Investigation into the dynamic collapse behaviour of a bulk carrier under extreme wave loads

Ocean Engineering 106 (2015) 115–127

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Investigation into the dynamic collapse behaviour of a bulk carrier under extreme wave loads Weijun Xu n, Wenyang Duan, Duanfeng Han College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 11 December 2014 Accepted 6 July 2015

The aim of this paper is to quantitatively evaluate the extent of collapse of a bulk carrier when the ship is subjected to extreme wave loads. A hydro-elastoplasticity theory, which was proposed by the present authors and takes into account the interaction between the large elasto-plastic deformation and the wave load evaluation, is applied to the ship’s structure with the assumption that a plastic hinge is formed in the midship region when the hull girder collapses in extreme wave conditions. The dynamic response of the hinge can be expressed by the relationship between the vertical bending moment and the curvature, which are obtained using nonlinear Finite Element Analysis (FEA). A comparative correct moment–curvature curve and a reasonable load evaluation are necessary for prediction of the severity of the collapse for the actual ship. A bulk carrier hull model with one frame space is constructed and analysed using an arc-length control method (Riks method). The geometric nonlinearity resulting from large deformations and the material nonlinearity are taken into account. The presence of an initial imperfection is considered using the consistent imperfection mode method in the FEA. A prediction of the extent of collapse for a bulk carrier subjected to an extreme wave load is carried out using the hydroelastoplasticity approach. This analysis clarifies the extent to which the hull girder may collapse in extreme wave conditions at an exceedance probability of 1/1000 in several short-term sea states. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Dynamic collapse behaviour Hydro-elastoplasticity approach Severity of collapse Extreme waves Exceedance probability Bulk carrier

1. Introduction The capacity of a ship’s hull girder to withstand a longitudinal bending moment must be addressed to assess the hull girder’s ultimate strength, and many researchers have devoted efforts to the study of the hull girder capacity. An evaluation of the post-ultimate strength behaviour can be estimated using a hierarchy of methods, including (a) simple “closed-form” formulations initially proposed by Caldwell (1965), (b) Smith methods based on Navier’s hypothesis and the average stress–average strain relationship for individual panels proposed by Smith (1977), (c) full nonlinear FEA first attempted by Chen et al. (1983) following attempts by Kut et al. (1985) and Valsgård et al. (1991), and (d) simplified FE methods such as the Idealized Structural Unit Method (ISUM), which were initially proposed by Ueda and Rashed (1984) to perform progressive collapse analysis on the transverse frame of a ship structure. In recent years, significant efforts have been devoted to methods (a), (b) and (d). Mansour and Thayamballi (1980) applied the simple “closed-form” formulations to analyse the limiting condition when the ship’s hull girder failed to perform its function, and the ultimate strength of the hull girder was determined. In this research, the

n

Corresponding author. Tel.: þ 86 451 8251 9902; fax:þ 86 451 8251 8443. E-mail address: [email protected] (W. Xu).

http://dx.doi.org/10.1016/j.oceaneng.2015.07.006 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

buckling and instability of the hull-stiffened plates, the fully plastic yield moments, and the shakedown moments were developed. Yao and Nikolov (1991, 1992) proposed a simple and practical analytical method to simulate the progressive collapse behaviour of ship’s hull subjected to longitudinal bending based on Smith’s method for estimating the load-carrying capacity of a ship’s hull, including the post-ultimate strength behaviour. More recent developments may be found in Yao et al. (2006). The ISUM rectangular plate and stiffened panel elements were developed by Ueda et al. (1995) and were implemented in ALPS/ISUM by Paik et al. (1996) based on the effective width concept. A new concept of the ISUM plate element that introduced the shape functions for deflection was proposed by Masaoka et al. (1998) and was further extended to the ISUM stiffened panel model by Fujikubo and Kaeding (2002). New ISUM elements have been developed by Fujikubo et al. (2002) to analyse the collapse behaviour for the double bottom structure in ships under combined thrusts and lateral pressure. The fundamental collapse modes and localised failures in the double bottom structure can be predicted with these elements, and the newest versions of the ISUM stiffened panel models have been successfully applied by Pei and Fujikubo (2005) to the analysis of progressive collapse of hull girder cross-sections. However, fewer studies are reported regarding the use of method (c), perhaps because the approach requires significant computational and time resources. The rapid development of computer technology has enabled the use of nonlinear finite

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element analysis to obtain accurate results within reasonable computation times together with proper modelling of the ship’s hull. Amlashi and Moan (2008) carried out the calculation in terms of the (1/2 þ1 þ1/2) hold tanks of the Capesize Bulk Carrier under double bottom bending using method (c). Shu and Moan (2009) extended the model to three cargo holds and four transverse bulkheads in the midship region to investigate the ultimate strength of the Capesize Bulk Carrier. The effect of the local lateral pressure loads was considered under hogging and alternate hold loading conditions, and the calculation results were compared with those from simplified methods. The evaluation of the ultimate strength of a hull girder is of undoubted importance for the safety of the ship, but from the viewpoint of risk-based design, the severity of hull girder collapse under wave conditions is also important because it is directly related to the consequence of the collapse. The risks may include the loss of the ship itself, the loss of cargo and lives, the occurrence of oil pollution, etc. Iijima and Fujikubo (2010) studied the target safety level of a ship’s hull girder based on risk optimisation, and the conclusion showed that the target safety level is affected by the scope of the risk model. Therefore, studies should focus on the extent of the failure with the increase of interest in risk-based design. The progressive collapse behaviour of a ship’s hull girder is commonly analysed in a quasi-static manner over a cross-section via displacement control, i.e., with monotonously increasing curvature. However, as Lehmann (2006) noted, the imposed forced displacement/rotation does not represent an actual loading condition, and the distributed pressure on a hull girder is the real load that acts on a ship. He stated that when the moment exceeds the maximum sustainable value, the calculation based on the input of curvature and the program that calculates the associated moment are not realistic, and in general, there is a dynamic snap-through effect to the next static equilibrium condition, a phenomena that indicates that the collapse behaviour of ship’s hull girder is so complicated that it is not decided by pathways or curvatures but by forces or moments. Instead, the behaviour can be followed only if the interaction between the collapsing structure and the loads is considered, as in Yao et al. (2009). The collapse analysis was performed under load control in Phase 1. However, the postultimate strength could not be completely solved because the load–deformation interaction has not yet been fully considered in the studies. Hydro-elastoplasticity is a research field in which the interaction between the fluid and the structure is considered in a range that includes the plastic deformation as well as the elastic deformation, as proposed by Kimura et al. (2010) and Xu et al. (2011a) for predicting the dynamic collapse behaviour of a hull girder in wave conditions. Iijima et al. (2011) detailed this approach and clarified the postultimate strength behaviour characteristics in wave conditions using numerical and tank test results. The important parameters related to the severity of the hull girder collapse were specified from the parametric dependencies study on the post-ultimate strength behaviour of a box-type hull girder in wave conditions carried out by Xu et al. (2011b, 2012). However, the hydro-elastoplasticity approach and the parametric dependencies study were based on the simplified boxtype hull girder, and more realistic ship and environment models are expected to follow. This paper primarily focuses on the investigation of the dynamic collapse behaviour of a bulk carrier under extreme wave loads. First, to obtain the capacity–displacement relationship of the bulk carrier, a ship’s model with one frame space is constructed and analysed using a displacement control method. In the capacity–displacement relationship, the post-ultimate strength behaviour over a large displacement range and unloading path are taken into account. Next, the hydroelastoplasticity analysis for the bulk carrier is carried out using the capacity–displacement relationship obtained from the finite element

analysis (FEA). The inertial forces and moments and the hydrostatic and hydrodynamic loads after deformation are evaluated based on nonlinear strip theory. The identification of the Response Amplitude Operator (RAO) of the vertical bending moment, the vertical displacement in heave motion and the rotational angle in pitch motion are carried out to validate the rationality of the present approach. Finally, a good illustration of the important influence of the significant wave height on the extent of collapse of the bulk carrier under a certain exceedance probability is clarified.

