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Ships and Offshore Structures ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tsos20 An analytical method for predicting the structural response of ship side structures by bulbous bow in oblique collision scenarios Zeping Wang, Chunyu Guo, Chao Wang, Gang Chen, Ying Xu & Qing Li To cite this article: Zeping Wang, Chunyu Guo, Chao Wang, Gang Chen, Ying Xu & Qing Li (2023): An analytical method for predicting the structural response of ship side structures by bulbous bow in oblique collision scenarios, Ships and Offshore Structures, DOI: 10.1080/17445302.2023.2247839 To link to this article: https://doi.org/10.1080/17445302.2023.2247839 Published online: 24 Aug 2023 Submit your article to this journal Article views: 28 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tsos20 SHIPS AND OFFSHORE STRUCTURES https://doi.org/10.1080/17445302.2023.2247839 An analytical method for predicting the structural response of ship side structures by bulbous bow in oblique collision scenarios Zeping Wanga,b, Chunyu Guob, Chao Wanga, Gang Chenc,d, Ying Xud and Qing Lic a College of Shipbuilding Engineering, Harbin Engineering University, Harbin, People’s Republic of China; bQingdao Innovation and Development Center of Harbin Engineering University, Qingdao, People’s Republic of China; cState Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, People’s Republic of China; dMarine Design and Research Institute of China, Shanghai, People’s Republic of China ABSTRACT ARTICLE HISTORY As more and more ships sail on the sea, the probability of collision between ships also increases At present, researches more on head-on ship collisions than oblique ship collisions According to statistical data, the probability of oblique ship collision is higher than that of head-on ship collision, and oblique ship collision may cause a wider range of structural damage Therefore, this paper studies the damage deformation of ship side structures in oblique collision scenarios In this paper, a simplified analysis method is proposed to predict the structural response of the struck ship’s side structures by bulbous bow in oblique collision scenarios and the analytical results match well with the numerical simulation results, which verifies the accuracy of the simplified analysis method The simplified analysis method can be used to assess the crashworthiness of ship side structures by bulbous bow in oblique collision scenarios Received November 2022 Accepted 11 June 2023 Introduction With the increase in ships sailing at sea, the risk of ship collision is also increasing Ship collision accidents lead to environmental pol lution and economic losses, so it is of great significance to carry out research on ship collisions According to statistics, the probability of oblique ship collision accidents is higher than that of head-on ship collision (Yamada et al 2016) However, a lot of researches have been carried out on head-on ship collisions, there are few researches on oblique ship collisions Therefore, it is necessary to carry out research on oblique ship collision The research methods of ship collision can be divided into four types: empirical formula method, experimental method, nonlinear finite element method, and simplified analysis method Minorsky (1958) was the first to put forward the empirical formula through statistical analysis of ship collision accidents Later, Woisin (1979), Pedersen and Zhang (1998) further revised and summarised the empirical formula By comparing with the experimental results in public literature, Zhang and Pedersen (2016) re-verified the accu racy of this method in the analysis of ship collision damage The experimental method has high accuracy in the research of ship collision problems, many scholars carried out ship collision and grounding tests Akita et al (1972) and Pedersen et al (1993) carried out experiments in order to design ship structures with sufficient strength to resist impact Amdahl (1983) and Wang et al (2000) conducted model tests to research the deformation mechanism of ship structures in ship collision and grounding scen arios Full-scale experiments were carried out by Tabri et al (2009) to validate the proposed theoretical model Villavicencio et al (2014) conducted experiments on a tanker side panel impacted by a knife edge indenter Liu and Soares (2019) carried out exper iments to research the influence of strain rate on laterally impacted steel plates Scaled experiments were conducted by Calle et al (2017) to verify the finite element analysis of impact tests on marine