CALCULATION OF EXPECTED SLIDING DISTANCE OF BREAKWATER CAISSON CONSIDERING VARIABILITY IN WAVE DIRECTION
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1 CALCULATION OF EXPECTED SLIDING DISTANCE OF BREAKWATER CAISSON CONSIDERING VARIABILITY IN WAVE DIRECTION SU YOUNG HONG School of Civil, Urban, and Geosysem Engineering, Seoul Naional Universiy, San 56-1, Shinlim-Dong, Gwanak-Gu, Seoul , Korea, KYUNG-DUCK SUH 1 School of Civil, Urban, and Geosysem Engineering & Research Insiue of Marine Sysems Engineering, Seoul Naional Universiy, San 56-1, Shinlim-Dong, Gwanak-Gu, Seoul , Korea, kdsuh@snu.ac.kr HYUCK-MIN KWEON Deparmen of Civil Engineering, Kyongju Universiy, San 4-1, Hyohyun-Dong, Kyongju-Si, Kyongsangbuk-Do 78-1, Korea, hmkweon@kyongju.ac.kr 1 Corresponding auhor, Temporal address unil January, 4: Deparmen of Civil, Consrucion, and Environmenal Engineering, Oregon Sae Universiy, Apperson Hall, Corvallis, Oregon , USA, Phone: , Fax: , kdsuh@snu.ac.kr or kdsuh@engr.ors.edu 1
2 ABSTRACT In his sudy, he reliabiliy design mehod developed by Shimosako and Takahashi in for calculaion of he expeced sliding disance of he caisson of a verical breakwaer is exended o ake ino accoun he variabiliy in wave direcion. The effecs of direcional spreading and he variaion of deepwaer principal wave direcion abou is design value were found o be minor compared wih hose of he obliquiy of he deepwaer design principal wave direcion from he shore-normal direcion. Reducing he significan wave heigh a he design sie by 6% o correc he effec of wave refracion when using Goda s model was found o be appropriae when he deepwaer design principal wave direcion was abou degrees. When we used he field daa in a par of he eas coas of Korea, aking he variabiliy in wave direcion ino accoun reduced he expeced sliding disance o abou one hird of ha calculaed wihou aking he variabiliy in wave direcion ino accoun, and he required caisson widh was reduced by abou 1 % a he maximum. Keywords: Breakwaer; caisson; expeced sliding disance; reliabiliy design; variabiliy in wave direcion; wave ransformaion model.
3 1. Inroducion In he convenional design of he caisson of a verical breakwaer, he required sizes of he caisson are calculaed from empirical formulas, wih a cerain margin of safey, so as o resis he design load relaed o a given reurn period. The convenional mehod is based on he force balance beween he wave loads and he resisance of he caisson, and no movemen of he caisson is allowed. Any small movemen of he caisson is considered o be damage. However, even if he caisson moves, he breakwaer can sill perform is funcion, unless he movemen is so grea as o sop he serviceabiliy of he breakwaer. Therefore, if we allow a cerain amoun of movemen of he caisson, a more economical design could be made. In he convenional design, i is assumed ha he lifeime of a breakwaer is he same as he reurn period of he design wave. In his case, he probabiliy of occurrence of wave heighs greaer han he design wave heigh during he lifeime of he breakwaer is abou 63 percen, which is larger han he probabiliy for a wave heigh greaer han he design heigh no o occur. For he breakwaer locaed ouside surf zone, he maximum wave heigh is usually aken o be 1.8 imes he significan wave heigh, bu a higher wave could appear especially when he sorm duraion is long. Moreover, errors are always involved in he compuaion of wave ransformaion and wave forces so ha he compued values could happen o be on he safe side. Considering all hese uncerainies, he convenional design uses a safey facor of 1. for sliding of a caisson, bu is reasoning is no so clear. If he design condiions are differen inside and ouside surf zone, he degree of sabiliy of he breakwaer should no be he same, even if we use he same safey facor. In order o cope wih he problems menioned above, reliabiliy design mehods or performance design mehods have been developed, which ake ino accoun he uncerainies of various design parameers and allow a cerain amoun of damage during he lifeime of a breakwaer. The reliabiliy or performance design mehods have been developed since he mid-198s, especially in Europe and Japan. In Europe, van der Meer (1988) presened a probabilisic approach for he design of breakwaer armor layer, and Burcharh (1991) inroduced parial safey facors in he reliabiliy design of rubble mound breakwaers. Recenly Burcharh and Sørensen (1999) esablished parial safey facor sysems for rubble mound breakwaers and verical breakwaers by summarizing 3
