Experimental and numerical investigations of the ultimate torsional strength of an ultra large container ship

Marine Structures 70 (2020) 102695

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Experimental and numerical investigations of the ultimate torsional strength of an ultra large container ship Qinghu Wang a, b, c, Chonglei Wang a, b, c, Jiameng Wu a, b, c, d, Deyu Wang a, b, c, * a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China The Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, China d Marine Design & Research Institute of China, Shanghai, China b c

A R T I C L E I N F O

A B S T R A C T

Keywords: ULCS Ultimate torsional strength Scale model Warping Shear buckling Rigidity distribution

Structures of ultra large container ships (ULCS) are characterized by large deck openings and low torsional rigidity. It is essential to comprehensively figure out their collapse behaviors under pure torsion with both model experiments and numerical simulations, making an evaluation of their ultimate torsional strength. In this paper, a similar scale model of a 10,000TEU container ship has been designed and manufactured first, in which both geometric similarity and strength similarity are taken into account. Next the collapse behaviors of the test model are detailedly illustrated with both experimentally and numerically obtained results. Then discussions on warping or shear buckling deformations involved in the collapse process of the structure are conducted with extended numerical simulations. Finally, the ultimate torsional strength of the true ship is eval­ uated according to the similarity theory. Results show that it is the yielding and shear buckling of the side shells that causes the failure of the hull girder under pure torsion. Further nonlinear finite element analysis demonstrates that it may either have warping or shear buckling deformations in the torsional collapse process of the hull girder with a large deck opening, depending on the local rigidity distribution of side shells, which has a significant effect on the ultimate torsional strength of the hull girder.

1. Introduction Compared with other conventional ships or ship-like offshore structures, the hull girder of container ships is featured by large deck openings and low torsional rigidity, especially for ultra large container ships (ULCS), of which the deck opening is larger, the hatch deformation is severer, the speed is higher and the stability requirement is more, leading to a more emphasis on the torsional capacity evaluation in structure design criterion of the ULCS. Research on ultimate loading capacity of ships and ship-like offshore structures has been deeply performed by researchers worldwide for decades, having obtained extensive achievements in theoretical principles, numerical simulations and experimental investigations, among which experimental investigations are considered to be the most convincing measurement of the collapse be­ haviors and the ultimate strength of structures. Experiments on true ship structures have rarely been conducted due to the huge cost. In most cases, scale models are designed in experiments as an alternative to represent the true ship, reflecting the collapse process and

* Corresponding author. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail address: [email protected] (D. Wang). https://doi.org/10.1016/j.marstruc.2019.102695 Received 16 June 2019; Received in revised form 17 September 2019; Accepted 9 November 2019 Available online 20 November 2019 0951-8339/© 2019 Elsevier Ltd. All rights reserved.

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failure mechanisms of the prototype. Compared with local components such as plates, stiffened plates and grillages in ship structures, ultimate strength experiments of hull girders in massive research of the ultimate strength are relatively fewer, however. An experiment of the ultimate torsional strength of a ship-type hull girder with a large deck opening was conducted by Sun and Soares [1], in which the effect of boundary conditions, initial deflections as well as welding residual stresses on the ultimate torsional strength of the hull girder was discussed. However, unlike the scale model designed to be composed of double hulls in this paper, the structure in the experiment conducted by Sun and Soares is a hull girder with a single hull, not a typical ultra large container ship structure and not a scale model of a true ship, thus unable to persuasively represent the failure mode of a true container ship. Gordo and Soares [2,3] had performed the ultimate strength experiments of hull box girders made of mild steel and high tensile steel under pure bending moment to study the collapse and post-collapse behaviors of box girders, developing some useful insights into identifying the governing parameters affecting the ultimate bending strength of 3D structures and the effect of residual stresses on the collapse be­ haviors of box girders. Later an experiment of three box girders made of mild steel under pure bending moment was also conducted by Gordo and Soares to study the effect of frame spacing variation on the ultimate bending strength of box girders, contributing to establishing the dependence on the column slenderness of the panel under compression [4]. Scale model tests of a ship’s hull girder had been conducted by Iijima and Fujikubo [5], studying the post-ultimate strength behaviors of hull girders under whipping loads induced by waves. Tanaka [6] had studied the ultimate strength performance of a 5250TEU container ship under combined loads with scale models, aiming at developing an analysis method of the ultimate strength of hull girders under combined loads. Liu performed an ultimate strength experiment of a SWATH ship subjected to transverse loads with a scale model designed with scaling laws and linear static analyses [7]. Collapse of the hull girder is a nonlinear process when external loads exceed the elastic limit state of the structure. Hence it is not accurate enough to design or validate the scale model in linear range only. From the literature review above, it can be seen that few model experiment investigations of the ultimate torsional strength of ULCS under pure torsion have ever been performed. The objective of this research is not only to make an evaluation of the ultimate torsional strength of a true ultra large container ship, but to experimentally and numerically answer how the structure of ULCS collapses under pure torsion, providing the literature with experimental data on the torsional collapse behaviors of the hull girder of ULCS. The design method of scale model, the experiment procedure and the numerical model presented in this paper could be referred when model experiment of the ultimate strength of other ships or ship-like structures is conducted. 2. Design and manufacture of the scale model As mentioned earlier, the whole collapse process of structures is nonlinear so that it is not accurate enough to design the scale model with geometric similarity only, especially for the structures with a complicated geometry configuration like ULCS. The scaling prin­ ciples between scale model and prototype, including geometric similarity and strength similarity, are illustrated in this section. 2.1. Geometric similarity Similarity of loads and geometric parameters has to be satisfied between scale model and prototype. The hull girder of ULCS could be considered as a thin-walled beam with varying cross sections in longitudinal direction. Geometric similarity principles between scale model and prototype could be derived with beam torsion theory, in which force F, torsional moment T, length of ship L, breadth molded B, depth molded D and thickness of components t are regarded as the primary parameters in scaling process. The formula of the torsional stress and deformation of the hull girder is presented as: 8 T > τ¼ > > > W P > > > > τ > > > <γ ¼ G (1) T > > > θ¼ > > GIP > > > > > > : φ ¼ TL GIP where τ is the shear stress of cross sections of the hull girder, γ is the shear strain. T is the torsional moment, WP is the section modulus in torsion and G is the shear modulus. θ is the rotation angle, ϕ is the torsional displacement and IP is the polar moment of inertia. Z Z � y2 þ z2 dA ¼ Izz þ Iyy IP ¼ ρ2 dA ¼ (2) A