2. Nonlinear finite element analysis The most advanced analysis tools currently available for residual and ultimate strength assessment of thin-plated hull structures are nonlinear finite element codes. This paper investigates the post-ultimate strength behaviour of a hull girder in a bulk carrier using nonlinear FEA. The analysis is carried out using the commercial nonlinear FE computer program ABAQUS, which is able to solve the unstable collapse and post-buckling of structures; both geometric nonlinearity and material nonlinearity are taken into account in the procedure of nonlinear finite element analysis. With the rapid development of computer power and technology, nonlinear finite element analysis for relatively complicated stiffened panel structures can be successfully performed to obtain relatively accurate results, and the computation time can be controlled to a reasonable extent with proper modelling of the cross-section of the hull girder. For simplicity and efficiency, certain approximations were adopted together with selected simplified methods, i.e., Smith’s method and the ISUM method. For analysis of the progressive collapse behaviour of a hull girder under a longitudinal bending moment, the vertical curvature of the hull is assumed to occur incrementally, and the corresponding incremental element strains are calculated based on the assumption that the plane section remains plane and that the bending occurs about the instantaneous elastic neutral axis of the crosssection. The nonlinear finite element method solves the postultimate strength problem with no such assumption, and the interaction between the structural components for different buckling modes can be automatically considered in the simulation. In the present study, the nonlinear finite element method is adopted for evaluation of the load-carrying capacity of the crosssection of a Bulk Carrier. The FE results can predict the characteristics of nonlinear structural collapse at both the local panel and the global hull girder levels. The collapse behaviour includes the reduction of the carrying capacity and the unloading of the recovered path, and the main purpose of calculation of the unloading path in the FEA is to find the unloading point related to the extent of collapse of the hull girder when it is subjected to extreme loads. This feature is available in ABAQUS by controlling the parameters in the loading step. Obtaining optimal values for the incrementation as well as the prescribed rotation of the end of the cross-section are highly important to control the convergence problem. The simulation consists of two steps. In the first step, an elastic buckling analysis (also known as a linear perturbation analysis) was performed on a perfect cross-section of the bulk carrier to obtain the buckling mode (eigenmode), which indicates the possible buckling mode of the cross-section. In the second step, a nonlinear analysis was carried out on the FE model using the modified Riks method. The material plasticity strains is assumed to be elastic-perfectly plastic and the geometric imperfections which have a relative small effect on the ultimate strength were also included in this analysis to obtain the ultimate failure loads and failure modes of the bulk carrier.

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3. Modified Riks method The nonlinear behaviours of structures have become an interesting topic of investigation, including the post-ultimate strength behaviour of the hull girder, which must follow the equilibrium path. In the procedure for solving nonlinear problems, singular points such as the limit or bifurcation points always exist in the identification and computation of the nonlinear behaviours of the hull girder. These phenomena must be examined and followed in the equilibrium path. Over several decades, several techniques have been presented in the literature to achieve such a solution pattern on the equilibrium path, of which the load-controlled Newton–Raphson method was the earliest method proposed, and the characteristic of this method is the constant load during the load step. However, this method fails near the limit point. To overcome these difficulties with limit points, displacement control techniques were introduced, and their characteristic is a constant displacement during the increment. Load control and displacement control were widely used in the analysis of the structural responses. However, for structure systems exhibiting snapthrough or snap-back behaviour, these techniques will not be used due to possible errors. To obtain a more general technique, we use the arc-length method for structural nonlinear analysis originally developed by Riks (1972, 1979) and Wempner (1971). Later, the arc-length was modified by several scholars and widely used in the nonlinear analysis of the structure, especially for nonlinear behaviour with snap-through or snap-back phenomena. In this study, Riks method is adopted to handle the snapthrough or snap-back problem in the collapse behaviour of the bulk carrier because this method is able to capture the imperfection proportionality factor, and the reduction phenomenon of the capacity of the structure can also be tracked. The convergence problem presents a difficulty in the use of this method, and thus reasonable prescribed parameters in the step options are applied since they are highly important to solve this problem.

4. Structural modelling for nonlinear FEA The analysis target in the present study is selected from the ISSC-2000 special task committee V1.2 Capesize Bulk Carrier, which was chosen for the benchmark calculation to evaluate the ultimate hull girder strength (Yao et al., 2000). 4.1. Vessel particulars The cross-section of the hull girder of the Capesize Bulk Carrier is shown in Fig. 1. The principal specifications of the bulk carrier and the dimensions of the longitudinal stiffeners of the bulk carrier are summarised in Tables 1 and 2. The hull girder is longitudinally stiffened except for the side shells between the top side tank and the hopper side tank, which are transversely stiffened with stiffener spaces of 870 mm. The longitudinal stiffener spacing is 880 mm except for the top side tank adjacent to the hatch with a longitudinal stiffener spacing of 870 mm. The transverse frame spacing in the bottom and hopper tank is 2610 mm and is 5220 mm in the deck and top side tank.

Fig. 1. Cross-section of ISSC-2000 Capesize Bulk Carrier.

Table 1 Principal specifications of the Capesize Bulk Carrier. Parameters (units)

Values

Parameters (units)

Values

Length (m) Breath (m) Depth (m) Design draft (m) Scantling draft (m) Block coefficient Speed (knots)

285 50 26.7 19.808 19.83 0.836 15.5

Cross-sectional area (m2) DWT (t) Neutral axis above base line (m) I (m4) vertical I (m4) horizontal Z (m3) deck Z (m3) bottom

5.582 180,000 11.024 682.110 1,796.763 42.965 61.872

Note: I¼ moment of inertia, Z ¼section modulus.

Table 2 Dimensions of the longitudinal stiffeners of the Capesize bulk carrier. No.

Dimensions

Type

σY (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

390  27 333  9þ 100  16 283  9þ 100  14 289  9þ 100  18 333  9þ 100  17 283  9þ 100  16 180  32.5  9.5 283  9þ 100  17 333  9þ 100  18 333  9þ 100  19 383  9þ 100  17 383  10 þ100  18 383  10 þ100  21 300  27

Flat-bar Tee-bar Tee-bar Tee-bar Tee-bar Tee-bar Bulb-bar Tee-bar Tee-bar Tee-bar Tee-bar Tee-bar Tee-bar Flat-bar

392 352.8 352.8 352.8 352.8 352.8 235.2 352.8 352.8 352.8 352.8 352.8 352.8 392

4.2. Material properties When attention is directed to the collapse behaviour of the hull girder, the unstable nonlinear problem becomes unavoidable, and it is important to construct the nonlinear material model in terms of the stress versus strain relationship. In the present study, the elasticperfectly plastic model is adopted to describe the nonlinear material properties. The material properties are given in Table 3.

4.3. FE modelling of the cross-section The FE model of the ISSC-2000 Capesize Bulk Carrier in the present study is illustrated in Fig. 2. This model is a one-frame space FE model that can represent the structure in the midship region of the bulk carrier. In the FE model coordinate system, the X-axis lies along the

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degrees of freedom is approximately 541,000. The mesh density used in the mesh region can be summarised as follows:

Table 3 Material properties of the Capesize bulk carrier. Material types

MS

HT32

HT36

HT40

Young’s modulus (N/m2) Poisson ratio Yield stress (N/m2)

2.06  1011 0.3 235.2  106

2.06  1011 0.3 313.6  106

2.06  1011 0.3 352.8  106

2.06  1011 0.3 392.0  106

Note: MS denotes mild steel, and HT denotes high tensile steel. The numbers following HT denote the nominal yield stress value.

