An investigation of ultimate strength for vloc stiffened panel structures

Modern Transportation June 2013, Volume 2, Issue 2, PP.23-38

An Investigation of Ultimate Strength for VLOC Stiffened Panel Structures Hung Chien Do1, 2#, Wei Jiang1#, Jianxin Jin1, Xuedong Chen1 1. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China. #Email: jiangw@mail.hust.edu.cn 2. Faculty of Naval Architecture, Ho Chi Minh City University of Transport, Ho Chi Minh city, 70000, Vietnam. #Email: dhchienvn@hotmail.com

Abstract Ultimate strength of ship and offshore structures is a very important issue that has drawn considerable attention from researchers, especially in the field of structural analysis applied to ship offshore, aircrafts, and base landings, etc. Finite Element Method (FEM) has been applied and developed to solve the complicated problem accurately. Particularly, with the help of aid tool and software, as well as Nonlinear Finite Element Method (NFEM), the ultimate strength of large model is improved significantly and accurately. The aim of this paper is to investigate the ultimate stiffened panel strength in the cargo hold areas for a very larger ore carrier built in China. The stiffened panel in the bottom, side and deck structures is a structural member which basically plays an important role in construction of ship and offshore.

Keywords: Ultimate strength; Very Large Ore Carrier; compressive load; lateral pressure; stiffened panel.

1 INTRODUCTION As marine nowadays has been developed quickly and has become the key industry with many ships built year by year in the world. Under the effect of global energy crisis, there have had many studies on the design of a new type of vessel. The market of shipping is very large thanks to the lowest cost in terms of logistics. In China, with the rapid development of economic, iron ore is necessary, and thus, carrier industry has been propelled to meet the huge shipping. As a result, a very large number of ore carriers have been built to import ore from foreign countries.

However, the ships are designed by means of traditional method known as approaching margin working stress.

Recently, ship and offshore structures have been performed by nonlinear finite element analysis with a large number of literatures concerning ultimate strength, ultimate limit state (ULS) aspects. These studies have been utilized in plates, stiffened plates and three cargo holds in mid-ship areas(Amlashi and Moan 2009). Estimating the ultimate strength of continuous plates are studied and developed by a simplified method which proposes formulae with accurate predictions (Fujikubo, Yao, et al. 2005) on ultimate strength compared to NFEM results. Assessment on ultimate strength for unstiffened plates surrounded by supporting members under combined uniaxial/biaxial compressive loads and lateral pressures has been performed based on a series of benchmark studies on the methods(Paik, Kim, and Seo 2008a). The rectangular plates under biaxial loadings also study the Elasto-plastic buckling behavior (Wang et al. 2009). The one-bay plate models from the 1/2 + 1 + 1/2 bay continuous plate is investigated by (Paik and Seo 2009a, b)to reveal the fact that the ultimate strength of unstiffened plate under biaxial compressive loads is significantly influenced by rotational restrain under lateral pressure actions. A new method to analyze the geometric nonlinear behavior of plates is developed by (Kee Paik et al. 2012), in which elastic large deflection or post-buckling of plates with partially restraint rotation and the torsional rigidity condition are applied.

Regarding the stiffened panel assessment, the results are obtained in order to continuously develop improved methods for prediction ultimate strength with accuracy and efficiency. To estimate the ultimate strength, there are - 23 www.ivypub.org/mt

some direct methods proposed by (Caldwell 1965, Mansour et al. 2008) and after simplified method (Smith’s method and the idealized structural unit method ISUM) is widely applied for analysis of hull girder under longitudinal bending loads only. By applying the mentioned methods to analysis, the results are rapidly obtained, but the accuracy depends on the average stress-strain relationship of individual structural members. The simplified method is developed (Yao 2003)for evaluation of the collapse strength for hatch covers with a folding type and a side sliding type of bulk carrier. (Paik, Kim, and Seo 2008b, c) performed ALPS/ULSAP method to determine the ultimate limit state of stiffened panel under uniaxial or biaxial compression and lateral pressures, the results are compared to ANSYS nonlinear finite element analysis(Inc. 2010).Related to the parameter effects on the collapse behavior of stiffened panels, shaped model in study with two (1 + 1) full bays was studied by FEM. In addition to the ultimate strength of plates, by means of nonlinear FE software to analyze two half bays plus one full bay (1/2 + 1 + 1/2 bay) model in the longitudinal direction by (Zhang and Khan 2009, Fujikubo, Harada, et al. 2005). A method for ultimate assessment via nonlinear finite element analysis is developed by (Paik and Seo 2009a, b). The influence of the stiffener’s geometry and boundary conditions on the ultimate strength of stiffened panels under combined thrust acting load including 3 bays, 1/2 + 1 + 1/2 bays, 1 + 1 bays and 1 bay, are analyzed by (Xu and Guedes Soares 2012).

In scope of this paper, we consider the difference of ultimate strength results between two half bays plus one full bay (1/2 + 1 + 1/2 bays) models under various parameters. In actual structures, a stiffened panel often includes plates, stiffeners, longitudinal girders and transverse floors. The model including 27 longitudinal stiffeners can be translated into 2 bays + 1 span (2B1S), 2 bays + 2 spans (2B2S), 2 bay + 3 spans (2B3S), 3 bays + 1 span (3B1S), 3 bays + 2 spans (3B2S) and 3 bays + 3 spans (3B3S) models. They are under uniaxial and biaxial compressive acting load with or without lateral pressure, satisfying simple supported boundary condition.

The aim of this study is to improve the economic efficiency in ship design and construction technology, the reducing ship hull structural weight is taken into account. The plates and stiffened panel are basic assembly structures in a ship. They are found in the deck, side and bottom structures, and even appear in the superstructures. In order to solve this problem, a hypothetical ship is studied and focused on the plates and stiffened panels in the cargo hold area(Do, Jiang, and Jin 2012). In this paper, the ultimate strength of the very large ore carrier is investigated with rectangular plates and stiffened panel object ship. By applying International Associate Classification Societies Common Structural Rules (IACS CSR) to bulk carrier class type and NFEM, the stiffened panel and plate models are designed and analyzed, in order to choose models complying with actual requirements (IACS 2006a, b).

