The Dynamic Response of the Large Containership's Bow Structure

Proceedings of the Twenty-seventh (2017) International Ocean and Polar Engineering Conference San Francisco, CA, USA, June 25-30, 2017 Copyright © 2017 by the International Society of Offshore and Polar Engineers (ISOPE) ISBN 978-1-880653-97-5; ISSN 1098-6189

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The Dynamic Response of the Large Containership’s Bow Structure under Slamming Pressures Bin Yang and De-yu Wang State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University Shanghai, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration Shanghai, China tests were conducted to study the dynamic behavior of the slamming loads on ships; finally, the slamming loads which have ideal half sine or quadratic function shapes, were applied on a stiffened plate selected from the bow flare region on which slamming acting, then the dynamic response and buckling of the stiffened plate was investigated. For these three different research techniques, very apparent limitations can be also found. The first and second methods aim to obtain time and space characteristic of slamming loads and the last method aims to dynamic response and collapse behavior of local ship structures. Unfortunately, these two methods can not reflect the dynamic response and collapse behavior of the bow flare structure under slamming loads due to too much simplified of the calculation model. For the last method, although a 3-D model of bow structure was established, some important influence parameters of slamming loads were not taken into account.

ABSTRACT The dynamic behaviors of the large containership’s bow structures subjected to slamming pressures with different influence parameters are studied based on nonlinear finite element method. The slamming pressure is characterized as symmetrical and non-symmetrical half sine impact load, traveling over the side shell of the bow at a constant speed. The yield failure criterion is used to determine the critical buckling strength of the bow structure under slamming loads in the present paper. Various influence parameters which are relevant to the dynamic response of the bow structure are discussed, including the impact duration, rise time of slamming pressures, load attenuation coefficient, the location of maximum slamming pressures and traveling speed of the slamming pressures over side shell of the bow flare. It can be found that these influence parameters have a great influence on the dynamic response of bow structure, and the results contribute to help the naval architect to design a ship which has a stronger capacity of resisting slamming.

For the analysis of the water impact, lots of tests and numerical calculations have been conducted by previous scholars. De Backer et al. (2009) conducted a new experimental study of the water impact on hemisphere and cone shapes with larger deadrise angles, in order to assess the slamming loads. These experimental results are used as references to validate 3-D numerical studies, which follow earlier work in two-dimensions. Yang and Qiu (2012) studied slamming problems on two-dimension and three-dimension bodies with constant water entry velocities and free-fall motions based on the CIP method. Wang and Guedes Soares (2014) investigated the hydrodynamic problem of 3-D bodies, including hemisphere and cones with different deadrise angles by applying the explicit finite element method.

KEY WORDS: Slamming pressure; bow structure; maximum stress response; large Container ship; impact duration.

INTRODUCTION

In terms of the experimental study of the dynamic response of similar models under slamming loads, some important papers can be found. Hermundstad and Moan (2005) presented a method for the prediction of slamming loads on ship hulls and model test results for a 120-m Ro– Ro vessel is used for validation. In the method, the relative vertical and roll velocities are obtained from nonlinear strip theory, they are used as input parameters of a generalized two-dimensional Wagner formulation which can compute the slamming loads. Wang and Guedes Soares (2016) carried out experimental and numerical study of the bow flare and bottom slamming of a chemical tanker in irregular waves, slamming occurrence probability and slamming loads were calculated. Hong (2014) investigated the characteristics of bow flare slamming, and the spatial and temporal distributions of slamming force are discussed detailed in regular and irregular waves. The main aim of these papers is to explain the dynamic behavior of slamming loads appeared on bow flare when the ship is encountering an extreme wave. Very few papers study the dynamic response of large containership’s