4 he resuls of he PIANC (Permanen Inernaional Associaion of Navigaion Congresses) Working Groups. The European reliabiliy design mehods belong o wha is called as Level 1 or Level mehod. On he oher hand, in Japan, Level 3 mehods have been developed, in which he expeced damage of breakwaer armor blocks (Hanzawa e al., 1996) or he expeced sliding disance of a breakwaer caisson (Shimosako and Takahashi, ; Goda and Takagi, ; Takayama e al., ) during is lifeime is esimaed. Mone Carlo simulaions are used o ake ino accoun he uncerainies of various design facors. Among he above-menioned Japanese auhors, Hanzawa e al. (1996) and Goda and Takagi () used Goda's (1975) model o calculae he wave ransformaion from deep waer o he design sie, which includes wave aenuaion due o random breaking. Unidirecional random waves normally inciden o a sraigh coas wih parallel deph conours were assumed so ha no wave refracion was involved. Shimosako and Takahashi () posulaed wave ransformaion including refracion as well as shoaling and breaking, bu hey also used Goda s (1975) model in he acual compuaion (Shimosako, 3). In real siuaions, direcional random waves wih variable principal wave direcions will be inciden o he shore. For more accurae compuaion of he wave heighs a he design sie, herefore, we should use more realisic wave ransformaion models aking ino accoun he variabiliy in wave direcion. Recenly Suh e al. () exended he mehod of Hanzawa e al. (1996) o include he effec of he variabiliy in wave direcion in he calculaion of he expeced damage of breakwaer armor blocks. In he presen sudy, by closely following Suh e al. s () approach, we exend he reliabiliy design mehod of Shimosako and Takahashi () for calculaion of he expeced sliding disance of he caisson of a verical breakwaer o ake ino accoun he variabiliy in wave direcion. The variabiliy in wave direcion includes direcional spreading of random direcional waves, obliquiy of he design principal wave direcion from he shore-normal direcion, and is variaion abou he design value. To calculae he ransformaion of random direcional waves over an arbirary bahymery including surf zone, we used Kweon e al. s (1997) model, which was also used by Suh e al. (). In he following secion, he mahemaical model o calculae he sliding disance of a caisson is described. In Sec. 3, he compuaional procedure for calculaing he expeced sliding disance of a caisson is explained. In Sec. 4, several compuaional examples are presened o compare he resuls of he presen sudy wih hose of previous 4
5 auhors and o illusrae he imporance of wave direcionaliy. The major conclusions hen follow.. Compuaion of Sliding Disance The disance of caisson sliding is calculaed wih he model presened by Shimosako and Takahashi (), which is summarized below for he sake of compleeness. Assuming ha he caisson sliding is small enough o neglec he wave-making resisance force behind he caisson, he equaion of moion describing caisson sliding is given by W g M a G d x d P F R (1) where W is he caisson weigh in he air, g he graviy, M a he added mass ( h' ), he densiy of sea waer, h ' he waer deph from boom of caisson o design waer level, horizonal wave force, coefficien, R x G he horizonal displacemen of caisson, P he W' U, he fricion F he fricional resisance force W ' he caisson weigh in waer, and U he uplif force. The sliding disance of he caisson can be calculaed by numerically inegraing he preceding equaion wice wih respec o ime. The horizonal force P () is calculaed by aking he larger value of a sinusoidal form P 1 represening sanding wave pressures and a riangular pulse indicaing impulsive pressures as shown in Figure 1, i.e., P 1 and max P P P 1, P are defined as follows: () P1 PP1max sin (3) T P 5
6 Pmax, P 1 Pmax, (4), P sin d : P P sin P (5) T 1 P 1max 1max P1maxT 1 T where he ime inerval indicaes he inerval saisfying 1, P ( ) P1 max sin( / T), P 1max he horizonal wave force calculaed by he Goda (1974) pressure formula considering only he parameer 1, P max he wave force calculaed by using he Takahashi e al.'s (1994) parameer * in place of in he Goda formula, T he wave period, and he duraion of he impulsive wave force. The parameer P is used o reduce he sinusoidally varying sanding wave force by he amoun increased due o he impulsive force. Similarly, U is calculaed as follows: maxu U U 1, (6) U1 UUmax sin (7) T 6
