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Ocean Engineering 34 (2007) 17061710A new expression for the direct calculation of the maximumwave force on vertical cylindersGiuseppe BarbaroDepartment of Mechanics and Materials, Via Graziella Loc. Feo de Vito, 89060 Reggio Calabria, ItalyReceived 19 April 2006; accepted 5 October 2006Available online 14 February 2007AbstractHere, an easy analytical solution for the direct calculation of the instant in which the maximum wave force on a support of an offshoreplatform is realized, and for the direct estimation of the aforementioned maximum force. The solution is obtained thanks to an artifice.The instant is expressed tmof the maximum force as limits of a succession tm0, tm1, tm2;.; and it is proved that in cases of practicalinterests the successions converge very quickly: tm tm1, less than negligible errors.The solution allows the estimate of useful synthesis to be arrived at in the preliminary phase of the project. In fact, it allows one toimmediately appreciate the effects of variations of the parameters in play: the sections of the cylinder, the depth of the sea floor and thecharacteristics of the waves.r 2007 Elsevier Ltd. All rights reserved.Keywords: Force; Cylinder; Wave1. IntroductionThe vertical cylinders in the sea typically function as asupport. It is concerned with, in the large majority of cases,circular-section cylinders that represent the fundamentalcomponents of the support structure of offshore jetties orplatforms with a reticular structure.Either due to the support of cylinders or to the support(leg) of the reticular platforms, the KeuleganCarpenter(KE number is usually greater than 2 so that thecalculation of the force can be undertaken with theformula of Morison et al. (1950). Furthermore, therelationship between the Reynolds (RE number andKeuleganCarpenter number normally surpass 104(excep-tions are made for cases of small cylinders) so that they canassume asymptotic values of inertia coefficient cinand ofdrag coefficient cdg(Boccotti, 1997). According to Sarp-kaya and Isaacson (1981), these asymptotic values are 1.85for cinand 0.62 for cdg. It concerns the substantial valueseven more recently confirmed by Sumer and Fredsoe(1997), even if there are some differences in the rule5oKEo20 where the asymptotic values of cinare shown tobe less than 1.85 and the asymptotic values of cdgareshown to be greater than 0.62.The instantaneous horizontal force on the cylinder isobtained by the integration of the unitary force (suppliedby Morisons formula) between the sea floor and thesurface of the water (this, naturally, for cylinders, as theyare in general the supports, which protrude from thesurface of the water). The maximum of this force is realizedfor an instant between the zero-up crossing and the crest ofthe wave, that is in the phase in which the component ofinertia and the component of drag have the same direction.(Actually, even in the interval comprising the zero-downcrossing and the concave the two components have thesame direction, but the total force is inevitably less than theinterval between the zero-up crossing and crest, in as muchas the portion of the loaded cylinder is less).The dependence of wave heights on the total force results inbeing rather complex, and therefore the isolation of themaximum of this force in the practice design is undertaken in anumerical manner. In this study, we will analyse this functionaldependence and we will arrive at obtaining an expression forthe direct calculation of the aforementioned maximum.ARTICLE IN PRESS/locate/oceaneng0029-8018/$-see front matter r 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.oceaneng.2006.10.013E-mail address: giuseppe.barbarounirc.it.Nowadays, with any PC, it is easy to obtain the totalmaximum force on a cylinder. Anyway, the analyticalsolution carries a doubtless advantage for the synthesis; anadvantage that is appreciated above all in the planningstage. In fact, in many cases, the analytical solution allowsone to see, simply and clearly, the effect of the variation ofthe various parameters in play: sections of the girder, depthof the sea-floor and characteristics of the waves.2. Analysis of the total forceWith reference to Fig. 1, the force per unit of length at adepth z isft ? cinrpR2gH2kfzsinot cdgr?Rg2H24o?2k2f2zcosotjcosotj,1where the first term in the right-hand side represents theinertia component and the second one the drag component,and where it is definedfz ? coshkd z?=coshkd.(2)Moreover, introducing the coefficients A and BA ? cinrpR2gH2k,3B ? cdgrRg2H24o?2k2.4The expression (1) can be rewritten in the formft ?Afzsinot Bf2zcos2ot.