In recently decades, with the development of global trade and the increase of shipping volume, the size of a container ship is getting larger. As a result, the large container ship usually has larger bow flare and stern areas to carry more cargos without decreasing the sailing speed. However, the larger bow flare and stern increases the probability and extent of slamming phenomenon, which results in higher risk of slamming failure. It was reported that slamming induced whipping can lead to local structural damage of ship (Storhaug, 2009). Therefore, it is necessary to take into account dynamic response of large container ships under slamming loads. In terms of bow flare slamming, there are three main research directions: firstly, some researchers simplified the bow flare as a 2-D or 3-D wedge which has a similar dead rise angle, and then a series of water impact experiments and numerical calculations was carried out to study the slamming loads on the wedge; secondly, similarity model

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bow structures under slamming loads. Therefore, enough attention should be paid to dynamic response analysis of bow structures. Some researchers select a stiffened plate from bow as study model, and the dynamic response analysis of the simplified model was studied. The ideal half-sine impact loads and triangular loads which are similar with the slamming loads are applied on the stiffened plates and flat plates to study their dynamic response. Srivastava, Datta and Sheikh (2003) investigated the dynamic characteristics of stiffened plates subjected to non-uniform harmonic in-plane edge loading by using finite element method. Paik (2003) conducted a series of dynamic collapse tests on flat plates under in-plane compressive loads and the effect of strain rate on the ultimate strength of ship plates is discussed. Ji and Wang (2014) studied the influence of impact loads shape on dynamic displacement response of across-stiffened plate subjected to in-plane impact loads based on Abaqus/explicit FEM code.

engineering problems.

NUMERICAL STUDY 10000TEU Container Ship Model Page Size The main aim of this paper is to compute the nonlinear dynamic response of a large containership’s bow flare subjected to slamming pressures. In this paper, the bow structure of a 1000TEU container ship is used as a numerical calculation model. The principal dimensions of a 1000TEU container ship are summarized in Table 1. Table 1 Principal dimensions of the 1000TEU container ship Item Length over all Length of freeboard: Length between perpendiculars: Breadth (Moulded): Depth (Moulded): Design Draught (Moulded): Lightship Weight (assumed):

Due to the complexity of bow structure, studying the dynamic behavior of the bow by using simplified stiffened plates or flat plates cannot explain the actual response of bow subjected to slamming loads. It is necessary to establish an explicit 3-D FE model of the bow structure to obtain the accurate dynamic response of the bow under slamming loads. Some researchers have done some researches on the dynamic behavior of the containership’s bow flare under slamming loads by using 3-D nonlinear finite element analysis. Yang (2012) established an explicit 3D finite element of 1700TEU containership’s bow structure model in MD Nastran, then the dynamic response and buckling of the bow subjected to slamming loads were investigated, one-time thickness deformation method and yielding failure method were used to accurately determine the critical buckling load of the bow structure subjected to slamming loads. Ren (2015) investigated the dynamic response of bow flare structures under slamming loads of the design condition. Firstly, a design wave is determined through a long-term analysis of the relative vertical motion between the hull and the wave; secondly, bow flare slamming pressure of the prescribed section motion is predicted by LS-DYNA software; finally, taking into account the geometric and material nonlinearity, dynamic response of the bow flare structure under slamming loads is calculated. Although the dynamic response of bow structure under slamming loads were evaluated by these papers, some influence parameters which have a great influence on the dynamic response of bow structures are are not discussed, such as amplitude of slamming loads, pulse duration, rise time of slamming loads, loads shape, loads attenuation coefficient, the occurrence location of maximum slamming loads and traveling speed of the loads over side shell of the bow.

Symbol L1 L2 L

Unit m m m

Value 337.0 321.8 320.0

B D d

m m m

48.2 27.2 13.0

M

t

38220.0

Slamming region and 3-D FE Model The bow flare slamming pressure is to be calculated for the side shell structure above the waterline in the area forward of 0.25L from the FP, as shown in Fig. 1 (LIoyd’s Register, (2009)). For bow flare slamming, the forebody structure from the watertight bulkhead to bow, that is the part-cabin of 0.3L region after the forward perpendicular, is to be modeled. Fig. 2 shows the 3D FE model of bow structure and slamming loads region. The model is about 100 meters long and consists of a 2-D plate and 1-D beam elements.