7 7 max max,, 1, U U U (8) sin : sin 1 max max max 1 T U U d T U U T U U (9) where max U denoes he uplif force calculaed from he Goda formula. The erm is relaed o he wave period as follows: k F (1) where he ime F and he consan k are given by.8,.4.8, 8.5 h H T h H T h H F (11) and.3 * 1 1 k (1) respecively. Here H is he wave heigh, and h he waer deph. 3. Procedure for Compuaion of Expeced Sliding Disance
8 In his secion, he procedure for compuing he expeced sliding disance is explained in conjuncion wih he compuaional flow char skeched in Figure. In general, he sliding of a breakwaer caisson is caused by large waves comparable o he design waves. Therefore, he annual maximum wave heigh is considered sufficien o be incorporaed ino he calculaion. The annual maximum offshore significan wave heigh H is randomly sampled from he exreme wave heigh disribuion (Weibull disribuion in his sudy), and he peak of sorm waves is assumed o coninue for hours. This wave heigh is furher given a sochasic variaion wih he normal disribuion having a mean H and sandard deviaion H. This variaion represens he uncerainy in he esimae of he exreme disribuion funcion owing o he limied sample size of exreme wave daa or he inaccuracy in wave hindcass. The mean and sandard deviaion are assumed o have he following relaions wih H (Takayama and Ikeda, 1994): e e (1 ), H H H e (13) H H H e where H and H denoe he bias and deviaion coefficien, respecively. The sample offshore wave heigh H c o be employed in he calculaion is hen deermined by a normalized random number based on Eq. (13). The corresponding significan wave period is deermined o yield a consan wave seepness (.3 in his sudy) in he offshore area: T se H c (14).3g This wave period may also conain uncerainy and hus is given a sochasic variaion wih he normal disribuion having a mean T s and sandard deviaion T s. The mean and sandard deviaion are assumed o have he following relaions wih T se: 8
9 T s ( 1 ) T, T T Tse (15) T s se s s where T s and T s denoe he bias and deviaion coefficien, respecively. The sample significan wave period T sc o be employed in he calculaion is hen deermined by a normalized random number based on Eq. (15). Offshore random direcional waves wih he direcional spreading parameer s max are assumed o be inciden wih he principal wave direcion P counerclockwise wih respec o he shore-normal direcion. The principal wave direcion is assumed o have a sochasic variaion wih he normal disribuion having a mean being he same as he design principal wave direcion D P and a sandard deviaion p. Unidirecional random waves normally inciden o he shore are simulaed by seing s, max, and p p D. The offshore direcional wave specrum was expressed as he produc of he Breschneider-Misuyasu frequency specrum and he Misuyasu-ype direcional spreading funcion (Goda,, Secion.3.). Wih he idal range of, ide level was assumed o vary sinusoidally beween LWL( ) and HWL( ). The effec of sorm surge was aken ino accoun by adding 1 % of he deepwaer wave heigh o he ide level. Once he offshore wave heigh, wave period, and ide level are deermined, he significan wave heigh a he locaion of he breakwaer should be calculaed. In order o ake ino accoun he effec of wave direcion on wave ransformaion, we use Kweon e al. s (1997) wave ransformaion model in he presen sudy. The significan wave heigh a he design sie H, calculaed by he wave ransformaion model, is also assumed o se have compuaional uncerainy, and hus is given sochasic variaion wih he normal disribuion as wih he offshore wave heigh. The mean H s and he sandard deviaion H s are assumed o have he following relaions wih H se : 9
10 H s ( 1 H ) H se, H H H se s (16) s s where H s and H s denoe he bias and deviaion coefficien, respecively. The sample wave heigh a he design sie H sc is deermined by a normalized random number based on Eq. (16). The Kweon e al. s (1997) model compues he mean wave direcion as well as he wave heigh a he design sie. The compued wave direcion is used as an inpu parameer in he calculaion of wave pressure using he Goda formula. Once he significan wave heigh a he locaion of he breakwaer is calculaed, he heighs of he individual waves during he sorm are randomly sampled by assuming he Rayleigh disribuion. An individual wave heigh greaer han he breaking wave heigh was reduced o he breaking wave heigh using he formula in Goda (, p. 81). The periods of he individual waves are given sochasic variaion wih he normal disribuion as wih he significan wave period. The mean T and he sandard deviaion T are assumed o