CONTACT Qing Li liqing5504@sjtu.edu.cn KEYWORDS Oblique ship collision; bulbous bow; crashworthiness; analytical method; numerical simulation structures Xu et al (2020) conducted the collision experiments of ship models in a water tank, with particular attention to structure in the collision region Considering the coupling effect of external dynamics and internal mechanics, the dynamic responses of ships during collision are studied The failure mode and deformation damage characteristics of ship’s side structure in collision region are also assessed Wang et al (2021) used the model test method and numerical simulation method to study ship-ship collisions The Coupled Eulerian-Lagrangian (CEL) was used to simulate the fluid-structure interaction for predicting structural deformation and ship motion during a normal ship-ship collision Meanwhile, a series of model tests were carried out to validate the numerical results However, actual ship tests or large-scale model tests require a large amount of funds For small-scale model tests, the non-linear behaviour will lead to scale effect, the test results may not be pre cisely converted to the real ship scale, and leading to some unex pected errors The nonlinear finite element method, which is considered as ‘numerical experiment’, can simulate ship collision accidents effec tively with the development of computational capacity Yamada and Endo (2008) studied the crashworthiness of ship structures in oblique collision scenarios Haris and Amdahl (2013) used the finite element program LS_DYNA to simulate several ship collision scenarios and verified the proposed analysis method, and the col lision force results of the relatively rigid bow and the rigid bow were compared, it was found that the two results were similar, as shown in Figure 1, which proved that the effectiveness of using a rigid bow in this study Yu et al (2013) and Hu et al (2011) con ducted numerical simulations of ship grounding scenarios, which verified the simplified analysis method proposed by Hong and Amdahl (2012) A benchmark study of an indenter impact a ship side structure by finite element numerical simulation was con ducted by Ringsberg et al (2018) Simonsen and Törnqvist (2004) simulated the crack propagation of large-scale shell structure State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China © 2023 Informa UK Limited, trading as Taylor & Francis Group Z WANG ET AL Figure Comparison results of force-indentation curves of the relative rigid and rigid bows (Haris and Amdahl 2013) through the finite element method The test of stiffened plates sub jected to impact was numerically simulated by Alsos et al (2009) Ehlers et al (2012) study the collision resistance of the X-core struc ture by applying the material relationship to the finite element model A numerical simulation method of structural impact con sidering the effect of strain rate was developed by Liu et al (2015) Rudan et al (2019) investigated the consequences of apply ing two different realistic ship collision modelling techniques invol ving fluid-structure interaction (FSI) analysis in LS_DYNA: arbitrary Lagrangian–Eulerian (ALE) technique and rigid body MCOL coupling, using LPG ship and a ferry collision scenario as a study case Then, results are compared and discussed Many other scholars also carried out research on ship collision and grounding problems through numerical simulation Nevertheless, due to modelling work and numerical simulation take a lot of time, sometimes it is not reasonable at the beginning of the design stage For these reasons, engineers are increasingly demanding for simplified design tools The simplified analysis method is based on the upper bound theorem to analyse the deformation mechanism of the structure, which has the advantages of reasonable accuracy and high efficiency The basic characteristic of the simplified analysis method is to propose a simplified theoretical model which includes the main deformation characteristics of the structure, and then the theoretical model is analysed to derive the analytical calculation formula In the past decades, many scholars put forward their sim plified analysis methods, which made contributions to this field Alexander (1960) creatively applied this method to the analysis of thin-walled structures Wang (1995) studied the deformation mechanism of ship structures in collision and grounding scenarios and proposed many analytical formulas The deformation mechan ism of ship bottom plate in ship grounding scenarios was researched by Simonsen (1997) and Zhang (2002), and analytical formulas