The present work focuses on the analysis of the dynamic response of bow structures under slamming induced loads which are given by simplified half-sine loads with millisecond order duration. Firstly, the slamming pressure which has a half sine wave shape and the millisecond order duration is presented, then the impact loads considering various influence parameters are applied on the 3-D FE model of the bow structure, finally, the dynamic response of the bow structure are studied based on 3-D nonlinear finite element method. The residual deformation of the bow structure would generate when the maximum stress reached the yield stress during slamming. The residual deformation weakens the material strength and rigidity seriously and may lead to the ship collapse under next slamming loads. Therefore, the yield failure criterion is used to determine the critical buckling strength of the bow structure in the present paper. The effects of various parameters, including the amplitude of slamming loads, pulse duration time, load shape, load attenuation coefficient, the occurrence location of maximum slamming loads and traveling speed of the slamming loads, are discussed in detail. The results obtained from the present paper offer an enormous amount of available information for the practical

Fig. 1. Extent of hull structure for slamming load prediction.

Fig. 2. Boundary conditions and slamming pressures region for FE model of bow structure.

Boundary Constraints

274

slamming loads can be given as tp= 0.14(s). Luo et al. (2012) indicated that the slamming load duration is a very important variable, and the slamming load duration changes with the sea state, navigational speed, geometry of the ship, etc. In order to investigate the influence of the slamming duration on the dynamic response of the bow structure systematically, the values of the slamming durations vary from 10ms to 1000ms.

To evaluate the effects of slamming on the dynamic response of the bow, the appropriate boundary constraints needs to be applied. Fig. 2 shows the boundary conditions that may be applied to the FE model of the bow. In the case of a symmetric structure, symmetric boundary conditions are applied on the centerline plane. The points A and B at the bottom of the after side of the model were fixed in three displacement degrees of freedom to ensure that all forces can be balanced. The boundary conditions of the model are summarized in Table 2. Due to the long enough FEM of the part cabin, so the boundary constraints at the points A and B have a little influence on the dynamic response of the bow structure under slamming pressures.

Slamming Loads Shape Many papers have demonstrated that it is reasonable to simplify the slamming loads as pulse loads with a half sine shape (Yang and Wang, 2016a, 2016b). The corresponding half sine function f1(t) is selected as load shape function, shown in Eq. (3). Then, slamming pressure P(t) is expressed as f1(t) multiplied by Pmax, shown in Eq. (4). The time history curve of slamming pressure is shown in Fig. 3.

Table 2 Boundary conditions of the calculation model. Location Points A and B

Translations Ux

Uy

Uz

0

0

0

Centerline plane

0

Rotations URx 0

URy

URz

 t  f1  t   sin   t   p 

(3)

0

 t  P  t   Pm f1  t   Pmax sin   t   p 

(4)

Slamming Pressure Amplitude On the basis of experimental and theoretical analysis (Stavovy and Chuang, 1976), the slamming pressure is heavily influenced by the geometrical shape of slamming region of bow structure such as bow flare angle and deadrise angle impact velocity etc. Von Karman (1929) was one of the first researchers who studied the hydrodynamic impact problem for seaplane landing. This work was followed by Wagner (1932) who proposed Wagner’s prediction formula based on momentum conservation theory. Wagner’s prediction formula is shown as follow:

Pmax 

 1 2  V 2 1   2 2  4 tan  

(1)

where Pmax is the maximum slamming pressure, V is the initial vertical slamming velocity, θ is the deadrise angle of the bow. According to some experimental study (Luo, Wang and Guedes Soares, 2012) on water impact of ship structures, the initial vertical impact velocity changes from 1m/s to 10m/s, and the deadrise angles of the boats of interest are in the range 5° to 45°, and the initial vertical slamming velocity is in the range 1m/s to10m/s. In this paper, in order to obtain conservative results, the values of vertical velocity and deadrise angles are determined to obtain the larger slamming amplitude which is usually produced under severe sea condition. The parameters applied in the paper are V=10m/s; θ=30°-60°.The maximum slamming pressure which is used in the present paper has a range from 0.9×10 5 to 4.197×105Pa according to the Eq. (1). Pmax =4.197×105Pa is selected as a calculating parameter in the paper.

