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Ocean Engineering 34 (2007) 17061710 A new expression for the direct calculation of the maximum wave force on vertical cylinders Giuseppe Barbaro Department of Mechanics and Materials, Via Graziella Loc. Feo de Vito, 89060 Reggio Calabria, Italy Received 19 April 2006; accepted 5 October 2006 Available online 14 February 2007 Abstract Here, an easy analytical solution for the direct calculation of the instant in which the maximum wave force on a support of an offshore platform is realized, and for the direct estimation of the aforementioned maximum force. The solution is obtained thanks to an artifi ce. The instant is expressed tmof the maximum force as limits of a succession tm0, tm1, tm2;.; and it is proved that in cases of practical interests 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 to immediately appreciate the effects of variations of the parameters in play: the sections of the cylinder, the depth of the sea fl oor and the characteristics of the waves. r 2007 Elsevier Ltd. All rights reserved. Keywords: Force; Cylinder; Wave 1. Introduction The vertical cylinders in the sea typically function as a support. It is concerned with, in the large majority of cases, circular-section cylinders that represent the fundamental components of the support structure of offshore jetties or platforms 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 the calculation of the force can be undertaken with the formula of Morison et al. (1950). Furthermore, the relationship between the Reynolds (RE number and KeuleganCarpenter number normally surpass 104(excep- tions are made for cases of small cylinders) so that they can assume asymptotic values of inertia coeffi cient cinand of drag coeffi cient cdg(Boccotti, 1997). According to Sarp- kaya and Isaacson (1981), these asymptotic values are 1.85 for cinand 0.62 for cdg. It concerns the substantial values even more recently confi rmed by Sumer and Fredsoe (1997), even if there are some differences in the rule 5oKEo20 where the asymptotic values of cinare shown to be less than 1.85 and the asymptotic values of cdgare shown to be greater than 0.62. The instantaneous horizontal force on the cylinder is obtained by the integration of the unitary force (supplied by Morisons formula) between the sea fl oor and the surface of the water (this, naturally, for cylinders, as they are in general the supports, which protrude from the surface of the water). The maximum of this force is realized for an instant between the zero-up crossing and the crest of the wave, that is in the phase in which the component of inertia and the component of drag have the same direction. (Actually, even in the interval comprising the zero-down crossing and the concave the two components have the same direction, but the total force is inevitably less than the interval between the zero-up crossing and crest, in as much as the portion of the loaded cylinder is less). The dependence of wave heights on the total force results in being rather complex, and therefore the isolation of the maximum of this force in the practice design is undertaken in a numerical manner. In this study, we will analyse this functional dependence and we will arrive at obtaining an expression for the direct calculation of the aforementioned maximum. ARTICLE IN PRESS 0029-8018/$-see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.10.013 E-mail address: giuseppe.barbarounirc.it. Nowadays, with any PC, it is easy to obtain the total maximum force on a cylinder. Anyway, the analytical solution carries a doubtless advantage for the synthesis; an advantage that is appreciated above all in the planning stage. In fact, in many cases, the analytical solution allows one to see, simply and clearly, the effect of the variation of the various parameters in play: sections of the girder, depth of the sea-fl oor and characteristics of the waves. 2. Analysis of the total force With reference to Fig. 1, the force per unit of length at a depth z is ft ? cinrpR2g H 2 kfzsinot cdgr ?Rg2 H2 4 o?2k2f 2zcosotjcosotj, 1 where the fi rst term in the right-hand side represents the inertia component and the second one the drag component, and where it is defi ned fz ? coshkd z?=coshkd.