Approximate Method For Probabilistic Presentation Of The Cross-Sectional Properties Of Shipbuilding Structural Profiles And Hull Girder

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1 Summary ABS TECHNICAL PAPERS th Internatonal Symposum on Practcal Desgn of Shps and Other Floatng Structures Houston, Texas, Unted States of Amerca 2007 Amercan Bureau of Shppng Approxmate Method For Probablstc Presentaton Of The Cross-Sectonal Propertes Of Shpbuldng Structural Profles And Hull Grder Lyuben D Ivanov Ge Wang A-Kuo Lee Amercan Bureau of Shppng, Houston-USA Presented at PRADS 2007, Houston, TX, USA, 1-5 October 2007 and reprnted wth the nd permsson of the 10th Internatonal Symposum on Practcal Desgn of Shps and Other Floatng Structures (PRADS) An s proposed for calculatng the and standard devaton of the geometrc propertes of shpbuldng structural profles and shp s hull grder when presented n probablstc terms. Its accuracy has been tested aganst results obtaned by the Taylor seres expanson. When appled for shpbuldng structural profles, the error s substantal. However, when appled for checng the probablty of meetng the exstng classfcaton socetes Renewal Crtera, the error reduces to 10%. When appled to hull grder geometrc propertes such as cross-sectonal area, secton modul and moments of nerta, the maxmum error s 7%. Introducton The probablstc presentaton of the hull structure s geometrc propertes s a necessary step for applcaton of the tme-varant relablty approach to shp structures desgn, mantenance and repar. It contrbutes to more accurate predcton of the hull structure behavor over tme. When the probablstc dstrbutons of the ntal scantlngs and corroson wastage are nown, the probablstc dstrbuton of the hull structure s geometrc propertes can be determned by Monte Carlo smulaton, Taylor seres expanson, composton of the dstrbuton laws of the consttuent varables, etc. These s requre many efforts and n some cases specal computer programs. A comprehensve wor on probablstc presentaton of the geometrc propertes of the shpbuldng structural profles and hull grder was done n ABS ([3], [4]). It showed that the truncated Gaussan dstrbuton fts well ther probablstc dstrbuton. Hence, two parameters are to be nown: value and standard devaton. The value can be calculated by the Taylor expanson,.e. ( ) Y = F x x = of x (1) Eq. (1) represents the so-called frst-order approxmaton. However, n ths case, the dfference between frst-order and second-order approxmaton s neglgble. It barely reaches % of the frst term [3]. Therefore, the applcaton of frst-order approxmaton s ustfed for use because of ts smpler form than that of second-order approxmaton and hgh accuracy. More dffcult s the calculaton of the standard devaton, σ Y. For ts calculaton, the followng approxmaton s proposed: Y ( ) ( ) σ = F x = x +σ F x = x (2) where x s the value of x and σ I s the standard devaton of any cross-secton s dmenson x. Applcaton of the for shpbuldng structural profles Frst, the verfcaton of the was performed for shpbuldng structural profles aganst results obtaned wth Taylor seres expanson. The accuracy of the Taylor expanson tself was checed aganst results obtaned wth Monte Carlo smulaton (the two s produced almost dentcal results [2]). A bulb plate wth cross-secton s dmensons gven n Table 1 was used n the example. The results for the standard devatons of some of the bulb plate s geometrc propertes are gven n Table 2. Table 1 Nomnal values of the cross-sectonal dmensons of the bulb plate (see Fg. 1) Parameter h b t w t 2 l p t p value Parameter R 2 R 3 R 4 R 5 β degree θ degree value Approxmate Method for Probablstc Presentaton of the Cross Sectonal 225