Fig. 3. Time history of the slamming pressure acted on the bow. The whole slamming process can be divided into two phases, load increasing and decreasing. In many cases, the duration of slamming load increasing which is called rise time is not equal to the duration of load decreasing. Therefore it is necessary to consider the influence of rise time on the dynamic response of the bow structure. The ratio of rise time and slamming duration, et, is defined as Eq. (5). When analyzing the influence of et on the dynamic response of the bow structure, nonsymmetrical half sine function f2(t) is selected as shape function of slamming pressure, shown in Eq. (6) (Ji and Wang, 2014). Accordingly, slamming pressure P(t) is described as f2(t) multiplied by Pmax, shown in Eq. (7).

Slamming Loads Duration For the time history of the slamming loads, Ochi and Motter (1973) proposed a simple formula to evaluate the slamming loads duration. They assumed that the time history of the slamming loads had a triangular varying from 0 to the maximum value and to 0 again during a period tp. According to experimental test and Froude scale law, the relationship between the period and the ship length was described as:

tp 

7.94 L 103

( s)

et 

(2)

Substituting the value of ship length into Eq. (2), the duration of

275

trise t  rise , trise  tdecline tp

t p  trise  tdecline

(5)

  t  sin     2et t p  f2 t        t  et t p    sin  2 1  e t  2  t p     P  t   Pmax f 2  t 

0  t  e t  t p

(6)

e t

t p

 t  tp  (7)

Spatial Distribution of Bow Flare Slamming Load

Fig. 4. The slamming region and the local coordinates system.

Some references (Hong et al., 2014; Yang et al., 2012) indicated that the amplitude of slamming loads at the different regions of the bow is different. In general, the amplitude of slamming loads gradually decreases from the position of the maximum slamming loads toward around. The position the maximum slamming load occurring changes with the sea state, navigational speed, geometry of the ship etc, so it is impossible to predict the position of the maximum slamming loads. In this paper, it is assumed that the coordinate of the maximum slamming pressure is (xmax, ymax), and the amplitude of slamming pressure decreases linearly from the maximum pressure along two directions. The region of slamming and local coordinates system is shown in Fig. 4. In the local coordinate system of the FE model, the origin denoted by O is on the left bottom of the slamming region, The X-axis runs along the direction of the ship length, the Y-axis lies along the direction of the shipping height. The spatial distribution function of slamming pressure action on the bow flare is shown in Eq. (8). Fig. 5 shows pressures distribution along the Y direction which is similar with along the X direction.

k1 x  k2 y  P0  k x  k y  2k y  P  2 2 0 0 P  x, y    1  k x  k y  2 k x  P 2 1 0 0  1   k x  k y  2 k x  2 k2 y0  P0 2 1 0  1

Fig. 5. Spatial distribution of slamming pressures on the bow flare.