have he following relaions wih T sc : T ( 1 ) T, T TTsc (17) T sc where T and T denoe he bias and deviaion coefficien, respecively. Theoreically, he oal sliding disance during he lifeime of a breakwaer should be calculaed by summing he sliding disances due o all he high waves during he lifeime. In he presen sudy, however, we assume ha he waves high enough o make a caisson slide appear once a year so ha he annual maximum wave heigh is sufficien o be incorporaed ino he calculaion. Therefore, he oal sliding disance is obained by repeaing he calculaion for he number of years of he breakwaer lifeime (usually 5 years). The process of one lifeime cycle is shown in Figure. This process is repeaed a large number of imes, and he expeced sliding disance is obained by aking he average of he oal sliding disance during each lifeime cycle. In order o ake ino accoun he sochasic variaion of various design parameers such as wave heigh, wave period, waer level, wave force, and fricion coefficien, he Mone-Carlo simulaion 1
11 mehod was used. Table 1 liss he design parameers employed in he presen sudy and heir bias and deviaion coefficien. 4. Illusraive Examples In his secion, we presen several compuaional examples o compare our resuls wih hose of previous auhors and o illusrae he imporance of wave direcionaliy. We consider only a plane beach, which is simple bu sufficien o illusrae he influence of wave direcionaliy. The common compuaional condiions are given below. The Weibull disribuion funcion wih he shape parameer k., scale parameer A.3, and locaion parameer B was used as he exreme disribuion of he offshore wave heigh, which gave a design deepwaer wave heigh wih a reurn period of 5 years o be 9. m. The deepwaer wave seepness was assumed o be consan a. 3 so ha he corresponding design wave period was 14. s. The bias and deviaion coefficien of various design parameers are given in Table 1, which are basically he same as hose used by Shimosako and Takahashi (). In he surf zone, he deviaion coefficien of wave force of obliquely inciden waves may be smaller han ha of normal incidence, because he impulsive breaking wave pressure wih larger deviaion han sanding wave pressure occurs only when he wave direcion is almos normal o he breakwaer. Unforunaely, however, here is no enough experimenal daa abou his. Therefore, we used he same value as ha used by Shimosako and Takahashi () regardless of wave direcion, because heir resuls are laer compared wih he presen model resuls. A idal range of. m was assumed, and waer dephs from 1 o 3 m a LWL a an inerval of m were examined. Seabed slopes of 1/5 and 1/ were used. The design wave heigh a each waer deph was deermined by compuing he wave heighs corresponding o H 9. m while changing he waer level from LWL o HWL and aking he larges wave heigh. The oal number of simulaions for he calculaion of expeced sliding disance was chosen o be 5 based on Shimosako and Takahashi (), who have shown ha a sable saisical resul can be obained by doing so. The breakwaer is assumed o be insalled parallel o he shoreline. The design significan wave heighs, maximum wave heighs and caisson widhs a differen waer 11
12 dephs are given in Table. These values are also ploed in Figure 3 for laer use. A consan mound berm widh of 8. m was used regardless of waer deph. The cres elevaion of he caisson was aken o be. 6 imes he design significan wave heigh a he locaion of he breakwaer. The waer deph on he rubble mound, d, was aken o be.65h. The heigh from he boom of he caisson o he op of he rubble mound was assumed o be. m so ha h ' d. m was used. The widh of he caisson was calculaed by he Goda formula wih he safey facor of 1.. In he following, he expeced sliding disance was calculaed for he caisson widh given in Table in each waer deph Unidirecional random waves normally inciden o plane beach Shimosako and Takahashi () compued he expeced sliding disance of he caisson of a verical breakwaer exposed o unidirecional random waves normally inciden o a plane beach using Goda's (1975) model. On he oher hand, Kweon e al. (1997) simulaed he unidirecional random waves on a plane beach by seing he direcional spreading parameer s o be 1 in heir hree-dimensional random breaking wave max model, showing ha heir resuls were in reasonably good agreemen wih Goda s. Herein we used Kweon e al. s model o compue he wave ransformaion, and compared he calculaed expeced sliding disance of he caisson wih Shimosako and Takahashi s resuls. I