1) A total of 9  26 shell elements in the plate bounded by the longitudinal stiffeners and transverse frames. 2) Four shell elements across the web of the longitudinal stiffeners. 3) Two shell elements across the flange of the longitudinal stiffeners. 4) A total of 28 shell elements across the height of the double bottom centre girder and side girders.

4.4. Initial imperfections of the FE model

Fig. 2. One-frame space FE model of the ISSC-2000 Capesize Bulk Carrier.

direction of the ship width, the Y-axis runs along the direction of the ship height, and the Z-axis runs along the direction of the ship length. In this unstable nonlinear study, the convergence problem is an inevitable phenomenon related to the mesh size, the mesh types and the total number of elements. Construction of a detailed model is time consuming in terms of manual work and additional details, and the computational time shows a steep increase with denser meshing of the FE model. A coarse mesh in the critical area enables the model to become too stiff to capture the local buckling, which results in an unrealistically high ultimate strength. Thus, the challenge is to find a balance between the modelling refinement and the required accuracy of the result. The ABAQUS/Standard includes general-purpose shell elements that provide robust and accurate solutions to most applications. A subset of these general-purpose shell element types is represented by the S3, S3R, S4 and S4R types. The mesh size, mesh distribution and element types adopted in the present study are based on the work of Shu and Moan (2012). The S4R shell element is adopted throughout the entire model. The S4R element type is a fournode doubly curved general-purpose shell, which offers reduced integration with hourglass control, uses six degrees of freedom per node and was chosen for use in the simulations. Simpson’s rule was chosen by default to calculate the cross-sectional behaviour of a shell. The shell element chosen assumes that a homogenous section contains five integration points through the thickness. To ensure that the localised deformation and failure modes of the structure can be captured in the collapsed areas, the shell element is finely meshed with a global size of 0.1 m. The total number of elements in the crosssection model is approximately 88,800, and the total number of

The initial imperfections in the marine structure due to welding residual stress and geometric imperfection can significantly affect the ultimate strength of the structure and should be taken into account in the ultimate strength assessment (Vhanmanea and Bhattacharyab, 2008). Generally, the welding residual stresses of 2bt/b¼0.1 can be assumed on the plates between stiffeners, and the initial deflection of A0/tp ¼ 0.01 in the plates, B0/a ¼0.001 and C0/a ¼0.001 in the stiffeners were assumed, respectively. bt is the width of the tensile residual stress region and tp the thickness of the plates (Yao et al., 2000). The parameters a, b, A0, B0, and C0 are indicated in Figs. 3 and 4. In the case of the post-ultimate strength behaviour, the initial imperfections have a negligible effect because the initial imperfections are rather small in the plastic deformation of the collapsed structures. However, the initial imperfections are useful for obtaining the correct buckling mode in the FEA, and thus the initial imperfections are also included in solving the convergence problem in the collapsing structure. In present FEA, the method of considering the initial imperfection involves applying the lowest order of the eigenvalue buckling mode as the least favourable imperfection, which is referred to as the consistent imperfection mode method. In the first analysis run perform an eigenvalue buckling analysis with Abaqus/Standard on the “perfect” crosssection of bulk carrier to establish probable collapse modes and to verify that the mesh discretizes those modes accurately. In the second analysis run use Abaqus/Standard or Abaqus/Explicit to introduce an imperfection in the deck structure of bulk carrier by adding these buckling modes to the “perfect” structure. The lowest buckling modes are frequently assumed to provide the most critical imperfections, so usually these are scaled and added to the perfect geometry to create the perturbed mesh. The imperfection thus has the form (Simulia, 2009)

Δxi ¼

M X

ωi ϕ i

ð1Þ

i¼1

0

0 0

Fig. 3. Assumed initial deformation in stiffened plates.

W. Xu et al. / Ocean Engineering 106 (2015) 115–127

119

Aft Fore Trans.frame

θ

MPC

θ

z0

MPC y

Fig. 4. Assumed welding residual stresses.

where ϕi is the ith mode shape and ωi is the associated scale factor. The lowest buckling mode should have the largest factor. The magnitudes of the perturbations used are typically a few percent of a relative structural dimension such as a beam cross-section or shell thickness. Herein, the scale factor of the mode can be taken as 10% of the thickness of the plate. 4.5. Boundary and loading conditions of the FE model The coordinate system of the cross-section is already defined in Section 4.3. Two different loading patterns are adopted in the FEA. If two nonlinear calculations are denoted as FEA-1 and FEA-2, the differences between the loading patterns can be explained as follows. In FEA-1, two ends of the cross-section are constrained using the MPC/beam option function. Pure bending moments are applied on the MPC points (independent points) using a prescribed rotation. Symmetric boundary conditions with respect to the XY-plane are adopted on the fore end, whereas a prescribed rotation with respect to the X-axis is exerted on the aft end. In FEA-2, two independent points are set as master nodes, and the nodes on the ends of the cross-section are set as slave nodes. Symmetric boundary conditions with respect to the XY-plane are applied on the master nodes, and the prescribed rotation with respect to the X-axis is also exerted on the master nodes. The motions of the slave nodes are dependent on the application of the equation z ¼z0 þ θyi, where z is the translation of the node on each end, z0 is the translation of the master node, θ is the rotation of the master node and yi is the coordinate of ith node to be constrained. The load condition and boundary conditions of the model are shown in Figs. 5 and 6. In the middle of the cross-section along the longitudinal direction, the two FEAs have the same constraint conditions. When the one-frame cross-section model is under pure bending moment loading condition, the transverse frame is almost of no use, so it is not modelled. Instead, boundary conditions are applied as follows: the translations of the nodes located on the transverse frame position are constrained along the X-axis and Y-axis as well as the rotations along the Z-axis. The boundary conditions of the model are summarised in Table 4. It needs to be noted that the majority of researchers adopt FEA-1 model to realize the loading process of cross-section structure. However, in the numerical simulation, the deflections along Y-axis near the ends of cross-section are liable to induce due to rotation of MPC point. However, for FEA-2, the deformation in the ends of cross-section can follow the realistic phenomenon of hull structure under the sagging condition. When the rotation is small, two types of models are almost same. However, differences of deformation pattern become obvious when the rotation is large.

z Frame space Fig. 5. Boundary and loading conditions for FEA-1.

Aft Fore Trans.frame z=z0+θyi θ

θ

yi z0

y z Frame space Fig. 6. Boundary and loading conditions for FEA-2.

Localization effect in FEA-1 model is more prominent. So the FEA-2 model was adopted for setting the loading condition in following studies.

5. Hydro-elastoplasticity theory 5.1. Theoretical model When a hull girder in extreme wave conditions is selected as a research object, its collapse behaviour can be analysed by constructing a hydro-elastoplasticity model. To effectively analyse the collapse behaviour of a hull girder in wave conditions, the entire ship hull is modelled as a two-rigid-body system connected by a nonlinear rotational spring. The rotational spring represents the nonlinear relationship between the displacement and the moment, including the linear bending rigidity path, the ultimate

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Table 4 Boundary conditions of the calculation model. Locations

Translations

Middle section of the model at the transverse frame position Fore end of the cross-section Aft end of the cross-section

Rotations

Ux

Uy

0

0

Uz

URx

URy

0

0 0

0 0

URz 0

Note: U denotes translation, UR denotes rotations and the subscripts denote the coordinate axis.

