( + ν )] in which Young s modulus E of he foundaion soil undeneah he caisson (ha is a an effecive veical sess aken as (Lunne e al., 997): E = E σ ' ' v + σv / ' σv in which E is he Young s modulus of he soil foundaion coesponding o an effecive sain kpa and σ is he addiional effecive sain wih espec o σ ' ' ' v = σv σv ' v (3). When no diec σ v ) has been ' ' σ v = σ v = measuemens ae available a he sie, values of E can be aken fom hose lised in lieaue (see, among ohes: Gazeas, 99 and Oumeaci e al. and efeences heein). In ou compuaions we assumed E = 35MPa, ha is ou soil foundaion o consis of medium dense sand. In he linea case, he mass, damping and siffness maixes fo he equaion of moion pesened in Eq. ae given by: mx mc + mhyd,x M = my = mc (4) m θ mc + mhyd,θ C C = C (5) C HRPP486

12 nsping k x,i K x nsping K = K y K yθ = nsping k y,i k y,i xi (6) i= K yθ K θ nsping nsping k y,i xi k y,i xi i= i= Alhough in a elaively simplified fashion, Equaions, and effecively chaaceise he linea behaviou of he soil foundaion of he beakwae while he supesucue sis on he ubble mound. Neveheless, duing mos violen impacs, wave induced hoizonal (shoewad) and veical (uplif) loads may indeed concu o paially up-lif he caisson fom is oiginal posiion so ha only pa of he foundaion paicipaes o esis sliding. In such cases, Equaions 9-4 do no povide anymoe a ealisic descipion of he dynamics of he sysem since hey equie ensile capaciy o be povided beween he sucue and he soil. In he following, an impoved model is inoduced ha allows accouning fo paial lifing of he caisson fom is soil foundaion Non-linea aspecs In he pesen model, he soil foundaion has been modelled by means of non-linea finie elemens (spings) o allow accouning fo he following non-linea feaues of foundaion soil: he inabiliy of caying axial ension; he dependency of he ficion beween he caisson and he foundaion soil on he effecive sain a he ineface. Beaing his in mind, he veical and hoizonal esponse of each sping has been coupled, esuling in he following expessions fo he siffness, especively in he veical and he hoizonal diecion: ( ) k y,i fo u y,i k y,i ( ) = ohewise (7) y,i ( ) k x,i fo y,i ( ) < and u( ) ux y,i k x,i = fo y,i ( ) < and u( ) > ux fo y,i ( ) (8) k y, i whee is he veical siffness of each sping i given by k y,i = K y / n sping ; k x,i is he hoizonal siffness expessed as a linea funcion of he veical foce y,i (), y, i is he veical foce acing on he i-h sping a es (gaviaional load); k x, i is he hoizonal siffness a es given by k x,i = K x nsping ; u x is he displacemen a incipien sliding coesponding o u x = µ s W K x. The esisance foce due o ficion is equal o µ S y, i ( ) a es and o µ D y, i ( ) duing sliding and µ s and µ d ae especively he saic and he dynamic ficion coefficiens, aken especively as.6 and.4. HRPP486

13 A epesenaion of he foce-displacemen elaionships fo boh he hoizonal and he veical diecions is shown in igue 4. Hoizonal load - x,i µ µ s y, i d y, i ( ) ( ) u x k = K x, i x n sping Hoizonal displacemen - u x,i Veical load - y,i k = K y, i y n sping Veical displacemen - u y,i igue 4 oce-displacemen elaionship of hoizonal (op) and veical (boom) esisance foces. The hoizonal moion (op) is descibed by means of a ficion model as a funcion of he veical foce: y,i (), he yield displacemen: u and he saic and dynamic ficion coefficiens: µ s and µ d. The veical x moion (boom) is descibed by means of a bilinea model, funcion of he veical displacemen u y,i (). HRPP486 3