(5)Integrating the ft per z in ?d;Z and making explicit theterm fz one hasFt ZZ?d?Acoshkd z?coshkdsinotdzZZ?dBcosh2kd z?cosh2kdcos2otdz,6defining the coefficientsA0?Acoshkd cinrpR2gH2k1coshkd,(7)B0?Bcosh2kd cdgrRg2H24o?2k21cosh2kd(8)one hasFt ? A0sinotZZ?dcoshkd z?dz B0cos2otZZ?dcosh2kd z?dz9and solving the integralsFt ? A0sinot1ksinhkd Z? B0cos2ot14kfsinh2kd Z? 2kd Zg.10Using the following linear approximations:sinhkd Z? sinhkd kZ sinhkd coshkdkZ(11)expression (10) becomesFt ?A0ksinotsinhkd coshkdkZ?B04kcos2otfsinh2kd cosh2kd2kZ 2kd 2kZg.12Substituting in (12) the values of A0and B0and using thefollowing definitions:W1? cinrpR2gH2tanhkd,13W2? cinrpR2gH24k,14W3? cdgrRg2H216o?2k1cosh2kdsinh2kd 2kd?,15W4? cdgrRg2H316o?2k21cosh2kdcosh2kd 1?.16Expression (12) of the total force on the cylinder Ft canbe rewritten in the formFt ? W1sinot ? W2cosotsinot W3cos2ot W4cos3ot.17The maximum of the function Ft does not change if thesign of the first two addends to the second member ischanged. Naturally, however, with such a change of sign,the maximum falls in the domain 0potpp=2. Inconclusion, the maximum of the function (17), or ratherthe maximum horizontal force on the cylinder, is equal tothe maximum of the functionFx W1x W2xffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2p W31 ? x2 W4ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2p1 ? x2.18For 0pxp1 , where, with evidence, x stands for sinot.Of the four terms in expression (18) of Fx, the firstterm expresses the inertia force under m.w.l, the second theARTICLE IN PRESSd2RzFig. 1. Reference scheme.G. Barbaro / Ocean Engineering 34 (2007) 170617101707inertia force above m.w.l , the third the component of dragunder m.w.l. and the fourth the component of drag abovem.w.l.Here, it is better not to consider the problem purely froma mathematical point of view. It is better, instead, to keeppresent the physical meaning of various terms that presentthemselves in the second member of (18). Doing so, onemanages on one hand to skirt round the mathematicalproblem that presents itself as rather complex, and on theother hand one can investigate the same mechanics of theforce on the cylinder.It is better to rewrite (18) in the formFx F1x F2x(19)definingF1x ? W1x W31 ? x2,20F2x ? W2ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2px W4ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2p1 ? x2,21where F1x is the force on the portion of the cylinderbetween the sea-floor and the average level.F2x is theforce on the portion of the cylinder between the averagelevel and the water surface.If the component of inertia is neatly predominantcompared to the component of drag, the maximum Fxis realized for x 1 (zero of the elevation of the wave). If,vice versa, the component of drag is neatly predominantover the force of inertia, the maximum of Fx is realizedfor x 0 (crest of the wave).F1x has a maximum in (0.1) if W1o2W3, otherwisethe maximum of F1x is realized for x 1. In cases ofpractical interest, if the maximum of F1x is realizedfor x 1, also the maximum of Eq. (19) is realized inx 1 or extremely near to x 1, so that one can rightlyassumeif W1X2W3: Fmax W1.(22)It concerns, as mentioned, cases in which the inertialcomponent is neatly predominant over the component ofdrag.We now come to the case in which W1o2W3. In thiscase the maximum of F1x is realized in x ? xm, or ratherW1? 2W3xm 0 ) xmW12W3.(23)Here, to derive the maximum of the total force, it is best togo back to the following series of functions:Fix W1x W2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2i?1q W31 ? x2W4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2i?1q1 ? x2,24with i 1;2;., xmprovided by (23) and xi, abscissa of themaximum of Fixxi12W1 W2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2i?1qW3 W4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2i?1q.(25)It can easily be verified that FixEq. (24)has the sameform as FxEq. (18) with the only difference being thatthe factorffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2pis substituted byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x2i?1q. Thesuccession of xiconverges and the value limit of thesuccession coincides with the abscissa of the maximum ofFx . In cases of practical interest, the convergence is veryfast, in as much as one can assume with a good degree ofcertainty that x1coincides with the limit of succession. As aresult, the desired maximum value of the functions on thecylinder, or rather the value maximum of Fx can beestimated as equal to Fx1.Or ratherif W1o2W3: Fmax W1x1 W2ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x21qx1 W31 ? x21 W4ffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? x21q1 ? x2126withx112W1 W2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? W1=2W32qW3 W4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ? W1=2W32q.