strength and the reduction of the load carrying capacity after the ultimate strength for loading until unloading. The aft and fore regions reside in different coordinate systems, as shown in Fig. 7. The right and left-hand bodies are referred to as body 1 and body 2, respectively, for convenience. In Fig. 7, F1w and F2w represent the external forces, F1R and F2R represent the restoring forces, M1w and M2w represent the dynamic external moments measured around the centre of gravity, M1R and M2R represent the restoring moments due to the hydrostatic force with respect to the centre of gravity, fint and Mint represent the reaction force and moment due to the spring, respectively, u1 and u2 are the vertical displacements of the two bodies’ centre of gravity, θ1 and θ2 show the rotational displacement of the respective bodies, and l1 and l2 represent the distance from the midship region to the centres of gravity of body 1 and body 2. 5.2. Formulations (Xu et al., 2012) The internal bending moment Mint is a function of the relative rotational angle θ(κ). The structural damping, which is proportional to the rate of the relative rotational angle, is considered in the theoretical model, and the internal bending moment can be expressed by Eq. (2). M int ¼ M int ðθÞ þ ck θ_

ð2Þ

or in its incremental form,

ΔMint ¼ kR U Δθ þck Δθ_

ð3Þ

where θ ¼ θ1  θ2, kR is the tangential stiffness of the function Mint(θ), and Ck is the structural damping coefficient. The equilibrium equations of motion can be obtained by reference to their centres of gravity as n o n o ½M 1  U€ 1 þ ½C 1  U_ 1 þ ½K 1 fU 1 g ¼ fF 1 g ð4Þ n o n o ½M 2  U€ 2 þ ½C 2  U_ 2 þ ½K 2 fU 2 g ¼ fF 2 g where

(

fU 1 g ¼ "

" ½C 1  ¼ "

( fU 2 g ¼

m111

m112

m121

m122

C 111

C 112

C 121

C 122

K 111

K 112

K 121

K 122

( fF 1 g ¼

)

θ1

½M 1  ¼

½K 1  ¼

u1

u2

)

" ½M 2  ¼

#

" ½C 2  ¼

#

M 1w M int  f int l1

m211

m212

m221

m222

C 211

C 212

C 221

C 222

" ½K 2  ¼

where mij (i, j¼1–2) represents the mass and inertia (including the added mass and inertia), Cij represents the hydrodynamic damping, and Kij represents the restoring force coefficient due to hydrostatic pressure. The superscripts indicate body 2 (aft) or 1 (fore). At the midship region, the compatibility condition with respect to the displacements can be constructed as u1 þ l1 θ1 ¼ u2  l2 θ2

3 m111 þ m211  m111 l1 þ m112  m111 þ m212 7 6 1 2 6 m  l 1 m1  l1 m121 þ m122 þ l1 m111  l1 m112  l2 m121 þ l1 l2 m111 7 11 5 4 21 m221  l2 m211 0 m222  l2 m212 3 2 8 9 C 1 þ C 211  C 111 l1 þ C 112  C 111 l2 þ C 212 Δu€ 2 > > < = 6 11 7 2 1 1 1 1 1 1 1 1 7 Δθ€ 1 þ 6 4 C 21  l1 C 11  l1 C 21 þ C 22 þ l1 C 11  l1 C 12 þ ck  l2 C 21 þ l1 l2 C 11  ck 5 > : € > ; 2 2 2 2 Δθ 2 C 21  l2 C 11  ck C 22  l2 C 12 þ ck 2

8 9 2 1 K þ K 211 Δu_ 2 > > < = 6 11 1 1 _ Δθ 1 þ 6 K 4 21  l1 K 11 > : _ > 2 Δθ 2 ; K 21  l2 K 211

K 211

K 212

K 221

K 222

)

( fF 2 g ¼

 kR

 K 111 l2 þ K 212  l2 K 121 þ l1 l2 K 111  kR K 222  l2 K 212 þ kR

8 9 8 9 > > > < F 1w þ F 2w > = <0 = Δθ1 ¼ M1w  F 1w l1  M int > > > > > : Δθ ; : M  F l ; : M > ; 2w 2w 2 int 2 t ¼ t 0 þ Δt t ¼ t0 38 9 2 1 2 1 1  m11 l1 þ m12  m111 þ m212 m11 þ m11 > u€ 2 > 7< € = 6 1 2 1 1 1 1 1 1 1 7 θ1 6 m  l m  l m þ m þ l m  l m  l m þ l l m 1 21 1 12 2 21 1 2 11 5 1 11 22 4 21 1 11 > :€ > ; m221  l2 m211 0 m222  l2 m212 θ2

2

C 111 þ C 211

C 221  l2 C 211

K 111 þ K 211 6 1 1 K 6 4 21  l1 K 11 K 221  l2 K 211

#

 K 111 l1 þ K 112 2  l1 K 121 þ K 122 þ l1 K 111  l1 K 112 þ kR

3 7 7 5

8 9 > < Δ u2 > =

6 1 1 6 4 C 21  l1 C 11

#

ð6Þ

By eliminating u1 and fint, the equations may be recast in an incremental form as follows:

2

θ2

#

F 1w  f int

ð5Þ

Fig. 7. Free body diagram for the two-rigid-body system.

 C 111 l1 þ C 112 2

 l1 C 121 þ C 122 þ l1 C 111  l1 C 112 0  K 111 l1 þ K 112 2

 l1 K 121 þ K 122 þ l1 K 111  l1 K 112 0

t ¼ t0

38 9 u_ > < 2> = 1 1 7 _  l2 C 21 þ l1 l2 C 11 7 5> θ 1 > : θ_ 2 ;t ¼ t C 222  l2 C 212 0 3 8 9  K 111 l2 þ K 212 u > > 2 7< =  l2 K 121 þ l1 l2 K 111 7 5> θ 1 > :θ ; 2 2 2 K 22  l2 K 12 t ¼ t0  C 111 l2 þ C 212

Once the initial conditions are given, the increments can be obtained in successive time steps using a numerical integration scheme such as the Newmark β method.

#

5.3. Evaluation method of external loads F 2w þ f int

M 2w þ M int  f int l2

)

The external loads are evaluated using a time-domain nonlinear method based on ordinary strip theory, which was developed by Fujino et al. (1984). Only the motions within a vertical

W. Xu et al. / Ocean Engineering 106 (2015) 115–127

plane are modelled herein, and more detailed introductions can be found in Iijima et al. (2011). The vertical load on a cross-section per unit length fdz may be given as follows: Z d dς dς ð7Þ f dz ¼ ½mHH  þ N HH þ pnz dl  w dt dt dt C0 where ðd=dtÞ ¼ ð∂=∂tÞ  Uð∂=∂xÞ, mHH is the added mass in the vertical direction per unit length, NHH is the damping force coefficient per unit length, ς is the relative wave elevation due to the vertical motion of the cross-section, p is the pressure evaluated assuming that the waves are undisturbed by the ship body, nz is the vertical component of the outward normal vector within the cross-section of the bulk carrier and w is the weight per unit length. The integration in Eq. (7) is performed over the instantaneous wetted surface C0 of the cross-section of the bulk carrier. The added mass and damping are also evaluated for the submerged portion below the instantaneously wetted surface. In Eq. (7), the impact loads due to the change of the added mass with respect to time are also included. The ship’s hull girder may be subjected to multiple large waves in severe and irregular seas; however, dynamic collapse behaviour due to a large single wave is pursued as a first step. The large single wave is realised using a focused wave technique. The focused wave can be modelled by the sum of regular elementary waves, and the wave elevation η for the longcrested wave spectrum S(ω) at x¼X and t ¼T can be given in following formula (Jensen and Capul, 2006).