14 4. Geneaion of ime-hisoy loads 4.. Non-beaking wave ime-hisoy loads Unde non-beak waves aack, ime-hisoy loads ae assumed o follow a slowly vaying, sinusoidal pah. o he j-h wave, he following expession is assumed fo he geneaion of boh hoizonal and veical foce ime hisoy loads (): π α qs qs+ sin fo < T j ( ) T j = (9) π qs sin fo T j T j whee T j is he wave peiod and qs+ and qs ae he quasi saic (in un hoizonal and veical) posiive (shoewad, uplif) and negaive (seawad, sucion) foce coesponding o pessue disibuion pediced using especively models by Goda () and Sainflou (98). Reducion coefficien α qs has been se equal o.8 accoding o ecommendaions by Kim e al. (4), based on ecen physical model ess a Univesiy of Kyoo. Level ams of oveuning momens due o foces in Equaion 9 ae evaluaed inegaing pessue disibuion by he coesponding heoies. 4.. Beaking wave ime-hisoy loads Typical beaking wave ime-hisoy load on a veical wall is ploed in igue, showing a high magniude - sho duaion peak supeimposed o a slowly vaying pulsaing load. Cuomo e al. () analysed beaking wave loads on veical walls ecoded duing lage-scale physical model ess. The auhos found ha beaking wave impac pessues migh well exceed hose pediced by mos esablished pedicion mehods and suggesed a new se of fomulae fo boh he impulsive and he pulsaing componens of he loading. Accodingly, beaking wave ime-hisoy loads ae evaluaed as follows. is, he ampliude of he impulsive ( x,max ) hoizonal componens of beaking waves loads is compued as (Cuomo e al. ): d d x,max = α max ρwg H L (3) b ( d ) s d Whee d s and L(d s ) ae he wae deph a he oe of he beakwae and he coesponding wave lengh elaed o T = T m, d is he wae deph in fon of he caisson wall, d b is he wae deph a beaking. Noe ha since Equaion 3 was deived o pedic andom wave impac foce a /5 significan level using wave heigh, H = H s (and assuming H /5 =.8 H s whee H s is he significan wave heigh) a educion faco α max = /.8 needs o be applied o be consisen wih he physics when using Equaion 3 wih monochomaic waves of heigh H. The posiive pulsaing componen ( qs+ ) of he load is again evaluaed using fomulaion by Goda (974), which has been found o give he bes epesenaion of shoewad pulsaing wave loads also unde beaking wave aack (Cuomo e al. ). The negaive pulsaing componen ( qs- ) is evaluaed accoding o Sainflou (98). HRPP486 4

15 The esuling foce ime-hisoy fo he j-h beaking wave load is evaluaed as: π max x,max, j ;qs+, j sin fo < T j π max x,max, j ;qs+, j sin fo < < ( ) = T j j π qs+, j sin fo < < T j T j π qs, j sin fo T j T j To ensue consisency beween impac maxima ( x, max ) and ise-ime ( ), in Equaion 3 he lae is evaluaed accoding o he pocedue descibed in secion 4... The veical (uplif) foce is sill obained using Equaion 3, bu eplacing qs+ and qs- wih hei coesponding veical componens evaluaed especively accoding o Goda (974) and Sainflou (98), while he following expession (Oumeaci e al., ) is used o evaluae he impac uplif foce y, max as a funcion of he hoizonal impac foce x, max : x,max b y,max =. 45 (3). 4H +. d + e b 7( ) in which H b is he beake heigh ( 4..), b is he caisson widh, d is he wae deph in fon of he caisson wall and e he deph by which he caisson is imbedded in he mound. Equaion 3 deives fom simplifying he expession fo uplif foces given in Oumeaci e al. () assuming a zeo uplif pessue a he shoewad end of he caisson. Leve ams of oveuning momens due o foces in Equaions 3-3 ae evaluaed inegaing pessue disibuion by he coesponding heoies (Goda, 974;Sainflou, 98; Oumeaci e al. ) up he wall Evaluaion of beake heigh and wae deph a beaking Beake heigh and wae deph a beaking play a majo ole in wave loading of caisson beakwae and hei coec evaluaion is a key issue when assessing he sabiliy of a caisson unde beaking wave aack. To evaluae wave loads a he sucue using he pesen model, he following infomaion is needed: wave heigh a he oe of he sucue (H s,oe ), he wae deph a beaking (d b ) and he limiing beake heigh (H b ) in fon of he sucue. In he pesen fomulaion H s,oe is evaluaed applying he Miche (944) cieion fo beaking while d b and H b ae evaluaed accoding o he beaking cieia descibed in Oumeaci e al. () Consisency beween impac maxima and ise ime Due o consevaion of momenum, he impulse is a finie quaniy and hus mos violen impac loads will necessay coespond o vey sho duaion and vice-vesa. Seveal aemps have been made o define a funcional elaion beween wave impac maxima and ise imes (see Cuomo (5) fo a eview) including analyical fomulaions (based on consevaion of momenum and compessibiliy of ai-wae mixue) and empiical elaions, usually in he fom: (3) HRPP486 5