2 ULTIMATE STRENGTH OF HYPOTHETICAL MODEL Ship bottom stiffened panel is subjected to combined biaxial compressive load and lateral pressure. Generally, the thickness of plate, the properties of stiffeners as well as longitudinal girder and transverse floor are determined by designer throughout Common Structure Rules. In this study, these particulars are applied in a hypothetical of very Large Ore Carrier (VLOC) built in China with deadweight of about 380,000 DWT, and known as the largest Bulk Carrier class; amid ship section and a cross sectional stiffener are shown in Fig. 1. The material, high tensile AH 36, is used for ship both of plating and component of structures, whose properties: Yield stress with  Y  355 MPa; Young’s modulus with E = 205.8 GPa and Poisson’s ratio   0.3 , are taken into account in ANSYS NFEM. Choosing plating and stiffened panel structures in the bottom of cargo hold is conveniently ultimate strength analysis because the structural shaped is hardly to be changed in three dimensions. This candidate method complies with requirement and guider of IACS CSR system.

TABLE 1 GEOMETRIC PROPERTY OF THE STIFFENED PANELS

Model

a mm

b mm

ns

tpmm

hwmm

twmm

bfmm

tfmm

2B1S / 3B1S

3660

915

4

21.80

470

10

162

16

2B2S / 3B2S

3660

915

8

21.80

470

10

162

16

2B3S / 3B3S

3660

915

12

21.80

470

10

162

16

Note: B = bay and S = span, i.e. 2B1S = 2-bay 1-span model - 24 www.ivypub.org/mt

2.1 Numerical model for analysis In the cargo hold of object ship of our study, the structures can be summarized as follows: 

The bottom with double hull structures including outer bottom and inner bottom, stiffeners along longitudinal edges, the distance of two stiffeners close to each other is b, similarly, the distance of two transverse floors is a .

The length of 2-bay and 3-bay panels is 2a and 3a, respectively (a and b are shown in Table 1).



In the port and starboard side, structural system is single hull with the adoption of transverse frame system.



On the deck, the longitudinal structural system and single deck are used with longitudinal stiffeners, girders and transverse frames.

Deck stiffened panels

Side plating

Inner bottom stiffened panels

Outer bottom stiffened panels

FIG. 1STIFFENED PANELS IN THE MIDSHIP

FIG. 2PLATE–STIFFENER AND STIFFENED PANEL

The procedure to determine the properties of plating and stiffened panel plate thickness can be specified into two steps. Firstly, by using the formula of CSR, the thickness of plates and the particular of stiffeners in case of mild steel are calculated. Secondly, the correction formulae are used to accurately determine the principal dimension for high tensile steel. In ship structural design, plate and stiffeners frequently welded to each other are seen as combination structures which are called continuous stiffened plate structure in Fig. 2(a) and plate-stiffener combination model in Fig. 2(b). The particulars of stiffened panel cross section are described in Table 1/Fig. 1, these models have the same dimensions of cross sectional stiffener (hw and tw is the height and thickness of web plates, bf and tf is the breadth and thickness of flange plates, respectively) and tp is thickness of plate, and the differences between six models are the number of stiffeners ns (ns = 4, 8, 12 stiffeners), longitudinal girders and transverse floors.

The principal parameters affect ultimate strength of plate and stiffened panels under compressive load are the plating and beam-column slenderness, defined as follows(Hughes, Paik, and Béghin 2010): the slenderness of plating,



b tp

 Yp E

(1)

Where, tp is the thickness of bottom plate and  Yp is the yielding stress of plates; and there is a difference between  Yp and  Y when the plate and the stiffeners are not made of unique type of material. In this study, the plating and the stiffeners are assumed to be made of same material type that means  Yp   Y  355 MPa. The column slenderness: 

a Y r E

(2)

I As

(3)

Where, r is the radius of inertia defined as follows,

r

- 25 www.ivypub.org/mt

I and As are the inertial moment and area, respectively, of cross-section including effective width p.

The effective width of plating element is defined: p 

2





1

(4)

2

2.2Initial imperfection The initial imperfection of plate in the form of initial deflections and residual stress is caused by welding during a complex fabrication process and they are subjected to significant uncertainty related to the magnitude and spatial variation. These initial imperfections are the important parts of the ultimate strength assessment accurately because they reduce the strength performance and in calculating, they should be a significantly influential parameter. Many literatures throughout theories as conducted Fourier analysis and measurements were performed at several point of plate, and provided a total description analysis of the deformed surface of plate. The initial distortions of the stiffeners are specified into two types depending on the direction of deflection such as y direction or z direction.

Concerning a column type, the former of initial distortion follows the high direction of stiffeners and then other type corresponding to a torsional initial distortion along sideways. In this paper, an equivalent initial imperfection is applied as initial distortion of the stiffened panels, as follows(Hughes, Paik, and Béghin 2010): The local panel with initial deflection: m y y sin (5) a b Where, w0 pl  0.05 2t p and m , the number of half buckling waves, dependent on ratio of a/b, generally, is equal to a/b, but if m is not an integer, it should be determined as a minimum integer which satisfies a condition as follows, wpl  w0 pl sin

a  m  m  1 b

(6)

The initial deflection of stiffeners as column type is:

x a The initial deflection of stiffeners depends on angular rotation about panel-stiffener in the side-ways: wc  w0c sin

ws  w0 s sin

(7)

x a

(8)

Where, w0c = w0s = a/1000 in (7)and(8).

T – type Longitudinal stiffeners

ns Tra

b

ors flo C se r e v

b

Bottom plate

y

x

A

z x

Lo ng

Lateral pressure p

itu din al s B tiff ene rs

x C’

a/2

y

a A’

FIG.3NUMERICAL OF STIFFENED PANEL

a/2

y

FIG.4 STIFFENED PANEL UNDER COMBINED LOADS

In finite element analysis (FEA), the initial imperfections are firstly calculated by formulae (5), (7) and (8), and after that the shapes of initial imperfections for each model are applied by ANSYS program design language (APDL).