The Durations of Slamming Loads Traveling over Bow Side

x  x0 , y  y0 x  x0 , y  y0 x  x0 , y  y0

It will be known that the slamming pressure cannot be applied to the region of bow structure simultaneously. Firstly, the bottom region of the bow subjected to slamming loads, and then slamming loads travels over the side immediately. The American Bureau of Shipping (American Bureau of Shipping, (2011)) suggested a simultaneous load factor which is used to compute the spatial distribution of instantaneous slamming pressures acting on the region of interest of the hull by multiplying the calculated maximum slamming pressure. However, the method suggested by American Bureau of Shipping regards the slamming process as quasi-static, so it can not reflect the actually dynamic behavior of slamming phenomena. The dynamic responses of boat bottom plates subjected to the slamming pressure, which is modeled as a high-intensity peak followed by a lower constant pressure and travels at a constant speed along the bottom plates, are studied analytically by Faltinsen (2005). The speed of the slamming load traveling over the bottom, c, is described as:

(8)

x  x0 , y  y0

where, P(x, y) is the spatial distribution of slamming pressure. P0 is the slamming pressure at the four edges of slamming region. xmax and ymax are the abscissa values and ordinate values of the slamming pressure peak, respectively. k1 is the growth rate of slamming loads along X direction from Y axis (x=0) to the x=xmax, k2 is the growth rate of slamming loads along Y direction from X axis (y=0) to the y=ymax. (9) k1  2  Pmax  P0  L

k2  2  Pmax  P0  H

(10)

where L and H are the length and width of the region of slamming, respectively. Pmax is the maximum slamming pressure amplitude acting on the bow flare. To study the influences of the position of the maximum slamming pressures on the dynamic response of bow structure, the following dimensionless quantities are introduced.

x0 

xmax , L

y0 

ymax H

c

V 1 V   2 tan  cos  2sin 

(12)

It is assumed that the c keep constant during the water entry. The time interval between the time of contacting water and time of the bow entry the water completely is described as:

(11)

T0 

The values of x0=xmax/L and y0=ymax/H were set as 0.1, 0.25, 0.5, 0.75 and 0.9. The value of k1 depends on k2 are 12.48Pa/mm and 18.66Pa/mm, respectively.

H c

(13)

Substituting the values of V and θ into Eq. (12), the traveling speed of the slamming load c can be evaluated and then the time interval T0=0.7s according to Eq. (13). Due to the irregularity of the cross section curve of the bow and unbalanced forces in the vertical direction, the vertical slamming velocity cannot keep same all the time after entering the water, so the c also vary every second and T0 is not a constant. However, considering the changing of vertical velocity will greatly increase the amount of calculation. Therefore, in the present paper, it is assumed that the vertical velocity which was used in each calculation is

276

constant, the influence of different time interval Td which represents the velocity of water entry on the dynamic response of the bow structure under slamming loads are investigated.