16 x,max b = a (33) Hee, consisency beween geneaed wave impac maxima and ise ime has been insued by means of he following simplified pocedue. o each wave heigh H i and peiod T i, he impac foce x,max,i and he quasi saic foce as+i, ae evaluaed especively accoding o Equaion 3 and Goda (974), ogehe wih he * * = T dimensionless impac foce = x,max qs+. The nomalised impac ise ime m is hen assumed o obey a log-nomal disibuion, ha is: * ( ) ( ) * ln µ f, µ, σ = exp (34) * σ π σ To ensue consisency beween impac maxima and ise imes, paamees σ and µ in Equaion 34 ae aken * as a funcion of as follows: ( * ) = a * µ b ( * ) = a * σ b µ µ (35) σ (36) σ whee he empiical paamees a µ =.34, b µ = -3.4 and a σ =.87, b σ = -.6 have been fi o daa gaheed duing physical model ess descibed in Cuomo e al. (). A ealisaion fo coesponding o a given * is hence andomly geneaed accoding o Equaion 34 wih paamees σ and µ given in Equaions 35 and 36 as a funcion of *. igue 5 shows he compaison of example geneaed (cicles) dimensionless impac maxima ( max / qs+ ) and ise ime ( / T m ), wih pedicion using Equaion 34 a 5% (black dos) and 95% (gey dos) nonexceedance pobabiliy levels. I migh be noiced ha consisency beween wave impac maxima and ise * imes is only paially insued by he pocedue descibed above as vaiabiliy in is only paially coupled wih impac maxima by means of exising funcional elaionship beween paamees σ and µin Equaions 35 and 36 and values of *. A deepe coupling beween vaiabiliy in * and * migh have been obained by consideing he join pobabiliy of impac maxima and ise-ime as suggesed in Cuomo e al. (9), based on join pobabiliy disibuion of impac maxima and ise imes. Alhough less igoous han he lae, he fome fomulaion has been found o give an efficien and inuiive epesenaion of he effecive vaiabiliy in impac maxima and ise imes and is heefoe been employed in he pesen calculaions. HRPP486 6

17 .4 p.3. max / qs /T m igue 5 Example impac maxima and ise imes (cicle) andomly geneaed accoding o pocedue in 4. using monochomaic waves H=.45m, T=3s, d=.4m. 5. Veificaion of he dynamic model and illusaive example calculaions 5.. Quasi-saic applicaion of a hoizonal foce exceeding he saic ficion esisance ( x >µ s y ) We applied a saic hoizonal (shoewad) foce on he seawad face of he caisson a he sill wae level (s.w.l.) hus geneaing a clockwise oveuning momen ha ends o oveun he caisson shoewads. Resuls ae shown in igue 6, in ems of (fom op o boom) hoizonal eacion, hoizonal displacemen, veical eacion and displacemen a each of he fou spings used o model he soil foundaion. om lef o igh gaphs efe o spings along he caisson base fom he seawad o he shoewad face. The effec of he oveuning momen is eviden in he boom panels, showing veical eacions linealy inceasing moving fom he shoewad owads he seawad end of he caisson. The effec of oveuning momen is also noiceable in he op panels, wih hoizonal eacions inceasing fom lef o igh, accodingly o he fac ha sping siffness inceases wih inceasing veical load. HRPP486 7