Actually, the evaluation of the ultimate strength is usually calculated in two steps: - 26 www.ivypub.org/mt

Step 1: Determining the first eigenvalue mode of buckling cause axial compressive load with linear analysis. When this step is completed, the initial imperfection is applied to the plate, column-type and side-way of stiffeners. In this step, the parameter of initial deflection formula in ANSYS depends on the result of first buckling linear analysis.

Step 2: After applying the initial imperfection completely to models (in this step, UPGEOM function is applied), the ultimate strength of numerical model is nonlinearly analyzed.

Hence, for convenient investigation, the effects of the boundary on the ultimate strength, the equivalent initial imperfections of the numerical model in the present paper are assumed as follows: 

Plate initial deflection wopl = b/200, the initial imperfection magnitude of local shaped plate.



Column-type initial deflection woc = a/1000, the initial imperfection magnitude of the stiffeners.



Side-ways initial deflection wos = a/1000, the initial imperfection magnitude of the stiffeners.

The numerical models built including the geometric and material nonlinearities, elastic–plastic large deflection are taken into account. During the linear as well as nonlinear analysis, the SHELL 181 element of ANSYS is used in the numerical model in Fig. 3, which is a four nodes element with six degrees of freedom at each node and can account for linear, large rotation and large strain nonlinear.

2.3 Boundary condition Generally, the stiffened panels are supported on the stronger member, and two types of boundary condition are frequently applied, namely simple supported and clamped. In the longitudinal structure system under combined axial compressive loads, when the acting load in the longitudinal edges is predominant, the effect of these two conditions is negligible as results of ultimate strength. When transvers axial compressive loads are predominant, the effect of boundary condition in the direction of longitudinal edges is significant. In scope of this study, for convenient calculation, the simply supported is adopted, with the 1/2 + 1 + 1/2 model of stiffened panels, including the numbers of 27 stiffeners and 2 bays as transverse floors in Fig.4. The parameters of this simple supported boundary condition are described as follows(Paik, Kim, and Seo 2008b), 

Along AA’ and CC’ edges, the symmetric condition is applied with ROTY = ROTZ = 0, all nodes and stiffeners nodes having an equal displacement in the x direction;



Along AC and AC’, the simply supported boundary condition is applied with UZ = 0 and ROTY = ROTZ = 0, including an equal displacement in the y direction for each edge;



Along the transverse floors (T1T1’, T2T2’ in Fig. 5 and T1T1’, T2T2’, T3T3’ in Fig. 6), the boundary condition is applied such that UZ = 0 for plate nodes, and UY = 0 for nodes of stiffener webs;



Along the longitudinal girder (L1L1’, L2L2’, L3L3’ in Fig. 5 and Fig. 6) with UZ = 0 for plate nodes.

C T2

C T3

L1'

L1'

T2

T1 A

L2'

L2'

T1 A

L1

L3'

L3' L1

C’

C’

L2

T3’ L2

T2' L3

T2' L3

T1' A’

T1' A’

FIG.52 BAYS + 3 SPANS PANEL MODEL

FIG.63 BAYS + 3 SPANS PANEL MODEL

Where UX, UY, and UZ are translation constraints in the x – coordinate, y – coordinate and z – coordinate, respectively; similarly, ROTX, ROTY and ROTZ indicate rotational constraint around the x – coordinate, y – - 27 www.ivypub.org/mt

coordinate and z – coordinate, respectively.

It is noticed that the boundary condition for 2 bays + 1 span and 2 bays + 2 spans is similarly applied to the case of 2 bays + 3 spans when 2 spans (L2L2’ and L3L3’) or 1 span (L3L3’) is excluded. In addition, the boundary condition for 3 bays + 1 span and 3bays + 2 spans is applied to similar boundary condition of 3 bays + 3 spans in Fig. 6.

The definition and application of boundary condition are the important parts in computation as well as assessment of collapse state behavior.

2.4 Load condition In this paper, numerical models are applied to compressive load as following procedure, Case 1: Acting the uniaxial compressive along edges in the x – coordinate (i.e. model under uniaxial x without lateral pressure p).

Case 2: Acting lateral pressure included compressive load in Case 1 (i.e. model under uniaxial x with lateral pressure p).

Case 3: Acting the biaxial compressive along edges in the x – coordinate and y – coordinate (i.e. model under biaxial with ratio of  x :  y  0.8 : 0.2 , without lateral pressure p).

Case 4: Acting lateral pressure included compressive load in Case 3 (i.e. model under biaxial with ratio of  x :  y  0.8 : 0.2 , without lateral pressure p).

The ratios of  x :  y (including 1.0:0.0, 0.9:0.1, 0.8:0.2, 0.7:0.3, 0.6:0.4, 0.5:0.5, 0.4:0.6, 0.3:0.7, 0.2:0.8, 0.1:0.9, and 0.0:1.0) are used in the design hull girder loading condition of the object ship. The lateral pressure p is determined by loading condition in operational assumption. In the full load condition, the hydro static pressure presses on outer bottom with the extreme magnitude, p = 0.23 MPa. Combined load conditions in this study are shown in Fig. 4.

2.5 Mesh model In evaluation of ultimate strength, the number of finite elements for rapidly and accurately obtained results can be determined especially using FEM and mesh is the computation strategy. Concerning mesh and elements are divided in the models (ISSC 2009, 2012), for the bottom plates in Fig. 4, the number of rectangular plate – shell elements along the breadth direction b and the length direction a are 6 and 20, respectively. In Fig. 5 and Fig. 6, for the stiffener web in the height direction and the flange in the breadth direction, the number of plate-shell elements is 4 and 2, respectively.

3 RESULT AND DISCUSSIONS 3.1 Ultimate strength of 2 bay stiffened panel model In this section, there are three considered model such as 2 bays + 1 span (1/2+1/2 span) – 2B1S, 2 bays + 2 spans (1/2 + 1 +1/2 span) – 2B2S and 2 bays + 3 spans (1/2 +1+1+1/2span) – 2B3S. The results obtain from ANSYS are shown in Fig. 25, Fig. 27 and Fig. 29. They are under longitudinal and transverse compressive load with effect of lateral pressures defined by full load condition of VLOC.