A

Under torsion, Geometric similarity between prototype and scale model could be expressed as:

2

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8 CT > > Cτ ¼ > > C > WP > > > > > C τ > > > Cγ ¼ < CG > CT > > Cθ ¼ > > C CIP > G > > > > > C CL > T > : Cφ ¼ CG CIP

(3)

where C denotes the similarity coefficient between prototype and scale model. Design of the cross section of scale model, including the height of neutral axis, the section modulus, the inertia moment about neutral axis and the polar inertia moment, could be guided with geometric similarity illustrated above. When it comes to the di­ mensions of local components of scale model, including longitudinal girders, stiffeners, transverse space between longitudinal girders or stiffeners, and longitudinal space of adjacent transverse frames, possible inconsistency of critical stress between prototype and scale model could occur if the test model is designed with geometric similarity only, however. 2.2. Strength similarity Strength similarity is a correction to the dimensions of local components which are playing an important role in collapse behaviors of the hull girder. Preliminary nonlinear finite element analysis has been performed to observe the collapse behaviors of the hull girder under pure torsion, numerically finding out that the yielding and shear buckling of the side shell predominantly contributes to the failure of the structure. To capture the main characteristics of collapse behaviors, shear strength similarity is chosen to be the gov­ erning function for plates in design. 2.2.1. Elastic shear buckling strength similarity for plates The elastic shear buckling strength for simply supported plates could be evaluated as [8]:

τE ¼ kτ

π2 E 12ð1

� t �2 v2 Þ b

(4)

where τE denotes shear buckling stress of plates under in-plane shear loading, υ is the Poisson’s ratio, t is the plate thickness, a and b are the length and width of the plate, respectively. kτ is the shear buckling coefficient which is defined as: 8 � �2 b > > > < 4 a þ 5:34 ​ ; ​ ​ ​ a=b � 1 kτ ¼ (5) � �2 > > > : 5:34 b þ 4 ​ ; ​ ​ ​ a=b < 1 a Elastic shear buckling strength similarity can be used to adjust the aspect ratio of plates in scale model to satisfy the strength similarity between scale model and prototype in elastic range during the collapse process. 2.2.2. Ultimate strength similarity for plates Unlike buckling due to axial compression, plates that suffer elastic shear buckling still possess a significant amount of post-buckling shear strength [9,10]. Ultimate strength similarity should be satisfied to improve the strength similarity between scale model and prototype in nonlinear range. The ultimate shear strength τu for plates could be given in the following formula [8]: � � 8 τE τE � 0:5 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​0< 1:324 > > τ τy > y > > > � �3 � �2 � � τu < τE τE τE ¼ 0:039 τE (6) þ 0:388 ​ ​ ​ 0:5 < 0:274 þ 0:676 �2 τy > τy τy τy τy > > > > > τE : >2 ​ ​ ​ 0:956 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​

τy

where τy ¼ p1ffi3ffiσy is the shear yielding strength. 2.2.3. Ultimate strength similarity for stiffened panels Compared with the longitudinal strength of hull girders under bending, stiffeners may not contribute as much to the torsional strength of hull girders under torsion. Stiffened panels are the main structural components to bear the compression when the hull girder is subjected to bending or torsion, thus the ultimate strength similarity of stiffened panels between scale model and prototype 3

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should be taken into account. Ultimate strength of stiffened panels under compression could be presented as [11]:

σu 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ σ y β0:28 1 þ λ3:2 where β is the plate slenderness ratio, and β ¼ bt

(7) qffiffiffi

E,

σy

λ ¼ πar

qffiffiffi

E,

σy

r¼

qffiffi

I A

is the radius of gyration of the stiffener including associated

full-width plating; I is the moment of inertia and A is the cross-sectional area of the stiffener including associated full-width plating. And it should be noted that the ultimate shear strength of stiffened panels is similar to that of plates when stiffeners are rigid enough [8]. With numerical simulations, the aspect ratio, the slenderness ratio of corresponding plates, the space between longitudinal girders and the span of transverse frameworks in scale model should be repeatedly adjusted according to strength similarity to numerically achieve the consistency of the collapse process between scale model and prototype. More detailed information and verifications about the similarity theory and model design are illustrated in Refs. [12,13], which are not stated again in this paper for the present study purpose. 2.3. Manufacture of the scale model In this paper, a 10,000TEU container ship is adopted as the object ship and the hull girder located in the mid-ship section is chosen to be the study case, of which length is 25.2 m, breadth is 48.2 m and height is 29.2 m. The principal characteristics of the object hull section are shown in Fig. 1 in conjunction with Tables 1 and 2, where Fig. 1 represents the typical transverse mid-ship section of the ULCS and Tables 1 and 2 show the main dimensions and most details of the ship. With a comprehensive consideration of loading capacity and load carrying capacity of the existing laboratory platform, a compartment between two neighboring transverse bulkheads is also adopted in the scale model of which length is 1.6 m, breadth is

Fig. 1. Transverse mid ship section of the 10,000 TEU container ship (all dimensions in mm). 4

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Table 1 Principal particulars of the object ship. Description

Value(m)

Length O.A. Length B.P. Breadth molded. Depth molded. Design draft Scantling draft

337 320 48.20 27.20 13.00 15.00

Table 2 Main details in the object ship. Description

Stiffeners (mm)

Plate Thickness (mm)

Upper deck 2ND deck 4791 platform Outer&Inner bottom shell Outer&Inner side shell Bilge side shell