2 t w Z l p R 4 R 5 b R 3 R 2 Fg. 1 Table 2 Standard devatons obtaned by the and Taylor seres One can observe n Table 2 that the error relatve to results obtaned by Taylor seres s substantal and ncreases wth the ncrease of the geometrc propertes ran (see also Fgs. 2). However one of the maor tass of the probablstc dstrbutons s to facltate the calculaton of the probablty that gven geometrc property wll exceed gven lmt (e.g., when the probablty of meetng gven Renewal Crtera s to be calculated). Therefore, the effect of the dfferences between the standard devatons obtaned by the two s on the probabltes P (y lmt y ; T) was also evaluated (see Table 3 and Fgs. 3 and 4). Table 3 resultng from applcaton of the when calculatng the probablty of meetng gven requrements Method Parameter Taylor Year Approxmate [%] A S A I Y SM A S = cross-sectonal area of the stffener alone A = total cross-sectonal area SM = secton modulus I Y1 = centrodal moment of nerta relatvely to the horzontal neutral axs Parameter A S I Y1 Perm. Shp s age [s] Reducton* % β θ t p t 2 h Y * There are no permssble reductons for A S and I Y1 n the classfcaton socetes Rules. The Rules [1] control only the web plate s thcness (and flange for bult-up T- bars). Its permssble reducton depends on the structure s type and locaton (t vares from 20% tll 30% of the as-bult thcness). Therefore, the followng approach was used n selectng permssble reductons for A S and I Y1 : Frst, the permssble reducton of the web plate s thcness s determned from the Rules. Second, the probablty that ths web plate thcness wll meet the requrements of the Rules at shp s age of 20 s s calculated. Then, the permssble reductons for A S and I Y1 are determned n such a way that the probablty of meetng these requrements s exactly the same as the prevously calculated probablty for the web plate s thcness (these permssble reductons were determned by an teratve procedure [3], [4]). Two cases were consdered: when the web plate s permssble reducton s 25% and 10%. Then, the correspondng permssble reductons for A S and I Y1 were calculated n the way mentoned before.. Only the probabltes P (y lmt y ; T) for the crosssectonal area of the bulb plate, A S, and the Centrodal Moment of Inerta relatvely to the horzontal Neutral Axes, I Y1, are shown n Fgs. 3 and 4 (y sgnfes any normalzed geometrc property,.e. y = Y/Y nom ). The probablty P (y lmt y ; T) for I Y1 s selected as an example because the dfference between ts standard devatons obtaned by the two s s maxmum among all other standard devatons of the geometrc propertes. At the same tme the dfference between the probabltes P (y lmt y ; T) for the cross-sectonal area of the bulb plate obtaned by the two s s mnmum. Thus, the whole range of dfferences s covered, whch facltates drawng conclusons for the applcablty of the. Based on Table 3, Fgs. 3 and 4, one can draw the followng conclusons for ths case: The maxmal dfference between the two probabltes P (y lmt y ; T) s around 15-16%. For case when the assumed permssble reducton of the web plate s thcness s 25% the produces more conservatve results than those obtaned wth Taylor seres. For case when the assumed permssble reducton of the web plate s thcness s relatvely small (e.g. 10%), the proposed produces more conservatve results for the frst part of the shp s lfetme whle for the second part of ts lfetme the trend s reverse. Strctly speang, the proposed s mathematcally ncorrect. However, the queston s how much ncorrect? To answer ths queston and to fnd out the lmtatons of the, the followng analyss was performed: 226 Approxmate Method for Probablstc Presentaton of the Cross Sectonal