were proposed Hong and Amdahl (2008), Liu and Soares (2016) proposed formulas for calculating the crushing resistance of web girders Buldgen et al (2012) established analytical formulas for the resistance of various super-elements to an oblique impact Firstly, the original super-elements method is briefly introduced Then, analytical calculations in oblique collision cases are per formed for the different super-elements involved in the procedure Finally, the formulations are validated by comparison with results provided by the classical nonlinear finite element method Buldgen et al (2013) proposed a new analytical formulation for estimating the impact resistance provided by inclined ship side panels Two different scenarios are treated They first deal with the case of an impact between the oblique plate and the stem of the striking ship, and then consider the situation where the inclined panel is impacted by the bulb For these two scenarios, an analytical formu lation relating the force and the penetration is provided and these developments are validated by comparing them to the results of finite elements simulations Liu (2017) proposed an analytical method to evaluate the ship structures subjected to collision Kim et al (2021) presented a procedure that simulates the influence of strongly coupled FSI effects on the dynamic response of ships involved in typical collision and grounding events The method couples an explicit 6-DoF structural dynamic finite element scheme with a hydrodynamic method accounting for (a) 6-DoF potential flow hydrodynamic actions; (b) the influence of evasive ship speed in the way of contact and (c) the effects of hydrodynamic resistance based on a RANS CFD model Kim et al (2022) pre sented a benchmark study that compares the structural dynamic response by explicit nonlinear FEA approaches and the seminumerical super-element method Simulations for typical accident scenarios involving passenger ships confirm that implementing the influence of hydrodynamic restoring forces in the way of contact may be useful for either collision or grounding Many other scho lars also proposed simplified analytical calculation formulas to pre dict the resistance of ship structures, which promote the development of simplified analysis methods However, most of the existing simplified analysis methods are suitable for head-on collision scenarios, there are few simplified analysis methods suit able for oblique collision scenarios Oblique ship collision accidents are statistically more frequent than that of head-on ship collision The difference between oblique ship collision and head-on ship collision is that oblique ship col lision causes a wider range of structural deformation in the length direction of the ship Nevertheless, compared with the simplified analysis method of head-on ship collision, less research has been done on oblique ship collision Some scholars studied the damage of ship side structure by bulbous bow in head-on ship collisions scenarios, such as Wang et al (2000), Simonsen and Lauridsen (2000), Lee et al (2004), in their research, the shapes of the bulbous bow or the struck plate are simplified, which may lead to errors in the simplified analysis results Therefore, this paper proposes a more reasonable and accurate simplified analysis method to obtain the structural response of ship side structures in bulbous bow obli que collision scenarios In this paper, six typical oblique collision scenarios are defined, and the proposed Johnson-Cook GTN model is used for numerical simulation by using the finite element program LS_DYNA to find out the main deformation characteristics of the side structures of the struck ship In addition, an assumption for the derivation of the analytical formulas is proposed Then, a simplified analysis method is proposed to predict the structural response of the struck ship’s side structures by bulbous bow in oblique collision scenarios The new simplified analysis method includes the deformation mechanism analysis of the side shell plating, the transverse frame and the web girder, and the resistance formulas are proposed, then integrated to evaluate the overall crashworthiness of the struck ship’s side structures Finally, the analytical calculation results are compared with the numerical simulation results, and the results match well, which proves the accuracy of the proposed simplified analysis method The simplified analysis method can be used to assess the crashworthiness of ship side structures by bulbous bow in oblique collision scenarios SHIPS AND OFFSHORE STRUCTURES Numerical