(2) Moreover, introducing the coeffi cients A and B A ? cinrpR2g H 2 k,3 B ? cdgrRg2 H2 4 o?2k2.4 The expression (1) can be rewritten in the form ft ?Afzsinot Bf 2zcos2ot. (5) Integrating the ft per z in ?d;Z and making explicit the term fz one has Ft Z Z ?d ?A coshkd z? coshkd sinotdz Z Z ?d B cosh2kd z? cosh2kd cos2otdz,6 defi ning the coeffi cients A0? A coshkd cinrpR2g H 2 k 1 coshkd ,(7) B0? B cosh2kd cdgrRg2 H2 4 o?2k2 1 cosh2kd (8) one has Ft ? A0sinot Z Z ?d coshkd z?dz B0cos2ot Z Z ?d cosh2kd z?dz9 and solving the integrals Ft ? A0sinot 1 k sinhkd Z? B0cos2ot 1 4k fsinh2kd Z? 2kd Zg.10 Using the following linear approximations: sinhkd Z? sinhkd kZ sinhkd coshkdkZ (11) expression (10) becomes Ft ? A0 k sinotsinhkd coshkdkZ? B0 4k cos2otfsinh2kd cosh2kd2kZ 2kd 2kZg.12 Substituting in (12) the values of A0and B0and using the following defi nitions: W1? cinrpR2g H 2 tanhkd,13 W2? cinrpR2g H2 4 k,14 W3? cdgrRg2 H2 16 o?2k 1 cosh2kd sinh2kd 2kd?,15 W4? cdgrRg2 H3 16 o?2k2 1 cosh2kd cosh2kd 1?.16 Expression (12) of the total force on the cylinder Ft can be rewritten in the form Ft ? W1sinot ? W2cosotsinot W3cos2ot W4cos3ot.17 The maximum of the function Ft does not change if the sign of the fi rst two addends to the second member is changed. Naturally, however, with such a change of sign, the maximum falls in the domain 0potpp=2. In conclusion, the maximum of the function (17), or rather the maximum horizontal force on the cylinder, is equal to the maximum of the function Fx W1x W2x ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 ? x2 p W31 ? x2 W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 ? x2 p 1 ? x2.18 For 0pxp1 , where, with evidence, x stands for sinot. Of the four terms in expression (18) of Fx, the fi rst term expresses the inertia force under m.w.l, the second the ARTICLE IN PRESS d 2R z Fig. 1. Reference scheme. G. Barbaro / Ocean Engineering 34 (2007) 170617101707 inertia force above m.w.l , the third the component of drag under m.w.l. and the fourth the component of drag above m.w.l. Here, it is better not to consider the problem purely from a mathematical point of view. It is better, instead, to keep present the physical meaning of various terms that present themselves in the second member of (18). Doing so, one manages on one hand to skirt round the mathematical problem that presents itself as rather complex, and on the other hand one can investigate the same mechanics of the force on the cylinder. It is better to rewrite (18) in the form Fx F1x F2x(19) defi ning F1x ? W1x W31 ? x2,20 F2x ? W2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 ? x2 p x W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 ? x2 p 1 ? x2,21 where F1x is the force on the portion of the cylinder between the sea-fl oor and the average level.F2x is the force on the portion of the cylinder between the average level and the water surface. If the component of inertia is neatly predominant compared to the component of drag, the maximum Fx is realized for x 1 (zero of the elevation of the wave). If, vice versa, the component of drag is neatly predominant over the force of inertia, the maximum of Fx is realized for x 0 (crest of the wave). F1x has a maximum in (0.1) if W1o2W3, otherwise the maximum of F1x is realized for x 1. In cases of practical interest, if the maximum of F1x is realized for x 1, also the maximum of Eq. (19) is realized in x 1 or extremely near to x 1, so that one can rightly assume if W1X2W3: Fmax W1.(22) It concerns, as mentioned, cases in which the inertial component is neatly predominant over the component of drag. We now come to the case in which W1o2W3. In this case the maximum of F1x is realized in x ? xm, or rather W1? 2W3xm 0 ) xm W1 2W3 .(23) Here, to derive the maximum of the total force, it is best to go back to the following series of functions: Fix W1x W2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? x2 i?1 q W31 ? x2 W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? x2 i?1 q 1 ? x2,24 with i 1;2;., xmprovided by (23) and xi, abscissa of the maximum of Fix xi 1 2 W1 W2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? x2 i?1 q W3 W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? x2 i?1 q.(25) It can easily be verifi ed that FixEq. (24)has the same form as FxEq. (18) with the only difference being that the factor ffiffi ffiffi ffiffi ffiffi ffiffi ffiffiffi 1 ? x2 p is substituted by ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffi 1 ? x2 i?1 q . The succession of xiconverges and the value limit of the succession coincides with the abscissa of the maximum of Fx . In cases of practical interest, the convergence is very fast, in as much as one can assume with a good degree of certainty that x1coincides with the limit of succession. As a result, the desired maximum value of the functions on the cylinder, or rather the value maximum of Fx can be estimated as equal to Fx1. Or rather if W1o2W3: Fmax W1x1 W2 ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffi 1 ? x2 1 q x1 W31 ? x2 1 W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? x2 1 q 1 ? x2 1 26 with x1 1 2 W1 W2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? W1=2W32 q W3 W4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 ? W1=2W32 q.