18 i,x (kn).5 i,x (kn).5 i,x (kn).5 i,x (kn) x -3.5 x -3.5 x -3.5 x -3.5 u x (m).5 u x (m).5 u x (m).5 u x (m) i,y (kn) - -3 i,y (kn) - -3 i,y (kn) - -3 i,y (kn) u y (m) x u y (m) x u y (m) x u y (m) x ime (s) ime (s) ime (s) ime (s) igue 6 Response o saically applied hoizonal foce (applied shoewad a s.w.l.) a locaions of he fou sping elemens locaed along he caisson-soil foundaion ineface. om op o boom: hoizonal eacion, hoizonal defomaion, veical eacion and veical defomaion. Geomeical and dynamic chaaceisics fo his example case as follows: l =.m, b =.78m, h =.m, d =.4m, K y =.9E+7 kn/m; K x =.3E+5 kn/m; D =.7E+ kns/m, W = 8.4 kn. 5.. Shock speca of a SDO sysem ( x <<µ s y ) The model has been applied o deive he "esponse specum" fo a single degee of feedom sysem subjec o pulse exciaion, ha is is "shock specum". The ime-hisoy load descibing he pulse shape is defined by he following iangula pah: x (,) = x,max < (37) > o diffeen values of, Equaion 37 descibes a symmeical iangula pah having magniude x,max and duaion. Resuls ae pesened in he ange < / T < 5 (T being he naual peiod of vibaion of he SDO) on he lef hand side of igue 7 in ems of he amplificaion faco defined as: max{ ux ( ) } Ψ = (38) u x HRPP486 8

19 whee u x () is he hoizonal displacemen and ux is he saic hoizonal displacemen defined as he aio of he load inensiy o he siffness of he sysem u = K. x Numeical esuls (cicles) ae almos supeimposed o he analyical soluion (dashed line) on he lef hand side of igue 7, demonsaing he efficiency of he model. Effec of sliding on he dynamic esponse of caisson beakwae is highlighed on he igh hand side of igue 7, showing he modificaion of he shock speca of he same caisson beakwae subjec o an hoizonal impulse having inensiy compaable o he saic sliding esisance of he caisson. A few conclusions can be deived by obseving plos on he igh hand side of igue 7: x fo << T he dynamic esponse of he sucue educes he foce effecively fel by he caisson so ha no-sliding occus even fo x max > µ W, (saic sliding esisance, SSR); s fo T he dynamic esponse of he sucue amplifies he foce effecively fel by he caisson so ha sliding occus even fo < µ W x, max s. x.4. analyic soluion model soluion Amplificaion faco Amplificaion faco - analyic soluion model soluion: x <<µ s y. model soluion: x <µ s y model soluion: x =µ s y / T - model soluion: x >µ s y - - / T igue 7 Shock specum fo caisson beakwae subjec o pulse exciaion. Response o he applicaion of an hoizonal foce having magniude smalle (lef) and lage (igh) han he saic sliding esisance (SSR). When compaed o pedicions by he saic fomulaion by Shimosako e al. (999), esuls using he pesen model show ha (igue 8): fo / T << he dynamic model pedics no-sliding even if > µ W x, max s ; fo / T he dynamic model pedics lage sliding han he saic appoach, in paicula, including he dynamic esponse of he sucue migh esul in sliding of he sucue even fo < µ W x, max s, HRPP486 9

