Influence of longitudinal and transverse bulkheads on ship grounding resistance and damage size

Transcription

1 Proceedngs of the ICCGS June, 2016 Unversty of Ulsan, Ulsan, Korea Influence of longtudnal and transverse bulkheads on shp groundng resstance and damage sze Martn Henvee 1), Krstjan Tabr 1), Mhkel Kõrgesaar 2), Annka Urbel 1) 1) Depth. of Mechancs, Faculty of Cvl Engneerng, Tallnn Unversty of Technology, Estona 2) Depth. of Appled Mechancs/Marne Technology, Aalto Unversty, Espoo, Fnland Abstract Ths paper presents mprovements to the smplfed shp groundng resstance and damage openng model for double bottom tankers of Henvee et al. (2013) by ncludng the effect of longtudnal and transverse bulkheads. The study s based on numercal smulatons of 90 groundng scenaros. The scenaros were constructed for three dfferent sze tankers, three rock szes and fve penetraton depths. Influence of the longtudnal bulkhead on the groundng resstance s descrbed va addtonal term. The effect of the transverse bulkheads on the groundng resstance s less profound and thus, ths nfluence s excluded from the smplfed formulas. A new approach for the calculaton of structural resstance coeffcent, that allows scalng of the groundng resstance accordng to the shp sze, s proposed based on the volume of deformed materal. Moreover, t s shown that Mnorsky s formula (Mnorsky, 1959) for shps collsons s also vald for shp groundng. Formulatons for the predcton of the sze of the damage openng were modfed to nclude the effect of the bulkheads. Keywords Shp groundng; Smplfed analytcal method; Groundng damage assessment. Introducton The paper presents a smple formula for a rapd predcton of groundng damage of double hull tankers. These smplfed formulaton are amed for rsk analyss studes where there s only lmted amount of nformaton avalable regardng the shps. Several smplfed models have been developed to descrbe a shp groundng accdents. The models ether base on a smplfed closed form expressons (Cerup-Smonsen et al. 2009), (Hong & Amdahl 2012) or on numercal smulatons (Alsos & Amdahl 2007). Precse numercal smulatons are too tme consumng for rsk analyses and requre detaled nput nformaton. On the other hand, smplfed models are often lmted to a certan sea bottom topology or to shp s structural confguraton. Moreover, often the methods requre that to some extent the damage mechancs are prescrbed: for example, the descrpton of contact energy s based on the fracture propagaton n the bottom platng. Smple formulaton based on a small number of parameters that descrbe the groundng resstance of a tanker n a groundng accdent was derved by Henvee et al. (2013). The longtudnal and transverse bulkheads contrbuton was omtted. The am of the current paper was to determne the effects of the longtudnal and transverse bulkheads to the average groundng resstance and to the damage sze. Large number of groundng scenaros wth three tankers ncludng longtudnal and transverse bulkheads are smulated for three rock szes at fve penetraton depths. Two transverse rock postons were selected for each groundng scenaro, one beng drectly under the longtudnal bulkhead and other between the bulkhead and the sde of the shp. Wth both rock postons, numercal groundng smulatons were conducted n dsplacement controlled manner at constant groundng velocty. For the each groundng smulaton, average horzontal groundng force was calculated and the values correspondng to the both rock postons were compared. The tankers used n the current paper are desgned to meet hgher strength requrements than tankers used n prevous studes (Henvee et al. 2013, Henvee & Tabr 2015). Thus, the unform pressure polynomal as the central element n the smplfed approach and the functon for the structural resstance coeffcent c T, that scales the resstance accordng to the shp sze, were updated usng the same approach as n Henvee et al. (2013). The structural resstance coeffcent c T s here evaluated based on the volume of the deformed materal. Furthermore, t s shown that the smple formula between the dsspated energy and the volume of damaged materal gven by Mnorsky (1959) s applcable also for shp groundngs. The effect of transverse and longtudnal bulkheads to the damage openng sze s studed and equatons for the outer and nner damage wdths are updated compared to Henvee & Tabr (2015). 99