simulations In order to investigate the deformation characteristics of the struck ship’s side structures in oblique collision scenarios and put forward assumptions for the analytical formulas derivation, firstly, numeri cal simulation is carried out by using the finite element program LS_DYNA 2.1 Finite element model In numerical simulation, a double-hull tanker was struck by an ellipsoid-shaped bulbous bow of a 60,000 DWT striking ship Figure shows the finite element models of the struck ship’s side structure and the striking bow, Table lists the details of the main components of the struck ship The proposed JohnsonCook GTN model is used for numerical simulation through using the finite element program LS_DYNA, complete details of the pro posed Johnson-Cook GTN model can be found in the paper by Wang et al (2020) The material of the striking bow is rigid, and the material parameters used for the side structure of the struck ship are listed in Tables and (Wang et al 2020) The material yielding stress of the deck is 355MPa, and the material yielding stress of the other side structural components is 235MPa The strik ing velocity is defined as m/s, and the friction coefficient is defined as 0.3 (Yamada and Endo 2008) Four-node quadrilateral Belytschko-Tsay element is used in the finite element model To obtain a reasonable numerical simulation time and capture the major deformation characteristics, the mesh size of the finite element model of the side structure is defined as 200 mm The side structure is restricted at both longitudinal ends by fixing the six degrees of freedom, the six degrees of freedom of the bow are not restricted so as to simulate the actual motion of the bow Table Main components and structural details of the tanker Item Length Breadth Depth Design draught Length of one compartment Spacing of double bottoms Spacing between longitudinal girders Spacing between transverse web frames Spacing between stiffeners The thickness of outer shell plating The thickness of inner shell plating The thickness of web girder The thickness of transverse web frame The thickness of stiffener The breadth of stiffener The thickness of the frame The thickness of the web The height of the web The thickness of the flange The height of the flange Value 288.0 m 65.0 m 29.4 m 22.0 m 35.0 m 3.38 m 7.2 m 5.0 m 0.9 m 21 mm 15 mm 12 mm 14 mm 13 mm 0.38 m 14 mm 18 mm 3m 13 mm 3m Table GTN parameters v 0.3 sy /E 1/857 q1 1.5 q2 1.0 q3 2.25 fc 0.01 ff 0.03 fN 0.03 1N 0.3 sN 0.1 Table Johnson-Cook parameters A(Pa) 2.45E + 08 B(Pa) 5.001E + 08 C 0.016 m 0.915 n 0.221 1˙ r 10−3 1–3; in case 4–6, the collision position is position 2, as the collision proceeds, the striking bow will pass through position 2.2 Collision scenario definition Figure is a schematic diagram of an oblique collision, defining the oblique collision angle Figure shows three typical collision pos itions, and the definition of six collision scenarios are shown in Table and Figure The collision position is position in case 2.3 The characteristics of oblique collision and assumptions Figure Finite element model of the side structure of the tanker and the bow Figure Definition of the collision angle b In the numerical simulation, an ellipsoid-shaped bulbous bow of a 60,000 DWT striking ship is chosen as the striking bow, the move ment of the striking bow is not restricted to ensure that the actual motion track of the striking bow can be simulated The damage deformation of the side structure of the struck ship and the motion track of the striking bow are related to the mass of the striking bow and the collision angle Through the numerical simulation results, it could be found that the movement direction of the striking bulbous bow changes little and basically maintains the original striking direction The characteristics of the oblique collision scenario owes to the limited effect of the deformation of the side plating on the striking bulbous bow, which restricts the sliding movement Z WANG ET AL interaction between the structural members of the struck ship, the overall crashworthiness of the side structure of the struck ship can be obtained by adding the resistance of the individual structural members of the side structure 3.1 Basic theory Figure Three typical collision positions Table Definition of collision positions in different collision scenarios Case Case Case Case Case Case Case Collision angle 30° 45° 60° 30° 45° 60° Impact starting position and passing position Position Position Position Positions and Positions and Positions and Jones (1972) summed up the simplified analysis method