0.1T0

0.629

0.734

0.830

0.797

0.728

0.785

0.823

0.3T0

0.722

0.533

0.473

0.741

0.702

0.817

0.820

0.5T0

0.931

0.651

0.364

0.484

0.710

0.813

0.814

COMPUTING RESULTS

0.7T0

0.772

0.809

0.580

0.403

0.691

0.781

0.805

0.9T0

0.451

0.773

0.759

0.368

0.665

0.688

0.681

1.0T0

0.703

0.818

0.733

0.325

0.639

0.772

0.785

1.3T0

0.809

0.480

0.410

0.356

0.559

0.685

0.687

1.5T0

0.527

0.426

0.512

0.360

0.521

0.677

0.688

1.7T0

0.599

0.736

0.570

0.453

0.496

0.665

0.686

2.0T0

0.546

0.652

0.451

0.420

0.458

0.632

0.676

Influence of Slamming Loads Duration on Dynamic Stress Response The nondimensional maximum stress response (σmax/σy) of bow under slamming loads with different impact duration and traveling time are summarized as follows in Table 3. σy is the yield strength of the material used in the large container ship and the value is 313.6MPa. The impact duration tp increases from 10ms to 1000ms, and the duration of slamming load traveling over bow side increases from 0.1T0 to 2T0 with x0=y0=1/2. As shown in Table 3, the results show that the maximum stress response of bow structure under slamming pressure changes with the varying of the slamming loads duration, but there is no reasonable change rule can be concluded. Therefore, the main aim of this paper is a deeper analysis of the occurrence rule of maximum stress. The maximum stresses in each row and in each column are highlighted in bold italic and in underline below. It is can be concluded that the duration of the slamming pressure which causes maximum stress for each row is less than 50ms, and the maximum stress changes a little when the duration of slamming pressure is more than 800ms. These results are opposite to the results in some references (Yang and Wang, 2016a, 2016b), which pointed out that the shorter duration of impact load, the higher is the dynamic buckling load at the same impact peak. The reason cause to this may be that the duration of slamming pressure approaches the natural frequency of local area in bow structure or is multiple of its. Therefore, it is a good way to keep the safety of ships traversing large waves to take some measures to make the impact duration larger than tp=50. For the each column in Table 3, the maximum stress usually happens when Td≤1.0T0, and the stress with Td=0.1T0 is maximal when tp≥50ms. Fig. 6 shows six history curves of maximum stress with tp=400ms, 800ms, 1000ms, Td=0.5T0, 1.0T0. As shown in Fig. 6, the stress response peak occurs before the slamming ends for the last five curves; on the contrary, the stress response peak occurs after the slamming ends for the first curve. There are no apparent fluctuations after slamming for these cases.

Influence of the Time of Slamming Pressures Traveling over Bow Side Influence of the duration of slamming load on the maximum stress of bow structure is shown in Table 3. The maximum stress in each column is highlighted in underlining. Two important rules found in Table 3 are that the maximum stress occurs for the case Td=0.1T0 when the impact duration tp≥50ms, and the maximum stress decrease with the increase of the time of slamming load traveling over the bow side when the impact duration tp≤200ms, respectively. However, it is hard to predict occurrence rule of maximum stress when the impact duration tp≤50ms, but it usually occurs when Td=0.1T0-1.0T0. Therefore, modifying the side shell of the bow appropriately to extend the time of slamming load traveling over the bow side is very useful to enhance the capacity of resisting slamming. The maximum value in Table 3 is 0.931 which is very close to the yield stress, the corresponding impact duration and traveling time are Td=0.5T0, tp=10ms. Naval architects should pay much attention to this case. If necessary, we can strength the area of the maximum stress occurs without increasing the difficulty of processing and assembling.

Influence of Ratio of Rise Time, et, on Dynamic Response of the Bow Structure Ji and Wang (2014) indicated that the rise time has an important role on dynamic response and buckling of stiffened plate. Due to the value of tp computed from Eq. (2) is equal to 140ms, the two most common cases in Table 3 which ships at sea usually encounter, are tp=100ms, Td=1.0T0 and tp=200ms, Td=1.0T0, respectively. The influence of the ratio of rise time on the dynamic stress response of the bow flare is studied under the two most common cases. Fig. 7 presents time history curves of maximum stress with 9 et values, et vary from 0.1 to 0.9, under the two most common cases, respectively. As shown in Fig. 7, both of the two curves have the same variation trend, it is observed obviously that the curve for the case tp=200ms is above on another one, which means it has a larger stress response. It has a minimum stress when et=0.5, of which load shape is same with the load shape described in Eq. (5). It is apparent that the maximum stress appears when et=0.1 for the two cases. The difference between the maximum and minimum stress for the case tp=100ms and tp=200ms are 0.191σy and 0.107σy. It can be concluded that the influence degree of rise time on the dynamic stress response increase with the decrease of the impact duration. It also can be found that the descending speed of the maximum stress from et=0.1 to et=0.5 is much larger than the case from et=0.9 to et=0.5. The maximum and minimum difference value between the two curves which appears when et=0.5 and et=0.1 are 0.137σy and 0.053σy, respectively. It means that the influence degree of impact duration on dynamic response increase with the increase of the rise time. It is

Fig. 6. Time history curves of maximum stress with et=0.5, x0=y0=0.5, k1=1. Table 3 The nondimensional maximum stress response of bow structure under different tp and Td with et=0.5, x0=y0=0.5. tp

Td

10

30

50

100

400

800

1000

277

evident that the variation of et has a little influence on the dynamic response when et vary from 0.375 to 0.75.