760 260 320 200 150 350

62 EH 15 AH 14 AH 23&16 AH 16&13 AH 19 AH

� 58 EH FB � 10 AH BP � 12 AH BP � 12 AH BP � 11 AH BP � 11/150 � 15 AH TB

1.81 m and depth molded is 1.134 m. Scantlings for manufacture of the scale model are presented in Fig. 2. Transition blocks are set up to transmit loads and reduce the effect of boundary conditions. And the transition blocks have to be rigid enough to hold the load transmission to make sure that the torsional moment is well transmitted to the test model. For this purpose, thickness of all components of the transition blocks is chosen to be 8 mm. 3. Experiment 3.1. Experiment setup As is shown in Fig. 3, the test model is clamped at one end by the reaction frame which is rigid enough to hold the required load, with two hydro-cylinders fixed at the other end, simultaneously providing a pair of forces of the same magnitude up and down to produce a pure torsional moment on the hull girder. Actual assembly of the experiment setup is presented in Fig. 4. 3.2. Arrangements of strain gauges Preliminary numerical analysis has been performed in order to search the regions of high possibility to yield in the hull girder. Based on the numerical results, several locations of three typical cross sections are chosen to be measured by single strain gauges or three-direction strain gauges, which are presented in Fig. 5. In consideration of the symmetry of test model and the anti-symmetry of the applied load, strain gauges are mainly arranged at one side and the ones attached to the other side are used to check the antisymmetry of structure response during loading process. Both the inner side shell and the outer side shell are divided into three parts which are the upper part, the middle part and the lower part respectively to be measured with single strain gauges to detect the strains of different parts of side shells. Locations close to the upper deck and bottom are detected with three-direction strain gauges. Several sites of inner bottom and outer bottom plates are detected with single strain gauges. To more clearly illustrate the arrange­ ments of the strain gauges in this experiment, a 3D sketch is given as Fig. 6. 3.3. Material properties tests Tensile tests for the standard specimens of the plates of different thickness have to be conducted before performing this experiment to accurately obtain the yielding strength of the material used to fabricate the test model. Plates with thickness of 8 mm, 5 mm, 4 mm and 3 mm were selected as tensile specimens according to the plates used in this model. Dimensions of the standard specimen are illustrated in Fig. 7. The load-displacement curves of the standard specimens in the tensile tests are shown in Fig. 8. It can be observed that the yielding strength of the specimens of all kinds is 300 MPa, which should be adopted in numerical analyses. 3.4. Experiment procedure Loading process is conducted with hydro-cylinders step by step with strains and deflections automatically read in every loading step. Deflections of the model are small in early stage of loading process and the structure is still in linearly elastic range. Loading increments can be a little bit larger at early stage. But when the structure gets into nonlinear stage, the loading increments should be smaller to capture the detailed progressive collapse characteristics. The stepped loading process in this experiment is shown in Fig. 9. This experiment is conducted in the following procedure. 5

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Fig. 2. Scantlings for manufacture of the scale model (all dimensions in mm).(a) Longitudinal section in center plane of the scale model. (b) Typical cross section of the scale model (section A-A). (c) The bulkhead of the scale model (section C-C).

(1) Small loads are applied to test the experiment setup, examining the performance of data acquisition system and test model. The load is set to be 50 kN with a loading rate of 20 kN/min, loading and unloading repeatedly to partially release the residual stress in the structure and to check the repeatability of test data and to observe the linear relations of loads and strains in elastic range. ~ (2) Loading process is conducted in three stages as shown in Fig. 9. Loading increment of 40 kN is adopted in 0 kN600 kN, with 20 ~ ~ kN loading increment in 600 kN1200 kN and 10 kN loading increment in 1200 kN1500 kN. Responding displacements and strains are acquired at every loading step until the structure collapses. 6

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Fig. 3. 3D sketch of the experiment setup.

(3) The structure is considered to reach the ultimate limit state when a sudden and considerable reduction of the applied load is observed with obvious buckling deformations appearing on side shells. Sharp noise can also be heard accompanying the occurrence of buckling deformations. After that even a small loading increment would cause continuous and larger deflections, indicating that the structure is unable to further bear more loads. The corresponding torsional moment at this time is considered to be the ultimate torsional moment of the structure. 3.5. Experiment results 3.5.1. Experimental load-displacement response curve of the model The experimentally obtained load-displacement curve of the model is presented in Fig. 10, in which the vertical coordinate denotes the loads applied by hydro-cylinders and the horizontal coordinate is the displacements of the loading point. Elastic limit state and ultimate limit state are respectively marked with point E and point U on the curve. It can be observed that the elastic limit strength and the ultimate limit strength of the test model is 850 kN and 1045 kN, respectively. Before the structure reaches its elastic limit state, the applied loads and the displacements of the loading point on the test model are in well linear relation. After that the nonlinear rela­ tionship between loads and displacements begins to occur with the slope of the curve decreasing gradually, which indicates that the rigidity of the structure has a continuous decrease. That means the same loading increments would cause larger and larger dis­ placements at this stage. More and more regions get into plastic state accompanied with the emergence of some noise. When the load reaches the ultimate limit state of the structure, a sudden and significant reduction of the applied load is observed with obvious shear buckling deformations appearing on side shells. 3.5.2. Symmetry examination of the experiment To check the symmetry of the model and the anti-symmetry of the applied load, two typical cross sections of section A and section C (shown in Fig. 5(a)) are chosen to examine the anti-symmetry of structure responses, by which the symmetry of the model and the antisymmetry of the applied load can be verified. Pairs of strain gauges (38,31), (37,32), (39,42), (40,41), (35,34) are selected in section A to plot the strain-loading curves, which are presented in Fig. 11(a) and Fig. 11(b). Similarly, pairs of strain gauges (43,50), (44,49), (54,51), (53,52), (45,48) are also selected to check the anti-symmetry feature of structure responses in section C, which are shown in Fig. 11(c) and (d). It needs to be noted that the strains shown in Fig. 11 are the normal strains (εx ), which are measured by the strain gauges arranged in the longitudinal direction of the hull girder to observe the warping effect. As is known, the strain gauges can detect the strains very well in the linear stage of structure responses. When collapse process of the structure gets into nonlinear stage, the strains at these measured points will have a sharper and shaper increase with the same loading increments or even present some disorders (shown in Fig. 11(a) and (c)), which is not good for the visualization and examination of the anti-symmetry feature of structure responses. As a result, to more explicitly display the anti-symmetry feature of structure responses, the obtained strains εx in the linear stage of the hull girder’s collapse process are presented in Fig. 11(b) and (d). The strain-loading curves of every two points in symmetry have been plotted in the same color to better display the symmetry feature. It could be observed from the strain-loading curves in both section A and section C that the structure responses at both sides of the structure are in good anti-symmetry, which gives a verification of the symmetry of the model and the anti-symmetry of the applied load. 3.5.3. Warping effect on the bottom plates To observe the warping effect on the bottom plates, warping strains (axial strain εx ) are measured with the strain gauges arranged in the longitudinal direction of the hull girder. The strain-loading curves of the measured points on both the outer bottom plate and the inner bottom plate are also plotted. As is shown in Fig. 12, the strain-loading curves of the measured points in section A, section B and section C are plotted in red, green and black, respectively. It can be seen from Fig. 12 that the stress distributions of both outer bottom 7