1) Ultimate strength of 2-bay stiffened panel without lateral pressure These collections of ultimate strength results are described in the Table2, in order to evaluate the difference of the results obtained from models, and a coefficient of variation (COV %) also known as “relative variability,” is equal to the standard deviation of a distribution divided by its mean which is included. When longitudinal compressive load is predominant  x :  y  1.0 : 0.0,  x :  y  0.9 : 0.1 and  x :  y  0.8 : 0.2 ), the ratio of longitudinal ultimate strength (LUS) and yielding stress  xu /  Y (in the x direction) with the acquired appropriate COV is very small (0.01%, 0.02% and 1.43%). While the difference of transverse ultimate strength (TUS) from these cases is significant (i.e. COV = 27.73%, 21.16% and 7.90%). In case of predominant transverse compressive load - 28 www.ivypub.org/mt

(  x :  y  0.0 :1.0,  x :  y  0.1: 0.9 and  x :  y  0.2 : 0.8 ), the values of COV are insignificant, and the maximum of COV is 6.42%. It is clear that the applied method complies with the case of the predominant longitudinal and transverse compressive load.

TABLE 2 ULTIMATE STRENGTH OF 2-BAY PANELS

Case of calculation

Without lateral pressure (p = 0) 2B1S

 x :  y  1.0 : 0.0  xu /  Y 0.7474  yu /  Y 0.0429  x :  y  0.9 : 0.1  xu /  Y 0.7498  yu /  Y 0.1539   x :  y  0.8 : 0.2  xu /  Y 0.7246  yu /  Y 0.1332  x :  y  0.7 : 0.3  xu /  Y 0.6804  yu /  Y 0.6680  x :  y  0.6 : 0.4  xu /  Y 0.5755  yu /  Y 0.7011  x :  y  0.5 : 0.5  xu /  Y 0.4594  yu /  Y 0.7042  x :  y  0.4 : 0.6  xu /  Y 0.3570  yu /  Y 0.6887  x :  y  0.3: 0.7  xu /  Y 0.3007  yu /  Y 0.6954  x :  y  0.2 : 0.8  xu /  Y 0.1705  yu /  Y 0.5494

 x :  y  0.1: 0.9  xu /  Y 0.1574  yu /  Y 0.3592  x :  y  0.0 :1.0  xu /  Y 0.1692  yu /  Y 0.2895

2B2S

2B3S

0.7475

0.7475

0.0747

COV

With lateral pressure (p = 0.23 MPa) 2B1S

2B2S

2B3S

COV

0.01%

0.6528

0.6530

0.6530

0.01%

0.0715

27.73%

0.4560

0.4612

0.5095

6.21%

0.7500

0.7502

0.02%

0.6160

0.6076

0.6076

0.80%

0.1064

0.1110

21.16%

0.5512

0.6469

0.6434

8.84%

0.7446

0.7403

1.43%

0.4453

0.4913

0.4907

5.55%

0.1485

0.1557

7.90%

0.7041

0.6493

0.6514

4.65%

0.5049

0.6727

16.01%

0.4706

0.4131

0.4157

7.49%

0.3444

0.5353

31.53%

0.3827

0.4362

0.4393

7.59%

0.2675

0.5500

36.81%

0.2065

0.2695

0.2105

15.42%

0.3337

0.6043

34.85%

0.4479

0.3645

0.4430

11.19%

0.1332

0.4334

53.02%

0.1306

0.1565

0.1147

15.74%

0.3917

0.6503

28.71%

0.3811

0.3198

0.4346

15.18%

0.0545

0.3365

67.81%

0.0608

0.0890

0.0654

21.10%

0.4845

0.6546

17.95%

0.3375

0.2031

0.3281

25.90%

0.1329

0.3090

40.14%

0.0198

0.0147

0.0398

53.61%

0.5879

0.7020

9.68%

0.4004

0.3883

0.4886

12.86%

0.1720

0.1910

6.42%

0.0781

0.0679

0.0254

48.92%

0.5637

0.6150

5.98%

0.4565

0.4215

0.2431

30.62%

0.1659

0.1661

3.04%

0.0993

0.1070

0.1075

4.43%

0.3926

0.3937

5.13%

0.3997

0.4417

0.4432

5.77%

0.1765

0.1793

2.99%

0.1449

0.1411

0.1149

12.21%

0.3180

0.3275

6.35%

0.4785

0.4699

0.3556

15.78%

In the contrast to compare these cases (in case of  x :  y  0.7 : 0.3, x :  y  0.6 : 0.4, x :  y  0.5 : 0.5,  x :  y  0.4 : 0.6 and  x :  y  0.3: 0.7 ), the difference of ultimate strength in both of x –direction and y–direction is significant. However, the distinction between results of 2B1S and 2B3S is very small, while the results derived from 2B2S are smaller than that from other two models (2B1S and 2B3S). According to the analysis of these models and application cases, when computing the ultimate strength of stiffened panel without lateral pressure, the reliable of result from 3 models can be obtained as the longitudinal or transverse compression is predominant, in other cases of load acting, the 2B1S and 2B3S models derived result with insignificant difference. Meanwhile, if the 2B2S model is used, it must be carefully considered.

The series of results obtain from 3 models 2B1S in case of under compressive load without lateral such as 2 bay + 1 span (1/2 +1/2 span) – 2B1S, 2 bay + 2 span (1/2 + 1 +1/2 span) – 2B2S and 2 bay + 3 span (1/2 +1+1+1/2 span) – 2B3S are shown in Fig. 7, Fig. 9, and Fig. 11(describing the relation of STRAIN and  xu /  Y – Ultimate strength along x – direction), in Fig. 8, Fig. 10 and Fig. 12 in which STRAIN and  yu /  Y are explained.

