【期刊】一个在垂直的圆筒的表示为极限力量的新的直接演算-外文文献.pdf

Ocean Engineering 34 2007 1706 1710 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 Keulegan Carpenter 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 Keulegan Carpenter 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 Morison s 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 barbaro unirc 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 f t cinrpR2g H 2 kf z sin ot cdgr Rg2 H2 4 o 2k2f 2 z cos ot jcos ot j 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 f z cosh k d z cosh kd 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 f t Af z sin ot Bf 2 z cos2 ot 5 Integrating the f t per z in d Z and making explicit the term f z one has F t Z Z d A cosh k d z cosh kd sin ot dz Z Z d B cosh2 k d z cosh2 kd cos2 ot dz 6 defi ning the coeffi cients A0 A cosh kd cinrpR2g H 2 k 1 cosh kd 7 B0 B cosh2 kd cdgrRg2 H2 4 o 2k2 1 cosh2 kd 8 one has F t A0sin ot Z Z d cosh k d z dz B0cos2 ot Z Z d cosh2 k d z dz 9 and solving the integrals F t A0sin ot 1 k sinh k d Z B0cos2 ot 1 4k fsinh 2k d Z 2k d Z g 10 Using the following linear approximations sinh k d Z sinh kd kZ sinh kd cosh kd kZ 11 expression 10 becomes F t A0 k sin ot sinh kd cosh kd kZ B0 4k cos2 ot fsinh 2kd cosh 2kd 2kZ 2kd 2kZg 12 Substituting in 12 the values of A0and B0and using the following defi nitions W1 cinrpR2g H 2 tanh kd 13 W2 cinrpR2g H2 4 k 14 W3 cdgrRg2 H2 16 o 2k 1 cosh2 kd sinh 2kd 2kd 15 W4 cdgrRg2 H3 16 o 2k2 1 cosh2 kd cosh 2kd 1 16 Expression 12 of the total force on the cylinder F t can be rewritten in the form F t W1sin ot W2cos ot sin ot W3cos2 ot W4cos3 ot 17 The maximum of the function F t 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 0po t pp 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 F x W1x W2x ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 x2 p W3 1 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 sin ot Of the four terms in expression 18 of F x 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 1706 17101707 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 F x F1 x F2 x 19 defi ning F1 x W1x W3 1 x2 20 F2 x 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 F1 x is the force on the portion of the cylinder between the sea fl oor and the average level F2 x 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 F x 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 F x is realized for x 0 crest of the wave F1 x has a maximum in 0 1 if W1o2W3 otherwise the maximum of F1 x is realized for x 1 In cases of practical interest if the maximum of F1 x 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 F1 x 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 Fi x W1x W2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 x2 i 1 q W3 1 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 Fi x 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 Fi x Eq 24 has the same form as F x Eq 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 F x 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 F x can be estimated as equal to F x1 Or rather if W1o2W3 Fmax W1x1 W2 ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffi 1 x2 1 q x1 W3 1 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 2W3 2 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 2W3 2 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 17 31 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 1706 17101708 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 17 31 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 Hmax H H m H m 5 Fig 4 Trend of the probability P Hmax 100years 4H and of the density p Hs h Hmax x for the district of Mazara del Vallo G Barbaro Ocean Engineering 34 2007 1706 17101709 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 sin ot 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 calc