2 Fnte element smulatons Ths chapter presents an overvew of numercal groundng smulatons. The prncples of numercal modelng and the post-processng of the analyss results are gven. FE models Three double hull tankers wth dfferent dmensons are modeled. The cross-sectons wth the man structural dmensons are gven n Fg. 1 and n Table 1. Herenafter we use superscrpts T150, T190 and T260 to denote the tankers. If the superscrpt s replaced by, t means that the descrpton s common to all three shps. Shpbuldng steel wth yeld stress of 285 [MPa] s used n the analyss. True stress-stran curve s presented n Fg. 3. ponts through ther thckness. The prevalng elementlength n the double bottom structure was around and [mm] elsewhere. Fner mesh n longtudnal and transverse bulkheads extend up to 4 [m] from the shp bottom. Standard LS-DYNA hourglass control and automatc sngle surface contact (frcton coeffcent of 0.3) s used for the dsplacement controlled groundng smulatons. The rgd rock frst moves to a requred penetraton depth and contnues to move at constant penetraton depth along the shp at a constant velocty of 10 [m/s]. The nodes at the forward and aft end of the models are fxed. Table 1: Man dmensons and parameters of the tankers Parameter T150 T190 T260 Length [m] Breadth [m] Draught [m] Depth [m] Desgn speed [kn] Deadweght [tdw] Double bottom heght [m] Outer platng thck. [mm] Tank-top thck. [mm] Grder spacng [m] Floor spacng [m] Classfcaton rules HCSR-OT Fg. 1: Tanker cross-sectons (dmensons not n scale). The correspondng fnte element models are presented n Fg. 2. The structure s modeled usng quadrlateral Belytschko-Ln-Tsay shell elements wth 5 ntegraton Fg. 2: FE model of the tanker. Materal falure was modeled wth the fracture crteron developed by Kõrgesaar (2015). Accordng to the crteron the fracture stran for shell element s calculated as a functon of stress state and element sze. 100

3 Fg. 3: True stress-stran curve. Groundng scenaros and rock locatons The groundng smulatons were conducted for fve dfferent penetraton depths (from 1.0 to 3.0 [m] wth 0.5 [m] spacng) and for three rocks. All the rocks are axsymmetrc wth parabolc cross-sectons gven by z=y 2 /a, wth a beng the parameter defnng the rock sze (Henvee & Tabr 2015). Rocks ranged from sharp rocks denoted as rock A (a=3) and rock B (a=6) to blunt shoal -type rock C (a=12). The groundng smulatons are done for two transverse rock locatons (see Fg. 4a): ) locaton B/4: between the longtudnal bulkhead and the shp sde.e. at B/4; ) locaton B/2: drectly under the central longtudnal bulkhead.e. at B/2. Fg. 4: Setup for FE calculatons: (a) rock postons, (b) ranges for the evaluaton of groundng forces. Horzontal rock travel starts two web frame dstances before the transverse bulkhead and termnates at two web frame dstances before the next bulkhead, see Fg. 4b. For each smulaton, tme hstores for the horzontal groundng force (Fg. 4b), deformaton energy and the volume of the deformed elements are obtaned. From each force tme hstores two average force values are evaluated, Fg. 4b: F B/2 (or F B/4 ) average force over the whole horzontal travel dstance ncludng the effect of the transverse bulkhead, see red sold lne n the fgure; B/2 (or B/4 ) average force over the reduced horzontal travel dstance excludng the effect of the transverse bulkhead, see red dashed lne n the fgure; The effect of longtudnal bulkhead can be determned by comparng the average forces B/2 and B/4. Smlarly, the effect of transverse bulkheads s determned by comparng the average forces F B/4 and B/4. Furthermore, to study the openng wdths n outer and nner bottom, the correspondng values are measured from each FE smulaton. Effect of longtudnal bulkhead The effect of longtudnal bulkhead to the groundng force s presented n Fg. 5 va comparson of average forces B/2 (longtudnal bulkhead contrbutes to the groundng resstance) and B/4 (no resstance contrbuton by longtudnal bulkhead). Fg. 5 presents the rato B/2 / B/4 as a functon of penetraton depth for dfferent rocks and shps. Fgure reveals that at low penetraton depths the longtudnal bulkhead ncreases the resstance about 10 % regardless of shp and rock sze. The nfluence of the bulkhead ncreases at hgher penetraton depths. For δ=3 [m] the maxmum force ratos for shps T150, T190 and T260 are 1.3 (30%), 1.46 (46%) and 1.36 (36%) respectvely. It should be noted that at δ>1 [m] the rato contnues to ncrease for rocks A and B, whle for the rock C the rato remans almost constant. As the rock C s relatvely large compared to the shp cross-sectons, the double sde starts to contrbute to the resstance at hgher penetraton depths. Thus, t can be concluded that wth large shoal-type rocks (a 12) the nfluence of the longtudnal bulkhead s small as t s partly compensated by the contrbuton from the double sde structure. Furthermore n Fg. 5a the rato decreases for rock C at >1.5 [m] due to the crushng of the shp sde that gves sgnfcant addtonal resstance. As the purpose was to determne the effect of longtudnal bulkhead, these scenaros are omtted n subsequent development of the term descrbng the effect of longtudnal bulkhead (Eq.1 and Fg. 6). In Fg. 6 the regresson curve s ftted through all the B/2 / B/4 ratos, gvng a term descrbng the effect of 101

4 the longtudnal bulkhead: B/2 B/4 = 0.105δ (1) In order to employ the obtaned relatonshp, we derve formula for the average force B/4 by usng the approach presented n Henvee et al. (2013). Thus, the rock drectly under the longtudnal bulkhead, the average groundng force can be calculated as B/2 = B/4 (0.105δ ), (2) where B/4 s average groundng force wthout the contrbuton from the longtudnal bulkhead. Fg. 6: The effect of longtudnal bulkhead to the average groundng force: The rato between average forces calculated at B/2 and B/4. Fg. 5: Increase of average groundng force due to the longtudnal bulkhead presented as a rato B/2 B/4. Dashed vertcal lne ndcates the double bottom heght. Fg. 7: The effect of transverse bulkhead to the groundng force presented as a rato F B/4 B/4. Dashed vertcal lne ndcates the double bottom heght. 102

5 Effect of transverse bulkhead To study the nfluence of transverse bulkheads we compare two average forces F B/4 and B/4. The average forces ratos F B/4 / B/4 are presented n Fg. 7 for three shps and for three rock szes. In Fg. 7 the force ratos F B/4 / B/4 for both rock postons reman almost constant and are approxmately equal to 1, whch means that bulkhead has only small nfluence to the average groundng force. Smlar behavor of the rato was observed also for F B/2 / B/2. As the nfluence of the transverse bulkhead to the average groundng force s small, ts contrbuton s not explctly presented n the smplfed equatons. Updated formulas for the groundng force The smplfed formula for the average horzontal groundng force F H was gven by Henvee et al. (2013) as F H = c T P A, (3) where c T s a coeffcent for shp and s characterzng shp s structural resstance and defned va blnear functon of shp length L, c T = f CT (L), P s the unform pressure polynomal descrbng the contact pressure as a functon of rock sze a and A s the projected contact area between the rock and the shp double-bottom (Henvee et al. 2013) gven n Appendx A. The current paper updates the functon for the structural resstance coeffcent c T and the unform pressure polynomal P usng the same procedure as presented n Henvee et al. (2013). The updated structural resstance coeffcent functon for c T takes the form (Fg. 8a) c T = f CT (L) = { 1375 L , f 150 L 190 [m] L , f L 190 [m] (4) In Fg. 8 the structural resstance coeffcents c T are presented for the tankers used n ths paper (Fg. 8a) and for those used n Henvee et al (2013) (Fg. 8b) and about 1.6 tmes ncrease n recognzed. Reasons for that are analyzed n the next secton, where the structural resstance coeffcent s connected to the volume steel materal. The updated form for the pressure polynomal was derved based on average forces B/4 and s as follows P (a) = a a (5) The contact force n groundng can now be calculated usng Eq. 3. If the rock s postoned drectly under the longtudnal bulkhead then Eq. 3 s to be multpled wth the term gven by Eq. 1gvng the average groundng force under the longtudnal bulkhead as F H = c T P A (0.105δ ). (6) Fg. 8: Functons f CT (L) for structural resstance coeffcent c T : (a) based on tankers used n current paper and (b) from Henvee et al. (2013). Structural resstance coeffcent as a functon of materal volume The dfference of structural resstance coeffcents n Fg. 8 s due to the dfferent desgn crtera used for the shps- the tankers n the current paper meet all the strength crtera accordng to HCSR-OT rule whle the tankers n Henvee et al. (2013) only satsfy the mnmum rule scantlng