2) Ultimate strength of 2-bay stiffened panel with lateral pressure In case of 2 bays stiffened panel without lateral pressure, the ultimate strength is obtained from three model methods, - 29 www.ivypub.org/mt

and the variability of these results can be accepted when the predominant of longitudinal or transverse thrust loading are applied along the edges of stiffened panels. In this subsection, three of these models can be also considered when lateral pressure is in application, which is calculated from extreme full load condition with pressure p = 0.23 MPa. The summary of the results is described in Table 2, and the difference of three model methods is negligible when the longitudinal compressive is predominant (i.e.  x :  y  1.0 : 0.0,  x :  y  0.9 : 0.1,  x :  y  0.8 : 0.2 and  x :  y  0.7 : 0.3 ), the two models of cases, 2B2S and 2B3S obtain results with a very small deviation. With uniaxial longitudinal compressive load, the COV value of three models is 0.01%, and under biaxial compressive load, the maximum value of COV is 8.84% less than 10%.

ULTIMATE STRENGTH OF 2 BAY - 1 SPAN PANEL, WITHOUT LATERAL PRESSURE

ULTIMATE STRENGTH OF 2 BAY - 1 SPAN PANEL, WITHOUT LATERAL PRESSURE

0.8

0.8

0.7

0.7

 x : y = 1.0:0.0

0.6

 x : y = 0.8:0.2  x : y = 0.7:0.3  x : y = 0.6:0.4

0.4

 x : y = 0.5:0.5

0.3

 x : y = 0.8:0.2  x : y = 0.7:0.3  x : y = 0.6:0.4

0.4

 x : y = 0.5:0.5

0.3

 x : y = 0.4:0.6  x : y = 0.3:0.7

0.2

 x : y = 0.9:0.1

0.5

 yu/ Y

 xu/ Y

0.5

 x : y = 1.0:0.0

0.6

 x : y = 0.9:0.1

 x : y = 0.4:0.6  x : y = 0.3:0.7

0.2

 x : y = 0.2:0.8

0

 x : y = 0.2:0.8

 x : y = 0.1:0.9

0.1

 x : y = 0.0:1.0 0

0.5

1

1.5

2

 x : y = 0.1:0.9

0.1 0

2.5

 x : y = 0.0:1.0 0

0.5

1

Strain (x10-3 )

2

FIG.8TUS OF 2B1S WITH p = 0

ULTIMATE STRENGTH OF 2 BAY - 2 SPAN PANEL, WITHOUT LATERAL PRESSURE

ULTIMATE STRENGTH OF 2 BAY - 2 SPAN PANEL, WITHOUT LATERAL PRESSURE

0.8

0.7

0.7

 x : y = 1.0:0.0  x : y = 0.9:0.1

0.6

 x : y = 0.8:0.2

 x : y = 1.0:0.0

0.6

 x : y = 0.9:0.1

 x : y = 0.7:0.3

0.5

 x : y = 0.6:0.4

 x : y = 0.8:0.2

0.5

 yu/ Y

 x : y = 0.7:0.3  x : y = 0.6:0.4

0.4

 x : y = 0.5:0.5

0.3

 x : y = 0.5:0.5

0.4

 x : y = 0.4:0.6  x : y = 0.3:0.7

0.3

 x : y = 0.2:0.8

 x : y = 0.4:0.6

 x : y = 0.1:0.9

0.2

 x : y = 0.3:0.7

0.2

 x : y = 0.0:1.0

 x : y = 0.2:0.8 0.1

 x : y = 0.1:0.9

0.1 0

 x : y = 0.0:1.0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

0

0.2

0.4

Strain (x10-3 )

0.8

0.7

0.7

 xu/ Y

 x : y = 0.7:0.3  x : y = 0.6:0.4  x : y = 0.5:0.5  x : y = 0.4:0.6  x : y = 0.3:0.7

 yu/ Y

 x : y = 0.8:0.2

0

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

0.5

1

1.5

1.6

1.8

2

 x : y = 0.9:0.1  x : y = 0.7:0.3

0.5

 x : y = 0.6:0.4

0.4

 x : y = 0.4:0.6

 x : y = 0.5:0.5  x : y = 0.3:0.7

0.3

 x : y = 0.2:0.8  x : y = 0.1:0.9

0.2

 x : y = 0.0:1.0

 x : y = 0.2:0.8 0.1

1.4

 x : y = 1.0:0.0

0.6

 x : y = 0.9:0.1

0.2

1.2

 x : y = 0.8:0.2

 x : y = 1.0:0.0

0.3

1

ULTIMATE STRENGTH OF 2 BAY - 3 SPAN PANEL, WITHOUT LATERAL PRESSURE

0.8

0.4

0.8

FIG.10TUS OF 2B2S WITH p = 0

ULTIMATE STRENGTH OF 2 BAY - 3 SPAN PANEL, WITHOUT LATERAL PRESSURE

0.5

0.6

Strain (x10-3 )

FIG.9LUS OF 2B2SWITH p = 0

0.6

2.5

Strain (x10-3 )

FIG.7LUS OF 2B1S WITH p = 0

 xu/ Y

1.5

2

2.5

0.1 0

0

0.5

Strain (x10-3 )

1

1.5

2

2.5

Strain (x10-3 )

FIG.11LUS OF 2B3S WITH p = 0

FIG.12TUS OF 2B3S WITH p = 0

(LUS = LONGITUDINAL ULTIMATE STRENGTH, TUS = TRANSVERSE ULTIMATE STRENGTH) - 30 www.ivypub.org/mt

In other cases, the variations in ultimate strength are significant, however, the deviation of results derived from 2B1S and 2B3S are hardly negligible. It is clear that the lateral pressure and the number of transverse floor as well as longitudinal girders affecting the ultimate strength with full structural model that the result obtained will be more accurate than the other model.

