Torsional hull girder response of containerships ... Thesis- Ng… · 1 Torsional hull girder...

1

Torsional hull girder response of containerships – feasible with Cargo Hold models?

Ngoc-Do NGUYEN

Master Thesis

presented in partial fulfillment of the requirements for the double degree:

“Advanced Master in Naval Architecture” conferred by University of Liege "Master of Sciences in Applied Mechanics, specialization in Hydrodynamics, Energetics and

Propulsion” conferred by Ecole Centrale de Nantes

developed at University of Rostock in the framework of the

“EMSHIP” Erasmus Mundus Master Course

in “Integrated Advanced Ship Design”

Ref. 159652-1-2009-1-BE-ERA MUNDUS-EMMC

Supervisor:

Prof. Robert Bronsart, University of Rostock Prof. Patrick Kaeding, University of Rostock Dr. Jörg Rörup, Germanischer Lloyd Mr. Helge Rathje, Germanischer Lloyd

Reviewer: Prof. Dario Boote, University of Genova

Rostock, February 2012

2 Ngoc-Do NGUYEN

Master Thesis developed at University of Rostock, Germany

CONTENTS

ABSTRACT ............................................................................................................................... 9

1. INTRODUCTION ............................................................................................................ 10

1.1. Current research objectives ....................................................................................... 10

1.2. Review of related work .............................................................................................. 10

1.3. Torsion of container ships ......................................................................................... 14

2. ENTIRE FE MODEL ANALYSIS .................................................................................. 20

2.1. General ....................................................................................................................... 20

2.2. Load cases generated from SHIPLOAD ................................................................... 21

3. CARGO HOLD FE MODEL ANALYSIS ...................................................................... 26

3.1. Cargo Hold Model in GL 2011 rules ......................................................................... 26

3.2. Cargo Hold Model in ABS rules ............................................................................... 28

3.3. Cago Hold Model in future HCSR rules ................................................................... 29

4. WORK WITH THE SMALL CONTAINERSHIP OF 2700 TEU .................................. 32

4.1. The procedure to create and work with Cargo Hold models ..................................... 32

4.2. FE model of 2700 TEU containership ....................................................................... 33

4.3. Cargo Hold 115-189 and Cargo Hold 78-189 ........................................................... 35

4.4. Influence of the constraint beams’ stiffness .............................................................. 37

4.4.1. Work with one type of constraint beams in one section ..................................... 37

4.4.2. Work with different types constraint beams in one section ................................ 43

4.5. Influence of nodal loads at end cross sections of Cargo Hold FE model .................. 45

4.6. Influence of the independent points’ position ........................................................... 46

4.7. Influence of longitudinal force .................................................................................. 48

5. WORK WITH THE BIG CONTAINERSHIP OF 11000 TEU ....................................... 51

5.1. FE model of 11000 TEU containership ..................................................................... 51

5.2. Cargo Hold 74-97 ...................................................................................................... 52

5.3. Influence of (Ax, Ay, Az) and (Ixx, Iyy, Izz) on Cargo Hold stiffness ............................. 54

5.4. Influence of the constraint beams’ stiffness in case of pure torsion and real-load case

....................................................................................................................................56

5.4.1. Pure torsion whose maximum value exists in the midship area ......................... 56

5.4.2. In real-load case and pure torsion whose minimum value exists in the midship

area .............................................................................................................................59

Torsional hull girder response of containerships - feasible with Cargo Hold models? 3

“EMSHIP” Erasmus Mundus Master Course, period of study September 2010 – February 2012

5.5. Influence of torsional moment in the fore and aft part on the midship area in case of

pure torsion ........................................................................................................................... 63

5.6. Independence of constraint beams’ stiffness on warping stress in Cargo Hold model

....................................................................................................................................65

5.7. Application of the suitable boundary condition of the big ship (11000 TEU) on the

small one (2700 TEU) .......................................................................................................... 67

6. CONCLUSIONS .............................................................................................................. 69

ACKNOWLEDGEMENTS ..................................................................................................... 71

REFERENCES ......................................................................................................................... 72

4 Ngoc-Do NGUYEN


LIST OF FIGURES

Figure 1: Characteristics of the 48000 dwt bulk carrier ........................................................... 11

Figure 2: Direct load calculations for the Entire model ........................................................... 12

Figure 3: Characteristics of the 176000 dwt bulk carrier ......................................................... 12

Figure 4: Stresses in main deck due to vertical bending .......................................................... 13

Figure 5: Stresses in main deck in oblique sea ......................................................................... 13

Figure 6: Stresses in main deck in roll condition ..................................................................... 14

Figure 7: Torsional moment due to wave force and effect of low position shear center ......... 15

Figure 8: Twisting and warping in midship area of 11000 TEU ............................................. 16

Figure 9: Shear flow over a closed thin-walled section ........................................................... 16

Figure 10: Warping rigidity of an open thin-walled section .................................................... 17

Figure 11: Idealized section and sectorial area diagram .......................................................... 17

Figure 12: High warping deflexion in midship due to torsion ................................................. 19

Figure 13: Boundary conditions in Entire models ................................................................... 21

Figure 14: Flow chart to determine global loads according to class rules ............................... 22

Figure 15: SHIPLOAD user interface ...................................................................................... 23

Figure 16: Definition of masses in SHIPLOAD ...................................................................... 24

Figure 17: Half-breadth Cargo Hold model in GL 2011 rules ................................................. 27

Figure 18: Support of the full-breadth Cargo Hold model in GL 2011 rules ........................... 28

Figure 19: Extent of Cargo Hold model according to ABS rules ............................................ 29

Figure 20: Boundary conditions of cago hold model in ABS rules ......................................... 29

Figure 21: Boundary conditions of Cargo Hold model in HCSR ............................................ 31

Figure 22: Working procedure in order to justify the boundary conditions ............................. 32

Figure 23: Entire FE model of 2700 TEU containership ......................................................... 34

Figure 24: Load diagrams of 7 considered loadcases of 2700 TEU containership .................. 35

Figure 25: Cargo Hold model 115-189 of 2700 TEU containership ........................................ 36

Figure 26: Cargo Hold 78-189 of 2700 TEU containership ..................................................... 36

Figure 27: Sections 115 of 2700 TEU containership ............................................................... 38

Figure 28: Sections 189 of 2700 TEU containership ............................................................... 38

Figure 29: Hatch diagonals whose the relative deflexions are considered .............................. 40

Figure 30: Relative deflexion of the hatch diagonals in Cargo Hold model using initial

constraint beams ....................................................................................................................... 40

Figure 31: Relative deflexions of the hatch diagonals in Cargo Hold model using BC 3 ....... 41



Figure 32: Difference of relative deflexions of the hatch diagonals between Entire and Cargo

Hold models using different constraint beams ......................................................................... 43

Figure 33: Relative deflexions of Cargo Hold models with and without end cross sections'

nodal loads ................................................................................................................................ 45

Figure 34: Cargo Hold model 115-189 with independent points at z = -9.94m ...................... 47