Søreide (1985) described the plastic mechanics theory that is called ‘upper bound theorem’, which can be used to obtain the resistance of the side structure under the impact of the striking bow, and the instantaneous resistance can be obtained by the following formula (Søreide 1985): F plastic · D˙ = E˙ m + E˙ b (1) where, Fplastic is the plastic force, D˙ is the striking velocity, the mem brane energy dissipation rate is E˙ m , and the bending energy dissipa tion rate is E˙ b (2) E˙ m = N0 1˙ avg dS S E˙ b = n M0 b˙ i li (3) i=1 where, the plastic membrane force is N0 , the average strain rate is 1˙ avg , the plastic bending moment is M0 , the curvature rate and the length of hinge number i are b˙ i and li , respectively s0 t (4) N0 = s0 t (5) M0 = where, the flow stress and the side plating thickness are s0 and t, respectively In oblique ship collision scenarios, the side plating, the web gir der, and the transverse frame are the main components to resist the impact Therefore, this paper studies the deformation mechanism of the side plating, the web girder, and the transverse frame in obli que collision scenarios 3.2 Deformation mechanism of the side plating in an oblique collision scenario Figure Definition of six oblique collision cases (a) Case 1: β = 30°; (b) Case 2: β = 45°; (c) Case 3: β = 60°; (d) Case 4: β = 30°; (e) Case 5: β = 45°; (f) Case 6: β = 60° of the striking bulbous bow The discovery of this characteristic helps us to make an assumption for the simplified analysis of struc tural deformation Therefore, an assumption is made as below based on the numerical simulation results, which is used for the simplified analysis in Section In the simplified analysis of damage deformation mechanism in oblique collision scenarios, it is assumed that the collision direction of the striking bow is unchanged during the collision process Therefore, the direction of the total collision force is consistent with the original collision direction Simplified analytical method The simplified analysis method can be used to quickly evaluate the crashworthiness of a struck ship Assuming that there is no According to the deformation mode of the side plating in numerical simulations, a theoretical deformation model of the side plating after oblique collision by a bulbous bow is proposed, and the theor etical deformation model is shown in Figure A rectangular plate with side lengths of 2a and 2b is struck by an ellipsoid-shaped bul bous bow, the equation of the ellipsoid-shaped bulbous bow is: x2 y2 z + + =1 lx2 ly2 lz2 (6) It is assumed that a rectangular plate is struck by an ellipsoid-shaped bulbous bow with an elliptic section, as shown in Figure The elliptic section is expressed as follows: x2 y2 + =1 lx2 ly2 (7) The deformation model of the rectangular plate is shown in Figure During the crushing process of the rectangular plate, SHIPS AND OFFSHORE STRUCTURES Figure Deformation model of the rectangular plate membrane energy dissipation rate is: ˙Em = N0 sideplate 2ab1 sin a1 a˙ + 2ab2 sin a2 a˙ cos2 a1 cos2 a2 N0 Figure A rectangular plate struck by an ellipsoid indenter where, N0 thickness sideplate sideplate = s0 (12) (13) is the plastic membrane force, is the side plating l1 )/tan a1 cos a1 l2 = lx sin b − (b2 − )/tan a2 cos a2 D = lx sin b − (b1 − (14) where, the side length b1, b2 of the plate can be expressed as: the strain of the left part of the plate can be expressed as: l1 − l1 /l1 = − cos a1 cos a1 (8) where, l1 is the instantaneous length of the left part of the deformed plate without bending, a1 is the instantaneous rotation angle of the left part of the deformed plate Therefore, the strain rate of the left part of the deformed plate is: sin a1 1˙ = a˙ cos2 a1 (9) Similarly, the strain of the right part of the deformed plate can be obtained as: 12 = − cos a2 (10) The strain rate of the right part of the deformed plate is: 1˙ = sin a2 a˙ cos2 a2 (15) b2 = x2 tan a2 + l2 cos a2 (16) where, x1, x2 are the coordinate values of the intersection point of the bending part and the straight part of the deformed plate in the x direction, as shown in Figure x1, x2 can be obtained as: ������������� tan2 a1 x1 = 1/ + (17) lx ly Figure A rectangular plate struck by an indenter (b1+ b2 = 2b) 11 = b1 = x1 tan a1 + l1 cos a1 ������������� tan2 a2 x2 = 1/ + lx ly (18) Then, the indentation is: ������������������������� cos4 a1 sin2 a1 cos2 a1 D = lx sin b + b1 tan a1 − 1/ + lx2 ly2 (19) The crushing velocity can be obtained as: − 1.5 sin 4a1 4cos3 a1 sin a1 ˙D = a˙ 1 cos a1 + sin a1 cos a1 − lx2 ly2 2ly2 lx2 b1 + cos a1 (20) (11) By substituting