20 fo / T >>, sliding pediced by he dynamic model is lowe han esimaes by he saic appoach due o he educion of effecive load due o moion of he sucue. x < µ s y x >µ s y. ( u x,max - u x ) / u x,saic ( u x,max - u x ) / u x,saic / T / T igue 8 Pemanen displacemen due o pulse exciaion having maxima slighly lowe (lef) and highe (igh) hen he saic sliding esisance, compaison of pedicions by saic appoach by Shimosako e al. (999, black maks) and he pesen model (whie maks) Dynamics of caisson beakwae subjec o non-beaking and beaking wave ime-hisoy loads Example ime hisoy loads fo he case of a single non-beaking and beaking wave ae epesened especively on he lef hand side of igues 9 and. om op o boom, hin solid lines epesen he hoizonal (shoewad) and veical (upwad) foces and he oveuning (clockwise) momen. Resuls ae shown in ems of eacions (lef) and defomaions (igh) fo he caisson as a whole (solid hick line) as well as fo spings a diffeen locaions along he ineface beween he caisson and is soil foundaion (doed line). The vaiaion in ime of he saic sliding esisance (SSR) is also shown in he op lef panel, highlighing insans of incipien sliding and elaive moion beween sucue and foundaion. As he wave eaches he sucue i applies a shoewad load o he caisson and he eacions of he foundaion elemens iniially incease all ove he ineface. As he load inceases, he caisson oaes shoewad and he seawad end of he caisson is up-lifed. A his ime, boh he hoizonal and veical eacions exeed by he foundaion elemens undeneah he seawad end of he caisson sa deceasing (up o zeo, when he conac beween he supesucue and he foundaion is los) while he elemens a he shoewad end of he ineface ae heavily loaded. Once he wae suface sas moving downwads, and a hough sas appeaing in fon of he caisson wall, he shoewad load deceases and he caisson sas oaing seawad. When he wave ough eaches he sucue, he loading pocess inves unil anohe wave eaches he sucue. If, a any ime duing he loading, he saic sliding esisance is exceeded by he eacion ha is equied o he caisson fo a es, he caisson slides. A his poin, he hoizonal beaing capaciy of he soil foundaion insananeously deceases as he incipien moion educes he ficion a he ineface beween he supesucue and he soil foundaion, as he caisson begins o move, he velociy (dashed-doed lines on HRPP486

21 he igh hand side of igues 9 and ) apidly inceases and eaches is maximum, hen decease unil he caisson suddenly sops (op igh panel). As highlighed in igue 9, he sliding of he caisson beakwae due o pulsaing wave pessue develops ove a significan amoun of ime and a single wave is able o dislocae he supesucue and o move i fo a consideable disance. Sliding due o beaking wave pessue follows a compleely diffeen pah, wih sudden and sho-lasing moion of he sucue (noe change of ime scale beween igues 9 and ). 8 Load Reacion S.S.R. Elemens -4 5 x -3 Cenoid (u x ) Cenoid (du x /d) x [kn] u x [m], du x /d [m/s] Load Reacion Elemens -4 4 x -4 Cenoid (u y ) Cenoid (du y /d) Elemens -4 y [kn] -5 - u y [m], du y /d [m/s] Load Reacion x -3 Cenoid (u θ ) Cenoid (du θ /d) θ [knm] 4 - u θ [ad],, du θ /d [ad/s] Time [s] Time [s] igue 9 Dynamic esponse of a caisson beakwae subjec o non-beaking wave loads exceeding he saic sliding esisance; noe iniial veical displacemen due o seling of he caisson unde self-weigh. Geomeical and dynamic chaaceisics fo his example case as follows: l =.m, b =.78m, =.m, d =.8m, K y =.4E+5 kn/m; J x =.5E+5 kn/m; D =4.8E+ kns/m, W = 8.4 kn. HRPP486