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Fig. 4. Actual assembly of the experiment setup. (a) Hull girder (b) Hydro-cylinders (c) Overall view.

plate and inner bottom plate are also in left and right anti-symmetry, which can be better displayed by the curves in the linear stage of the structure responses shown in Fig. 12(b) and (d). From the experimentally obtained strains shown in Fig. 12(a) and (c), it can be observed that the resulting strains of the bottom plates stay at a low level in the whole collapse process except several special locations like strain gauges of No. 6 and No. 25 which are close to the diagonal edge of the outer bottom plate, implying that bottom plates are in 8

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Fig. 5. Strain gauges arrangement of the test model: (a) Locations of typical cross sections. (b) Strain gauges at section A. (c) Strain gauges at section B. (d) Strain gauges at section C.

low damage and not the main support members when the hull girder is subjected to torsion. Strain-loading curves of all selected points are in well linear relations before the elastic limit strength and start to behave in nonlinear relations after the elastic limit strength. When external loads approach the ultimate limit strength, strains are larger and larger with the same loading increments till largely flow at the ultimate limit state. From the distribution of the resulting strains at different measured points, it can be seen that, just as what is shown is Fig. 12(b) and (d), the bottom plates have been deformed in the longitudinal direction of the hull girder by warping effect. 9

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Fig. 6. 3D sketch of strain gauges arrangements.

Fig. 7. Dimensions of the standard specimen (all dimensions in mm).

Fig. 8. Load-displacement curves of the standard specimens.

3.5.4. Strain-loading curves of the side shells Strain-loading curves of the measured points on both outer side shell and inner side shell are presented in Fig. 13, in which curves representing the strain-loading responses of section A, section B and section C are respectively plotted in black, red and blue. It needs to be noted that the strains shown in Fig. 13 are the oblique strains (ε45� ), which are measured by the strain gauges that are specially � arranged at an angle of 45 to the longitudinal direction of the hull girder. Compared with the bottom plates which are in low damage, 10

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Fig. 9. Stepped loading process in the experiment.

Fig. 10. Load-displacement curve of the test model.

most regions of side shells have yielded, which can be seen from Fig. 13(a) and (c). It can also be observed from Fig. 13(b) that the curves are divided into two distinguished clusters, one is the curves of strain gauges of No. 1, No. 13 and No. 22 which represent strainloading responses of the upper part of side shell, and the other one is the curves of strain gauges of No. 2, No. 14, No. 23, No. 3, No. 15 and No. 24 which represent strain-loading responses of the middle and lower part of side shell. Obviously, stress of the upper part keeps considerably lower than that of the middle and lower part during the collapse process. The side shells exhibit a stepped yielding feature with the middle and lower part yielding earlier than the upper part, which will be stated again with numerical analysis in Section 4.2.2. What should also be noted is that the stress distribution of side shells is almost identical at these three typical cross sections during the collapse process, which can be observed by comparing the curves in different color groups. The curves have been grouped in different colors, each color representing a cross section. It is easy to find out that the three curves in each color are identically distributed in Fig. 13(b), indicating that the stress states of the corresponding measured points at each cross section are identical. That’s to say, the stress state of side shell at one typical cross section could represent that of the whole side shell when the hull girder is under pure torsion, which has been experimentally confirmed in this study. As is shown in Fig. 13(c) and (d), inner side shell displays a similar stress distribution and collapse feature with the outer side shell, of which the middle and lower part suffers a significantly higher stress level than the upper part. In addition, the measured strains on inner side shell are opposite in direction with that of the outer side shell, indicating that, for the hull girder with double side shells, the outer side shell and the inner side shell are in opposite stress state when the hull girder is under pure torsion. 3.5.5. Three-direction strain loading curves of the upper part of the side shells Section A and Section C are close to the transverse bulkheads and experience an abrupt section change, which may suffer a complicated stress state under the warping and shear effect, especially for the upper part of the side shells. Therefore, three-direction 11

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Fig. 11. Anti-symmetry examination of structure responses. (a) The whole loading process (section A); (b) The linear stage only (section A); (c) The whole loading process (section C); (d) The linear stage only (section C).

strain gauges are employed to measure the strains of the upper part of the side shells in section A and section C. Considering the symmetry of the hull girder structure, the three-direction strain gauges on the side shells of one side of the hull girder are chosen to display the experimentally obtained strains, which are shown in Fig. 14. It can be seen from the strains measured by the three-direction strain gauges that the upper parts of the side shells in section A and section C experience both a large axial strain (εx ) and a large oblique strain (ε45� ), which means that the regions which are close to the bulkheads are significantly affected by both warping and shear effect. By the comparison of the normal strains εx in section A (No. 39) and section C (No. 51), it can be observed that the upper parts of the side shells in section A and section C are in opposite axial stress state under the warping effect, with tension in section A and compression in section C. And this phenomenon can also be clearly seen in the contour of the normal stress σ x of the hull girder (see Fig. 19(b)). The oblique strains ε45� on the outer side shell and the inner side shell are also in opposite direction. 3.5.6. Collapse mode of the model Plastic deformations of the model are presented in Fig. 15, in which bottom plates, longitudinal girders, transverse frameworks and stiffeners are not displayed because these members are in low damage and not visibly deformed. Due to the anti-symmetry of structure responses, outer side shell and inner side shell at one side of the hull girder are chosen to display the permanent deformations of the model, from which it can be observed that the structure mainly collapses at side shells, especially at the outer side shell with obvious shear buckling waves on the panels (Fig. 15(a)). Diagonal buckling will be induced in a plate which is subjected to a pure shear [10], tension and compression acting perpendicularly along two diagonals, which means that the side shells mainly fail in the shear failure mechanism, causing the collapse of the hull girder under pure torsion. 4. Nonlinear finite element analysis 4.1. Numerical model An experimentally validated numerical model could serve as an extension and a supplement of experiments. As shown in Fig. 16, a finite element model is created with HYPERMESH and ABAQUS in this study. Factors that affect the results of numerical simulations 12