Figure 35: Relative deflexions of Cargo Hold models with different independent points ...... 47


Figure 37: Longitudinal force Nx in the load case A9 –tor m of the Cargo Hold Model ......... 49

Figure 38: Relative deflexions of Cargo Hold models with and without adjustments of Nx ... 50

Figure 39: Entire FE model of 11000 TEU containership ....................................................... 51

Figure 40: Load diagrams of 7 considered loadcases of 11000 TEU containership ................ 52

Figure 41: Cargo Hold model 74-97 of 11000 TEU containership .......................................... 53


Figure 43: Local coordinates of constraint beams ................................................................... 55

Figure 44: Pure torsional moment similar to A9 +tor m loadcase applied in 11000 TEU ship

.................................................................................................................................................. 57

Figure 45: Deflexions of Cargo Hold models (hold 1) (in % of Entire Model’s deflexion)

having different values of Ay, Ixx and Izz of constraint beams ................................................... 58

Figure 46: Deflexions of Cargo Hold models (hold 2) (in % of Entire Model’s deflexion)

having different values of Ay, Ixx and Izz of constraint beams ................................................... 59

Figure 47: Pure torsional moment similar to A9 -tor m loadcase applied in 11000 TEU ship

................................................................................................................................................. .61

Figure 48: Considered points for the comparison of longitudinal stress .................................. 62

Figure 49: Two pure torsional moments similar to A0 +tor a loadcase ................................... 63

Figure 50: Two pure torsional moments similar to A9 +tor m loadcase.................................. 64

Figure 51: Loadcase LC1 Tor1 of the 48000 DWT bulkcarrier .............................................. 66

6 Ngoc-Do NGUYEN


LIST OF TABLES

Table 1: Wave selection criteria in SHIPLOAD ...................................................................... 25

Table 2: Support of the full-breadth Cargo Hold model in GL 2011 rules .............................. 26

Table 3: Boundary conditions for Cargo Hold model in HCSR .............................................. 31

Table 4: Main characteristics of 2700 TEU containership ....................................................... 34

Table 5: Homogeneous constraint beams' characteristics of the Cargo Hold model 115-189

................................................................................................................................................ ..39

Table 6: Relative deflexion of the hatch diagonals in Cargo Hold model using different sets of

constraint beams ....................................................................................................................... 41

Table 7: Constraint beams using linear relation between plate thickness and (A, I) ................ 44

Table 8: Constraint beams using exponential relation between plate thickness and (A,I) ....... 44

Table 9: Deflexions of Cargo Hold models using different sets of constraint beams in one

section ....................................................................................................................................... 45

Table 10: Relative deflexions of Cargo Hold models with and without end cross sections'

nodal loads ................................................................................................................................ 46

Table 11: Relative deflexions of Cargo Hold models with different independent points ........ 48

Table 12: Relative deflexions of Cargo Hold models with and without adjustments of Nx ..... 50

Table 13: Main characteristics of 11000 TEU containership ................................................... 51

Table 14: Homogeneous and unhomogeneous constraint beams' characteristics of the Cargo

Hold model 74-97 ..................................................................................................................... 55

Table 15: Relative deflexions of Cargo Hold models using standard boundary conditions .... 56

Table 16: Boundary conditions with the same constraint beams for both end cross sections . 58

Table 17:Constraint beams’ characteristics of the Cargo Hold modell 74-97 ......................... 60

Table 18: Relative deflexions of Cargo Hold models using different sets of boundary

conditions in real-load cases .................................................................................................... 60

Table 19: Relative deflexions of Cargo Hold models using different sets of boundary

conditions in 2 pure-torsion loadcases ..................................................................................... 61

Table 20: Longitudinal stress in the Cargo Hold model using different boundary conditions in

real-load cases .......................................................................................................................... 62

Table 21: Comparison of the deflexion and warping stress in Entire FE models analysing the

effect of the torsion at the fore and aft part .............................................................................. 64

Table 22: Two boundary conditions (very soft and very stiff) applied to the 11000 TEU ship

.................................................................................................................................................. 65



Table 23: Longitudinal stress in the two Cargo Hold models of 11000 TEU ship

corresponding to two different boundary conditions ............................................................... 65

Table 24: Two boundary conditions (very soft and very stiff) applied to the 48000 DWT ship

.................................................................................................................................................. 66

Table 25: Longitudinal stress in the two Cargo Hold models of 48000 DWT ship

corresponding to two different boundary conditions ............................................................... 67

Table 26: Constraint beams' characteristics of the Cargo Hold model 115-189 (2700 TEU) . 67

Table 27: Relative deflexions of Cargo Hold models with different boundary conditions (2700

TEU) ......................................................................................................................................... 68

8 Ngoc-Do NGUYEN


Declaration of Authorship

I declare that this thesis and the work presented in it are my own and have been generated by

me as the result of my own original research.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the exception

of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made clear

exactly what was done by others and what I have contributed myself.

This thesis contains no material that has been submitted previously, in whole or in part, for

the award of any other academic degree or diploma.

I cede copyright of the thesis in favour of the University of Rostock.

Date: Signature



ABSTRACT

The analysis of warping stresses and hatch opening deformations is an essential part of the

containerships’ structural design. These vessels also call for detailed investigations of the hull

girder response to torsion and to hull girder bending in order to predict fatigue damage and

buckling capacity.

The global strength analysis which takes into account all hull girder sectional load

components, i.e, vertical bending, horizontal bending, vertical shear, horizontal shear and

torsion is to obtain a reliable description of the overall hull girder stiffness, to calculate and

assess the global stresses and deformations of all primary hull members for specified load

cases resulting from realistic loading conditions including the wave-induced forces and

moments. This method is able to verify the structural adequacy of the longitudinal and

transverse primary structure, particularly the influence of torsional moment on side shell

longitudinals, radii of the hatch corners as well as face plates and horizontal girders of the

transverse bulkheads.

The current Cargo Hold model is able to carry out a global strength analysis by calculating

and assessing the combined stresses and deformations in the midship area for specified load

cases in head and following sea only, including wave-induced loads as well as vertical hull

girder bending.

This study aims to analyze the feasibility of using the Cargo Hold model to analyze the

torsional effects in oblique sea on containerships by justifying and developing a new set of

boundary conditions. The two criteria used to adjust the boundary conditions are the relative

deflexions of hatch diagonals and the longitudinal stress in the coaming and bilge areas.

These criteria are compared between Cargo Hold finite element (FE) model and Entire FE

model in different loading cases. The GL computer program SHIPLOAD, which is a linear

frequency-domain hydrodynamic load generation program based on strip theory is used to

calculate ship accelerations and wave-induced pressures corresponding to different load cases.

Two ships of different sizes (2700 TEU and 11000 TEU) have been analysed. By numerical

analyses, it is shown that the current boundary constraint beams should be stiffer in the case

where high torsional moment exists in the midship area in order to have a correct relative

deflexion. Also, the warping stresses are shown to be independent of the constraint beam

stiffness.