(27)The errors which occur when applying expressions (26) and(27) for the estimation of Fmaxin cases of practical interest,are within 1.1%.3. The data used in the applicationThe data used in the applications are taken from thebuoy of Mazara del Vallo, which belongs to the ReteOndametrica Nazionale (RON) of the Servizio Idrogra-fico e Mareografico Nazionale (SIMN), active since July1989.The records are normally acquired for a period of 30minevery 3h and with shorter intervals in the case ofparticularly significant heavy seas. The buoy is in deepwater.Fig. 2 shows, referring to the Mazara buoy, a serious ofstorms with a level of significant wave height for the period1731 December 1997. From the aforementioned figure, itis possible to reveal the presence of some significant heavyseas. The most intense, recorded on the 28th December,presents a maximum value of significant height equal to3.5m.4. Application at the district of Mazara del ValloThe characteristic parameters of the district of Mazaradel Vallo, located in the Sicilian Channel areu 1:256;w 1:012m.Now let us consider the reticular platform of Fig. 3 placedin that district at a depth of 150m and let us estimate themaximum force of the elements of support of dimensionsequal to R 2m.Let us fix a project life L 100 years and a value of 0.10of the probability P that during L the event to assume atARTICLE IN PRESSG. Barbaro / Ocean Engineering 34 (2007) 170617101708the base of the project is realized at least once. From thegraphics in Fig. 4, with the aforementioned data, one caninfer the maximum wave height expected Hmax 16m andthe significant height of the sea state h 8m in which themaximum wave of 16m is realized in the district subjectedto study.As a result, the period of the highest wave in that localityis equal to (Boccotti, 2000)Th 24:55ffiffiffiffiffi84gs 12s.Therefore, the wave of the project for the structure in Fig. 3in the district of Mazara del Vallo will beHmax 16m;Th 12s.For those conditions we haveREKE 3:33 ? 105.So that one can assume the asymptotic values cin 1:85,cdg 0:62.Using Eqs. (13)(16) one hasW1 187:7t;W2 41:9t;W3 40:2t;W4 17:9t.In this case, W1is greater than 2W3and therefore thecomponent of inertia neatly prevails over that of drag, andthe Fmaxcan be estimated directly through the very simpleARTICLE IN PRESSMazara del Vallo(17-31 Dicembre 1997)00.511.522.533.541717181920212122232425252627282829303131Hs (m)Fig. 2. A series of storms with a levels of significant height recorded in the district of Mazara del Vallo (Sicilian Channel) in the period 1731/12/97.Fig. 3. The support structure of a reticular platform.00.250.50.75105101520250204060801001200101520250.1168P(Hmax(100 anni)H)- p(Hs=h;HmaxH)H (m)H (m)5Fig. 4. Trend of the probability PHmax100years4H? and of the densitypHs h;Hmax x for the district of Mazara del Vallo.G. Barbaro / Ocean Engineering 34 (2007) 170617101709relation (22). Therefore, the maximum force exercised onthe project wave, in the district of Mazara del Vallo, on thediagonals of the platform result:Fmax 187:7t.Now we shall pass to a support pole of ray R 0:25m ofthe jetty in Fig. 5, as always, placed at Mazaro del Vallo ata depth d 15m, and we will estimate the maximum forceof it.Resulting the coefficient of diffraction in the position ofthe jetty equal to 0.25, the height of the wave of the projectresults as equal to 4m. Also in this case resulting condition:REKE 1:13 ? 104.Onecanassumetheasymptoticvaluescin 1:85,cdg 0:62.From the Eqs. (13)(16) one hasW1 0:709t;W2 0:199t;W3 0:357t;W4 0:176t.As W1is less than 2W3, one has to fall back on Eqs. (26)and (27). The value of x1, results equal to 0.97 and themaximum force results equal toFmax 0:76t,x1 0:97 means that the value of sinot for which it isverified that the maximum of force is equal to 0.97; orrather it means that the maximum force has a phase anglearcsin 0:97 76?in regard to the crest of the wave. Weare in a condition in which the drag component prevailsbut the inertia component is not negligible (one shouldremember that the maximum of drag force is realized incorrespondence to the crest of the wave and the maximumof inertia force is realized in correspondence to the zero ofthe wave).5. ConclusionsIn this paper, a new expression for the direct calculationof the maximum force is proposed, produced by the waveson vertical cy