ULTIMATE STRENGTH OF 2 BAY - 1 SPAN PANEL, WITH P = 0.23 MPa

ULTIMATE STRENGTH OF 2 BAY - 1 SPAN PANEL, WITH P = 0.23 MPa

0.8

0.7

0.7

0.6

 x : y = 1.0:0.0

 x : y = 0.7:0.3  x : y = 0.6:0.4  x : y = 0.5:0.5

0.3

 x : y = 0.8:0.2  x : y = 0.7:0.3  x : y = 0.6:0.4

0.4

 x : y = 0.5:0.5

0.3

 x : y = 0.4:0.6 0.2

 x : y = 0.9:0.1

0.5

 yu/ Y

 xu/ Y

 x : y = 0.8:0.2 0.4

 x : y = 1.0:0.0

0.6

 x : y = 0.9:0.1

0.5

 x : y = 0.3:0.7

 x : y = 0.4:0.6  x : y = 0.3:0.7

0.2

 x : y = 0.2:0.8

 x : y = 0.2:0.8 0.1

0

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

0.5

1

1.5

2

2.5

 x : y = 0.1:0.9

0.1 0

3

 x : y = 0.0:1.0 0

0.5

1

1.5

2

FIG.13LUSOF 2B1S, p = 0.23 ULTIMATE STRENGTH OF 2 BAY - 3 SPAN PANEL, WITH P = 0.23 MPa

ULTIMATE STRENGTH OF 2 BAY - 2 SPAN PANEL, WITH P = 0.23 MPa 0.7

0.6

0.6

 x : y = 1.0:0.0

 x : y = 1.0:0.0

 x : y = 0.9:0.1

0.5

 x : y = 0.9:0.1

0.5

 x : y = 0.8:0.2

 x : y = 0.7:0.3

0.4

 yu/ Y

 xu/ Y

 x : y = 0.8:0.2  x : y = 0.6:0.4  x : y = 0.5:0.5

0.3

 x : y = 0.7:0.3

0.4

 x : y = 0.6:0.4  x : y = 0.5:0.5

0.3

 x : y = 0.4:0.6 0.2

 x : y = 0.4:0.6 0.2

 x : y = 0.3:0.7

 x : y = 0.3:0.7

 x : y = 0.2:0.8 0.1

 x : y = 0.2:0.8 0.1

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

0.5

1

1.5

2

2.5

0

3

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

0.5

1

Strain (x10-3 )

1.5

2

3

FIG.16TUSOF 2B2S, p = 0.23 MPa

ULTIMATE STRENGTH OF 2 BAY - 2 SPAN PANEL, WITH P = 0.23 MPa

ULTIMATE STRENGTH OF 2 BAY - 3 SPAN PANEL, WITH P = 0.23 MPa

0.7

0.7

0.6

0.6

 x : y = 1.0:0.0

 x : y = 1.0:0.0

 x : y = 0.9:0.1

0.5

 x : y = 0.9:0.1

0.5

 x : y = 0.8:0.2

 x : y = 0.8:0.2

 x : y = 0.7:0.3

0.4

 yu/ Y

 xu/ Y

2.5

Strain (x10-3 )

FIG.15LUSOF 2B2S, p = 0.23

 x : y = 0.6:0.4  x : y = 0.5:0.5

0.3

 x : y = 0.7:0.3

0.4

 x : y = 0.6:0.4  x : y = 0.5:0.5

0.3

 x : y = 0.4:0.6 0.2

 x : y = 0.4:0.6

 x : y = 0.3:0.7

0.2

 x : y = 0.3:0.7

 x : y = 0.2:0.8 0.1

0

3

FIG.14TUSOF2B1S, p = 0.23 MPa

0.7

0

2.5

Strain (x10-3 )

Strain (x10-3 )

 x : y = 0.2:0.8 0.1

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

0.5

1

1.5

2

2.5

3

0

 x : y = 0.1:0.9  x : y = 0.0:1.0 0

Strain (x10-3 )

0.5

1

1.5

2

2.5

3

Strain (x10-3 )

FIG.17LUS OF 2B3S, p = 0.23

FIG.18TUS OF 2B3S, p = 0.23 MPa

The comparisons between the results obtained without lateral pressure (in Fig. 7, Fig.9 and Fig.11) and with lateral pressure (in Fig. 13, Fig. 15 and Fig. 17) the ultimate strength (in Table 2) along the longitudinal edge (x – direction) decreases about 15% when acting of later pressure. However, the ultimate strength along the transverse edge (y– direction) is significantly increased; which is described in Fig. 14, Fig. 16 and Fig. 18. It is certain that the - 31 www.ivypub.org/mt

translations of combined load direction take an important role in the capacity of ultimate strength, particularly from x– direction to y– direction.

In addition, the strain values different when models reach ultimate limit state are also discussed. With the appearance of hydrostatic pressure, the model is quicker to reach buckling state. In comparison with longitudinal ultimate strength in case of without lateral pressure (in Fig. 7, Fig. 9 and Fig. 11), the maximum value of strain is 0.0018 and with lateral pressure (in Fig. 13, Fig. 15 and Fig. 17), the maximum value strain is about 0.0017. It is clear that the effect of lateral not only decreases the longitudinal ultimate strength, but also increases the possibility to reach ultimate limit state faster. However, in some cases, the later pressure can increase the capacity of transverse ultimate strength.

3.2 Ultimate strength of 3-bay stiffened panel model In section 3.1, the series of results are obtained from computation of 2-bay models, the structural form in calculating as well as the appearance of lateral pressure affect ultimate strength capacity. In this section, this problem should be illustrated more clearly especially with longitudinal ultimate strength, and the collection of 3-bay stiffened panels include 3 bays + 1span (3B1S), 3 bays +2 spans (3B2S) and 3 bays+ 3 spans (3B3S). The summaries of results obtained from these models are shown in Fig. 26, Fig. 28 and Fig. 30.

TABLE 3 ULTIMATE STRENGTH OF 3-BAY PANELS

Case of calculation

 x :  y  1.0 : 0.0  x :  y  0.8 : 0.2  x :  y  0.6 : 0.4  x :  y  0.4 : 0.6  x :  y  0.2 : 0.8

 x :  y  0.0 :1.0

Without lateral pressure (p = 0)

With lateral pressure (p = 0.23 MPa)