10 Ngoc-Do NGUYEN


1. INTRODUCTION

1.1. Current research objectives

The marine industry has seen a revolution in container transport with a very rapid growth of

vessel sizes. At present, container vessels carrying 14000 TEU have been built. Such vessels

can have a length of 400 m, a breadth of 55 m and a draft of 14 m. With their large deck

openings, these vessels have a low torsional rigidity. The low position of the shear centers

makes the additional torsional moment due the horizontal forces become important. The

torsional response then plays an important role in the total structural response of the vessel

hull and needs to be taken into account in detailed investigations.

The current global strength FE analysis of Cargo Hold structures is intended to verify the

stress level, buckling and deflection of hull girder and primary supporting members under the

applied static and dynamic loads (CSR, Chapter 7, Sec 2/ Page 8). However, this current

model is only able to carry out the analyses including vertical bending moment and vertical

shear force in case of head and following seas. In case of oblique sea and beam seas, the

horizontal bending moment, horizontal shear force and torsional moment become important

and need to be taken into account.

The future HCSR (Harmonized Common Structural Rules) for bulk carriers and tankers

recommend the use of Cargo Hold models for the global strength analysis. The model covers

three Cargo Holds. Its set of boundary conditions consists of rigid links and end constraint

beams at both ends. The rigid links transfer the constraints of movement from the independent

points to the longitudinal structures and the end constraint beams are to simulate the warping

rigidity of the cut-out structures (CSR, Chapter 7, Sec 2/ Page 5). The cut-out structures are

the parts which are located outside the Cargo Hold model’s limit. Details can be found in the

3rd

chapter. Even though these boundary conditions (BC) work well with bulk carriers giving

realistic warping deflections and torsional stresses, verifications need to be done before using

this set of BC for containerships due to much less torsional rigidity of these vessels compared

to bulk carriers.

1.2. Review of related work

Dr. Jörg Rörup from the Rules Development Department of GL has used the set of BC

recommended in HCSR to carry out the Cargo Hold analysis for two bulk carriers: 48000

DWT (figure 1) and 176000 DWT (figure 3). The three hold FE models are generated with



the homogeneous constraint beams having the moments of inertia Ixx=Iyy=Izz=1/25 of the

vertical hull girder moment of inertia of fore/aft end cross sections and the cross section areas

Ax=Ay=Az=1/10 of the bottom plating areas at fore/aft ends.

Figure 1: Characteristics of the 48000 dwt bulk carrier

(Study on Cargo Hold model applied for bulk carrier of Dr. Jörg Rörup)

The loads applied in the Entire FE model are based on direct calculation (figure 2). Identical

mesh is used for the Entire and Cargo Hold models. The direct loads are first generated in the

Entire FE model for 15 different load cases. They are then transferred into the Cargo Hold FE

model to ensure that the same local loads exist in the two models in the midship area. The

same global behavior of the two models are assured by adjusting the shear forces, the bending

and torsional moments at the end cross sections according to the given values from the Entire

model.

The results show that for both two types of bulk carriers, using the current BC, even in case of

following, oblique seas and roll condition, similar stresses on the primary structures in FE

Entire and FE Cargo Hold models can be obtained (figure 4 and 5).

12 Ngoc-Do NGUYEN


Figure 2: Direct load calculations for the Entire model


Figure 3: Characteristics of the 176000 dwt bulk carrier




Figure 4: Stresses in main deck due to vertical bending


Figure 5: Stresses in main deck in oblique sea


14 Ngoc-Do NGUYEN


Figure 6: Stresses in main deck in roll condition


1.3. Torsion of container ships

The torsional theory applied to container ships is described in the book ‘Torsion and shear

stresses in ships’ of M.Shama.

Regarding the torsional moment applied, it consists of the static and the wave-induced part.

Static torsion depends on how the cargos and tanks are arranged. Wave-induced torsion

depends on wave characteristics including wave length, wave phase which is reflected by the

position of the crest and finally the heading angle of the wave. Normally, for containerships

the wave-induced torsion is much more important than the static one. The wave-induced

torsion itself is set up by hydrostatic force coming from the different level between port and

starboard sides and hydrodynamic force including the inertia forces due to the accelerations as

a result of the ship’s motions in waves (figure 7). At the midship where large opening sections

exist, the shear centers locate far below the hull bottom. The sum of the horizontal forces then

does not pass through the shear center and in consequence creates additional torsions on the

hull structure.



Figure 7: Torsional moment due to wave force and effect of low position shear center

(‘Torsion and shear stresses in ships’of M.Shama)

Regarding the torsional resistance, this is composed of St. Venant torsional moment Ts and

warping torsional moment Tw. The sum of the two should be equal to the torsional moment

applied:

T = Ts+Tw

The St. Venant torsion is dominant in beams and closed sections meanwhile the warping

torsion is more important in case of open thin-walled sections. In a close thin-walled section,

both types of torsion are essential. In case of containerships, the midship section consists of

thin-walled multi-cell box-girders and both torsional effects should be considered (figure 8).

The next part will give a general formula of torsional effect according to beam theory taking

into account both kinds of torsion.

The St. Venant torsion is related to the twisting rigidity G.Jt with the hythothesis that one

plane section remains plane during the deformation which is suitable for beams and closed

sections. The St. Venant torque is given by:

Ts = G.Jt.d∅/dx

With:

Jt: twisting torsion constant

∅: the angle of twist

16 Ngoc-Do NGUYEN


x: length coordinate

Figure 8: Twisting and warping in midship area of 11000 TEU

For a thin-walled closed section, the twisting constant:

24 / /t

J A ds t= ∫�

With:

A: The enclosed area of the section (figure 9)

∫� : The integral along the periphery of the closed section

t : thickness of the wall

Figure 9: Shear flow over a closed thin-walled section


This torque creates the St. Venant shear stress τs in the structure. The formula to calculate the

shear stress for a thin-walled section is:

τs = Ts.t/Jt



However, in reality, during the torsion, the section of a thin-walled does not remain plane

anymore and will be warped and deformed out of the plane. It is then twisted and warped. The

warping deformation depends on the warping rigidity E.Jw of the section. It gives:

Tw = -E.Jw. d3∅/d

3x

With: Jw: warping torsion constant ; E: Young modulus

For a thin-walled closed section, the warping constant:

2

w

A

J w dA= ∫

With: 0

( ) .

s

w s r ds= ∫ as described in figure 10.

Figure 10: Warping rigidity of an open thin-walled section


In a defined loading condition, different structures on a section will have different warping

deformation depending on the w-distribution of that section. The w-distribution of one

idealized ship section is shown in the figure 11:

Figure 11: Idealized section and sectorial area diagram


18 Ngoc-Do NGUYEN


The upper part of the section where locates the deck and the coamings are expected to have

high warping deformation when high-value torsional moment exists and high warping stress

when zero-crossing torsional moment exists in this area.

Due to warping, it exists in the structure the shear and warping stresses τw and σw respectively.