ηðX; T Þ ¼

n  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ui Sðωi ÞΔω cos ðki X þ ωi T Þ þ ui Sðωi ÞΔω sin ðki X þ ωi T Þ i¼1

ð8Þ where ωi is the wave circular frequency of the ithe elementary wave, ki is the corresponding wave number and ui and ui are independent sets of independent zero-mean Gaussian variable with unit variance. From the deterministic and probabilistic viewpoints, the probability that the wave elevation exceeds the threshold value of wave elevation that can cause hull girder collapse at a defined location in time and space can be obtained using the First Order Reliability Method (FORM). The corresponding formula is given as Pf ¼ Ф(  β), where Ф(  ) is the standard normal distribution function, and β is the reliability index. In the FORM analysis, the rather useful property that the reliability index (β) is strictly inversely proportional to the significant wave height is useful for determining the reliability index in later calculations (Jensen, 2012).

in the same figure. It can be observed that the ultimate strength of the cross-section is approximately 1.44  1010 N m, and the stiffness of the structure is approximately 1.39  1014, which are similar to the results obtained in ISSC 2000 (Yao et al., 2000). It is noted that when the bending moment reaches the ultimate strength, the plastic region formed in the cross-section only spreads in the local region and not over the entire structure. The load-carrying capacity sharply decreases when the curvature of the cross-section reaches approximately 1.12  10  4 m  1 at Point (a). When the curvature reaches approximately 1.34  10  4 m  1 at Point (b), the load carrying capacity decreases to 1.28  1010 N m, and the slope of the load-carrying capacity curve becomes slightly more flat. When the curvature reaches approximately 2.32  10  4 m  1 at Point (c), the bending moment and the curvature curve obtained by present FEM are terminated at this point because the nonlinear calculation on the cross-section of the bulk carrier by FEM has converged. The corresponding loadcarrying capacity decreases to 1.05  1010 N m, and the ratio of the reduction of the loading-carrying capacity to the ultimate strength, also known as the drop factor, is approximately 0.3. When the force residual for the iteration is zero at every degree of freedom in the calculation model, the structure will be in equilibrium. In the nonlinear problem, the force residual for the iteration will never reach exactly zero, and thus Abaqus/Standard compares it to a tolerance value. If the force residual is less than this force residual tolerance at all nodes, then Abaqus/Standard accepts the solution as in equilibrium. Thus, the calculation of bending moment and the curvature with a large deformation range makes the iteration step very small, which leads to very time consuming calculation. The maximum bending moment that the cross-section can carry is equal to the fully plastic bending moment (Amlashi and Moan, 2008). To quantitatively evaluate the extent of collapse of the bulk carrier, the bending moment and the curvature curve obtained from FEM were selected as the input data for calculation via the hydro-elastoplasticity approach. When the input curve of the bending moment and the rotational angle are determined, the load-carrying capacity corresponding to the last point in the input curve will be presumed to follow a constant value in the theoretical curve. The deformed shape and Von Mises stress distribution at two steps are found in Fig. 9. The left figure corresponds to Point (a) in Fig. 7, whereas the right figure corresponds to Point(c). From Fig. 9(a), it can be observed that the plastic deformation has not formed at step 65, and the Von Mises stress distribution on the surface of deck plate includes the effect of the initial imperfections. The deformation is primarily induced by an initial

Present FEM ISSC-Yao ISSC-Chen ISSC-Rigo ISSC-Cho ISSC-Masaoka

10

6. Dynamic collapse behaviour of the Capesize bulk carrier

1.6x10

To predict the dynamic collapse behaviour of the hull girder, the relatively correct post-ultimate strength behaviour of a generic ship under a pure bending moment under sagging conditions is necessary. The modified Riks method was implemented to predict the post-ultimate strength behaviour, which can be interpreted by the relationship between the bending moment and the curvature of the cross-section. Nonlinear FEA was carried out on a Mouse Computer with an Intel Core (TM) with 3.2 GHz (12 CPU) and 32 GB RAM. Beyond the ultimate strength, the FE calculation becomes more time consuming, especially for the Riks method. The relationship between bending moment and the curvature in solid line with solid triangle symbol is illustrated in Fig. 8. The calculation results in ISSC 2000 (Yao et al., 2000) are also plotted

Bending moment (N.m)

10

6.1. Relationship between the bending moment and the rotational angle of the cross-section

121

1.4x10

10

1.2x10

(a) (b) (c)

10

1.0x10

9

8.0x10

9

6.0x10

9

4.0x10

9

2.0x10

0.0 0.0

-5

5.0x10

-4

1.0x10

-4

1.5x10

-4

2.0x10

-4

2.5x10

Curvature (1/m) Fig. 8. Comparison of relationship between the vertical bending moment and the curvature of the cross-section of bulk carrier under sagging conditions by different methods.

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Fig. 9. Plot contours on the deformed shape and stress distribution in different steps. (a) Step 65, the Von Mises stress distribution corresponding to point (a) in Fig. 8. (b) Step 115, the Von Mises stress distribution corresponding to point (c) in Fig. 8.

imperfection. When the vertical bending moment exceeds the ultimate strength, the deformation concentrates on a local area on the deck plate, which includes the initial imperfection and the elastic and plastic deformations. From Fig. 9(b), it can be observed from the deck plate that the high Von Mises stress distributes in a local area near the mid-section of the model, whereas the low Von Mises stress distributes in another area at step 115. This occurs because the stress is redistributed; when a large plastic deformation is formed, the deformation can represent the extent of collapse of the bulk carrier when it is subjected to a pure bending moment. To apply the hydro-elastoplasticity approach to predict the collapse extent of a generic ship’s hull girder in extreme waves, the correct post-ultimate strength behaviour must be given as an input condition for evaluation. For the numerical simulation, the curvature of the structure is expressed as the rotational angle

when the structure begins to collapse. When the ship’s hull girder collapses due to the extreme wave loads, the extent of collapse predicted by the hydro-elastoplasticity approach not only depends on the intrinsic moment–curvature relationship but is also related to the external force/moment. In the hydro-elastoplasticity approach, the still-water bending moment was not considered, and therefore the post-ultimate strength behaviour for evaluation should be calculated by subtracting the component of the stillwater bending moment, and the load carrying capacity of the ship hull obtained by the hydro-elastoplasticity approach can be added back for evaluation. The longitudinal distribution of the still-water bending moment at any transverse section shown in Fig. 10 can be considered. In this situation, the MSW is the designed still-bending moment located in the midship region in hogging or sagging conditions. In this paper, the value of the still-water bending moment for the sagging condition is taken as 4.38  109 N m from

W. Xu et al. / Ocean Engineering 106 (2015) 115–127

123

Fig. 10. Preliminary still-water bending moment distribution. Fig. 11. Distribution factor FM. Table 5 Distribution factor FM.

Table 6 Real wave conditions for calculations.

Hull transverse section location

Distribution factor FM

0 rx r 0.4L 0.4L r x r0.65L 0.65L o x rL

2.5x/L 10 2.86(1  x/L)

the following formula (IACS, 2009), M SW;S ¼ 175CL2 BðC B þ 0:7Þ10  3  M WV;S

ð9Þ

where L is the length of the ship in meters, B is greatest moulded breadth of the ship in meters, CB is the block coefficient with a value of CB ¼0.8511 in this work, and C is a wave parameter taken equal to   300  L 1:5 C ¼ 10:75  for90 r L r 300 m 100 C ¼ 10:75

Cases Significant wave height Hs (m)

λ/L Circular frequency (rad/s)

Cases Significant wave height Hs (m)

λ/L Circular frequency (rad/s)

1 2 3 4 5 6 7

0.3 0.4 0.5 0.6 0.7 0.8 0.9

8 9 10 11 12 13

1.0 1.2 1.4 1.6 1.8 2.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.849 0.735 0.658 0.600 0.556 0.520 0.490

1.0 1.2 1.4 1.6 1.8 2.0

0.465 0.424 0.393 0.368 0.347 0.329

for300 r L r 350 m

and M WV ;S is the vertical wave bending moment in kN m defined in the following formula, M WV ;S ¼ 110F M f P CL2 BðC B þ 0:7Þ10  3 where FM is the distribution factor defined in Table 5 (see also Fig. 11), and f P is a coefficient corresponding to the probability level taken equal to fp¼ 1.0 for strength assessments corresponding to the probability level of 10  8. fp¼ 0.5 for strength assessments corresponding to the probability level of 10  4.