22 5 Load Reacion S.S.R. Elemens -4.5 x Cenoid (u x ) Cenoid (du x /d) -4 6 x [kn] Load Reacion Elemens -4 u x [m].5 - x Cenoid (u y ) Cenoid (du y /d) Elemens -4 x du x /d [m/s] y [kn] u y [m] du y /d [m/s] x Load Reacion 3 x -4 Cenoid (u θ ) Cenoid (du θ /d) θ [knm] Time [s] u θ [ad] Time [s] 5 5 du θ /d [ad/s] x -4-5 igue Dynamic esponse of a caisson beakwae subjec o beaking wave loads exceeding he saic sliding esisance; noe iniial veical displacemen due o seling of he caisson unde self-weigh. Geomeical and dynamic chaaceisics fo his example case as follows: l =.m, b =.78m, h =.m, d =.4m, K y =.5E+5 kn/m; K x =.6E+5 kn/m; D =6.E+ kns/m, W = 6.7 kn. 6. Compaison wih measuemens fom physical model ess 6.. Physical model ess. Expeimens a Po and Aipo Reseach Insiue (PARI) wee pefomed in he 5m long, 3m wide and.5m deep wave flume of he Maiime Sucues Division. A schemaic epesenaion of he expeimenal seup is epoduced in igue. Two ses of ess wee caied ou especively unde pulsaing and beaking wave aack. Duing he lae se of expeimens, waves wee foced o beak in fon of he sucue by means of an exa-bem buil in fon of he sucue (dashed line in figue) which educed fuhe he wae deph in fon of he wall (fom.8m o.4m). Expeimens ae descibed in deail in Shimosako e al. (994) and Shimosako and Takahashi (999). Regula waves wee geneaed in he ange m ( H,... 4 in, Tables 3 and 4) while he wave HRPP486

23 peiod was kep consan and equal o 3.4s. Effec of caisson weigh was esed in he ange of kn ( W,,...4 in Table 3) fo non beaking waves and kn ( W,,... 5 in Table 4) fo beaking waves. o scale, m 4 4, ,4 igue (unis: cm). Expeimenal seup used in he small-scale physical model ess pefomed a PARI Table 3 Summay of egula wave condiions and geomeical configuaions adoped fo physical model ess a PARI unde non-beaking wave aack. H T h s d W m s m m kn H = W = 7.8 H = W = 7.8 H 3 = W = 7.8 H = W = 7.5 H = W = 7.5 H 3 = W = 7.5 H 4 = W = 7.5 H = W 3 = 8. H = W 3 = 8. H 3 = W 3 = 8. H 4 = W 3 = 8. H = W 4 = 8.9 H = W 4 = 8.9 H 3 = W 4 = 8.9 H 4 = W 4 = 8.9 HRPP486 3

24 Table 4 Summay of egula wave condiions and geomeical configuaions adoped fo physical model ess a PARI unde beaking wave aack. H T h s d W m s m m kn H = W =.3 H = W = 3.7 H = W = 3.7 H 3 = W = 3.7 H 4 = W = 3.7 H = W 3 = 6. H = W 3 = 6. H 3 = W 3 = 6. H 4 = W 3 = 6. H = W 4 = 7.8 H = W 4 = 7.8 H 3 = W 4 = 7.8 H 4 = W 4 = 7.8 H = W 5 = 9.4 H = W 5 = 9.4 Tess wee caied ou by aking measuemens of 5 consecuive waves (N z = 5) fo a given es condiion. Tes condiions ae summaised in Table 3 and 4 especively fo physical model ess caied ou using nonbeaking and beaking wave condiions. Compaison of pedicions wih measuemens fom physical model ess a PARI. Numeical simulaions wee pefomed using ime-hisoy loads geneaed accoding o he pocedue descibed in 4- and 4.. o each expeimen (ha is fo each wave condiion and geomeical configuaion) ealizaions wee simulaed. In his secion, esuls fom numeical simulaions ae compaed wih measuemens a PARI fo expeimens unde boh non-beaking and beaking wave condiions. Alhough he physical model ess wee pefomed using monochomaic waves, some vaiabiliy of wave heigh a he oe of he sucue was obseved duing esing. To accoun fo his effec, waves in he numeical simulaions have been assigned a andom vaiabiliy of abou % of he nominal wave heigh. Sliding disances measued duing he expeimens ae compaed wih hose pediced by he model, in ems of he mean, maximum and minimum oal (ove a N z = 5 waves es) sliding disance. Example calculaion In ode o faciliae applicaion of he pocedue descibed in his pape, an example calculaion is pesened in his secion fo one of he configuaions esed duing he physical model ess a PARI. Inpu daa ae summaised in Table 5. The dynamic chaaceisics of he soil foundaion, in ems of global hoizonal and veical siffness and global damping can be calculaed using Equaions, and, and ae summaised in Table 6. HRPP486 4