3 The corroson wear affects the thcness of the steel plates, web plates, flanges of bult-up T-bars or bulb head of rolled profles. The wdth of the steel plates s not affected. The heght and the flanges /bulb heads wdth s nsgnfcantly changed. Consequently, one can conclude that: The cross-sectonal area of the plates and bult-up T- bars s a lnear functon of the thcness. The cross-sectonal area of rolled stffeners s very close to lnear functon of the thcness and can be treated as lnear one wth confdence (see, e.g., [5]). The geometrc propertes of hgher ran geometrc propertes such as Statc Moments, Moments of Inerta, etc. are also very close to lnear functons of the area (hence, the thcness). The reason for t s that the dstance from the centrod of each structural component to gven axs for comparson does not change much due to corroson wear. The change of ths dstance s few orders of magntude smaller that that of the cross-sectonal area due to corroson wear. Ths s a nown fact and one can rely on t to assume that the geometrc propertes of stffeners are ether lnear (e.g. cross-sectonal area) or very close to lnear functons of the thcness. Let s represent any non-dmensonal geometrc property, Y, n the most general form: Y = a1x1+ a2x ax = ax (3) = 1 where the geometrc nterpretaton of any coeffcent a s the tangent of the angle between the tangent lne to Y and any axs X. The mathematcal nterpretaton of a s the frst dervatve of Y relatvely to each x. If eq. (2) s appled to eq. (3), one can derve the followng expresson for the standard devaton of Y: Y, approx ( ) ( ) a a a = 1 = 1 = 1 (4) σ = x + σ x = σ where σ Y, approx s the standard devaton of Y derved wth the proposed. If the varance of Y s calculated by the frst-order Taylor, the followng formula s vald: σ = 1 2 = a D (5) where σ s the standard devaton of Y derved wth Taylor seres for statstcally ndependent parameters. Eq. (5) can be rewrtten the followng way: 2 = a a a = 1 (6) σ σ 2 σσ The two standard devatons obtaned by the proposed and Taylor seres wll be equal f the second term under the square root n eq. (6) s equal to zero,.e. σy, approx = σ f aa σσ = 0 σy, approx σ f aa σσ 0 (7) The term aa σσ n Eq. (7) s practcally never equal to zero, whch s that σ. σ Y, approx The cumulatve dstrbuton functons of Y calculated by the two s are shown n Fg. 2, whch s very convenent for explanng the dfferences between the probabltes P ( y lmt y ; T ) obtaned by the two s. Let s assume that for shp s age = 25 s, the permssble reducton of I Y1 s (1- α)i Y1,nom. Then, the probablty that gven y (n ths case I Y1 ) wll be above the assumed permssble lmt can be calculated wth the equaton: P ( Y 1-α ) = 1 F( Y = α) (8) where: P( Y 1-α) s the probablty that I Y1 wll be greater than (1-α)I Y1,nom F(Y= α) s the cumulatve dstrbuton functon of I Y1 calculated for Y = α. There are three possble values for α (see Fg. 2), for whch the rato between the probabltes P( Y 1-α ) derved by the proposed and Taylor seres s: 1 α < y then 1 proposal 1 α 2 y ( α1 ) < Taylor ( α1 ) P y 1- P y 1- = then proposal 1 α 3 y ( α2 ) = Taylor ( α2 ) P y 1- P y 1- > then proposal ( α3 ) > Taylor ( α3 ) P y 1- P y 1- In other words, when the permssble reducton of y s greater than the reducton of the value of Y, the proposed produces more conservatve results than Taylor seres. The two s produce the same or almost the same results when the permssble reducton s equal or close to the reducton of the value of Y. When the permssble reducton s smaller than the reducton of the value of Y, the proposed produces results n the non-conservatve sde. Applcaton of the for the hull grder geometrc propertes The accuracy of the proposed was also tested n calculatons of the Hull Grder geometrc propertes. The standard devatons of some of these geometrc propertes obtaned by the two s are gven n Tables 4 10 for 25K DWT bul carrer wth Approxmate Method for Probablstc Presentaton of the Cross Sectonal 227