requrements and thus, present very conservatve approach n means of structural resstance. Clearly, the latter tankers contan less steel. To determne ther dfferences, we calculate the volume of deformed materal V def (materal where plastc stran ε p > 0.01 ) for four numercal smulatons: two conducted wth T190 tankers and two wth T260 tankers, see Table 2. Table 2: Ratos of V def and c T for dfferent tankers. Scenaro [m] V def [m 3 ] Rato (V def ) Rato (c T ) T190,rock A * T190,rock A ** T260,rock A * T260,rock A ** = = = = 1.61 * tanker used n Henvee et al. (2013); ** tanker used n the current study. The results gven n Table 2 show that the volume of deformed materal for the current tankers s 1.5 and 1.4 tmes hgher for T190 and T260 tankers, respectvely. Ths ndcates a possble correlaton between the structural resstance coeffcent and the volume of deformed steel materal. If such correlaton exsts, the c T values presented by Eq. 4 can be used as a bass to evaluate a c T j value for any shp j once the steel volumes V mat and V j mat are determned: c T c j = V mat (a, δ) j T V mat (a, δ) c T j = V mat j (a, δ) V mat (a, δ) c T, (7) where V mat (a, δ) and V mat j (a, δ) are approxmatons 103

6 for the steel volume to be deformed per unt length n a certan groundng scenaro defned va rock sze a and penetraton depth. A routne to approxmate ths volume s presented n detal n Appendx A, whch also presents the V mat values for the tankers (Table. A1) used n the current paper. In the calculaton procedure the volume V mat ncludes the contrbutons from the double bottom structural members, whch are n drect contact wth the rock. [m] for whch the normalzed value s Ths s due to rapd and local ncrease n steel volume as the penetraton slghtly above the double bottom heght (h db =1.4 [m] for T150). Ths effect dmnshes as the penetraton ncreases further. Thus, t s suggested to use h db for the evaluaton of the V mat n Eq. 7. Shp Table 3: Normalzed materal volumes. V mat T260 V mat penetraton [m] T T T In Fg. 10 the normalzed volumes from Table 3 and the c T values from Fg. 8 are presented as a functon of shp sze. All the values are normalzed wth respect to the correspondng value of the largest tanker T260. The comparson shows that the average volume behaves smlar to the structural resstance coeffcent. Thus, the structural resstance coeffcent can be determned va the steel volumes by usng Eq. 7. It should be noted, that for the comparson n Eq. 7, the same a and values should be used both for V mat (a, δ) and V mat j (a, δ) and for h db. Moreover, as the lnk between the V mat and the structural resstance exsts, there should also be a relatonshp between the steel volume and the deformaton energy. Ths relatonshp s studed n the next secton. Fg. 9: Averaged volume of deformed materal compared for shps T150,T190 and T260: a) Rock A; b) Rock B and c) Rock C. In Fg. 9 averaged steel volume V mat s presented for all the smulated scenaros wth poston B/4. Two patterns can be recognzed. Frst, for each shp the averaged volume ncreases proportonally wth the penetraton depth. Ths holds for all the rocks. Ths ndcates that the rato of average volumes at each penetraton depth s constant between any two shps. Ths s also presented n Table 3, where the averaged steel volumes are normalze wth respect to the volume of the largest tanker T260. The Table 3 reveals that the normalzed values are constant for each tanker, except for T150 at =1.5 Fg. 10: Comparson of normalzed steel volume and structural resstance coeffcents gven. Relatonshp between the dsspated energy and volume of deformed materal It was shown by Mnorsky (1959) that there s lnear correlaton between the volume of the deformed materal and the energy dsspated durng the shps collson. In Luukkonen (1999) the performance of Mnorsky s equaton together wth several other smplfed models was analyzed wth respect to real groundng accdents. Although the correlaton between the deformed materal 104