The second term will contribute in the longitudinal stress of the structure. They are calculated

by the formulas:

τw = Tw.Sw/(Jw.t)

σw=w.Mw/Jw

with: Mw: bi-moment = -EJw. d2∅/d

2x;

Sw: sectorial static moment

w: sectorial coordinate

E: modulus of elasticity

In total, the combination of twisting and warping torsions gives:

G.Jt.d∅/dx - E.Jw. d3∅/d

3x = T

In this equation, the angle of twist ∅ is the variable. By solving this equation, the twisting

angle ∅ can be found and one can then calculate the St. Venant torsion and the warping one.

The shear stresses τs, τw and warping stress σw can then be deduced. In order to solve this

equation, one needs to know the torsion distribution along x-axis as well as the torsional

constraints. There are there types of torsional constraints:

- Fixed end i.e., No warping:

∅= 0 and d∅/dx = 0

- Free end i.e., free warping:

d2∅/d

2x = 0

- Constrained end i.e., constrained warping:

d∅/dx = Te.(1-f)/(G.Jt)

with: Te : torsional moment at the end of the member

f : constraint factor 0 < f < 1.

The value of f depends on the ship length, the rigidity of ship sections and is difficult to

determine. Also, due to the complexity of the ship structure, its sections varie along the ship

length. It is consequently not possible to determine the exact torsional rigidity (G.Jt and E.Jw)

of one section as well as the exact positions of the shear centers in order to calculate the

additional torsions due to the horizontal forces.



These three reasons make it impossible to have an exact solution for the torsion phenomenon.

The fore and the aft parts of the containerships are normally closed section structures and

have high twisting and warping rigidity. In fact, warping normally does not exist in these

areas. In consequence, with similar torsions, these parts warp much less than the midship as

seen in figure 12. The fore and aft parts then have different behavior from the midship part in

torsion. This again emphasizes the necessity of taking the torsional rigidity of the cutout parts

which are the fore and aft ones into consideration in Cargo Hold model analyses.

Figure 12: High warping deflexion in midship due to torsion

Concerning the longitudinal stress, the warping stress can play an important role in large

opening sections. Theoretically, the total longitudinal stress consists of 3 parts with σx1:

longitudinal stress due to vertical bending moment, σx2: longitudinal stress due to horizontal

bending moment; σx3: longitudinal stress due to warping.

σx=σx1+σx2+σx3

The two first stresses can be accurately calculated from the simple formulas:

1

.VX

yy

M z

Iσ = and 2

.H

X

zz

M y

Iσ =

with: MV; MH: The vertical and horizontal bending moment

Iyy; Izz: The moment of inertia against vertical and horizontal bending moments of the

section

y, z: The coordinates of the calculation point.

20 Ngoc-Do NGUYEN


2. ENTIRE FE MODEL ANALYSIS

2.1. General

This part describes how the Entire FE models of two containerships were created. Those

models follow the GL guidelines of “Global Strength Analysis”. Here below quotes some

important points.

- The entire FE model is used to calculate and assess the global stresses and deformations

of all primary hull members for specified load cases resulting from realistic loading

conditions and the wave-induced forces and moments. More information concerning the

loadcases applied can be found in the part 2.2.

- The FE tool used for calculation is GL Poseidon and GL Frame.

- The FE model represents the entire ship including the deckhouse.

- Due to its coarse mesh, this model is not able to carry out local effects such as bending of

stiffened plates under water pressure.

- Primary structural members such as shell, inner bottom, girders, web frames, horizontal

stringers and vertical girders of transverse bulkheads are normally idealized by 4-node

plane stress elements.

- Secondary stiffening members are normally idealized by 2-node truss.

- Unless otherwise noticed, the boundary conditions used for FE model are as in the figure

13 with six supports arranged at the ends. They are high-stiffness spring elements. Their

locations can vary along x-axis without influencing the accuracy of the relative deflexions

and stresses levels in primary structures.



Figure 13: Boundary conditions in Entire models

(“Global Strength Analysis” Guidelines)

2.2. Load cases generated from SHIPLOAD

This part describes how the loads are generated in the Entire FE models. The principles of

computation of wave-induced loads with the focus on equivalent regular wave approach are

presented in details by Schellin et al. in part 4.4 of the book ‘Ship structural analysis and

design’ of Paik et al. 2010.

Classification society rules require the ship to withstand given global loads (rule-based)

including shear forces, bending moments and torsional loads. In order to assess the global

structural integrity of containerships, the equivalent regular wave approach was developed.

This is a compromise between the rule-based load approach and the physical-load approach.

This method is implemented in the software SHIPLOAD. The important point which will be

justified later is that if the ship can resist all the design load cases from SHIPLOAD, it will

satisfy the rule-based loads requirements as well. Taking the Entire FE model as an input, this

tool computes the nodal loads on the model corresponding to different realistic extreme

loadcases. Regarding the study of this thesis, it allows the analysis of the Entire and Cargo

Hold models in different loadcases instead of only one given by the envelope curves. The

general computation procedures are described in the figure 14 and the graphic interface is

shown in figure 15.

22 Ngoc-Do NGUYEN


Figure 14: Flow chart to determine global loads according to class rules

(‘Computation of wave-induced loads’ – Schellin et al.2010)

Step 1: From the Entire FE model, the program calculates the light weight of the ship and the

hull form for the hydrostatic and hydrodynamic computations.

Step 2: Different loading conditions including the distribution of masses and tanks are defined

by the user based on the stability booklet. Different types of masses are used as in figure 16.

Step 3: From the given loading conditions, the user selects 2 conditions that give respectively

the maximum and minimum still-water bending moment. Based on experience, they are the

most dangerous still water conditions when a wave arrives.

Step 4a: The critical wave heights are first determined. Those are corresponding to the

heading and following waves that result the maximum wave bending moments in hogging

(VBMWHhog) and sagging (VBMWHsag) condition according to the rules. These are rule-based

equivalent design wave heights.

Step 4b: A very large number of waves are investigated corresponding to different wave

lengths, wave phases and heading angles. These parameters range from 0.35 to 1.2 times ship

length for the wave lengths, from 0 to 180 degrees at 30 degrees intervals for wave headings

1. GENERATION OF FE MESH

2. GLOBAL LOADING CONDITIONS

3. GROUPED MASSES ADDED

TO FE MODEL

4. DESIGN WAVE CONDITIONS

5. EQUIVALENT REGULAR WAVES

6. ENVELOPE CURVES OF GLOBAL

LOADS

An FE mesh is generated to match the

structural properties of the hull

A set of global loading conditions are

selected for the structural analysis

Grouped masses are added to the FE model

Appropriate wave conditions for loads are

obtained

Equivalent regular waves are selected for

an assessment of extreme global loads

Longitudinal distributions of global loads

yield envelope curves



angles and the number of wave phases reaches 50. In total, the program works with about

9500 different waves.

Concerning the algorithm, SHIPLOAD is based on strip-theory code with added viscous roll

damping according to Blume and correction for non-linear hydrodynamic pressure according

to Hachmann (GL SHIP LOAD for Strength Analysis of Containerships).