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(caption on next page) 13

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Fig. 12. Strain-loading curves of bottom plates. (a) The whole loading process (outer bottom); (b) The linear stage only (outer bottom); (c) The whole loading process (inner bottom); (d) The linear stage only (inner bottom).

such as mesh size and boundary conditions have been investigated in Refs. [12,14], which are not elaborated again in this paper. For stiffened panel structures which are fabricated by fusion welding, initial deflections are inevitable and could have a significant effect on the collapse behaviors of the hull girder. For the numerical model, the initial deflections have been imposed into the hull girder members by modeling the shape and magnitude of the initial deflections. Corresponding to true ship structures, three types of initial deflections are taken into account, which are the thin-horse mode for local plates between stiffeners, the column-type initial deflections for plate related stiffeners and the sideways initial deflections for independent stiffeners, respectively [14–16]. For the thin-horse mode of initial deflections of the local plates wop , �mπx� �πz� wop ¼ w0 sin sin (8) a b where w0 is the amplitude of initial deflections, a is the distance between two adjacent transverse frames and b is the distance between two adjacent longitudinal stiffeners. m is the number of buckling half waves, which is taken as an integer corresponding to a/b which has to satisfy . pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a b � mðm þ 1Þ (9) The amplitude of the initial deflections of the local plates w0 can be defined as follows [17]: 8 < 0:025β2 t ​ for ​ slight ​ level w0 ¼ 0:1β2 t ​ for ​ average ​ level : 0:3β2 t ​ for ​ severe ​ level

(10)

in which t is the plate thickness; E is Young’s modulus; σY is the yielding stress of material and β is the plate slenderness parameter which is defined by rffiffiffiffiffi b σY β¼ (11) t E

Fig. 13. Strain-loading curves of side structures. (a) The whole loading process (outer side shell); (b) The linear stage only (outer side shell); (c) The whole loading process (inner side shell); (d) The linear stage only (inner side shell). 14

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(caption on next page) 15

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Fig. 14. Strain loading curves of the measured points on the upper part of side shells. (a) NO.37 (section A, outer side shell); (b) NO.39 (section A, inner side shell); (c) NO.49 (section C, outer side shell); (d) NO.51 (section C, inner side shell).

It should be noted that the average level is adopted for the initial deflections in the numerical model. For the column-type initial deflections of the plate related stiffeners wos , �πx� � πz � a wos ¼ sin sin 1000 a 2b

Fig. 15. Collapse mode of the test model. (a) Outer side shell. (b) Inner side shell. (c) Overall view. 16

(12)

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Fig. 16. Finite element model of experiment.

For the sideways initial deflections of independent stiffeners due to angular rotation about panel-stiffener νos , �π x� ay νos ¼ sin 1000hw a

(13)

in which hw is the web height of stiffeners. Boundary and loading conditions are identical to those illustrated in experiment setup in Section 3.1. Four-node doubly curved shell element S4R and three-node triangular shell element S3 are used for the mesh of the numerical model. The material properties are taken from the standard specimen tensile tests, of which the yielding stress is 300 MPa. Based on the material tests, the material properties in numerical simulations are assumed to be elastic–perfectly plastic material with yielding stress of 300 MPa, Young’s modulus of 206 GPa and Poisson’s ratio of 0.3. 4.2. Numerical results 4.2.1. Comparison of experimental and numerical load-displacement response curves Experimental and numerical load-displacement response curves are presented in Fig. 17, from which it can be observed that the numerical model could excellently predict the elastic limit strength of the test model, although there exist some discrepancies in the early stage of the loading process as a result of the slip of jigs. When external loads exceed elastic limit strength, collapse process gets into plastic stage, resulting a larger discrepancy of the overall collapse process between numerical analysis and experiment. However, the ultimate strength of the numerical model is very close to that of test model, with 1087 kN for the numerical model and 1045 kN for the test model, of which the error is only 3.8%. This phenomenon, which is that good agreement of the ultimate strength between numerical model and test model can be achieved but large discrepancies still exist in the intermediate process of collapse, can also be found in other research when numerical model and test model are compared [7], indicating that the numerical model can well predict the special points like elastic limit state and ultimate limit state, but may perform not so well when it comes to predicting the in­ termediate process of collapse, especially for the nonlinear stage. It is known that the overall collapse process of the structure is the gradual accumulation of the failures of local support members, of which the failure modes can be significantly affected by inevitable manufacturing defects such as geometrical imperfections and welding residual stresses which are hard to be quantitatively simulated