7 and the dsspated energy was recognzed, sgnfcant varatons occurred for all the models. Obvously, the dfferences were partly due to poor reportng of the real accdents, e.g. the groundng velocty and the descrpton of the groundng scenaro. Here the am s to develop lnear relatonshp between the steel volume V mat and the absorbed energy based on numercal smulatons, where the groundng scenaro s well defned. We use the steel volume V mat (Appendx A) to approxmate the volume of the deformed materal V def. For each groundng smulaton the volume of deformed materal V def was calculated for two dfferent levels of equvalent plastc strans: ε p > 0.01 and ε p > 0.1, whch are plotted aganst the dsspated energy n Fg. 11 for the poston B/4. The dsspated energy ncludes the contrbuton from frcton. In the fgure, both the energy E and the steel volumes are presented per unt length. For that the deformaton energy E absorbed durng the rock travel over the horzontal dstance L h (Appendx A) was dvded wth L h to obtan E. For both plastc strans a strong lnear correlaton can be notced. Clearly, the amount of deformed materal depends how one defnes the deformed materal. It s nterestng to note that for ε p > 0.01 the obtaned dependency s very smlar to the one shown by Mnorsky (1959). To mantan the smlarty to Mnorsky s classcal relatonshp, we derve the relatonshps based on ε p > 0.01, gvng the deformaton energy E per unt length as: E = 38.11(1.07V mat ) [MJ m] for rock at B/ (1.26V mat 0.016) [MJ m] { for rock at B/2 where V mat unt s [m 3 /m]. (8) Fg. 11: Averaged energy vs volume of deformed materal per unt damage length n case of B/4: (a) ε p > 0.01; (b) ε p > 0.1. Fg. 12: Average force calculated wth Eq. (3) and Eq. (8), (ep>0.01). It should be noted, that the energy per unt length, E gven by Eq. 8, has a unt of [MJ/m] and actually represent the average groundng force. Thus, t can be drect- 105

8 ly compared to the average groundng force gven by numercal smulatons and wth Eq. 3 and Eq. 6. For the poston B/4, ths comparson s gven n Fg. 12, where empty crcles present the average groundng force from numercal smulatons, flled crcles present the energy per unt length from numercal smulatons, sold lnes present Eq. 3 and dashed lnes present Eq. 8. Good correlaton exsts between the equatons and the numercal smulatons, except for Eq. 8 and tanker T260, where the devaton s about 15-20%. Dependng on the avalable nformaton for groundng scenaro ether Eq. 3, Eq. 6 or Eq. 8 can be used for the calculaton of the average groundng force. If the rock sze, penetraton depth and the shp scantlngs are avalable then Eq. 8 can be used to take nto account the resstance of the specfc shp. However, f such detaled data for shp s not avalable then Eq. 3 or Eq. 6 can be employed usng the penetraton depth, rock sze, shp length and double bottom heghts as varables. Sze of the damage openng In ths chapter the effect of transverse and longtudnal bulkheads to the damage openng wdth s studed. The damage openng formulas developed n Henvee & Tabr (2015) are updated accordngly. The damage openng formulas gve the dmensons of the openng wdths and should be used together wth a crtera defnng whether the falure n the nner bottom occurs. Frst, the formulas for the openng wdths are updated followng the updated crtera for crtcal penetraton depth. Damage wdth n outer and nner bottom In each numercal smulaton the average openng wdth was measured for the outer and nner bottom and the measurements are presented n Fg. 15. Fg. 15 presents the damage wdths only for the poston B/4. In B/2 the behavor and the damage dmensons were smlar meanng that the effect of longtudnal bulkhead to the average nner openng wdth s modest. Moreover, observatons from FE smulatons showed that a notceable ncrease n openng wdth n the nner bottom occurred locally at the vcnty of the transverse bulkhead and ths has only mnor effect on the average wdth. The numercal smulatons showed that the groundng damages wth respect to the shp sze were relatvely local and concentrated to the vcnty of the ntrudng rock, see Fg.13b. In s nterestng to notce that the presence of the transverse and longtudnal bulkheads contrbuted to the localzaton of the damage. In the smulatons wthout the bulkheads (Henvee & Tabr, 2105) the stffness of the double bottom was lower and, especally n the case of larger rocks, the resultng damage was global deformaton of the whole double bottom, see Fgure 11a. When the bulkheads are ncluded, the domnatng deformaton mode s a combnaton from local tearng and global crushng n case of all three rocks. Fg. 13: Comparson of bottom damages: (a) tanker wthout the bulkheads (Henvee & Tabr 2015) (b) tanker wth bulkheads. In Henvee & Tabr (2015) the equaton for the damage openng