Fig. 12. RAO of vertical bending moment of the Capesize Bulk Carrier.

6.2. Prediction of the extent of collapse of a bulk carrier by the hydro-elastoplasticity approach 6.2.1. Identification of a response amplitude operator (RAO) from the calculation data The identifications of the RAO for the calculations are performed under small regular wave conditions. The cases for the identification are listed in Table 6. The wave steepness Hs/λ (Hs: significant wave height, λ: wave length) is held constant at 1/285 throughout these calculations. In this calculation, Fig. 12 shows the RAO of the vertical bending moment, and Fig. 13 shows the RAO of the vertical displacement in heave motion and the rotational angle in pitch motion in the following wave at zero forward speed. In Fig. 12, Mw is the vertical bending moment induced by the wave, η is the wave amplitude, and λ is the wavelength. It can be observed that the RAO is largest when the wavelength is close to the ship length, and the vertical bending moment induced by the wave varies significantly with the amplitude variation. This sensitivity could be explained by the fact that the wave frequency is close to the first natural frequency of the hull girder, which can easily cause the structural system to encounter resonance phenomena. When the ratio of the wavelength to the ship length is 0.5, the RAO of the vertical bending moment suddenly decreases because the wave-induced vertical bending moment decreases while the wave

Fig. 13. RAO of the vertical displacement in heave motion and of the rotational angle in pitch motion.

amplitude continues its monotonic increase with the increase in the wavelength. The reason for this behaviour can be interpreted in terms of the encountering wave, which symmetrically exerts force on the

W. Xu et al. / Ocean Engineering 106 (2015) 115–127

individual bodies of the hull girder and causes the vertical bending moment in the midship region to decrease due to the interneutralisation of the wave-induced bending. In Fig. 13, u is the average value of the vertical displacement of body 1 and body 2, and θ is the rotational angle of body 1 or body 2 with respect to its centre of gravity. It can be observed that the RAO of the heave motion has two characteristic points that must be addressed. When the ratio of λ/L is equal to 1.0 and 0.5, the RAO of the heave motion becomes smaller due to the inter-neutralisation of the wave pressure when it is integrated along the length of the hull girder. The calculation of the heave motion for the entire hull girder is taken as the average of the heave motions of body 1 and body 2 in the time series. When the ratio of λ/L is approximately 0.7, the RAOs of the heave motion and pitch motion show inverse change tendencies. When the ratio of λ/L is larger than 1.0, the RAOs of the heave motion and pitch motion increase with the increase in the ratio of λ/L. The former has a more weak dependence on the wavelength than the latter.

0.12 0.10

Rotational angle (rad)

124

0.08 0.06 0.04 0.02 0.00 0

20

40

60

80

100

120

140

160

Time (s) Fig. 14. Time history of rotational angle in the midship region of the hull girder when the significant wave height is 15.0 m.

10

1.6x10

Hydro-elastoplasticity approach FE method

10

1.4x10

Bending moment (N.m)

6.2.2. Prediction of the severity of collapse of the bulk carrier The severity of collapse is defined as the extent of collapse of the hull girder, and in this work, it is expressed by the magnitude of the plastic rotation at the collapsed section, i.e., the residual difference of the pitch angle between body 1 and body 2 as shown in Fig. 7. The severity of collapse is calculated for a focused wave. A focused wave technique is adopted using FORM (Jensen and Capul, 2006), and the target-focused wave is generated from the ISSC wave spectrum with a significant wave height Hs0 of 15 m with a mean period of T0 of 13.5 s. The time series of the focused wave is scaled to compare the severity of collapse under wave motion with various amplitudes. The probability distribution of the individual wave heights in a stationery sea state may be well approximated by a Rayleigh distribution. It is assumed that the ship collapses in an extreme wave at an exceedance probability of 1/ 1000 in the short-term sea state. To this end, the reliability index β ¼3.7 is selected for generating an equivalent focused wave. The time history of the relative rotational angle located in the midship region of the hull girder when the significant wave height is 15.0 m is plotted in Fig. 14. The severity indicates that the permanent deformation induced at approximately 80 s by the focused wave is approximately 9.45  10 2 rad. Fig. 15 shows the relationship between the bending moment and the rotational angle as calculated by the hydroelastoplasticity method and the FE method. It can be observed that the collapse behaviour obtained from the FEM and the hydro-elastoplasticity approaches show good agreement. The small discrepancies depend on the sampling division of the mean encountered wave period, and the larger the number, the smaller the discrepancies. When the load-carrying capacity decreases to approximately 1.06  1010 N m, the rate of decrease becomes smaller, and the load-carrying capacity approaches a constant. This behaviour occurs because the load-carrying capacity is a function of the deformation. The deformation is determined by the wave loads and the moment–curvature relationship, including the ultimate strength, drop factors and reduction slope. When the above-mentioned factors have been determined, the collapsed structure will enter a selfequilibrium state after the deformation is formed unless other disturbances (i.e., a subsequent wave series) are introduced to upset the equilibrium state. The load-carrying capacity approaches a constant because it has been determined due to the permanent deformation. It should be noted that the severity of the collapse cannot be reflected correctly from the FEM results due to convergence problems; however, we can provide clarification of the deterministic and probabilistic aspects in irregular seas from the hydro-elastoplasticity approach. When the significant wave height is less than 14.5 m, almost no severity of the wave-induced collapse occurs. For the case of a significant wave height of 14.5 m, the time history of the wave amplitude and the rotational angle in the midship region of the hull girder are given in Figs. 16 and 17, respectively. At 80s, the

10

1.2x10

10

1.0x10

9

8.0x10

9

6.0x10

9

4.0x10

9

2.0x10

0.0 0.0

-4

4.0x10

-4

8.0x10

-3

1.2x10

-3

1.6x10

-3

2.0x10

Rotational angle (rad) Fig. 15. Relationship between the bending moment and rotational angle in the FE method and numerical method when the significant wave height is 15.0 m.

trough can be treated as a focused wave acting on the hull girder. It is noted that only the wave-induced bending moment in the trough is larger than the ultimate strength of the hull girder, whereas that of the crest is smaller than the ultimate strength, which cannot cause structural collapse. The severity of damage to the hull girder is observed as approximately 2.21  10  5 rad. The bending moment obtained by the hydro-elastoplasticity approach is approximately 9.40  109 N m. When the still-water bending moment of 4.38  109 N m is added, the load carrying capacity of the hull girder in the midship region is approximately 1.38  1010 N m, as shown in Fig. 18. The ultimate strength of the hull girder obtained by Abaqus is approximately 1.44  1010 N m, which is quite close to the result obtained from the hydroelastoplasticity approach. The corresponding moment–rotational angle curve can be found in Fig. 19. The moment–rotational angle curve obtained from the FEM was also plotted in the same figure for comparison.

6.2.3. Significant wave height dependence of the severity of collapse Fig. 20 shows the relationship between the severity of collapse and the non-dimensional significant wave height, marked as S  Hsi/Hs0. In the figure, the chain dotted line with solid triangles shows the present numerical (bulk carrier) results. The Hsi

W. Xu et al. / Ocean Engineering 106 (2015) 115–127

10

125

Hydro-elastoplasticity approach FE method

10

Bending moment (N.m)

1.4x10

Wave elevation (m)

5

0

-5

-10

10

1.2x10

10

1.0x10

9

8.0x10

9

6.0x10

9

4.0x10

9

2.0x10

-15

0

20

40

60

80

100

120

140

0.0

160

-4

0.0

-4

4.0x10

2.0x10

Time (s)

-4

-4

6.0x10

8.0x10

-3

1.0x10

Rotational angle (rad)

Fig. 16. Time history of the wave amplitude when the significant wave height is 14.5 m.