Step 5: The waves are selected based on different criteria using an evaluation function of

forces i.e. shear forces, torsional moment and bending moments with different factors. Search

restrictions are used to limit the number of waves used for each time of search. The searching

criteria are as defined in the table 1. There are in total 43 load cases calculated in a full

calculation of SHIPLOAD. This study limits the number of load cases to 7. They are

considered as typical loadcases (LC) to check the accuracy of FE models. In reality, the wave

effects on the ship in rolling and upright condition are different. That is why the additional

roll angle is set up. In the first three loadcases, the ship is in upright condition meanwhile roll

motion exists in the last four loadcases.

Figure 15: SHIPLOAD user interface

24 Ngoc-Do NGUYEN


Figure 16: Definition of masses in SHIPLOAD

LC 1 (A0 hogg m): The wave creates maximum vertical bending moment at the midship. The

torsion in this loadcase consists only of the static part due to the asymmetric distribution of

cargos and tanks. It is much smaller than in the other loadcases and consequently, the

distortion or warping stress is not of interest in this case. This estimation will be justified in

the results.

LC 2 (A0 -hor m): The wave creates minimum horizontal bending moment at the midship.

LC 3 (A0 +tor a): The wave creates maximum torsion at the aft part.

LC 4 (A5 -tor a): The wave creates minimum torsion at the aft part and the ship has the roll

angle equaling to 50% of the maximum role angle defined by rules.

LC 5 (A5 +warp m): The wave creates minimum torsion at the fore part and the ship has the

roll angle equaling to 50% of the maximum role angle defined by rules.

LC 6 (A9 +tor m): The wave creates maximum torsion at the midship and the ship has the roll


LC 7 (A9 -tor m): The wave creates minimum torsion at the midship and the ship has the roll


The last two loadcases have a high absolute torsional moment in the midship region and

theoretically give high distortions of the ship hull and moderate warping stresses in this area.

On the contrary, the loadcases where zero-crossing point of the torsional moment is located in

STEEL, PIPING,

FLOATING

FLOORS

ELEMENT

MASSES

EQUIPMENT

BOX MASSES

DECK

CONTAINERS,

HOLD

CONTAINERS

CONTAINER

MASSES

FUEL OIL,

WATER

BALLAST

TANK MASSES

LOADING CONDITION 1:

MAX. STILL-WATER

VERTICAL BENDING

MOMENT

LOADING CONDITION 2:

MIN. STILL-WATER

VERTICAL BENDING

MOMENT



the midship tend to have low distortions and high warping stresses on the longitudinal

structural members. The comparison criteria then should be selected correspondingly for each

loadcase and each Cargo Hold model to avoid the sensitivity of the results.

Table 1: Wave selection criteria in SHIPLOAD

Title

Selector Factors Search Restriction

Type

X/Lpp Shear Torsion BM W-dir add.roll angle > 66%

from to Horiz. Vert. Horiz. Vert. from to from to

A0 hogg m Max 0,4 0,6 0 0 0 0 1 0 360 0 12

A0 -hor m Max 0,4 0,6 0 0 0 -1 0 0 360 0 6

A0 +tor a Max 0,2 0,3 0 0 1 0 0 0 360 0 6

A5 -tor a Max 0,2 0,3 0 0 -1 0 0 0 360 0 12

A5 +warp m Max 0,65 0,75 0 0 -1 0 0 0 360 0 12 A5 +tor a

A9 +tor m Max 0,48 0,52 0 0 1 0 0 0 360 12 20

A9 -tor m Max 0,48 0,52 0 0 -1 0 0 0 360 12 20

The load diagrams of 7 interested load cases of 2700 TEU and 11000 TEU ships are as in the

figure 24 and 40.

26 Ngoc-Do NGUYEN


3. CARGO HOLD FE MODEL ANALYSIS

There is so far no Cargo Hold model that works well with containerships in all sea states.

This part gives a summary of the Cargo Hold models used in different rules such as GL, ABS

and future HCSR.

3.1. Cargo Hold Model in GL 2011 rules

The GL 2011 rules describe the use of Cargo Hold model for vertical bending analysis. This

can be found in Chapter 5-Structural rules for containerships, section 22-Cargo Hold

Analysis.

The vertical bending moment comes from the static still-water and the wave-induced loads.

The stresses and deformation in the midship area especially in the bottom area can be

calculated for specified realistic loading conditions. The vertical bending moment normally

acts symmetrically on the port and starboard and allows the use of half-breadth model (figure

17). The model expands over a two-hold length starting and ending at two bulkheads. The use

of the full-breadth model is of course also possible.

In this Cargo Hold model, no horizontal or torsional moment is considered. The boundary

condition for a full-breadth model consists of supports at the model ends (table 2 and figure

18).

Table 2: Support of the full-breadth Cargo Hold model in GL 2011 rules

(GL 2011-Chapter 5: Structural rules for containerships- Section 22: Cargo Hold Analysis)

Location Translation Rotation

δx δy δz θx θy θz

Aft End

Intersection of

Centerline and outer

bottom

Fix Fix Fix - - -

Intersection of

Centerline and deck - Fix - - - -

Fore End

Intersection of

Centerline and outer

bottom

- Fix Fix - - -

Where: - no constraint applied (free)



And because the model is in equilibrium, the sum of reaction forces acting on these supports

in x, y and z directions should be equal to zero. In case of half-breadth model, additional

boundary conditions need to be applied in the plane where y-coordinate is zero such as: zero

deformation in y-direction.

The end cross sections are kept plane by using the rigid links in x-direction. The hull girder

bending moment and shear force are adjusted at these positions.

Figure 17: Half-breadth Cargo Hold model in GL 2011 rules


28 Ngoc-Do NGUYEN


Figure 18: Support of the full-breadth Cargo Hold model in GL 2011 rules


3.2. Cargo Hold Model in ABS rules

ABS specifies the use of Cargo Hold model in the guidance notes ‘Safehull finite element

analysis of hull structures’ in 2004.

The recommended extent for the Cargo Hold model is three hold length with the middle hold

located within 0.4L amidships. This is higher required compared to the GL 2011 rules which

require only two-cargo-hold length. There is also a short extension beyond the bulkheads at

the ends.

For the containerships, ABS uses spring supports at the two end cross sections to balance the

loads applied in the model. Those supports are placed at the two transverse bulkheads near the

ends (figure 19). With this set up boundary conditions, the regions near the supports will be

highly deflected and the border effect needs to be carefully checked before accepting the

results.

The warping constraint from the cut off parts of the ship is modelised by the constraint beams

at the two end sections. Those beams are located at the same position with the longitudinal

structure such as the bottom and side platings, longitudinal girders.

