An analytical method for predicting the structural response of ship side structures by bulbous bow in oblique collision scenarios

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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

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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

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

ARTICLE HISTORY

Received 9 November 2022 Accepted 11 June 2023

KEYWORDS

Oblique ship collision; bulbous bow;

crashworthiness; analytical method; numerical simulation

1 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

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

CONTACT Qing Li liqing5504@sjtu.edu.cn State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China https://doi.org/10.1080/17445302.2023.2247839

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 modellapply-ing 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 semi- numerical 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

Figure 1 Comparison results of force-indentation curves of the relative rigid and

rigid bows (Haris and Amdahl 2013 ).

2 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 2 shows the finite element models of the struck ship’s side

structure and the striking bow, Table 1 lists the details of the

main components of the struck ship The proposed Johnson-

Cook 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 2 and 3 (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 3 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

2.2 Collision scenario definition

Figure 3 is a schematic diagram of an oblique collision, defining the

oblique collision angle Figure 4 shows three typical collision

pos-itions, and the definition of six collision scenarios are shown in

Table 4 and Figure 5 The collision position is position 1 in case

1–3; in case 4–6, the collision position is position 2, as the collision proceeds, the striking bow will pass through position 3

2.3 The characteristics of oblique collision and assumptions

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

Figure 2 Finite element model of the side structure of the tanker and the bow.

Table 1 Main components and structural details of the tanker.

Spacing between longitudinal girders 7.2 m Spacing between transverse web frames 5.0 m

The thickness of outer shell plating 21 mm The thickness of inner shell plating 15 mm The thickness of web girder 12 mm The thickness of transverse web frame 14 mm

The thickness of the flange 13 mm

Table 2 GTN parameters.

0.3 1/857 1.5 1.0 2.25 0.01 0.03 0.03 0.3 0.1

Table 3 Johnson-Cook parameters.

2.45E + 08 5.001E + 08 0.016 0.915 0.221 10−3

Figure 3 Definition of the collision angle b

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 3

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

3 Simplified analytical method

The simplified analysis method can be used to quickly evaluate the

crashworthiness of a struck ship Assuming that there is no

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

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):

where, F plastic 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

˙E m=

􏽚

S

N0˙1avg dS (2)

˙E b=

􏽘n

i=1

M0b˙i l i (3)

where, the plastic membrane force is N0, the average strain rate is

˙1avg , the plastic bending moment is M0, the curvature rate and

the length of hinge number i are ˙b i and l i, respectively

M0=s0t2

N0=s0t (5) 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

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 6 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 bulbul-bous bow is:

x2

l2 +y2

l2

y

+z2

l2

z

It is assumed that a rectangular plate is struck by an ellipsoid-shaped bulbous bow with an elliptic section, as shown in

Figure 7 The elliptic section is expressed as follows:

x2

l2+y2

l2

y

The deformation model of the rectangular plate is shown in

Figure 8 During the crushing process of the rectangular plate,

Figure 4 Three typical collision positions.

Table 4 Definition of collision positions in different collision scenarios.

Case Collision angle Impact starting position and passing position

Case 1 30° Position 1

Case 2 45° Position 1

Case 3 60° Position 1

Case 4 30° Positions 2 and 3

Case 5 45° Positions 2 and 3

Case 6 60° Positions 2 and 3

Figure 5 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°

the strain of the left part of the plate can be expressed as:

11= l1

cos a1

− l1

/l1= 1 cos a1

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:

˙11= sin a1

Similarly, the strain of the right part of the deformed plate can be

obtained as:

12= 1 cos a2

The strain rate of the right part of the deformed plate is:

˙12= sin a2

By substituting Equations (9) and (11) into Equation (2), the

membrane energy dissipation rate is:

˙E m=N 0 sideplate 2ab1

sin a1

cos2a1a˙1+2ab2

sin a2

cos2a2a˙2

(12)

where, N 0 sideplate is the plastic membrane force, t p is the side plating thickness

D =l x sin b − (b1− l1

cos a1

)/tan a1

=l x sin b − (b2− l2

cos a2

)/tan a2 (14)

where, the side length b1, b2 of the plate can be expressed as:

b1=x1tan a1+l1cos a1 (15)

b2=x2tan a2+l2cos 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 8 x1, x2 can be obtained as:

x1=1/

�������������

1

l2+tan2a1

l2

y

􏽳

(17)

x2=1/

�������������

1

l2+tan2a2

l2

y

􏽳

(18) Then, the indentation is:

D =l x sin b + b1tan a1− 1/

�������������������������

cos4a1

l2 +sin2a1cos2a1

l2

y

􏽳

(19) The crushing velocity can be obtained as:

˙

D = ˙a1

􏼠 1 2

cos4a1

l2 +sin2a1cos2a1

l2

y

sin 4a1

2l2

y

− 4cos3a1sin a1

l2

+ b1

cos2a1

􏼡

(20) where, ˙a1, ˙a2 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

Figure 6 A rectangular plate struck by an ellipsoid indenter.