For the analysis of the influence of impact durations and traveling speed of slamming pressure on the dynamic response, the duration of the slamming pressure which causes maximum stress for each row is less than 50ms, and the maximum stress changes a little when the duration of the slamming pressure is more than 800ms. It has been proven that traveling speed also has a great influence on the dynamic response of bow structure under slamming pressure, the stress response at Td=0.1T0 is maximal when tp≥50ms. For the analysis of the influence of rise time, et, on the dynamic response of the bow structure, the stress response is minimum when et=0.5, which means the symmetric load shape leads to smaller stress response comparing with the unsymmetrical load. The stress response is maximal when et=0.1. The difference between the maximum and minimum stress for the case tp=100ms and tp=200ms are 0.191σy and 0.107σy. It implies that the influence degree of rise time on the dynamic stress response increase along with the decrease of the impact duration. The influence degree of impact duration on dynamic response increases with the increase of the rise time. The stress response changes a little when et vary from 0.375 to 0.75.

Fig. 7. Dimensionless maximum stress versus rise time with Td=1.0T0, x0=y0=0.5.

Influence of the position of maximum slamming pressure The position of the maximum slamming loads acting on side shell of the bow region can be represented as x0=xmax/H and y0=ymax/Y. The influences of the coordinate position of maximum slamming pressure on the dynamic stress response are shown in Table 4. Although the change of the maximum stress with x0 or y0 is irregular, they have same variation trend for every row or every column. It can be found that the maximum stress in bow structure happens at x0=0.25 for each row, and for each column, the maximum stress happens at y0=0.75. The minimum stress in bow structure happens at x0=0.10 for each row, and for each column, the minimum stress happens at y0=0.25. The maximum stress in Table 4 is more than one and a half times the minimum stress, which explains that the position of maximum slamming pressure has a great influence on the dynamic stress response of the bow structure. It can be concluded that when the slamming pressure amplitude occurs the upper-left of slamming region, the slamming pressure will lead to the largest stress response.

The position of the maximum slamming pressure is an important influence parameter for the dynamic response of bow structure under slamming pressure, the stress response for each row or each column has same variation trend. The maximum stress response happens at x0=0.25 and y0=0.75 for each row and each column, respectively.

Table 4 The maximum nondimensional stress response of bow structure under different x0 and y0.

ACKNOWLEDGEMENTS

x0

0.10

0.25

0.50

0.75

0.90

0.10

0.433

0.545

0.445

0.450

0.504

0.25

0.410

0.495

0.432

0.423

0.475

0.50

0.533

0.565

0.500

0.437

0.480

0.75

0.604

0.625

0.562

0.511

0.536

0.90

0.520

0.543

0.501

0.458

0.485

y0

Load attenuation coefficient also plays an important role in the dynamic response of bow structure under slamming pressure. The stress response of bow structure linearly decreases with the increasing of the load attenuation coefficient. The longer impact duration leads to larger stress response due to larger total energy acting on the bow structure. The results obtained from the present paper are very useful in the field of shipping engineering and contribute to help the naval architect to design a ship which has a stronger capacity of resisting slamming.

The work presented in this paper has been carried out under the cosupport provided by the Ministry of Education and Ministry of Finance of China (Grant No. 201335), and by NSFC(51239007). The authors would like to acknowledge the co-support. The present work is supported by the Chinese Government Key Research Project KSHIP-II Project (Knowledge-based Ship Design Hyper-Integrated Platform) No 201335. The authors would like to acknowledge the Project support.