25 Table 5 Geomeical and geoechnical chaaceisics of an example configuaion esed duing physical model ess a PARI. Caisson lengh (in wave diecion) l. m Caisson widh (ohogonal o wave diecion) b.78 m Caisson weigh in wae W 8.4 kn Young s modulus a σ v = kpa E kpa Shea modulus a σ v = kpa G.3 5 kpa Poisson s aio ν.33 - Soil foundaion densiy ρ s 3 kg/m 3 Wae deph a he wall d.8 m Table 6 Dynamic chaaceisics of an example configuaions esed duing physical model ess a PARI. Caisson weigh in wae W 8.4 kn Mass M 464 kg Veical siffness Hoizonal siffness K y.4 5 kn/m K x.5 5 kn/m Damping C 4.8 kns/m Naual peiod of vibaion (undamped) T n. s Non-beaking wave aack. Time-seies of wave-foces and coesponding eacions and defomaions ae shown in igue fo a egula waves es. When sliding occus unde non-beaking waves aack, he saic sliding esisance is exceeded fo a significan amoun of ime and he esuling pemanen displacemen is lage ( 5.3). uhemoe, since pulsaing wave foces ae less vaiable han impacs, a lage numbe of waves wihin a single som migh induce sliding of he supesucue. This is he case when waves exceeding he wave condiion assumed in he deeminisic design each he beakwae wihou beaking. Such condiion is paiculaly dangeous fo caissons and usually esuls in significan sliding of caissons unis o failue of he beakwaes. Boh he afoemenioned effecs ae clealy visible in he hoizonal displacemen ime-hisoy shown in he op-igh panel of igue. HRPP486 5

26 x [kn] Load Reacion S.S.R. Elemens u x [m] Load Reacion Elemens -4 4 x -4 Cenoid Elemens -4 y [kn] -5 - u y [m] Load Reacion 6 x -4 θ [knm] 4 u θ [ad] Time [s] Time [s] igue Dynamic esponse of caisson beakwaes subjec o non-beaking wave aack, esuls fom a single ealizaion using N z =5 waves, geomeical and dynamic chaaceisics fo his example case ae as in igue 9. Resuls fom simulaions unde non-beaking wave aack ae summaised in igue 3, in ems of maximum, mean and minimum oal sliding disance (ove ealizaions of each N z = 5 waves es) as a funcion of he inciden wave heigh and of he caisson weigh. om lef o igh esuls efe o ess un wih inceasing caisson weigh in he ange 6.9 kn (W ) 8.9 kn (W 5 ) and inceasing wave heigh in he ange H =.48m -.69m. As expeced, esuls fom numeical simulaions confim ha fo a given caisson weigh, he sliding disance inceases fo inceasing wave heighs while fo a given wave heigh, he sliding deceases wih inceasing caisson weigh. When compaed o measuemens fom physical model ess, he model seems o give a good pedicion of oal sliding disances (op panel in igue 3) when he caisson weigh is elaively small (W ), while he model oveesimaes sliding when he caisson weigh inceases (W W 4 ). HRPP486 6

27 5 4 Toal sliding disance W W W 3 W 4 Sliding [cm] H [m] Pesen model (max) Pesen model (mean) Pesen model (min) Shimosako e al. 999 Expeimens 5 4 H H H 3 H 4 Sliding [cm] Caisson weigh in wae [kn] igue 3 Sliding unde non-beaking wave aack, compaison of measuemens fom physical model ess (*) and pedicion by he pesen model (dos) and Shimosako e al. 999 (cicle). When compaed o pedicions by Shimosako e al. (999), boh appoach seem o capue he same oveall end, alhough pedicions by he wo models ae significanly diffeen, wih he pesen model geneally giving a safe esimaion of oal sliding disance. Beaking wave aack. Time-seies of wave-foce and coesponding eacions and defomaions ae shown in igue 4 fo a egula waves es. Two majo sliding evens ae clealy disinguishable in he ohewise slowly vaying hoizonal displacemen ime-hisoy (op-igh panel). When sliding occus unde beaking waves aack, he saic sliding esisance is exceeded fo a small amoun of ime and he esuling pemanen displacemen is heefoe small ( 5.3). uhemoe, since impulsive wave foces ae exemely vaiable, duing a som only a small numbe of waves migh be able o cause he supesucue o slide. Neveheless, if he idal excusion is significan (as i is in he case of he Noh Euopean and Japanese coass) heavy beaking migh fequenly occu a he sucue, and impac magniude migh hen be amplified by he dynamic esponse of he sucue, wih he effecive load esuling in significan sliding of he caisson. HRPP486 7