Fig. 19. Relationship between the bending moment and rotational angle in the FE method and the numerical method when the significant wave height is 14.5 m.

0.25

2.5x10

0.20

1.5x10 1.0x10

Severity (rad)

Rotational angle (rad)

2.0x10

5.0x10 0.0 -5.0x10

0.15

0.10

0.05

-1.0x10 -1.5x10 0

20

40

60

80

100

120

140

160

Time (s) Fig. 17. Time history of rotational angle in the midship region when the significant wave height is 14.5 m.

0.00 0.9

1.0

1.1

1.2

Hsi / Hs0 ( M

1.3

1.4

1.5

/M )

Fig. 20. Relationship between the severity of collapse and the significant wave height.

10

Bending moment (N.m)

1.6x10

10

1.2x10

9

8.0x10

9

4.0x10

0.0 9

-4.0x10

0

20

40

60

80

100

120

140

160

Time (s) Fig. 18. Time history of the bending moment in the midship region of the hull girder when the significant wave height is 14.5 m.

indicates the significant wave height, and Hs0 denotes the original significant wave height of 15.0 m. The mean period is taken as a constant of 13.5 s. The Hsi is varied as 14.4 m, 14.5 m, 14.6 m, 14.8 m and 15.0 m. When this value exceeds 15.0 m, the variation is changed from 15.0 m to 22.0 m on the unit interval. When the significant wave height reaches 14.5 m, the maximum vertical

Fig. 21. Assumed relationship between the vertical bending moment and the rotational angle.

bending moment approaches the ultimate strength, and the severity of collapse is not significant. When the significant wave height is increased to 14.6 m, the severity of collapse suddenly becomes serious. The rotational angle in the midship region of the hull girder increases from 2.21  10  5 rad to 8.01  10  2 rad. The latter value corresponds to a change in the fore and aft end draft of

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Table 7 Comparison of the characteristics of the capacity model and load model in the numerical method for different ships.

Capacity model

Parameters

Numerical (box-shape ship)

Numerical (bulk carrier)

(1) Ultimate strength Mu (N m)

9.17  109 0.20

13.90  109 0.67

 M BC (2) Capacity drop factor α ¼ Mu M u  (3) Sharpness of drop slope γ ¼ k2 =k1 

Load model

(1) Original significant wave height Hs0 (m) (2) Mean period T0 (s)

approximately 5.7 m. When the significant wave height is further increased, the rate of increase of the extent of collapse flattens out and approaches a nearly constant value. The relationships between the severity of collapse and the nondimensional amplitude of the dynamic external moment obtained from the analytical solution and the numerical simulation for a box-shape ship are also plotted in Fig. 20. The solid line with solid squares and the dashed line with solid circles indicate the analytical (box-shape ship) and numerical (box-shape ship) results, marked as S Mexti/M0 and SHsi/ Hs0, respectively. The Mexti displays the different amplitudes of the dynamic external moment, and M0 denotes the original amplitude of the dynamic external moment in the analytical solution. Several simplifications have been applied to the moment–rotational angle curve for derivation of the analytical (box-shape ship) solution and the numerical (box-shape ship) simulation. The moment–rotational angle relationship is approximated by piece-wise linear curves, as illustrated in Fig. 21. The Mu is the ultimate strength or collapse moment at Point A. Before Point A, the curve is approximated by a straight line with the assumption that neither buckling nor yielding occurs before Point A. The slope is given by k1 ¼EI/h, where EI is the bending rigidity of the hull girder, and h is the distance between transverse members. Point A is followed by path AB with slope k2, and the path reaches B. Along path BC, the capacity is assumed to be constant with slope k3. The slope of the unloading path CD is assumed to be the same as the original stiffness along path OA. Additional details can be found in Xu et al. (2012). Several influential parameters that characterise the capacity and load models in the numerical method for the box-shape ship and the bulk carrier are summarised in Table 7. The capacity model indicates the ultimate strength Mu, the capacity drop  factor  α ¼ ðM u  M BC Þ=M u and the sharpness of the drop slope γ ¼ k2 =k1  just after the ultimate strength is reached. In contrast, the load model indicates the generated equivalent focused wave corresponding to the reliability index β ¼ 3.7. It is shown that the capacity drop factor and the sharpness of the drop slope of the bending moment–rotational angle curve in the bulk carrier are much larger than those of the box girder model. It can be observed from Fig. 20 that the relationships between the severity of collapse and the non-dimensional amplitude of the dynamic external moment obtained from the present bulk carrier model and those obtained for the box-shape ship have similar tendencies; i.e., the severity of failure, or the plastic rotational angle, is significantly induced when the applied bending moment exceeds the ultimate capacity even slightly, and it increases nearly linearly with the increase in the significant wave height. This tendency, found from the analytical solution in Xu et al. (2012), was confirmed in the numerical analysis of the real ship model as well as that of the box-girder ship. In terms of parameter (2) of the capacity model in Table 7, the capacity drop factor in the analytical solution is smaller than that in the present study. The severity of collapse is dependent on the drop factor, and a larger drop factor induces large severity. In terms of (3), the severity of collapse is also dependent on the sharpness of the drop slope, and the larger sharpness of the drop slope induces larger severity. These results agree with those found in Iijima et al. (2011) and Xu et al. (2011b, 2012). In terms of the load model in Table 7, the significant wave heights are nondimensionalised in Fig. 20. The effects of the original mean period

0.10 12 13.9

1.52 15 13.5

T0 on the severity in different methods show no difference because the similar original mean periods are selected for calculation individually. Therefore, the discrepancies between different methods are mainly attributed to the different capacity models, the capacity drop factors and the sharpness of the drop slope. It should be noted that the quantitative contribution of these characteristics to the severity of collapse could be clarified in future research. More accurate relationships between the bending moment and rotational angle based on fewer assumptions are also expected from future work.

7. Conclusions and remarks The post-ultimate strength behaviour of a Capesize Bulk Carrier obtained via FEA was selected in the first step of the analysis. Next, the hydro-elastoplasticity approach was applied to quantitatively evaluate the extent of collapse when the structure is subjected to extreme wave loads. The results were compared with the theoretical predictions from an analytical solution derived in previous research. The following conclusions can be drawn from this work: 1. The post-ultimate strength behaviour of the one-frame space model of the bulk carrier obtained with the Riks method was acceptable for predicting the severity of damage to the hull girder because the large deformation is suitable for use of the hydro-elastoplasticity approach in the second step. 2. The load-carrying capacity of the hull girder in the bulk carrier under consideration rapidly decreases after reaching the ultimate strength. The decrease rate becomes flatter with the development of plastic deformation. The reduction ratio of the load carrying capacity to the ultimate strength, also known as the drop factor, is approximately 0.67 for the considered range of curvature. 3. The severity of collapse of the bulk carrier is approximately 9.45  10  2 rad when the hull encounters a wave train at an exceedance probability of 1/1000 in irregular seas with a significant wave height of 15 m and a mean period of 13.5 s. 4. The relationship between the bending moment and large curvature obtained in the present model is based on certain assumptions. Future work should focus on analysis based on fewer assumptions for prediction of the severity of damage to the hull girder. The quantitative contribution of the capacity drop factors and sharpness of the drop slope on the severity of collapse also must be clarified.

Acknowledgments The authors express their sincere thanks Professor Fujikubo, M. and Iijima, K. of Osaka University give us so great help in the research work. The authors were partially funded by China Postdoctoral Science Foundation (grant number 2013M541347), the Scientific Research Foundation for Returned Overseas Chinese Scholars of Heilongjiang Province, China (grant number LC2013C18), and Heilongjiang Postdoctoral Foundation (grant number LBH-Z13061).