The same flexural stiffness and shearing areas are assumed for all the three directions and

identical beams are used for the aft and fore ends. Their properties are:

Ixx = Iyy = Izz = 1/3 of the vertical moment of inertia of the hull girder amidships

Ax = Ay = Az = 1/10 of the bottom plating cross-sectional area amidships



Figure 19: Extent of Cargo Hold model according to ABS rules

(‘Safehull finite element analysis of hull structures’ - 2004)

Aft end

Fore end

Figure 20: Boundary conditions of cago hold model in ABS rules

(‘Safehull finite element analysis of hull structures’ - 2004)

3.3. Cago Hold Model in future HCSR rules

The future harmonized common structural rules (HCSR) describe in details the use of Cargo

Hold model analysis for tankers and bulk carriers (‘Harmonized common structural rules-Part

30 Ngoc-Do NGUYEN


1, Chapter 07, Section 02’). The extent of the model is a three-hold length which is the same

as in ABS and larger than the two-hold length in GL 2011 rules in order to better reduce the

border effect. It is also to achieve a better adjustment of forces i.e. shear forces, bending

moments and torsional moment in the Cargo Hold model that the three hold model is

recommended (HCSR, Part1, Chapter 7, Section 2, Draft Rule Text, Page 2, 2.1.1.c).

The boundary conditions in the Cargo Hold model consist of the point constraints, the rigid

links and the end constraint beams.

The point constraints are to give the translation and rotation constraints on the structure. They

are located at the location of independent points which are the intersection between the

neutral axis and the centerline of the section. An additional x-constraint is applied by a point

constraint at the intersection between the centerline and the inner bottom at fore end to limit

the translation of the model around x-axis.

The rigid links are the imaginary connections in y-translation, z-translation and x-rotation

between the nodes at the end cross sections and the independent points so that the constraints

and torsional moment applied on the sections can be transferred to those points. There is no

rigid links in x-translation, y-rotation and z-rotation. This is different from ABS rules where

rigid links do not exist at all. Using the rigid links, the end cross sections will be less

deformed and the border effect will be smaller.

As in ABS rules, the end constraint beams are applied at both ends of the model to simulate

the warping constraints of the cut off structures. They are identical for both ends and have the

properties:

Ixx = Iyy = Izz = 1/25 of the vertical hull girder moment of inertia of the fore/aft end cross

section based on the net FE model.

Ax = Ay = Az = 1/10 of the bottom plating area at fore/aft end.

There are differences between these properties and the ones in ABS. If in ABS, the end

constraint beams properties depend only on the amidship section, in HCSR they depend

directly on the end sections. Also, the end constraint beams are much stiffer in the ABS rules

with the ratios of 1/3 and 1/10 for I and A against 1/25 and 1/10 in HCSR. Using the HCSR

rules, careful attention should be paid on the hull form at the end sections whose bottom

platings will decide on the shear area of the constraint beams. The smaller the difference

between the two sections is, the more realistic the Cargo Hold model is. This can be assured

by choosing the extent of the model in the region where the hullform varies slowly.



Figure 21: Boundary conditions of Cargo Hold model in HCSR

(‘Harmonized common structural rules-Part 1, Chapter 07, Section 02’)

(On the left: position of the independent points

On the right: Constraint end beams and independent points in FE Cargo Hold model)

Table 3: Boundary conditions for Cargo Hold model in HCSR

(‘Harmonized common structural rules-Part 1, Chapter 07, Section 02’)

Location Translation Rotation

δx δy δz θx θy θz

Aft End

Cross section - Rigid

Link

Rigid

Link

Rigid

Link - -

Independent point - Fix Fix endTM−

- -

Cross section End beam

Fore End

Cross section - Rigid

Link

Rigid

Link

Rigid

Link - -

Independent point - Fix Fix Fix - -

Intersection of Centerline

and inner bottom Fix - -

Cross section End beam

Where: - no constraint applied (free)

32 Ngoc-Do NGUYEN


4. WORK WITH THE SMALL CONTAINERSHIP OF 2700 TEU

4.1. The procedure to create and work with Cargo Hold models

The figure 22 describes a loop of calculation with the Cargo Hold models starting from the

global geometric file to the comparison of results between the Cargo Hold and the Entire FE

model.

Figure 22: Working procedure in order to justify the boundary conditions

Step 1: The complete structure of the ship is required to generate the FE model. GL Poseidon

is a software to modelise ship structure.

Step 2: The Entire FE model has to have the same mesh with the Cargo Hold FE model in the

midship area where the comparisons of deflexions and stresses will be done.

Step 3: Cargo Hold FE models are generated from the geometric model in Poseidon.

Boundary conditions concerning the position of the independent points can be changed at this

1. GEOMETRIC MODEL IN POSEIDON

2. ENTIRE FE MODEL IN GL FRAME

3. CARGO HOLD FE MODEL IN GL

FRAME

4. LOAD GENERATION OF THE

ENTIRE FE MODEL IN SHIPLOAD

5. LOAD TRANSFER FROM THE

ENTIRE FE MODEL TO THE CARGO

HOLD FE MODEL

6. ADJUSTMENT OF THE FORCES IN

THE CARGO HOLD FE MODEL

7. COMPARISON BETWEEN THE

ENTIRE FE AND CARGO HOLD FE

MODEL



step. The others such as stiffness of end constraint beams can be changed in GL Frame, a

module to work with FE model of Poseidon.

Step 4: Loads are generated using GL Shipload as described in part 2.2

Step 5: The nodal loads in the Entire FE model are transferred to the Cargo Hold FE model

using a script developed at GL. This script will remember the positions of the nodes in the

first FE model, look for the corresponding ones in the second model based on their positions

and give these nodes the forces as in the Entire FE model. By doing the transfer of nodal

loads, we make sure that the two models have the same local loads in the area of Cargo Hold

model.

Step 6: The global effects need to be taken into account by adjusting the forces in the Cargo

Hold model. The adjustments are done based on the following rules:

- Longitudinal force is not required to be adjusted between the Cargo Hold and Entire FE

models. Part 4.7 will discuss more about this rule.

- The vertical shear force, vertical bending moment do not effect the horizontal shear force

and the horizontal bending moment. The vertical and horizontal effects are independent. The

adjustments will be done separately for the forces and bending moments in each direction.

- The adjustments of the shear forces need to be done first by applying bending moments at

the end cross sections.

- Concerning the adjustments of vertical and horizontal bending moments, they are done after

adjusting the shear forces by applying the longitudinal axial nodal forces to all hull girder

longitudinal members.

- The torsional moment is adjusted in the end by applying a torsional moment MT-end at the

independent point of the aft part. Thanks to the rigid links, this adjustment moment will be

transferred to all the longitudinal members of the end cross sections. This adjustment moment

is equal to the difference between the target torsional moment and the sum of the torsional

moment due to the original local loads and the additional torsional moments induced by the

horizontal reaction loads.

Step 7: Two main criteria of comparison are the relative deflexion of hatch diagonals in the

midship and the longitudinal stress at the coamings and at the bilge area.

4.2. FE model of 2700 TEU containership

The main characteristics of the ship are shown in table 4.