Fig. 17. Experimental and numerical load-displacement response curves. 17

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when numerical analysis is conducted. For the structures with a complicated geometry configuration like ULCS, the collapse process is rather complicated. It is hard to simulate the failure process of corresponding local support members and the interactions between them, causing the difficulties in achieving good agreement of the intermediate process of collapse between numerical model and test model. However, the global rigidity of the structure may not be considerably affected by above assumed reasons, making it possible to accurately predict the ultimate strength of test model with numerical analysis that can be used in engineering design. The issues illustrated in this section can be studied in future research quantitatively and detailedly. 4.2.2. Comparison of experimental and numerical warping strains Strain gauges of several locations which are suffering a large axial stress (warping stress) state are chosen to plot the axial strainloading curves in the linear stage of the structure response. Correspondingly, the numerically obtained strains of the element at the position of the specific strain gauge are extracted. The comparison of the experimentally obtained warping strains and the numerically obtained warping strains is shown in Fig. 18. The warping strains of the selected points at the elastic limit state of the structure are presented in Table 3, in which εex and εnum denotes the experimentally obtained warping strains and the numerically obtained warping strains, respectively. It can be seen from Fig. 18 that there exist quite a large discrepancy of the warping strain-loading curves between experiment and numerical analysis, which means that the numerical analysis may perform not so well in accurately predicting the resulting strains of a specific point of the structure. In fact, the resulting strains of an element are evaluated by the average strains of the nodes attached to this element. As a result, it may be difficult to accurately evaluate the resulting strains of a specific point in numerical simulations. However, the numerical simulation can well predict the overall stress distributions and the collapse behaviors of the structure. 4.2.3. Collapse mode analysis with numerical results Compared with experiments, numerical results could visualize the collapse mode of the structure in a more convenient and detailed way. Von-Mises stress distributions of the hull girder at elastic limit state and ultimate limit state are shown in Fig. 19(a). It can be observed from Fig. 19(a(1)) that the hull girder reaches its elastic limit state after the whole middle and lower part of the double side shells yields. However, most regions of the upper part of the double side shells are still at a low stress level at this elastic limit state. And the double bottom plates are also in low damage except for the bilge part which locates at the corner of cross sections. As the loading process further continues, more and more regions of the upper part of the double side shells begin to yield and the shear buckling deformations begin to occur on the panels of the middle and lower part of the double side shells. The hull girder finally collapses under the yielding and shear buckling of the double side shells (shown in Fig. 19(a(2))). The normal stress σx distributions of the hull girder at elastic limit state and ultimate limit state are presented in Fig. 19(b), from which it can be observed that there exist large axial stresses (warping stresses σ x ) in the regions which are close to the transverse bulkheads on the upper part of the side shells and the bilge part of the double bottom plates, with compression at one end and tension at the other one. This axial stress distribution indicates that the side shells have been deformed in the longitudinal direction by warping effect. The bottom plates are also deformed in the similar way by warping effect, which has been illustrated by the experimentally obtained results shown in Fig. 12(b) and (d). The normal stress σy distributions of the hull girder at elastic limit state and ultimate limit state are presented in Fig. 19(c). Like the distributions of the normal stress σx , the normal stress σ y also manly exists in the regions that are close to the transverse bulkheads on the upper deck and the bilge part of the double bottom plates. The normal stress distributions shown in Fig. 19(b) and (c) indicate that the upper part of the double side shells, the bilge part of the double bottom plates and the upper deck are playing an important role in carrying the warping stresses induced by warping effect, especially for the regions which are close to the transverse bulkheads.

Fig. 18. Experimental and numerical warping strains at several specific points in the linear stage of the structure response. 18

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Fig. 19. Stress distributions of the hull girder at elastic limit state and ultimate limit state. (a) Von-Mises stress; (b) normal stress x σ; (c) normal stress y σ; (d) shear stress xy τ. 19

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Table 3 Warping strains of the selected points at elastic limit state. No.

εex

39 37

Error

εnum

592.7 � 10

6

799.2 � 10

6

529.2 � 10

6

291.1 � 10

6

35

1060.3 � 10

42

402.6 � 10

Note: ‘Error’ is evaluated by ðεnum

6

6

35% 45%

773.8 � 10

6

27%

872.3 � 10

6

117%

εex Þ=εex .

Fig. 20. Stepped yielding features of outer side shell. (a) Non-uniform thickness plates. (b) Uniform thickness plates. 20

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The shear stress τxy distributions of the hull girder at elastic limit state and ultimate limit state are presented in Fig. 19(d), from which it can be observed that the vertical members such as the double side shells and the bilge part of the double bottom plates are suffering a high shear stress level at both elastic limit state and ultimate limit state, indicating that the hull girder collapses in the shear failure mechanism. As is known, the horizontal members such as the bottom plates and deck plates are the main components to carry the longitudinal bending stresses (axial tension or compression) under longitudinal bending moments. And the vertical members like side structures are the main components to carry the shear stresses induced by torsion. The hull girder in this research collapses mainly in the shear failure mechanism, of which the shear stresses are dominating the torsional failure process (shown in Fig. 19(d)), not the normal stresses induced by warping effect (shown in Fig. 19(b) and (c)). As shown in Fig. 13(b), the stepped yielding feature of side shells has been illustrated with experimentally measured stress of typical cross sections in Section 3.5.4, which is that the middle and lower part of side shells stays at a considerably higher stress level and yields earlier than that of the upper part. Here in this section, the stepped yielding feature is discussed again with visualized numerical results. It can be observed from Fig. 20(a) that the yielding of side shells exhibits a stepped feature during the collapse process. And it should be noted that the side shells of the test model are designed to be composed of stepped plates with the thickness of 8 mm for upper part, 4 mm for middle part and 3 mm for lower part to try to satisfy the strength similarity illustrated in Section 2.2. In fact, the side shells of a true ULCS are also composed of non-uniform thickness plates which are arranged in the similar way of the model design in this experiment, of which the upper part tends to be manufactured with thicker plates than those of middle and lower part to guarantee the rigidity required by a comprehensive consideration of design criterions. To figure out whether it is the stepped distribution of plate thickness that causes the stepped yielding feature of side shells, another numerical model with side shells composed of plates with a uniform thickness of 5 mm is also created to qualitatively examine the effect of local distribution of plate thickness on the stepped yielding feature. As can be seen from Fig. 20(b), the side shell composed of uniform thickness plates also takes the stepped yielding feature, indicating that it is not the stepped distribution of plate thickness that causes the side shell yielding in a stepped feature, as long as the variation of thickness is not too large which will barely happen in the design and manufacturing of side shells. However, it should be noticed that there exists a distinguished difference of the deformation shape at ultimate limit state between two numerical models. For the side shell with stepped thickness plates, shear buckling waves are observed on the lower part of the panel accompanied with no apparent warping deformations (see Fig. 20(a)). In contrary, for the side shell with uniform thickness plates, warping deformations occur at the top corner of one end of the side shell with no obvious shear buckling deformations (see Fig. 20(b)). It needs to be further investigated why there exist two distinguished deformation modes of the torsional collapse behaviors of a hull girder with a large deck opening under pure torsion, which will be qualitatively discussed in the next section. 5. Discussions on warping or shear buckling deformations It is known that warping and shear effects are normally developed when a thin-walled beam with open cross sections is subjected to torsion [18–20]. Correspondingly, it has been experimentally and numerically found that there are two deformation modes at ultimate limit state of the collapse process of the hull girder with a large deck opening subjected to a pure torsional moment. In this section, corresponding to the model used in this experiment, numerical models with the side shells composed of plates with different thickness distributions are respectively created to qualitatively study the deformation modes involved in the collapse process of the model, of which the plate thickness arrangements are shown in Table 4. Numerically obtained collapse modes of model No. 1 ~ No. 4 are presented in Figs. 21~24, respectively. Von-Mises contours are shown in Figs. 21(a)~24(a), to see where the deformations form. Maximum out-of-plane displacements induced by warping or shear and the dimensionless progressive collapse process of the models are respectively presented with the curves in different colors in Fig. 21(b)~24(b), to observe when the deformations occur. For model No. 1, obvious warping deformations occur in the regions shown in Fig. 21(a), of which all locations are close to the abrupt section changes while no apparent shear buckling deformations are observed on the panel, however. This can also be quan­ titatively seen in Fig. 21(b), from which it can be found that the out-of-plane displacements induced by warping increase faster and faster when external loads exceed elastic limit strength, and an even more greatly sharper increase rate emerges when external loads approach the ultimate limit strength. The structure collapses rapidly during the sharp increase of warping deformations while the outof-plane displacements induced by shear stay almost zero in the whole collapse process.