Figure 7 A rectangular plate struck by an indenter (b1+ b2 = 2b).

Figure 8 Deformation model of the rectangular plate.

resistance of the rectangular plate can be obtained as:

F p xoy=˙E m

˙

D

=

N 0 sideplate 2ab1 sina1

cos 2a1+2ab2 sina2

cos 2a2

˙

a2

˙

a1

1

2

cos 4a1

l2 + sin 2a1cos 2a1

l2

sin 4a1

2l2 − 4cos 3a1sina1

l2

+ b1

cos 2a1

(21)

Similarly, when the equation of the cross section is:

x2

l2+z2

l2

z

The calculation method of the instantaneous resistance of

rectangu-lar plate is the same as the above method, which can be expressed

as:

F p xoz=˙E m

˙

D

=

N 0 sideplate 2ba1 sinu1

cos 2u1+2ba2 sinu2

cos 2u2

˙

u2

˙

u1

1

2

cos 4u1

l2 + sin 2u1cos 2u1

l2

sin 4u1

2l2 − 4cos 3u1sinu1

l2

+ a1

cos 2u1

(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:

F p=1

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:

F S=m · F p·sin b (25) The energy dissipation due to friction can be obtained as:

E f =

􏽚

S

(F S+F p·cos b) · DsdS (26)

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:

F p=1.51s0t 1.5 l 0.5 n(sin((n − 2)p/2n)) 0.5( 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 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 5 The finite element model, the defor-mation model, and the defordefor-mation 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)

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 << (b1, b2), so the bending energy dissipation rate is:

˙E b=4M0(b1+b2) ˙a (29)

The geometric relation between the instantaneous indentation δ and the instantaneous rotation angle α can be expressed as:

The relationship between the striking velocity and the angular bending rate ˙a is:

˙

8H

������������������

1 − 1 − d

8H

Integrating the Equation (29) for the first folding process, α

increases from 0 to π / 2, the total bending energy dissipation can

Table 5 Scantling of the finite element model.

be obtained as:

E b=2pM0(b1+b2) (32) For the membrane energy dissipation, the minimum energy

dissi-pation is possible to be achieved when the plate is stretched in

length direction (Simonsen and Hasan 1999) When the first fold

is formed, the mean strain and strain rate of the outer fibre are:

1OA=1 2

d

b1

􏼒 􏼓2

(33)

˙1OA= d

The strain rate of other plastic hinge lines can be obtained as:

˙1OB= d

The mean strain rate in length direction over the entire height range

of 8H is:

˙1avg=11 16

d

Therefore, the total membrane energy dissipation rate is:

˙E m=11

2N0H

1

b1

+ 1

b2

When the first fold is formed (δ = 8H ), the total membrane energy

dissipation can be obtained by integrating the Equation (38):

E m=176N0H3 1

b1+

1

b2

(39)

The relationship between the bulbous bow displacement d′and the

instantaneous indentation δ is:

The instantaneous resistance of the web girder during the

Figure 9 Deformation of the web girder in numerical simulation (β = 30°) (a) Finite element model (b) Deformation model (c) Deformation process of the cross-section.

Figure 10 Deformation of the web girder in numerical simulation (β = 45°) (a) Finite element model (b) Deformation model (c) Deformation process of the cross section.

Figure 11 Deformation of the web girder in numerical simulation (β = 60°) (a) Finite element model (b) Deformation model (c) Deformation process of the cross section.

formation of the first fold can be expressed as:

F girder(d′) = ˙E b+ ˙E m

˙d

= s0t2(b1+b2)

8H′sin b

������������������

1 − 1 − d

′

8H′

􏽳

+11

2 s0tH

′d′sin2b 1

b1

+ 1

b2

(41)

where, 0 ≤ d ≤ 8H′

The mean resistance of the web girder during the formation of

the first fold is:

F m first=E b+E m

8H′

=ps0t2(b1+b2)

16H′ +22s0t(H′)2sin b 1

b1+

1

b2

(42) According to the upper bound theorem, when the mean resistance

is minimised, then the energy dissipation is minimised That is:

∂F m first

By substituting Equation (42) into Equation (43), H ’ can be

obtained as:

H′=0.165( b1b2t

sin b)