REFERENCES American Bureau of Shipping (2011). Guide for slamming loads and strength assessment for vessels. Backer G D, et al. (2009). “Experimental investigation of water impact on axisymmetric bodies,” Applied Ocean Research, 31(3), 143-156. Faltinsen O (2005). “Hydrodynamics of high-speed marine vehicles,” Cambridge University Press. Hermundstad O A and Moan T (2005). “Numerical and experimental analysis of bow flare slamming on a Ro–Ro vessel in regular oblique waves,” Journal of Marine Science and Technology, 10(3), 105-122. Hong, SY, Kim, KH, Kim, BW, Kim, YS (2014). “Experimental Study

CONCLUSIONS In this paper, the dynamic response of large containership’s bow structure under slamming pressure is numerically investigated, by applying a 3-D nonlinear finite element method. The maximum stress used to evaluate the safety margin is selected as the main detected dynamic response of the bow structure. Various parameters affecting the dynamic response of the bow structure have been investigated. A multitude of available results are obtained as follow:

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on the Bow-Flare Slamming of a 10,000 TEU Containership,” Proc 24th Int Offshore and Polar Eng Conf, Busan, Korea, 816-823. Ji Z H, Wang D Y (2014). “Influence of load shape on dynamic response of cross-stiffened deck subjected to in-plane impact,” Thin-Walled Structures, 82, 212-220. LIoyd’s Register (2009), “Rules and Regulations for the Classification of Naval Ships”. Luo H, Wang H and Soares C G (2012). “Numerical and experimental study of hydrodynamic impact and elastic response of one free-drop wedge with stiffened panels,” Ocean Engineering, 40(2),1-14. Ochi, M.K. and Motter, L.E (1973). “Prediction of slamming characteristics and hull response for ship design,” Transactions SNAME, 81, 144-190. Paik J K, Thayamballi A K (2003). “An experimental investigation on the dynamic ultimate compressive strength of ship plating,” International Journal of Impact Engineering, 28, 803-811. Ren HL, Yu PY and Wang Q (2015). “Dynamic Response of the Bow Flare Structure Under Slamming Loads,” 34th International Conference on Ocean, Offshore and Arctic Engineering, Newfoundland, Canada, V003T02A014. Srivastava A K L, Datta P K, Sheikh A H (2003). “Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading,” Journal of Sound & Vibration, 262(5), 1171–1189. Stavovy A B and Chuang S L (1976). “Analytical determination of slamming pressures for high-speed vehicles in waves,” Journal of Ship Research, 20(4). Storhaug, G. (2009). “The 4400TEU container vessel MSC Napoli broke its back, but did whipping contribute?,” Hydroelasticity in Marine

Technology, 233-243. Von Kármàn, T (1929). “The Impact on Seaplane Floats during Landing,” Technical Report, NACA TN, 321. Wagner H (1932). “Über Stoß‐und Gleitvorgänge an der Oberfläche von Flü ssigkeiten,” ZAMM ‐ Journal of Applied Mathematics and Mechanics/Zeitschrift f ü r Angewandte Mathematik und Mechanik, 12(4), 193-215. Wang S and Soares C G (2014). “Numerical study on the water impact of 3D bodies by an explicit finite element method,” Ocean Engineering, 78(3), 73-88. Wang S and Soares C G (2016). “Experimental and numerical study of the slamming load on the bow of a chemical tanker in irregular waves,” Ocean Engineering, 111,369-383. Yang B and Wang D Y (2016a). “Buckling strength of rectangular plates with elastically restrained edges subjected to in-plane impact loading,” ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science, 203-210. Yang B and Wang D Y (2016b). “Dynamic buckling of stiffened plates with elastically restrained edges under in-plane impact loading,” ThinWalled Structures, 107,427-442. Yang Q and Qiu W (2012). “Numerical simulation of water impact for 2D and 3D bodies,” Ocean Engineering, 43(4), 82-89. Yang S H, Lee Y J and Chien H L (2012). “Dynamic Buckling Strength Assessment of Containership’s Bow Structures Subjected to Bow Flare Slamming,” ASME 2012 International Mechanical Engineering Congress and Exposition, 727-733.

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