4 the followng man dmensons (see Fg. 5): L BP = 172 m B = m D = m Note: The values of the geometrc propertes n the two s are the same. The dfference between the two standard devatons (except for the plastc secton modulus) s not bg due to the followng reasons (see Fg. 6): There are much more parameters n the Hull Grder that act n opposte drecton (.e. wth sgn + or - ), whch reduces substantally the second term n Eq. (6),.e. the products 2 aa σσ may well balance each other. In addton, what s left from the term 2 aaσσ does not have a strong effect on the dfference between the standard devatons because of the rootng procedure. Ths leads to even smaller dfferences between the probabltes of meetng gven requrements when compared wth the results for structural profles (see Tables 4 10 and Fgs. 7,8, 9). Table 4 Comparson between the results for the standard devatons of A obtaned by Taylor and 2 Taylor 2 2 % Table 5 Comparson between the results for the standard devatons of SM dec obtaned by Taylor and m. 2 Taylor m. 2 m. 2 % Table 6 Comparson between the results for the standard devatons of SM btm obtaned by Taylor and m. 2 Taylor m. 2 m. 2 % Table 7 Comparson between the results for the standard devatons of SM sde obtaned by Taylor and m. 2 Taylor m. 2 m. 2 % Table 8 Comparson between the results for the standard devatons of I Y1 obtaned by Taylor and m 2 2 Taylor m 2 2 m 2 2 % I Y1 = centrodal moment of nerta relatvely to the horzontal neutral axs of the hull grder Table 9 Comparson between the results for the standard devatons of A sh obtaned by Taylor and 2 Taylor 2 2 % A sh s the shear area of the mdshp at Neutral Axs 228 Approxmate Method for Probablstc Presentaton of the Cross Sectonal

5 Table 10 Comparson between the results for the standard devatons of plastc SM, SM P, obtaned by Taylor and Method m. 2 Taylor m. 2 m. 2 % Concluson An s developed that can be used to calculate the numercal characterstcs of the probablstc dstrbuton of the geometrc propertes of shpbuldng structural profles and shp s hull grder. When used for shpbuldng structural profles, the error (compared to results obtaned by the Taylor seres expanson ) s substantal (t may reach 100%). However, when used to calculate the probablty that gven structural profle meets the exstng classfcaton socetes Renewal Crtera, the error resultng from ts applcaton s less than 10%. There are no avalable accurate analytcal s for probablstc presentaton of the geometrc propertes used n calculaton of the shear, plastc and torson strength of shpbuldng structural profles. Only numercal s such as Monte Carlo smulaton could be appled at present but even then one should have analytcal soluton n order to apply the, whch s not an easy tas, Therefore, the proposed may become attractve because of ts smplcty, acceptable accuracy n some cases and ease n the applcaton. When the s used to calculate the standard devaton of the hull grder elastc geometrc propertes, the maxmum error s 7% (excepton s the poor accuracy when appled to plastc hull grder secton modulus). Therefore, the can be used wth confdence for probablstc presentaton of the geometrc propertes of the hull grder. References 1. Amercan Bureau of Shppng Rule Requrements for Survey after Constructon, 2001, part 7, Appendx, Secton 4 2. Joe Foster The Applcaton of Monte Carlo Methods to the Analyss of Hull Grder Stresses, Amercan Bureau of Shppng Techncal Report R&R a, December 2003, 53 pages 3. Lyuben D Ivanov A probablstc assessment of all Hull Grder geometrc propertes at any shp s age, Amercan Bureau of Shppng, Techncal Report R&R , 166 pages, Houston December 2002, 4. Lyuben D Ivanov Probablstc presentaton of the geometrc propertes of shpbuldng structural profles when assessng elastc bendng strength, vol. 1 Theory and vol. 2 Numercal examples (340 pages), Amercan Bureau of Shppng, Techncal Report RD , Houston, December, Sdney Berecoa, Lyuben D Ivanov - Analytcal to swtch from requred net secton modulus nto requred gross secton modulus for shpbuldng structural profles, Marne Structures and Offshore Technology (Brazl), vol. 1, No. 3, December 2005, pp When used to calculate the probablty that hull grder elastc secton modulus meets the exstng classfcaton socetes Renewal Crtera, the error resultng from ts applcaton s even smaller than 7%. Approxmate Method for Probablstc Presentaton of the Cross Sectonal 229