was expeced ha hese wo resuls would no show grea difference because he wave models used gave similar resuls. The parameers expressing he uncerainies in he compuaion of wave ransformaion were. and H s.1 as wih Shimosako and Takahashi. H s Figure 4 compares he expeced sliding disance a differen waer dephs beween he presen model and he Shimosako and Takahashi's () model. A small difference is observed for waer dephs greaer han m because of he difference beween he wo wave ransformaion models used, bu he overall rend wih respec o he waer deph is quie similar. Noe ha a safey facor of 1. was used in all he compuaion of caisson sliding disance hereafer including Figure Influence of variabiliy in wave direcion 1
13 The primary purpose of he presen sudy is o examine he influence of he variabiliy in wave direcion upon he compuaion of he expeced sliding disance of a caisson, which was no included in Goda's (1975) model. For his purpose, we carried ou he compuaion for he eigh cases lised in Table 3. Case 1 is for unidirecional waves normally inciden o he beach as in Goda's (1975) model. Case includes he effec of direcional spreading. The spreading parameer s equal o was used, which corresponds o he deepwaer wave seepness of.3 max (Goda,, p. 35). Case 3 is for unidirecional waves inciden a wih respec o he shore-normal direcion, including only he effec of wave refracion. Case 4 examines he effec of he variaion of he principal wave direcion. ( ) and p D ( p ) 15 were used. For Cases 1 o 4,. 6 and. 1 were used. Cases o 4, H s however, included a fracion of he effecs of refracion and direcional spreading ha were ignored in Goda s (1975) model. Therefore, he bias mus be smaller han.6, e.g.,.4. However, how small was uncerain, so he value of.6 was used wihou change. Cases 5 o 8 included all of he variabiliy in wave direcion parly considered in Cases o 4. To examine he influence of he principal wave direcion, he expeced sliding disance was calculaed in Cases 5 o 7 wih he deepwaer principal wave direcion of 1,, and 3 degrees, respecively. Case 8 represened he ypical condiions beween Uljin and Pohang in he eas coas of Korea as given by Suh e al. (). In Cases 5 o 8, all of he variabiliy in wave direcion was included, so no bias was assumed in he compuaion of wave ransformaion, i.e.,. was used. However, he compuaional error mus sill exis, so. 1 was kep he same. Figure 5 shows a comparison of he expeced sliding disance a differen waer dephs beween Case 1 and. In Case where he effec of direcional spreading is included, he wave heigh a he locaion of he breakwaer becomes smaller compared wih ha of he unidirecional waves in Case 1. Therefore, he expeced sliding disance in Case is smaller han in Case 1. The difference of expeced sliding disance beween he wo cases becomes smaller as waer deph decreases, because he effec of direcional H s H s H s 13
14 spreading disappears as he waves propagae oward he shore. Figure 6 shows a comparison beween Case 1 and 3. The heigh of obliquely inciden waves is smaller han ha of normally inciden waves owing o wave refracion. Thus, he expeced sliding disance in Case 3 is smaller han in Case 1. Figure 7 shows a comparison beween Case 1 and 4. Again due o he effec of wave refracion, he expeced sliding disance in Case 4 is compued o be smaller han in Case 1. Figure 6 and 7 show ha he effec of wave refracion diminishes wih decreasing waer deph. This is probably because, in shallow waer, he maximum wave heigh is resriced by he waer deph so ha he wave hrus has an upper limi. Comparison of Figures 5 o 7 shows ha he effec of direcional spreading is almos same as ha of variaion of principal wave direcion, bu he effec of wave refracion is greaer han hese wo effecs even for a relaively small deepwaer wave inciden angle of degrees. Figures 8, 9 and 1 show comparisons beween Case 1 and Cases 5, 6 and 7, respecively, which examine he influence of he principal wave direcion on he expeced sliding disance when all he variabiliy in wave direcion is aken ino accoun. In Cases 5, 6 and 7, he deepwaer principal wave direcion was 1, and 3 degrees, respecively. As seen in Figure 8, when he deepwaer principal wave direcion was 1 degrees, he expeced sliding disance calculaed wih he variabiliy in wave direcion aken ino accoun is greaer han ha calculaed wihou aking he variabiliy ino accoun. On he conrary, when he principal wave direcion was 3 degrees, he opposie occurs as shown in Figure 1. On he oher hand, he expeced sliding disances calculaed wih and wihou aking he variabiliy in wave direcion ino accoun almos coincide each oher when he principal wave direcion was degrees, as shown in Figure 9. From he resuls given in Figures 8 o 1, we can say ha he bias.6 employed o ake ino accoun he variabiliy in wave direcion is suiable H s when he deepwaer design principal wave direcion is abou. Noe ha he bias.6 was used in Case 1 bu. in Cases 5 o 7. A value smaller han H s.6 in magniude (e.g.,. 4 H s ) should be used when p D is smaller han, or vice versa. More compuaions may be needed for differen wave condiions and 14