(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 application The data used in the applications are taken from the buoy of Mazara del Vallo, which belongs to the Rete Ondametrica Nazionale (RON) of the Servizio Idrogra- fi co e Mareografi co Nazionale (SIMN), active since July 1989. The records are normally acquired for a period of 30min every 3h and with shorter intervals in the case of particularly signifi cant heavy seas. The buoy is in deep water. Fig. 2 shows, referring to the Mazara buoy, a serious of storms with a level of signifi cant wave height for the period 1731 December 1997. From the aforementioned fi gure, it is possible to reveal the presence of some signifi cant heavy seas. The most intense, recorded on the 28th December, presents a maximum value of signifi cant height equal to 3.5m. 4. Application at the district of Mazara del Vallo The characteristic parameters of the district of Mazara del Vallo, located in the Sicilian Channel are u 1:256;w 1:012m. Now let us consider the reticular platform of Fig. 3 placed in that district at a depth of 150m and let us estimate the maximum force of the elements of support of dimensions equal to R 2m. Let us fi x a project life L 100 years and a value of 0.10 of the probability P that during L the event to assume at ARTICLE IN PRESS G. Barbaro / Ocean Engineering 34 (2007) 170617101708 the base of the project is realized at least once. From the graphics in Fig. 4, with the aforementioned data, one can infer the maximum wave height expected Hmax 16m and the signifi cant height of the sea state h 8m in which the maximum wave of 16m is realized in the district subjected to study. As a result, the period of the highest wave in that locality is equal to (Boccotti, 2000) Th 24:55 ffi ffi ffi ffi ffi 8 4g s 12s. Therefore, the wave of the project for the structure in Fig. 3 in the district of Mazara del Vallo will be Hmax 16m;Th 12s. For those conditions we have RE KE 3:33 ? 105. So that one can assume the asymptotic values cin 1:85, cdg 0:62. Using Eqs. (13)(16) one has W1 187:7t;W2 41:9t;W3 40:2t;W4 17:9t. In this case, W1is greater than 2W3and therefore the component of inertia neatly prevails over that of drag, and the Fmaxcan be estimated directly through the very simple ARTICLE IN PRESS Mazara del Vallo (17-31 Dicembre 1997) 0 0.5 1 1.5 2 2.5 3 3.5 4 17 17 18 19 20 21 21 22 23 24 25 25 26 27 28 28 29 30 31 31 Hs (m) Fig. 2. A series of storms with a levels of signifi cant 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. 0 0.25 0.5 0.75 1 0510152025 0 20 40 60 80 100 120 010152025 0.1 16 8 P(Hmax(100 anni)H) - p(Hs=h;HmaxH) H (m) H (m) 5 Fig. 4. Trend of the probability PHmax100years4H? and of the density pHs h;Hmax x for the district of Mazara del Vallo. G. Barbaro / Ocean Engineering 34 (2007) 170617101709 relation (22). Therefore, the maximum force exercised on the project wave, in the district of Mazara del Vallo, on the diagonals of the platform result: Fmax 187:7t. Now we shall pass to a support pole of ray R 0:25m of the jetty in Fig. 5, as always, placed at Mazaro del Vallo at a depth d 15m, and we will estimate the maximum force of it. Resulting the coeffi cient of diffraction in the position of the jetty equal to 0.25, the height of the wave of the project results as equal to 4m. Also in this case resulting condition: RE KE 1:13 ? 104. Onecanassumetheasymptoticvaluescin 1:85, cdg 0:62. From the Eqs. (13)(16) one has W1 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 the maximum force results equal to Fmax 0:76t, x1 0:97 means that the value of sinot for which it is verifi ed that the maximum of force is equal to 0.97; or rather it means that the maximum force has a phase angle arcsin 0:97 76?in regard to the crest of the wave. We are in a condition in which the drag component prevails but the inertia component is not negligible (one should remember that the maximum of drag force is realized in correspondence to the crest of the wave and the maximum of inertia force is realized in correspondence to the zero of the wave). 5. Conclusions In this paper, a new expression for the direct calculation of the maximum force is prop