28 5 Load Reacion S.S.R. Elemens -4 x -3.8 x [kn] 5 u x [m] y [kn] Load Reacion Elemens u y [m] x -4 Cenoid Elemens Load Reacion 4 x -4 θ [knm] 5 u θ [ad] Time [s] Time [s] igue 4 Dynamic esponse of caisson beakwaes subjec o beaking wave aack, esuls fom a single ealizaion using N z =5 waves, geomeical and dynamic chaaceisics fo his example case ae as in igue. Resuls fom simulaions unde beaking wave aack ae summaised in igue 5, in ems of maximum, mean and minimum oal sliding disance (ove ealizaions of each N z = 5 waves es) as a funcion of he inciden wave heigh and of he caisson weigh. om lef o igh esuls efe o ess un wih inceasing caisson weigh and inceasing wave heigh. No supising, sliding deceases wih inceasing caisson weigh, neveheless, fo a given caisson weigh inceasing of sliding wih incoming wave heigh is no monoonic. This is due o non-linea effec inoduced by wave beaking in boh wave impac maxima (Equaion 3) and ise imes. Indeed, due o he apid vaiaion of he wae deph in fon of he sucue he highes waves will beak befoe eaching he sucue, esuling in elaively low maxima; on he ohe hand, waves eaching he wall wihou beaking will necessay have a lowe heigh and hus less enegy, esuling again in elaively low impac maxima. The highes impac maxima acually esuling fom waves ha cul jus befoe hiing he sucue bu no exacly a he wall. HRPP486 8

29 Toal sliding disance.6 W W W 3 W 4 W 5 Sliding [cm] H [m] Pesen model (max) Pesen model (mean) Pesen model (min) Shimosako e al. 999 Expeimens.6 H H H 3 H 4 Sliding [cm] Caisson weigh in wae [kn] igue 5 Sliding unde beaking wave aack, compaison of measuemens fom physical model ess (*) and pedicion by he pesen model (dos) and Shimosako e al. 999 (cicle). Moeove, whils he same impac maxima migh heoeically coespond o diffeen combinaion of he incoming wave heigh and wae deph (Equaion 3), he coesponding impac ise imes will be lage fo waves ha ae lage a he oe of he sucues, which would necessaily esul in a lage quasi-saic load. Theefoe, as fa as he sabiliy of he caisson is concened, waves slighly highe han he beake heigh coesponding o he wae deph in fon of he wall H b = H b (d) will be moe ciical (and he mos ciical oveall indeed) han hose slighly lowe han H b since he impacs induced by he fome will ansmi a lage impulse. This is confimed in igue 5, wih maximum sliding disance coesponding o wave heighs anging beween.54m and.569m, he wae deph in fon of he wall being d =.4m. Conclusions and fuhe wok A non-linea dynamic model fo he dynamics of caisson beakwaes subjec o wave foces has been pesened. Diffeences o pevious fomulaions include boh he geneaion of ime-hisoy loads o use in he analysis and he descipion of he non-linea feaues of he ineacion beween he supesucue and he foundaion soil. In paicula, impac pessue maxima and ise imes ae deived fom daa ecoded a high sample ae duing lage scale physical model ess, which ensues a good descipion of peak pessue evoluion in ime. The effecive eacion a he ineface is deived by esolving in ime he dynamic equaion of moion in which sliding is also accouned fo by means of a non-linea model assumed fo he behaviou of he soil foundaion. HRPP486 9