15 D p seabed slopes o obain a more reliable relaion beween and H s. Figure 11 shows a comparison beween Case 1 and 8. Because he deepwaer principal wave direcion is very large a 48 in Case 8, he significan wave heigh a he locaion of he breakwaer is calculaed o be very small due o he effec of severe wave refracion. Therefore, he expeced sliding disance in Case 8 is very small compared wih ha in Case 1. The difference from Case 1 is prominen even in smaller waer dephs, which is seen a lile in Figure 1 bu is hardly seen in Figures 8 and 9, where he deepwaer principal wave direcion is relaively small and so is wave refracion. When he seabed slope is 1/, he expeced sliding disances are very small in waer dephs smaller han abou 16 m for all he cases shown in Figures 4 o 11. As shown in Figure 3, he caisson widh is quie large in hese waer dephs of 1/ beach slope hough H decreases wih decreasing waer deph because of wave breaking. I max seems ha inside he surf zone of a seep beach he convenional design mehod is oo conservaive in he viewpoin of expeced sliding disance. Anoher feaure seen in Figures 4 o 11 is ha he expeced sliding disance of 1/ slope is smaller han ha of 1/5 slope in waer dephs smaller han abou 18 m, he reverse happens in waer dephs beween 18 and 6 m, and he reverse happens again in greaer waer dephs. This also seems o be relaed o he caisson widhs shown in Figure 3. On he beach of 1/ slope he caisson widhs are relaively large in smaller waer dephs, while on he 1/5 slope beach relaively large caisson widhs are needed in he middle waer dephs of 16 o m. Again i seems ha he convenional design mehod is conservaive so ha a smaller expeced sliding disance is calculaed when he caisson widh is relaively large. However, he influence of seabed slope is no clear and furher invesigaion is needed. Figure 1 shows he raio of B 3, he caisson widh designed for he expeced sliding disance o be 3 cm, o B d, he caisson widh designed wih he convenional mehod, as a funcion of waer deph, for Cases 1 and 8. In his figure, he relaive caisson widh B 3 /Bd smaller han 1. means ha he presen design mehod is more economical han he convenional one, or vice versa. Even in Case 1 where he variabiliy in wave direcion was no aken ino accoun, he presen design mehod is more economical han he convenional one in waer dephs smaller han abou 5 m (or 15
16 H D / h.37 ), while he reverse is rue in deeper waer. However, since H D / h is larger han.37 for mos ordinary design condiions, he presen design mehod generally gives a more economical cross secion. In Case 8 where he variabiliy in wave direcion was aken ino accoun, B 3 /Bd is smaller han 1., indicaing ha he presen design mehod is more economical han he convenional one in all he waer dephs examined. Oher cases show similar rends excep ha he difference from Case 1 is differen for each case. For example, B 3 /Bd is almos same beween Case 1 and 6 as shown in Figure 13. Noe ha he expeced sliding disance was almos same beween hese wo cases (see Figure 9). 5. Conclusions In his sudy, he deformaion-based reliabiliy design mehod developed by Shimosako and Takahashi () for calculaing he expeced sliding disance of he caisson of a verical breakwaer was exended o ake ino accoun he variabiliy in wave direcion. On he whole, he effec of direcional spreading or he variaion of he deepwaer principal wave direcion abou is design value is no so significan, bu he effec of he obliquiy of he design principal wave direcion from he shore-normal direcion is relaively imporan so ha he expeced sliding disance ends o decrease wih increasing obliquiy of principal wave direcion. Especially in he case where he field daa in he eas coas of Korea were used, he expeced sliding disance calculaed wih he variabiliy in wave direcion aken ino accoun was reduced o abou one hird of ha calculaed wihou aking he variabiliy ino accoun. Reducing he significan wave heigh a he design sie by 6 % o correc he effec of wave refracion negleced by assuming unidirecional waves normally inciden o a coas wih sraigh and parallel deph conours seems o be appropriae for he deepwaer design principal wave direcion of abou degrees. A smaller or larger reducion should be used for he deepwaer principal wave direcion smaller or larger, respecively, han degrees. I may also be possible o propose a relaionship beween he deepwaer design principal wave direcion and he bias of wave ransformaion hrough more compuaion in he fuure. If we design he caisson wih he allowable expeced sliding disance of 3 cm, in 16