3B1S

3B2S

3B3S

COV

3B1S

3B2S

3B3S

COV

 xu /  Y

0.7432

0.7437

0.7436

0.03%

0.6315

0.6263

0.6357

0.75%

 yu /  Y

0.0448

0.0456

0.0456

1.08%

0.5377

0.4620

0.5519

9.34%

 xu /  Y

0.7068

0.7288

0.7121

1.61%

0.4139

0.4233

0.4108

1.56%

 yu /  Y

0.1059

0.2304

0.1557

38.21%

0.6785

0.6619

0.6671

1.27%

 xu /  Y

0.5546

0.3154

0.5536

28.90%

0.1856

0.1038

0.2105

33.49%

 yu /  Y

0.6911

0.1877

0.5758

54.43%

0.2536

0.6873

0.4430

47.13%

 xu /  Y

0.3545

0.0752

0.3217

60.95%

0.0219

0.1088

0.0566

70.10%

 yu /  Y

0.6956

0.3857

0.6331

28.67%

0.3205

0.6827

0.5749

35.36%

 xu /  Y

0.2413

0.2432

0.2262

3.93%

0.0581

0.0543

0.0637

8.07%

 yu /  Y

0.6853

0.6864

0.6645

1.81%

0.3200

0.3135

0.3506

6.05%

 xu /  Y

0.1152

0.1196

0.1139

0.03%

0.0958

0.0845

0.0817

8.58%

 yu /  Y

0.5844

0.5914

0.5780

1.08%

0.2752

0.2293

0.2253

11.39%

1) Ultimate strength of 3-bay stiffened panel without lateral pressure Table 3 details the longitudinal ultimate strength obtained from three models in each appropriate case under compressive load along the longitudinal and transverse edges. The difference of these values is small as the longitudinal thrust load is predominant (  x :  y  1.0 : 0.0 and  x :  y  0.8 : 0.2 ), and the maximum value of COV is 1.61%, particularly in the uniaxial compressive load, the COV is 0.03%. According to the analyses of results and the accuracy of ultimate strength derived from the 3 bay panel models are higher than 2 bay panel models. In uniaxial compressive load, 2-bay models give the average value ratio of ultimate strength and yields stress  xu /  Y  0.7475 and this value is obtained from 3 bay model of  xu /  Y  0.7435 .

In case of the transverse thrust load is predominant (i.e.  x :  y  0.0 :1.0 and  x :  y  0.2 : 0.8 ), the COV of longitudinal is also good agreement, and the maximum value of COV is 3.93%, meanwhile, this value of COV in 2 bay panels model is 48.92%. Obviously, the 3 bay models give much more accurate results than 2 bay models in the appropriate case behavior. Actually, in the assessment of ultimate strength, a suitable choice of calculation defines the accurate results. Concerning the transverse ultimate strength in Table 3, these models give also better results than - 32 www.ivypub.org/mt

2 bay panel models, the value of COV derive is small in the case of the longitudinal and transverse compressive load are predominant, in  x :  y  0.0 :1.0 and  x :  y  0.2 : 0.8 , the maximum of COV is 1.81%.

ULTIMATE STRENGTH OF 3 BAY - 1 SPAN PANEL, WITHOUT LATERAL PRESSURE

ULTIMATE STRENGTH OF 3 BAY - 1 SPAN PANEL, WITH P = 0.23 MPa

0.8

0.7

0.7

0.6

0.6

0.5

 xu/ Y

 xu/ Y

0.5 0.4

 x : y = 1.0:0.0

0.3

0.4

 x : y = 1.0:0.0

0.3

 x : y = 0.8:0.2

 x : y = 0.8:0.2 0.2

 x : y = 0.6:0.4

0.2

 x : y = 0.6:0.4

 x : y = 0.4:0.6

0

 x : y = 0.4:0.6 0.1

 x : y = 0.2:0.8

0.1

 x : y = 0.0:1.0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

 x : y = 0.2:0.8  x : y = 0.0:1.0 0

0.5

1

Strain (x10-3 )

1.5

2

2.5

3

Strain (x10-3 )

FIG.19 LUS OF3B1S, p = 0

FIG.20TUS OF3B1S,p = 0.23 MPa

ULTIMATE STRENGTH OF 3 BAY - 2 SPAN PANEL, WITHOUT LATERAL PRESSURE

ULTIMATE STRENGTH OF 3 BAY - 2 SPAN PANEL, WITH P = 0.23 MPa

0.8

0.7

0.7

0.6

0.6

0.5

 xu/ Y

 xu/ Y

0.5 0.4

 x : y = 1.0:0.0

0.3

0.4

 x : y = 1.0:0.0

0.3

 x : y = 0.8:0.2

 x : y = 0.8:0.2 0.2

 x : y = 0.6:0.4

0.2

 x : y = 0.6:0.4

 x : y = 0.4:0.6

0

 x : y = 0.4:0.6 0.1

 x : y = 0.2:0.8

0.1

 x : y = 0.0:1.0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

 x : y = 0.2:0.8  x : y = 0.0:1.0 0

0.5

1

Strain (x10-3 )

1.5

2

2.5

Strain (x10-3 )

FIG.21LUS OF3B2S,p = 0

FIG.22LUSOF 3B2S,p = 0.23 MPa

ULTIMATE STRENGTH OF 3 BAY - 3 SPAN PANEL, WITHOUT LATERAL PRESSURE

ULTIMATE STRENGTH OF 3 BAY - 3 SPAN PANEL, WITH P = 0.23 MPa

0.8

0.7

0.7

0.6

0.6

0.5

 xu/ Y

 xu/ Y

0.5 0.4

 x : y = 1.0:0.0

0.3

0.4

 x : y = 1.0:0.0

0.3

 x : y = 0.8:0.2

 x : y = 0.8:0.2 0.2

 x : y = 0.6:0.4

0.2

 x : y = 0.6:0.4

 x : y = 0.4:0.6

0

 x : y = 0.4:0.6 0.1

 x : y = 0.2:0.8

0.1

 x : y = 0.0:1.0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

 x : y = 0.2:0.8  x : y = 0.0:1.0 0

0.5

Strain (x10-3 )

FIG.23LUS OF 3B3S,p = 0

1

1.5

2

2.5

3

Strain (x10-3 )

FIG.24LUS OF 3B3S,p = 0.23 MPa

In the other case, when the difference of longitudinal and transverse compressive load is insignificant (i.e.  x :  y  0.6 : 0.4 and  x :  y  0.4 : 0.6 ), similar to 2 bay panel model, the value of COV is very big especially 3B2S gives the long ultimate strength much lower than 3B1S and 3B3S. The difference of 2B1S and 3B1S are about 3.6% and 0.7% of the longitudinal ultimate strength as the  x :  y  0.6 : 0.4 and  x :  y  0.4 : 0.6 , respectively, and in consideration of transverse ultimate strength, these differences are 1.4 % and 1.0 %. In regard to 2B3S and 3B3S, the error results of  x :  y  0.6 : 0.4 are 0.7% and 4.7% for longitudinal and transverse ultimate strength, - 33 www.ivypub.org/mt

respectively. Concerning the error results of  x :  y  0.4 : 0.6 , these values are 4.4 % and 3.3 %.