wdths were gven separately for two rock sze ranges due to the domnant global crushng modes n the case of large rocks (a 12). Here, the deformaton modes were smlar for all the covered rock szes and the equatons can be presented for a sngle range coverng all the rocks (3 a 12). Analyss revealed that wthn the range of penetraton depths 1.0 to 3.0 [m] the behavor of the openng wdth n the outer bottom generally follows the rock wdth. Smlar observatons as n Henvee & Tabr (2015) can be made: ) the openng wdth n the nner bottom grows smlarly to the openng wdth n the outer bottom; ) onset of falure n the nner bottom s delayed by b h db compared to that n the outer bottom, where constant b Smulatons revealed that the outer bottom falure was observed roughly at 0.5 [m]. Thus, the smplfed formulas for the predcton of openng wdths n the outer and nner bottom are as follows: openng wdth n the outer platng D out (a, δ) = 2 a δ [1.6δ 0.8] for δ 1[m] = { a δ for δ>1[m] (9) 106

9 for 150 [m] L 260 [m], 3 a 12 openng wdth n the nner platng D n (a, δ, h db ) = 2 a(δ 0.75h db )[1.6(δ 0.75h db ) 0.8] { for δ 1 [m] a(δ 0.75h db ), for δ > 1[m] for 150 [m] L 260 [m], 3 a 12. (10) Comparson between the measured openng wdths and the calculatons usng the above equatons are presented n Fg. 15. The fgure shows that Eq. 9 slghtly underestmates the wdth of the damage openng n the outer platng especally n the case of larger penetraton depths. The devaton s about 20%. For the nner platng openng, the Eq. 10 alone, see dashed lnes n Fg. 15, sgnfcantly overestmates the damage wdth for lower penetraton depths, whle for hgher values the predcton s reasonable. Thus, a crteron s requred to defne the onset of the falure n the nner platng. Ths crtera s presented n the next secton. Fg. 14: Fracture crteron for the nner bottom. Crtcal penetraton depth for the nner bottom falure The crtcal penetraton depth f defnes whether the nner bottom s thorn open as Eq. 10 alone mght predct nner bottom falure prematurely, see Fg. 15. Updated form for the crtcal penetraton depth s derved n a smlar manner to Henvee & Tabr (2015). In the dervaton, the smulatons wth the rock poston B/2 are used. The crtcal penetraton depth δ f obtaned from the numercal smulaton was dvded wth the correspondng double bottom heght h db, provdng the relatve crtcal penetraton depth. These ratos are presented n Fg. 14 as a functon of the rato between the rock sze and the shp breadth - a/b. The regresson lne through the measured ponts forms the crteron as follows: δ f = 0.75 a h db B δ f = (0.75 a (11) B ) h db. The nner bottom damage occurs once the penetraton depth s hgher than gven by Eq. 11. After the crtcal penetraton depth s reached, the wdth of the openng n the nner bottom can be evaluated usng Eq. 10, see Fg. 15. Conclusons Smplfed formulas for the calculaton of average groundng force gven n Henvee et al. (2013) have been updated to consder the contrbuton from the transverse and longtudnal bulkheads. The contrbuton s studed va seres of numercal groundng smulatons. The analyss of numercal smulatons showed that the longtudnal bulkhead substantally ncreases the average groundng force. If the ntrudng rock s drectly under the longtudnal bulkhead the groundng force can be up to 50 % hgher compared to the stuaton when the rock s between the bulkhead and the shp sde. Ths nfluence s ncluded n the smplfed formulas va addtonal term dependng on the penetraton depth. Analyss also revealed that, n general, the transverse bulkhead has small nfluence to the average groundng force and thus ts contrbuton n not explctly ncluded n the equatons, whle ts nfluence s mplctly ncluded n the structural resstance coeffcent. A new approach was proposed for the predcton of resstance coeffcent c T based on the approxmaton of the volume of the deformed materal. It was shown that the structural resstance coeffcent s proportonal to the volume of deformed materal. Even though the calculaton requres detaled nformaton of shp's double bottom structure, t provdes analytcal measure to develop more shp-specfc estmate for c T. Moreover, smulatons revealed that a lnear relatonshp exsts between the volume of the deformed materal and the energy absorbed n groundng.e. Mnorsky's relatonshp, though slghtly modfed, s applcable also for shp groundngs. Equatons to predct the volume of the deformed materal n a