17 waer dephs smaller han abou 5 m, even wihou aking he variabiliy in wave direcion ino accoun, he widh of he caisson could be reduced up o 3 percen compared wih he convenional design. When we used he field daa in he eas coas of Korea and ook ino accoun he variabiliy in wave direcion, he required caisson widh was reduced by abou 1 percen a he maximum, and a smaller caisson widh was required han he convenional design in he whole range of waer deph (1 o 3 m). Acknowledgemen This work was suppored by he Brain Korea 1 Projec. References Burcharh, H. F. (1991). Inroducion of parial coefficien in he design of rubble mound breakwaers. Proc. Conf. on Coasal Srucures and Breakwaers, Ins. of Civil Engrs., London, pp Burcharh, H. F. and Sørensen, J. D. (1999). The PIANC safey facor sysem for breakwaers. Proc. In. Conf. Coasal Srucures 99, A. A. Balkema, Spain, pp Goda, Y. (1974). A new mehod of wave pressure calculaion for he design of composie breakwaer. Proc. 14h In. Conf. on Coasal Engrg., American Soc. of Civil Engrs., Copenhagen, pp Goda, Y. (1975). Irregular wave deformaion in he surf zone. Coasal Engrg., Japan, 18, pp Goda, Y. (). Random Seas and Design of Mariime Srucures, nd edn., World Scienific, Singapore. Goda, Y. and Takagi, H. (). A reliabiliy design mehod of caisson breakwaers wih opimal wave heighs. Coasal Engrg. J., 4(4), pp Hanzawa, M., Sao, H., Takahashi, S., Shimosako, K., Takayama, T. and Tanimoo, K. (1996). New sabiliy formula for wave-dissipaing concree blocks covering horizonally composie breakwaers. Proc. 5h In. Conf. on Coasal Engrg., 17
18 American Soc. of Civil Engrs., Orlando, pp Kweon, H.-M., Sao, K. and Goda, Y. (1997). A 3-D random breaking model for direcional specral waves. Proc. 3rd In. Symp. on Ocean Wave Measuremen and Analysis, American Soc. of Civil Engrs., Norfolk, pp Shimosako, K. (3). Personal communicaion. Shimosako, K. and Takahashi, S. (). Applicaion of deformaion-based reliabiliy design for coasal srucures. Proc. In. Conf. Coasal Srucures 99, A. A. Balkema, Spain, pp Suh, K. D., Kweon, H.-M. and Yoon, H. D. (). Reliabiliy design of breakwaer armor blocks considering wave direcion in compuaion of wave ransformaion. Coasal Engrg. J., 44(4), pp Takahashi, S., Tanimoo, K. and Shimosako, K. (1994). A proposal of impulsive pressure coefficien for he design of composie breakwaers. Proc. In. Conf. Hydro- Technical Engrg. for Por and Harbour Consucion, Yokosuka, Japan, pp Takayama, T. and Ikeda, N. (1994). Esimaion of encouner probabiliy of sliding for probabilisic design of breakwaer. Proc. Wave Barriers in Deepwaers, Por and Harbour Research Insiue, Yokosuka, pp Takayama, T., Ikesue, S.-I. and Shimosako, K.-I. (). "Effec of direcional occurrence disribuion of exreme waves on composie breakwaer reliabiliy in sliding failure." Proc. 7h In. Conf. on Coasal Engrg., American Soc. of Civil Engrs., Sydney, pp van der Meer, J. W. (1988). Deerminisic and probabilisic design of breakwaer armor layers. J. Waerway, Por, Coasal and Ocean Engrg., American Soc. of Civil Engrs., 114, pp
19 Table 1. Esimaion errors of design parameers. Design parameers Offshore wave heigh Bias Deviaion coefficien..1 Sorm surge..1 Remarks Sandard is 1% of offshore wave heigh Wave ransformaion -.6 or..1 Significan wave period..1 Period of individual waves..1 Wave force..1 Goda formula considering * Fricion coefficien..1 Sandard is =.6 19
20 Table. Design wave heighs and widhs of caisson in differen waer dephs. Sea Bed Slope 1/ h (m) H s (m) H max (m) B (m) Sea Bed Slope 1/ Sea Bed Slope 1/5 h (m) H s (m) H max (m) B (m) Sea Bed Slope 1/
21 Table 3. Tes condiions. Case No. s max ( p ) D (deg.) ( p ) (deg.) H s
22 P P max P 1max p P 1max P () P 1max sin(/t) P 1 () P()=max{P 1 (),P ()} T/ Figure 1. Wave force profile for calculaion of sliding disance.