Equations (9) and (11) into Equation (2), the where, a˙ , a˙ are the angular bending rate, respectively Since the bending energy dissipation accounts for a small pro portion, there will be no big errors in predicting the total energy absorption without considering it Therefore, the instantaneous Z WANG ET AL resistance of the rectangular plate can be obtained as: Fp xoy = E˙ m D˙ Table Scantling of the finite element model sin a1 sin a2 a˙ 2ab + 2ab sideplate cos2 a1 cos2 a2 a˙ = − 1.5 cos4 a1 sin2 a1 cos2 a1 sin 4a1 4cos3 a1 sin a1 b1 + + − l2x l2y 2l2y lx2 cos2 a1 N0 (21) Similarly, when the equation of the cross section is: x2 z + =1 lx2 lz2 (22) The calculation method of the instantaneous resistance of rectangu lar plate is the same as the above method, which can be expressed as: Fp xoz = E˙ m D˙ sin u1 sin u2 u˙ 2ba1 + 2ba cos2 u1 cos2 u2 u˙ = − 1.5 cos u1 sin u1 cos u1 sin 4u1 4cos3 u1 sin u1 a1 + − + lx2 ly2 2l2y lx2 cos2 u1 N0 sideplate (23) Finally, a simplified linear method (Haris and Amdahl 2012) is used to obtain the instantaneous resistance of the rectangular plate impacted by an ellipsoid indenter: Fp = (F p xoy + Fp xoz ) (24) 3.2.1 The frictional force and the energy dissipation due to friction During the oblique collision process, friction occurs between the striking bow and the side plating of the struck ship The frictional force between the side plating and the bulbous bow can be expressed as: FS = m · Fp · sin b Value 1635 mm 6000 mm 450 mm mm 10 mm mm 3.3 Deformation mechanism of the web girder In order to research the deformation mechanism of web girders under different collision angles, the collision process of web girders is numerically simulated Three typical oblique collision angles are selected, and the finite element model of the bulbous bow striking the web girder under the scenarios of 30°, 45°, and 60° is established for numerical simulation The main data of the finite element model are listed in Table The finite element model, the defor mation model, and the deformation process of the cross-section of the web girder under the scenarios of 30°, 45° and 60° are illus trated in Figures 9–11 It can be seen from Figures 9–11 that the folding height ratio of web is similar at different oblique collision angles, so the web folding height ratio in the theoretical model is proposed based on the numerical simulation results In the oblique collision scenario, the impact force can be divided into a tangential force and a vertical force The vertical force leads to a two-dimensional folding deformation of the web, while the tan gential force has little influence on the folding deformation Based on the numerical simulation results of the web girder under different oblique collision angles, a theoretical deformation model of the web girder is proposed, as shown in Figure 12 When the web girder is struck by the indenter, the length of one side is b1, and the length of the other side is b2 = b−b1 Figure 13 shows the folding deformation process of the cross section of the web girder The middle wrinkle is five-thirds the height of the uppermost wrinkle in the first fold, which means BC = 5/3AB, when the crushing height is 8H, the first fold is completed The crushing height of the second fold is 6H The relationship between the crushing distance H’ in the oblique collision direction and the crushing height H in the vertical direction is: H = H ′ sin b (28) (25) The energy dissipation due to friction can be obtained as: Ef = (FS + Fp · cos b) · DsdS (26) S 3.2.2 Perforating model of the side plating With the increase of the indentation of the bulbous bow, the side plating will gradually enter into the ultimate bearing state, and the rupture will occur Although the resistance of the side plating will decrease obviously, it still has a certain bearing capacity For this reason, Wang et al (2000) put forward a formula for predicting the resistance of the plate after rupture by analysing the defor mation process of the plate, and the analytical calculation formula of the deformation resistance of the plate after rupture is as follows: 1.5 0.5 Item Deck height Overall length Width of side plating Deck plate thickness Side plating thickness Stiffener thickness 0.5 Fp = 1.51s0 t l n(sin((n − 2)p/2n)) ( tan w + m) (27) where n is the number of cracks after the plate ruptures, l is the length of each crack, w is the semiapex angle of the bulbous bow, m is the friction coefficient 3.3.1 Deformation mechanism of the first fold In the first fold, the crushing height increases from zero to 8H According to the assumption that 8H