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References Amlashi, H.K., Moan, T., 2008. Ultimate strength analysis of a bulk carrier hull girder under alternate hold loading condition—a case study—Part 1: Nonlinear finite element modelling and ultimate hull girder capacity. Mar. Struct. 21, 327–352. Caldwell, J.B., 1965. Ultimate longitudinal strength. Trans. R. Inst. Naval Arch. 107, 411–430. Chen, Y.K., Kutt, L.M., Piaszczyk, C.M., Bieniek, M.P., 1983. Ultimate strength of ship structures. SNAME Trans. 91, 149–168. Fujikubo, M., Kaeding, P., 2002. New simplified approach to collapse analysis of stiffened plates. Mar. Struct. 15 (3), 251–283. Fujikubo, M., Olaru, D.V., Yanagihara, D., Setoyama, Y., 2002. ISUM approach for collapse analysis of double-bottom structures in ships. In: Proceedings of the 12th International Offshore and Polar Engineering Conference, Kitakyushu. Fujino, M., Yoon, B.S., Kawada, J., Yoshino, I., 1984. A study on wave loads acting on a ship in large amplitude waves. J. Soc. Naval Arch. Jpn. 156, 144–152. IACS. 2009. Common structural rules (CSR) for bulk carrier. In: International Association of Classification Societies, Chapter 4, Section 3. Iijima, K., Fujikubo, M., 2010. Investigations into the target safety level of hull girder ultimate strength. In: Proceedings of the Fourth International Maritime Conference on Design for Safety, Trieste, Italy. Iijima, K., Kimura, K., Xu, W., Fujikubo, M., 2011. Hydro-elastoplasticity approach to predicting the post-ultimate strength behavior of a ship’s hull girder in waves. J. Mar. Sci. Technol. 16, 379–389. Inc Simulia, 2009. Abaqus Analysis user’s Manual, Version 6.9-3. Simulia Inc, Providence. 〈http://www.simulia.com〉. Jensen, J.J., Capul, J., 2006. Extreme response predictions for jackup units in second order stochastic wave by FORM. Probab. Eng. Mech. 21, 330–337. Jensen, J.J., 2012. Probability of parametric roll in random seaways. In: Thor, I. Fossen, Henk, Nijmeijer (Eds.), Parametric Resonance in Dynamical Systems. Springer, pp. 91–106. Kimura, K., Iijima, K., Wada, R., Xu, W., Fujikubo, M., 2010. Dynamic collapse behavior of ship’s hull girder in waves. In: Advanced Maritime Engineering Conference and Fourth Pan Asian Association of Maritime Engineering Societies Forum, Singapore, pp. 261–266. Kut, L.M., Piaszczyk, C.M., Chen, Y.K., Liu, D., 1985. Evaluation of the longitudinal ultimate strength of various ship hull configurations. SNAME Trans. 93, 33–53. Lehmann, E., 2006. Discussion on report of Committee III. 1: ultimate strength. In: Proceedings of 16th ISSC. 3, Southampton, UK, pp. 121–131. Mansour, A.E., Thayamballi, A., 1980. Ultimate strength of a ship’s hull girder plastic and buckling modes. In: Ship Structure Committee Report, SSC-299. Masaoka, K., Okada, H., Ueda, Y., 1998. A rectangular plate element for ultimate strength analysis. In: Second International Conference on Thin-Walled Structures, Singapore, pp. 469–476. Paik, J.K., Thayamballi, A.K., Che, J.S., 1996. Ultimate strength of ship hulls under combined vertical bending, horizontal bending and shearing forces. SNAME Trans. 104, 31–59. Pei, Z., Fujikubo, M., 2005. Application of idealized structural unit method to progressive collapse analysis of ship’s hull girder under longitudinal bending. In: Proceedings of the Fifteenth International Offshore and Polar Engineering Conference, Seoul, Korea. pp. 766–773. Riks, E., 1972. The application of Newton’s method to the problem of elastic stability. J. Appl. Mech. 39, 1060–1065.

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Riks, E., 1979. An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct. 15, 529–551. Shu, Z., Moan, T., 2009. Assessment of the hull girder ultimate strength of a bulk carrier using nonlinear finite element analysis. Analysis and design of marine structures, In: Second International Conference on Marine Structures, Lisbon, Portugal. Shu, Z., Moan, T., 2012. Ultimate hull girder strength of a bulk carrier under combined global and local loads in the hogging and alternate hold loading condition using nonlinear finite element analysis. J. Mar. Sci. Technol. 17, 94–113. Smith, C.S., 1977. Influence of local compressive failure on ultimate longitudinal strength of a ship’s hull. In: Proceedings of the PRADS, A-10, Tokyo, Japan, pp. 73–79. Ueda, Y., Rashed, S.M.H., 1984. The idealized structural unit method and its application to deep girder structures. Comput. Struct. 18 (2), 277–293. Ueda, Y., Rashed, S.M.H., Paik, J.K., 1995. Buckling and ultimate strength interaction in plates and stiffened panels under combined inplane biaxial and shearing forces. Mar. Struct. 8 (1), 1–36. Valsgård, S., Jorgensen, L., Boe, A.A., Thorkildsen, H., 1991. Ultimate hull girder strength margins and present class requirements. In: Proceedings of the Society of Naval Architects and Marine Engineers Symposium on Marine Structural Inspection, Maintenance and Monitoring, March B1-19, Arlington, USA. Vhanmanea, S., Bhattacharyab, B., 2008. Estimation of ultimate hull girder strength with initial imperfections. Ships Offshore Struct. 3 (3), 149–158. Wempner, G.A., 1971. Discrete approximation related to nonlinear theories of solids. Int. J. Solids Struct. 7, 1581–1599. Xu, W., Iijima, K., Fujikubo, M., 2011a. Hydro-elastoplasticity approach to ship’s hull girder collapse behavior in waves. In: Guedes Soares, C., Fricke, W. (Eds.), Adv. Marine Struct.. Taylor & Francis Group, Hamburg, Germany, pp. 239–247. Xu, W., Iijima, K., Fujikubo, M., 2011b. Investigation into post-ultimate strength behavior of ship’s hull girder in waves by analytical solution. In: Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2011, Rotterdam, Netherland, OMAE2011-49617, pp. 455–462. Xu, W., Iijima, K., Fujikubo, M., 2012. Parametric dependencies of the post-ultimate strength behavior of ship’s hull girder in waves. J. Mar. Sci. Technol. 17, 203–215. Yao, T., Nikolov, P.I., 1991. Progressive collapse analysis of a ship’s hull under longitudinal bending. J. Soc. Naval Arch. Jpn. 170, 449–461. Yao, T., Nikolov, P.I., 1992. Progressive collapse analysis of a ship’s hull under longitudinal bending (2nd report). J. Soc. Naval Arch. Jpn. 172, 437–446. Yao, T., Astrup, O.C., Caridis, P., Chen, Y.N., Cho, S.-R., Dow, R.S., Niho, O., Rigo, P., 2000. Special task committee VI. 2: ultimate hull girder strength. In: Proceedings of the 14th International Ship and Offshore Structures Congress, Oct. 2–6, Nagasaki, Japan. Yao, T., et al., 2006. Report of Committee III. 1: Ultimate strength. In: Proceedings of the 16th ISSC, 1, Southampton, UK, pp. 356–437. Yao,T., Fujikubo, M., Iijima, K., Pei, Z., 2009. Total system including capacity calculation applying ISUM/FEM and loads calculation for progressive collapse analysis of ship’s hull girder in longitudinal bending. In: Proceedings of the 19th International Offshore and Polar Engineering Conference, Osaka, Japan, pp. 706–713.