23 Offshore wave heigh disribuion, Sea bed slope, Waer deph : h, Tidal range : [Yearly maximum wave rain of he i-h year] Offshore waves : H ( i), Ts ( i) Saisical parameers : Sorm duraion : (i), H H denoes random number generaion H e ( i) Design for wave heigh of reurn period of 5 years Offshore wave heigh by exreme disribuion and probabiliy of non-exceedance ( i = 1 o 5 ) ( ) H c i T sc (i) (i) H T ( i),, e H H se( i ),, T s T s ~, phase : ~ ( i) ( i).1h ( i) ( ) ( i ) s max P ( i ) c ( P) D, ( P) Subrouine S( i) S( i 1) S S for one lifeime sample Expeced value of S can be calculaed by repeaing above process more han 5 imes (a) Compuaional flow Figue. Flow char for compuaion of oal sliding disance wihin one lifeime. 3
24 Wave deformaion by Kweon e al.(1997) Hse ( i), ( i) H sc (i) H se( i ),, H s H s Rayleigh disribuion for H (individual wave heigh) considering wave breaking Random selecion of H T T sc ( i ),, T T Wave force P e and U e P c U c c Pe Ue,, P P P,,,, P Sliding disance by individual wave Sum of T (i) Sliding disance is accumulaed for sorm duraion (i) Sliding disance in a sorm (b) Subrouine Figure. Coninued. 4
25 H s, H max and B (m) 3 Slope = 1 / 5 B 15 H max 1 H s Waer deph (m) Figure 3. Design wave heighs and widhs of caisson in differen waer dephs. 5
26 H s, H max and B (m) 3 Slope = 1 / 5 5 B 15 H max 1 H s Waer deph (m) Figure 3. Coninued. 6
27 Expeced Sliding Disance (m) Presen model (Slope=1/5) Shimosako and Takahashi(1999) (Slope=1/5) Presen model (Slope=1/) Shimosako and Takahashi (1999)(Slope=1/) Deph (m) Figure 4. Comparison of expeced sliding disance beween presen model and Shimosako and Takahashi s () model. 7
28 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case (Slope=1/5) Case 1 (Slope=1/) Case (Slope=1/) Deph (m) Figure 5. Change of expeced sliding disance due o effec of direcional spreading. 8
29 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 3 (Slope=1/5) Case 1 (Slope=1/) Case 3 (Slope=1/) Deph (m) Figure 6. Change of expeced sliding disance due o effec of wave refracion. 9
30 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 4 (Slope=1/5) Case 1 (Slope=1/) Case 4 (Slope=1/) Deph (m) Figure 7. Change of expeced sliding disance due o effec of variaion of principal wave direcion. 3
31 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 5 (Slope=1/5) Case 1 (Slope=1/) Case 5 (Slope=1/) Deph (m) Figure 8. Change of expeced sliding disance due o all effecs of direcional spreading, wave refracion, and variaion of principal wave direcion: 1. p D 31
32 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 6 (Slope=1/5) Case 1 (Slope=1/) Case 6 (Slope=1/) Deph (m) Figure 9. Same as Figure 8 bu for. p D 3
33 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 7 (Slope=1/5) Case 1 (Slope=1/) Case 7 (Slope=1/) Deph (m) Figure 1. Same as Figure 8 bu for 3. p D 33
34 Expeced Sliding Disance (m) Case 1 (Slope=1/5) Case 8 (Slope=1/5) Case 1 (Slope=1/) Case 8 (Slope=1/) Deph (m) Figure 11. Same as Figure 8 bu for use of field daa beween Uljin and Pohang in eas coas of Korea. 34
35 Case 1 (Slope=1/5) Case 8 (Slope=1/5) Case 1 (Slope=1/) Case 8 (Slope=1/) 1 B 3 /B d H D /h Figure 1. Comparison of relaive caisson widh as funcion of waer deph beween Case 1 and 8. 35
36 Case 1 (Slope=1/5) Case 6 (Slope=1/5) Case 1 (Slope=1/) Case 6 (Slope=1/) 1 B 3 /B d H D /h Figure 13. Same as Figure 1 bu for beween Case 1 and 6. 36
Using Rates of Change to Create a Graphical Model. LEARN ABOUT the Math. Create a speed versus time graph for Steve s walk to work.
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