certan groundng scenaro were derved based on the structural confguraton of the double bottom. The longtudnal and transverse bulkheads nfluence also the damage openng sze durng the groundng over large rocks. In the case of smaller rocks, the nfluence of the bulkheads on the openng sze was modest. Acknowledgement Ths research work has been fnancally supported by the BONUS STORMWINDS (Strategc and Operatonal Rsk Management for Wntertme Martme Transportaton System) project, by the research grant IUT1917 from Estonan Scence Foundaton and by Tallnn Unversty of Technology project B18 (Tool for drect damage calculatons for shp collson and groundng accdents). Ths help s here kndly apprecated. 107

10 Fg. 15: Openng wdth n the nner and the outer bottom: FE smulatons vs. equatons. References Alsos, H and Amdahl, J On the resstance of tanker bottom structures durng strandng. Marne Structures 20: pp Henvee, M., Tabr, K and Kõrgesaar, M A smplfed approach to predct the bottom damage n tanker groundngs., Proc. 6 th Internatonal Conference on Collson and Groundng of Shps and Offshore Structures, Trondhem, Norway. Henvee, M and Tabr, K A smplfed method to predct groundng damage of double bottom tankers. Marne Structures 43: pp Hong, M and Amdahl, J Rapd assessment of shp groundng over large contact surfaces. Shps and Offshore Structures 7( 1): pp Kõrgesaar, M Modellng ductle fracture n shp structures wth shell elements. Ph.D. thess, Aalto Unversty, Helsnk, Luukkonen, J Damage of shp bottom structures n groundng accdent [n Fnnsh], M-239 report. Espoo: Helsnk Unversty of Technology. Mnorsky, V.U An analyss of shp collson wth reference to protecton of nuclear power plants. Journal of Shp Research 3: pp 1-4. Smonsen, B.C., Törnqvst, R and Lützen, M A smplfed groundng damage predcton method and ts applcaton n modern damage stablty requrements. Marne Structures 22: pp Appendx A Procedure for the calculaton of steel volume: In transverse and longtudnal drecton, the structural members contrbute to the total steel volume only f n drect contact wth the rock, see red area n Fg. A1. In longtudnal drecton the steel volume s evaluated over the horzontal length L h that s the length of one tank compartment and s symmetrc wth respect to the transverse bulkhead. The steel volumes are calculated wth the followng steps: () Equvalent thcknesses for the nner and outer plate, grders and floors are calculated as follows t eq = t pl + n A stf D, where t pl s the plate thckness, n s the number of stffeners on the plate and D s the plate wdth. If the platng conssts of several plates wth dfferent thcknesses then the equvalent value for the t pl s calculated as 108

11 t pl = d t, D where d s the wdth of -th plate and t s the correspondng thckness. () Determne the length for the longtudnal members and the number for the transverse members: The length of longtudnal members (nner and outer plate, grders) s taken as L h. Number of the transverse members s equal to the number of floors nsde the length L h. () Takng nto account the poston of the rock wth respect to the structural members, calculate the total volumes for the structural members: Outer plate Inner plate Grders V out_pl = 2 a δ t eq L h, V n_pl = 2 a (δ h db ) t eq L h, V gr = δ d t gr L h, where, t gr s the equvalent thckness of a grder, δ d s the heght of the deformed part of a grder whch s gven as: δ d = δ Y2 a, where, Y s the horzontal dstance from the tp of the rock to the grder, see Fg. 1A. If entre grder s damaged then δ d = h db. Floors V floor = n t eq A floor, where n s the number of floors, t eq s equvalent thckness of the floor and A floor s equal to the contact area A between the floor and the rock gven by A floor = A = = { 4 3 a [δ( a δ(3 2 ), f δ h db ) (δ h db ) (3 2 ) ], f δ > h db (v) Total volume of materal V mat for a scenaro s sum of all ndvdual volumes of structural members V mat = V out_pl + V n_pl + V gr + V floor [m 3 ].. Fg. A1. The prncple scheme for the calculaton the steel volume of deformed materal V mat. Table A1 present the V mat and V def values for the groundng scenaros smulated n ths paper. The presented values are calculated for B/4. Table. A1: Steel volumes [m 3 m], (ε p > 0.01). [m] V mat V def V mat V def V mat V def Rock A (a=3) Rock B (a=6) T150 Rock (a=12) T T C (v) Volume per meter s calculated as V mat = V mat L h [m 3 m]. 109