Therefore, the instantaneous resistance and the mean resistance of

the web girder during the formation of the first fold can be

obtained

3.3.2 Deformation mechanism of the subsequent folding

After the formation of the first fold, as the increase of the

indenta-tion, a second fold will be formed, the characteristic height of the

subsequent fold is 6H Similarly, the energy dissipation can be

divided into the membrane energy dissipation and the bending

energy dissipation

The first fold will be stretched and absorb energy during the

mation of the second fold, so the resistance formula during the

for-mation of the second fold still retains the membrane instantaneous

resistance formula of the first fold, the instantaneous resistance can

be expressed as:

F membrane(d′) =11

2s0tH

′sin2b(2d′− 8H′) 1

b1+

1

b2

(45)

In the second fold, the bending energy dissipation rate is:

E 2b=pM0(b1+b2) (46) The instantaneous resistance caused by bending deformation can

be expressed as:

F bending(d′) =ps0t

2(b1+b2)

The instantaneous resistance of the web girder during the

for-mation of the second fold is:

F girder(d′) = F membrane(d′) + F bending(d′)

=11

2 s0tH

′sin2b(2d′− 8H′) 1

b +

1

b

+ps0t2(b1+b2)

24H′sin b

Similarly, when the fold of the web girder upon the nth (n ≥ 2), the

instantaneous resistance of the web girder can be obtained as:

F girder(d′) =11

2 s0tH

′

sin2b(nd′− (n − 1)(3n

+2)H′) 1

b1

+ 1

b2

+ps0t2(b1+b2)

24H′sin b (49) The flange is usually fitted on the web girder, the resistance of the web flange is calculated by the beam theory (Hong and Amdahl

2008):

F flange(d′) = 4s0at f

b1+b2

where, a is half of the width of the flange plate, b1 + b2 is the length

of the flange plate, t f is the thickness of the plate

Ignoring the coupling between different structures, the total resistance of the web girder with flange plate can be expressed as:

F total(d′) = F girder(d′) + F flange(d′) (51)

3.4 Deformation mechanism of the transverse frame

When the bulbous bow strikes the transverse frame, the main struc-tural deformation mode of the frame is bending deformation There

is a collision angle between the bulbous bow and the transverse frame, so the collision force can be divided into a vertical force and a tangential force The deformation mode of the transverse frame subjected to oblique collision is similar to the deformation mode of the transverse floor of the double-bottom structure when the ship is grounded Therefore, on the basis of the research of Hong and Amdahl (2008) on the deformation of the transverse floor when the ship is grounded, the deformation model of the trans-verse frame in oblique collision scenario is proposed Assuming that the transverse frame is impacted by a linear load of length d, the tan-gential force causes horizontal displacement of the edge of the trans-verse frame, which is shown in Figure 14 The vertical force leads to a two-dimensional folding deformation, which is shown in Figure 15 The horizontal displacement can be obtained by the formula (Hong and Amdahl 2008):

u = 2h tan u (52)

where, u is the horizontal displacement of the edge, h is half of the vertical compression distance, θ is half of the compression wave

angle

where, δ is the indentation, b is the oblique collision angle (Hong

and Amdahl 2008)

The bending energy dissipation can be expressed as:

The indentation δ is 8H, so the membrane energy dissipation is:

E m=176N0d3 2

b+

d

b2

(56) The total energy dissipation can be obtained as:

E t=E b+E m=2pM0(d + 2b) + 176N0d3 2

b+

d

b2

(57)

Based on the upper bound theorem (Søreide 1985), when the

energy dissipation is minimised, that is:

∂E t

Substitute Equation (57) into Equation (58), d is:

d = ptb

3

The total energy dissipation is:

The relationship between the vertical force F T frame and the

tangential force F V frame is:

By integrating Equation (52) through Equation (61), we can obtain

the tangential force F T frame and vertical force F V frame:

u + d · tan b

=

ps0t2

2

pt

352(d ′ sin b)3b

3 +b

+ 176 · s0t(d′ sin b)3· ptb

352(d ′ sin b)3+

1

b

(1.0836d′sin b + 0.0652) · tan (0.47b − 0.0024b2 ) + d′· sin b · tan b

(62)

FV frame=FT frametan b

=

ps0t2

2

pt

352(d′sin b)3b

3 +b

+ 176 · s 0t(d′ sin b)3· ptb

352(d′sin b)3+

1

b

(1.0836d′sin b + 0.0652) · tan (0.47b − 0.0024b2 ) + d ′ · sin b · tan b

· tan b

(63)

Figure 12 A deformation model of the web girder after collision.

Figure 13 Folding deformation process of the cross section (the black dots

rep-resent the location of the plastic hinges). Figure 14 The cross section of transverse frame before and after collision.