FIG.25VON-MISES STRESSES OF 2 – BAY

FIG.26VON-MISES STRESSES OF 3-BAY MODELS

(WHEN LONGITUDINAL COMPRESSIVE LOAD IS PREDOMINANT - AMPLIFICATION FACTOR OF 25)

- 34 www.ivypub.org/mt

FIG.27 VON-MISES STRESSES OF 2-BAY

FIG.28 VON-MISES STRESSES OF 3-BAY MODELS

(MODELS ARE UNDER BIAXIAL COMPRESSIVE LOAD - AMPLIFICATION FACTOROF 25) - 35 www.ivypub.org/mt

FIG.29 VON-MISES STRESSES OF 2-BAY

FIG.30 VON-MISES STRESSES OF 3-BAY MODELS

(WHEN TRANSVERSE COMPRESSIVE LOAD IS PREDOMINANT – AMPLIFICATIONFACTOR OF 25)

Otherwise, these error results of 2B2S and 2B3S are 17.9% and 43.8% for  x :  y  0.6 : 0.4 appropriate ratio of  xu /  Y and  yu /  Y ; for  x :  y  0.4 : 0.6 , these errors are 38.1% and 20.4% of  xu /  Y and  yu /  Y , respectively. Following this analysis, when an assessment on the ultimate strength of stiffened panel is made, these models 3B1S and 3B3S give good agreement results, and in actual calculation, they are also applied to a structural stiffened panel of the deck and bottom. If the 3B2S model is used for calculation in the case that the longitudinal or - 36 www.ivypub.org/mt

transverse compressive load is predominant, while in another case, the results are obtained with large error and it should be contrasted to carry out the experience. The longitudinal ultimate strengths in this case are shown in Fig. 19, Fig. 21 and Fig. 23. Dealing with the effect of lateral pressure, these models will be considered in the following subsection.

2) Ultimate strength of 3-bay stiffened panel with lateral pressure p = 0.23 MPa In the present study, three models 3B1S, 3B2S, and 3B3S under combined biaxial compressive load and lateral pressure p = 0.23 MPa are considered. The ultimate strengths of these three stiffened panels are shown in Table 3 and Fig. 20, Fig. 22 and Fig. 24. The error of longitudinal ultimate strength is insignificant when the compressive along longitudinal edges or transverse edges of these panels appropriate  x :  y  1.0 : 0.0,  x :  y  0.8 : 0.2 ,  x :  y  0.2 : 0.8 and  x :  y  0.0 :1.0 . This maximum value of COV is 8.58%, especially when structural is under only compressive load in the x-direction, often called uniaxial compressive, COV = 0.75%, i.e. a very small value. In these cases, the difference of transverse ultimate strength values is also negligible, when the panel is under combined biaxial compressive load with lateral pressure p = 0.23 MPa, the maximum value of COV is 6.05 % less than 10 %, and this error is adopted.

Following these results, the capacity of ultimate strength when lateral pressures take part in the combination of load acting is reduced about 15.1% - 24.9% in the case of uniaxial compressive, and in case of structures under biaxial compressive, this reduction is 41.9% - 75.2%, which is referred to Table 3. In the transverse ultimate strength aspect, when the lateral pressure is applied to structures, the ratio of  yu /  Y increases significantly when the longitudinal compressive load is predominant. In case of  x :  y  0.4 : 0.6 and  x :  y  0.6 : 0.4 , the error of ultimate strength obtained from these three models is very large. However, the average difference of value between with and without lateral pressure for transverse ultimate strength is negligible. In the general calculation of ultimate strength in ship structural longitudinal system, the longitudinal compressive load is always predominant, and the ratio of  x :  y frequently is 1.0:0.0 to 0.7:0.3. Because with the ship having length above over 90 meters the system of structures in the main hull is frequently longitudinal. The predictable results of ultimate strength applied to longitudinal structural system can be obtained with small error values.

4 CONCLUSION In the investigation of ultimate strength for VLOC as well as the large bulk carrier, the necessity to carry out experiments of stiffened panel is not neglected. However, with the larger model in the actual process of predicting, the ultimate limit state is very complicated, because it requires the giant size of equipment that is not real. To solve this problem, the performances of finite element method in nonlinear analysis are applied and operated. Designers can build a large model, division a lot of elements, with the power and resources of the computer unit processor (CPU), which needs complicated requirements. In the course of calculation, the designers know how to build a suitable for models as well as strategy simulation and it plays an important role in accurate results. In this paper, the problem is solved, the conclusions are as follows: 1) Concerning the uniaxial compressive load, the ultimate strength is obtained from the stiffened panel with insignificant error.

2) During the investigation on biaxial combined load with or without lateral pressure, the accuracy has been obtained from the case of the predominant longitudinal or transverse compressive load.

3) The lateral pressure applied to the stiffened panel should reduce about 15% capacity of longitudinal ultimate strength; meanwhile, the transverse ultimate strength is significantly increased. This conclusion shows that the lateral pressure takes an important part in capacity of ultimate strength of ship structures.

4) In comparison between 2 and 3 bays stiffened panels, the 3 bay models give more accuracy than 2 bay models with the same condition of boundary and acting load.

5) By applying NFEM to this study, the results of ultimate strength allow predicting ultimate limit state without experiments when condition of equipment is not demanded for actual requirements.

Although the NFEM is coded by ANSYS, the results are obtained from a series of models in outer bottom with high reliability; and these models in the other area of ship structures (i.e. on the deck structures, in the inner bottom structures, inside as well as in bulkhead structures) should be analyzed by other FEM software in the future.

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