6 Mean Value α2 Nomnal Value [-] α1 α 2 α I Y1 / I Y1, nom c.d.f. (Taylor) Fg. 2 Cumulatve dstrbuton functons (c.d.f.) obtaned by the proposed and Taylor seres. The example s gven for the Centrodal Moment of Inerta relatvely to the horzontal Neutral Axs of a bulb plate. The shp s age s 25 s α 1 c.d.f Probablty of meetng gven Renewal Crtera requrements [ - ] shp's age [s] As (Taylor) As (proposed ) IX1 (Taylor) IX1 (proposed ) Fg. 3 Comparson between the results obtaned for the probablty P ( y lmt y ; T ) wth Taylor seres and the proposed. The permssble reducton of the web plate's thcness s assumed = 25% whle the correspondng permssble reducton of A S s = 21.90% and that of I X1 s = % 230 Approxmate Method for Probablstc Presentaton of the Cross Sectonal

7 Probablty of meetng gven Renewal Crtera requrements [ - ] shp's age [ s ] As (Taylor) As (proposed ) IX1 (Taylor) IX1 (proposed ) Fg. 4 Comparson between the results obtaned for the probablty P ( y lmt y ; T ) wth Taylor seres and the proposed. The permssble reducton of the web plate's thcness s assumed = 10% whle the correspondng permssble reducton of A S s = 8.80% and that of I Y1 s = 9.00 % Z 1 Z ε G G 1 H α w C.L. H 1 I F Horzontal central axs Y 1 O 1 E e Y N M L K J α h O A B C D γ B.L. Y Fg. 5 Mdshp secton of 25K DWT bul carrer Approxmate Method for Probablstc Presentaton of the Cross Sectonal 231

8 [ - ] I Y1 / I Y1, nom Taylor ( T = 0 ) Approxmaton ( T = 0 ) Taylor ( T = 25 s) Approxmaton ( T = 25 s ) Fg. 6 Cumulatve dstrbuton functons (c.d.f.) obtaned by the proposed and Taylor seres. The example s gven for the Centrodal Moment of Inerta relatvely to the horzontal Neutral Axs of a 25K DWT bul carrer for T = 0 (.e., as-bult shp) and T = 25 s (.e., shp s age of 25 s) Probabltes of meetng gven Renewal Crtera requrements [%] shp's age [s] Ash(approx) Ash(Taylor) SM(d,p,Taylor) or M(P,Taylor) SM(d,p,approx) or M(P,approx) Fg. 7 Probabltes of meetng the Renewal Crtera requrements for the cross-sectonal area, A, dec Secton Modulus, SM dec, and bottom Secton Modulus, SM btm. The permssble reducton of A s 12.91%, of SM dec 10% and of SM btm 13.48% 232 Approxmate Method for Probablstc Presentaton of the Cross Sectonal

9 Probabltes of meetng gven Renewal Crtera requrements [%] shp's age [s] SM(sde,approx) SM(sde, Taylor) I(Y1,approx) I(Y1,Taylor) Fg. 8 Probabltes of meetng the Renewal Crtera requrements for the horzontal Secton Modulus, SM sde, and the central moment of nerta relatve to horzontal Neutral Axes, I Y1. The permssble reducton of SM sde s 11.70%, of I Y % Probabltes of meetng gven Renewal Crtera requrements [%] shp's age [s] A(aprox) A(Taylor) SM(d, approx) SM(d, Taylor) SM(btm, approx) SM(btm, Taylor) Fg. 9 Probabltes of meetng the Renewal Crtera requrements for Shear Area, A sh, and the plastc Secton Modulus, SM P (or plastc ultmate Bendng Moment, M P ). The permssble reducton of A sh s 14.64%, of SM P 12.48% Approxmate Method for Probablstc Presentaton of the Cross Sectonal 233