34 Ngoc-Do NGUYEN


Table 4: Main characteristics of 2700 TEU containership

Length btw. perpendiculars (m): 212.5 Scantling draught (full load cond.)(m): 12

Length of water line at T(m): 216.7 Block coefficient: 0.61

Breadth (m): 31 Dead weight at T(tons): 48900

Scantling length(m): 210.2 Depth (m): 19.7

Figure 23: Entire FE model of 2700 TEU containership

Load case A0 hogg m

Load case A0 – hor m

Load case A0 + tor a

Load case A5 –tor a



Load case A5 +warp m

Load case A9 + tor m

Load case A9 –tor m

Figure 24: Load diagrams of 7 considered loadcases of 2700 TEU containership

4.3. Cargo Hold 115-189 and Cargo Hold 78-189

Two Cargo Hold models were generated for the small containership: the first one (CH 115-

189) of two holds covering from frame 115 till frame 189 and the second one of three holds

from frame 78 till frame 189 (CH 78-189) as in figure 23, 25 and 26. They both start and end

at bulkheads.

As the first calculations, both two models use coarse mesh in order to reduce the calculation

time. The length of the mesh at the midpart of the Cargo Hold is about 3 meters.

The relative delexion is defined as the difference between the distances of 2 hatch diagonals

after and before deformation. In the FE model, this relative deflexion is measured by using

truss elements connecting the involved hatch diagonals. In this case, there are 4 truss elements

giving the relative deflexion of 8 hatch diagonals (figure 29). The truss elements have a very

low rigidity due to their small sectional area and do not influence the rigidity of the model.

The relative deflexion is calculated based on the stress obtained in the truss elements as in the

following formulas:

E

σε = With:

0

L

Lε

∆= = the relative deflexion (m); ∆L: The elongation of the truss;

L0: The initial length of the truss; σ: The tension stress in the truss; E: Young modulus

36 Ngoc-Do NGUYEN


Then: The relative deflexion 0.L LE

σ∆ = (m)

Figure 25: Cargo Hold model 115-189 of 2700 TEU containership

Figure 26: Cargo Hold 78-189 of 2700 TEU containership



4.4. Influence of the constraint beams’ stiffness

This part tries to find suitable constraint beams regarding the deflexion of the hatch diagonals

and the warping stress. In order to do so, some boundary conditions have been worked with

and the best one was selected at first based on the comparison of the relative deflexions of the

hatch diagonals in the last two loadcases (Load case 6: A9 +tor m and load case 7: A9 –tor m)

(figure 24). These two loadcases have a high torsional moment at the midship and

consequently induces high deflexion of the hull. The longitudinal stress of coaming element

in the Cargo Hold model using this set of boundary conditions will be compared with the

Entire model. We consider that the adjustments of the shear forces, bending moments and

torsional moment are accurate enough. It means that at one section, the horizontal and vertical

bending moments are almost the same in the two FE models. Consequently, the longitudinal

stresses induced by these bending moments are the same in the two models and the difference

of longitudinal stresses only comes from the warping stress (from part 1.2). Then the

longitudinal stress will be used as a criterion of comparison for the warping stress. Two

approaches have been considered here: Using one type of homogeneous constraint beams in

one section and another one using different types of homogeneous constraint beams in one

section. “Homogeneous” means that the beam has the same characteristics in the three

directions: A=Ax=Ay=Az and I=Ixx=Iyy=Izz. The stiffness of one constraint beam can be

characterised by only two parameters: A and I. The first approach is recommended in the

future HCSR rules thanks to its simplicity meanwhile the second one seems to be physically

more appropriate since the constraint beams represent the warping rigidity of the cutout part

and this warping rigidity depends on the thickness of the plate which varies in one section.

This second approach tries to take the difference of stiffness of plates into account.

4.4.1. Work with one type of constraint beams in one section

This part compares the results of four different Cargo Hold models using four different sets of

boundary conditions (table 5). Due to the difference of the end cross sections (figure 27 and

28), the characteristics of the constraint beams differ between the section 115 and 189

regarding the shear area since it depends on the thickness and the breadth of the outer bottom

plate.

38 Ngoc-Do NGUYEN


Figure 27: Section 115 of 2700 TEU containership

Figure 28: Section 189 of 2700 TEU containership



Table 5: Homogeneous constraint beams' characteristics of the Cargo Hold model 115-189

(BC: Boundary Condition)

Initial BC BC 1 BC 2 BC 3

Aft part

(Section 115)

A (m2) 0.023 0.046 0.023 0.046

I (m4) 7.794 7.794 15.588 15.588

Fore part

(Section 189)

A (m2) 0.015 0.029 0.015 0.029

I (m4) 7.159 7.159 14.319 14.319

The initial constraint beams’ characteristics are calculated based on future HCSR. Starting

from this, the first (BC 1) and second (BC 2) sets of constraint beams have the shear area (A)

or moment of inertia (I) twice higher respectively. The third set (BC 3) of constraint beams

has both shear area and moment of inertia twice higher than the initial one.

Using the initial constraint beams, in the loadcase 6, the difference of relative deflexions of

hatch diagonals between the Cargo Hold and Entire models is about 30.5% in the hold 1

(using trusses 1 and 2 (figure 29)) and 53.5% in the hold 2 (using trusses 3 and 4). These

values are correspondingly 18.5% and 32.5% in the loadcase 7 (figure 30, 31 and table 6).

The initial Cargo Hold model seems to be softer than the Entire model. That is why

modifications of A and I have been made to stiffen the model later. Beside this difference, in

one loadcase, the errors of deflexion are also not the same for the two holds. Hold 2 of the

model is relatively stiffer than hold 1 with the difference of 21% (53.5% - 32.5%) in loadcase

6 and 14% (32.5% - 18.5%) in loadcase 7.

The last set of constraint beams with high stiffness gives the best correlation between the

Entire model and the Cargo Hold one regarding the deflexions. In the loadcase 6, the

difference of relative deflexions between the two models is reduced to 12.5% in the hold 1

and 29.5% in the hold 2. In the loadcase 7, those are correspondingly 1.5% and 11%. Besides,

even though this stiffer set of constraint beams has improved the errors of the Cargo Hold

model; it did not correct successfully the difference of stiffness between 2 holds. Hold 2 is

still much stiffer than the hold 1 with the difference of 17% (29.5% - 12.5%) and 9.5% (11% -

1.5%) in the corresponding loadcases 6 and 7, compared to 21% and 14% at the beginning.

The comparison of the relative deflexions in these four Cargo Hold models is shown in

figures 32.