Table 4 Plate thickness arrangements of different models. No.

Upper part

Middle part

Lower part

1 2 3 4

5 mm 4 mm 4 mm 8 mm

5 mm 7 mm 8 mm 4 mm

5 mm 4 mm 3 mm 3 mm

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Fig. 21. Collapse mode of model NO.1. (a) Contour of von-Mises stress. (b) Maximum out-of-plane displacements induced by warping or shear.

For model No. 2 and No. 3, of which the upper and lower parts are made of thinner plates than those of model No. 1, in addition to the warping deformations which are similar to those in model No. 1 (see Fig. 22(a) and Fig. 23(a)), mild and obvious shear buckling deformations are respectively observed on lower parts of the side shells of two models, indicating that the side shells with thinner lower parts are prone to shear buckling. It can also be observed in Figs 22(b) and 23(b) that the out-of-plane displacements induced by shear display a significant increase trend in the collapse process of model No. 2 and No. 3. Side shells of model No. 4 are composed of plates with the thickness arrangements that are most similar to those of true ULCS, of which the upper deck and upper part of side shells are manufactured with considerably thicker plates than those of middle and lower part. Accompanied with few warping deformations, obvious shear buckling deformations are observed on lower parts of the side shells (see Fig. 24(a)), indicating that the upper deck and upper part of side shells which are manufactured with thicker plates could help resist warping deformations. It can be seen from Fig. 24(b) that the out-of-plane displacements induced by shear increase rapidly in collapse process while the displacements induced by warping stay at a low level. 22

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Fig. 22. Collapse mode of model NO.2. (a) Contour of von-Mises stress. (b) Maximum out-of-plane displacements induced by warping or shear.

It is known that a change in plate thickness is inherently changing the rigidity of the plate since plate thickness is a decisive factor of the rigidity of a plate. In this section, this paper has demonstrated a reason why there will exist different deformation modes in collapse process of a hull girder with a large deck opening under pure torsion, which is that there may exist different local rigidity distributions of side shells. As mentioned above, warping and shear effects are normally coexisting in a thin-walled beam with open cross sections which is subjected to torsion. When upper deck and upper part of side shells are rigid enough to withstand the warping effect, shear buckling will dominate in deformation modes, like model No. 4. On the other hand, if upper deck and upper part of side shells are not rigid enough, warping deformations may predominantly exist in deformation modes, like model No. 1. However, if both upper deck, upper part of side shells and lower part of side shells are not rigid enough, shear buckling and warping deformations will co-dominate in deformation modes, like model No. 3. It has been demonstrated above that local rigidity distributions of side shells could have a significant effect on the deformation modes of them in collapse process. To study the effect of local rigidity distributions of side shells on the ultimate torsional strength of 23

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Fig. 23. Collapse mode of model NO.3 (a) Contour of von-Mises stress. (b) Maximum out-of-plane displacements induced by warping or shear.

the hull girder, curves representing dimensionless progressive collapse process are plotted in Fig. 25, in which MTi and MTU1 denotes the applied torsional moment of model No. i and the ultimate torsional strength of model No. 1, respectively. As shown in Fig. 25, there exist some discrepancies of the ultimate torsional strength of different models, of which the model with side shells of uniform rigidity has a larger ultimate torsional strength than those with side shells of non-uniform rigidity and the ultimate torsional strength tends to decrease with the increase of the degree of the non-uniformity of rigidity. For a true ultra large container ship which is the prototype of the test model in this experiment, plate thickness arrangements of side shells can be observed in Fig. 1, of which the upper deck and the side shells that are close to the upper deck are respectively manu­ factured with plates of 62 mm and 56 mm, then plates of 30 mm are used as the transition regions of thickness variations, and the rest of side shells are made of plates of 16 mm. The numerically obtained collapse mode of the hull girder of the true ultra large container ship under pure torsion is presented in Fig. 26, from which it can be observed that shear buckling waves occur on the panels of the side 24

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Fig. 24. Collapse mode of model NO.4. (a) Contour of von-Mises stress. (b) Maximum out-of-plane displacements induced by warping or shear.

shells at ultimate limit state, especially on the middle and lower part of the side shells. 6. Evaluation of the ultimate torsional strength of true ship As is illustrated in Section 2, the test model in this experiment is designed through the combination of geometric similarity and strength similarity, in which strength similarity can provide the local strength correction to improve the similarity between scale model and true ship. And the method had been numerically verified by achieving the consistency of the converted progressive collapse process between scale model and true ship [12]. However, to precisely convert the ultimate torsional strength of a test model into that of a true ship, there still exist some difficulties. One of the most difficult points is how to quantitatively control welding imperfections such as initial deflections and welding residual stresses which are currently almost impossible to achieve the similarity between the scale model and the prototype, which 25

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Fig. 25. Comparisons of the ultimate torsional strength of different models.

Fig. 26. Collapse mode of a true 10,000TEU container ship under pure torsion (only shown 1/2 FE model).

should be further investigated in future study. As shown in Fig. 1, the configuration of a true ship is far more complicated than that of the test model both in local components and material properties as a result of the limitation of the budget and the capacity of the existing testing platform, possibly causing some discrepancies in collapse behaviors and ultimate strength between the true ship and the test model. Compared with the true ship, the stiffeners in the scale model are more sparsely arranged. If the stiffeners in the scale model are arranged under the scale of geometrical dimensions, the distance between the stiffeners in the scale model will be too small. Consequently, it will be quite difficult to manufacture the scale model and the effect of the welding induced deflections and residual stresses on the hull girder’s rigidity will be largely enhanced. It is known that the effective breadth of the plate panels could have a significant effect on the bending stiffness of the hull girder [21]. Although the effect of the effective breadth of plate panels on the torsional collapse behaviors of the hull girder is still not figured out, it is one of the most difficult points to quantitatively assess the effect of the effective breadth of plate panels on the converting process and evaluation of the ultimate torsional strength between the scale model and the true ship. The scaling effect of the similarity theory used to design the scale model of the true ship is still unknown, leaving some uncertainties in the converting process between test model and true ship, which also needs to be investigated with massive numerical simulations and corresponding experiments in future study. Based on the existing similarity theory and the model experiment, the ultimate torsional strength of a true 10,000TEU container ship is evaluated with Cτ ¼ CCWT in Eq. (3). p

Yp Cτ is assumed to be Cτ ¼ σYm , in which σYP and σYm are the yielding stress of the materials of true ship and test model, respectively.

σ

For true ship, σ YP is 355 MPa for AH materials and 390 MPa for EH materials. And 355 MPa is adopted in scaling process because most regions of the side shells are manufactured with AH materials. Of course, the scaling effect of materials can be further investigated to make appropriate corrections to this assumption. W

CWp is assumed to be CWp ¼ Wmp , in which Wp and Wm are the section modulus in torsion of true ship and test model, respectively.

According to the model design, CWp is 3704 in this experiment.

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CT is assumed to be CT ¼ Tmp , in which Tp and Tm are the ultimate torsional strength of true ship and test model, respectively. In this

experiment, the ultimate torsional strength of test model is 2436.94 kNm (1045 kN � 2.332 m). So the ultimate torsional strength of the true ship could be evaluated by (14)

Tp ¼ Cτ � CWp � Tm

which is 1:068 � 107 ​ kNm (355 300 � 3704 � 2436:94 ​ kNm). Although there exist some unresolved difficulties in the converting process of the ultimate strength between test model and true ship, this model experiment has experimentally provided a better understanding for the collapse process and failure mechanisms of ultra large container ships under pure torsion, which could be referred in the design, manufacturing and optimization of the structures of ultra large container ships. 7. Conclusions In this paper, the torsional collapse behaviors and the ultimate torsional strength of a 10,000TEU container ship have been detailedly investigated through both model experiment and numerical simulations. Based on the experimental and numerical analysis conducted in this paper, the following conclusions can be drawn: (1) With the symmetry examination of experiment, it can be observed that the stress-loading responses of the structure are in good anti-symmetry at typical cross sections, indicating that the model and the applied loads are respectively in good symmetry and anti-symmetry in this experiment. (2) Results obtained from both model experiment and numerical analyses indicate that the main support members for the hull girder of an ultra large container ship subjected to pure torsion are the double side shells, not the double bottoms instead. It is the yielding and shear buckling of side shells that causes the collapse of the hull girder. (3) Both experimental and numerical results indicate that the side shells exhibit a stepped yielding feature which is that the middle and lower part tends to yield earlier than the upper part, which is also verified with numerical simulations of the model with side shells composed of uniform thickness plates to prove that it is not the stepped thickness distribution of side shells that causes the stepped yielding feature of them. (4) It has been experimentally confirmed that the structure responses of the side shell at one typical cross section could represent those of the whole side shell when the hull girder is under pure torsion. For the double side shells, inner side shell and outer side shell are in opposite stress state when the hull girder is under pure torsion. (5) With the comparison of experimentally and numerically obtained load-displacement response curves, it can be observed that the numerical model can well predict the elastic limit strength and the ultimate limit strength of the test model, while there exist some discrepancies in the intermediate process of collapse. (6) It may either have warping or shear buckling deformations in the collapse process of the hull girder under pure torsion, depending on the local rigidity distributions of side shells, which could also significantly affect the ultimate torsional strength of the hull girder. The non-uniform rigidity of side shells tends to decrease the ultimate torsional strength of the hull girder. (7) Even though there still exist some unresolved difficulties in converting the ultimate strength of a test model to that of a true ship, the ultimate torsional strength of a true 10,000TEU container ship has been evaluated based on the existing similarity theory and model experiment. In future work, better methods for designing the scale model and controlling the welding imperfections should be proposed to incorporate more complete similarity to improve the scaling process of the ultimate strength and collapse behaviors between test model and true ship. Scaling effect of the existing similarity theory should also be studied to determine the uncertainties of the converting process between scale model and prototype under different scaling factors to make a more comprehensive understanding of scaling process. Further ultimate strength experiments of container ships under combined loads are encouraged to be conducted to obtain a more complete understanding of the ultimate strength and collapse behaviors of the hull girder of container ships. Acknowledgements The authors would like to gratefully acknowledge the Chinese Government Key Research Project KSHIP-II Project (Knowledgebased Ship Design Hyper-Integrated Platform, with No. 201335), and the National Natural Science Foundation of China (No. 51809168, No. 51979163 and No. 51809167), by which this paper is supported. References [1] [2] [3] [4]

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