40 Ngoc-Do NGUYEN


Figure 29: Hatch diagonals whose the relative deflexions are considered

Figure 30: Relative deflexion of the hatch diagonals in Cargo Hold model using initial constraint

beams



Figure 31: Relative deflexions of the hatch diagonals in Cargo Hold model using BC 3

Table 6: Relative deflexion of the hatch diagonals in Cargo Hold model using different sets of

constraint beams

Relative Deflexion (mm)

Entire FE model Cargo Hold Model Difference (%)

Initial BC

L6-T1 54.8 70.8 29

L6-T2 -50.6 -66.6 32

L6-T3 40.5 60.6 49

L6-T4 -34.3 -54.3 58

L7-T1 -48.8 -58.0 19

L7-T2 53.3 62.6 18

L7-T3 -34.3 -46.6 36

L7-T4 41.1 53.1 29

BC 1

L6-T1 54.8 65.9 20

L6-T2 -50.6 -61.8 22

L6-T3 40.5 56.1 38

42 Ngoc-Do NGUYEN


L6-T4 -34.3 -49.9 45

L7-T1 -48.8 -53.7 10

L7-T2 53.3 58.2 9

L7-T3 -34.3 -42.6 24

L7-T4 41.1 49.3 20

BC 2

L6-T1 54.8 68.0 24

L6-T2 -50.6 -63.8 26

L6-T3 40.5 57.6 42

L6-T4 -34.3 -51.5 50

L7-T1 -48.8 -55.5 14

L7-T2 53.3 60.1 13

L7-T3 -34.3 -44.1 29

L7-T4 41.1 50.6 23

BC 3

L6-T1 54.8 61.3 12

L6-T2 -50.6 -57.1 13

L6-T3 40.5 51.4 27

L6-T4 -34.3 -45.3 32

L7-T1 -48.8 -49.6 2

L7-T2 53.3 54.0 1

L7-T3 -34.3 -38.4 12

L7-T4 41.1 45.1 10



Figure 32: Difference of relative deflexions of the hatch diagonals between Entire and Cargo Hold

models using different constraint beams

4.4.2. Work with different types constraint beams in one section

The results from the last part show that even though good correlation of the deflexions of the

two models can be obtained in one specific hold and one specific loadcase; they still differ in

the other holds and loadcases. That is why this part tries a second approach to determine the

characteristics of the constraint beams.

We continue to use homogeneous constraint beams but different types of them will be applied

in one section depending on the thickness of the plates they represent. Because the thickness

of plates in one section varies a lot (figure 27 and 28), it is not possible to take all of them into

account separately. They are then categorized into three groups:

- First group: Plates having small thickness of about 8mm which are some longitudinal

stiffeners of the double bottom.

- Second group: Plates having thickness of about 14, 17, 18 and 19 mm which are

mostly the side shell and the bottoms.

- Third group: Plates having thickness of 36 and 42 mm which are mostly located on the

top of the side shells and the coaming.

The average thickness of the second and third group is about 2 and 4 times higher than the

first one respectively. The last part 4.4.1 concluded that the boundary conditions need to be

stiffer by multiplying the shear area and the moment of inertia of the constraint beams by a

44 Ngoc-Do NGUYEN


factor of 2 to have proper results. Since the second group of plates is the most popular one.

The shear area (A) and moment of inertia (I) of this group will be multiplied by two and will

be taken as the reference. The other groups which have the averaged thickness

correspondingly twice lower and twice higher will have A and I defined by two approaches:

- Linear relation between the plate thickness and (A, I) of constraint beams. A and I of

the first group are then half of the second group’s. A and I of the third group are twice

of the second group’s (table 7).

Table 7: Constraint beams using linear relation between plate thickness and (A, I)

A(m2) I(m

4)

First group

Section 115 0.023 7.794

Section 189 0.015 7.159

Second group

Section 115 0.046 15.588

Section 189 0.029 14.319

Third group

Section 115 0.092 31.176

Section 189 0.058 28.638

- Exponential relation between the plate thickness and (A,I) of constraint beams. A and I

of the first group are then 1/4 of the second group’s. A and I of the third group are four

times of the second group’s (table 8).

Table 8: Constraint beams using exponential relation between plate thickness and (A,I)

A(m2) I(m

4)

First group

Section 115 0.012 3.897

Section 189 0.007 3.580

Second group

Section 115 0.046 15.588

Section 189 0.029 14.319

Third group

Section 115 0.184 62.352

Section 189 0.117 57.275

The best results of relative deflexions using different approaches are shown in table 9

including the absolute and relative values (compared to Entire FE model). It can be seen that

using different sets of constraint beams in one section did not help to correct the difference of

stiffness between two holds of the Cargo Hold model. Hold 2 is always relatively stiffer than

hold 1 of about 16% in loadcase 6 and 9% in loadcase 7.



Table 9: Deflexions of Cargo Hold models using different sets of constraint beams in one section


Entire FE model Cargo Hold model

One set of BC for one section Linear relation

Exponential

relation

(mm) (mm) (%) (mm) (%) (mm) (%)

L6-T1 54.8 61.3 12 60.1 10 58.9 8

L6-T2 -50.6 -57.1 13 -56.0 11 -54.8 8

L6-T3 40.5 51.4 27 50.3 24 49.4 22

L6-T4 -34.3 -45.3 32 -44.2 29 -43.2 26

L7-T1 -48.8 -49.6 2 -48.5 -1 -47.4 -3

L7-T2 53.3 54.0 1 53.0 -1 52.0 -3

L7-T3 -34.3 -38.4 12 -37.6 10 -36.7 7

L7-T4 41.1 45.1 10 44.1 7 43.2 5

4.5. Influence of nodal loads at end cross sections of Cargo Hold FE

model

Theoretically, the nodal loads at the end cross sections (ECS) have no effect on the shear

forces and bending moments. This part will justify the necessity of deleting the nodal loads at

ECS by comparing the relative deflexion of hatch diagonals in two CH models: one with and

another one without the nodal loads. The FE model used for this work is the CH 115-189 of

the 2700 TEU containership with the standard boundary conditions as defined by HCSR.

Figure 33: Relative deflexions of Cargo Hold models with and without end cross sections' nodal loads

(E-C-S: End cross-section; L i - T j: Corresponding to Loadcase i – Truss j)

46 Ngoc-Do NGUYEN


The figure 33 and table 10 show the results in the loadcase 6 (A9 + tor m) and loadcase 7 (A9

–tor m) which have the highest relative deflexions (figure 24).

The results show that regarding relative deflexion, the two CH models have a difference

smaller than 1.1 %. It is then not quite important in the first steps to delete the nodal loads at

the ECS.

Table 10: Relative deflexions of Cargo Hold models with and without end cross sections' nodal loads


Without E-C-S nodal loads With E-C-S nodal loads Difference (%)

L6-T1 70.4 70.8 0.6

L6-T2 -66.2 -66.6 0.7

L6-T3 60.1 60.6 0.7

L6-T4 -53.9 -54.3 0.8

L7-T1 -58.5 -58.0 -0.8

L7-T2 63.1 62.6 -0.7

L7-T3 -47.1 -46.6 -0.9

L7-T4 53.7 53.1 -1.1

4.6. Influence of the independent points’ position

Most of the sections of containerships have the shear center located below the outer bottom.

In the case of the 2700 TEU ship, the midship section’s shear center is at the position z = -

9.94m and the neutral axis at z = 8.6m. The distance between the shear center and the neutral

axis is about 18.5m. The horizontal reaction force which goes through the shear center

becomes very sensitive because it will create an important additional torsion on the

independent points due to the long lever of 18.5m. Even though the adjustments are done by a

script in excel which gives a high accuracy, the influence of horizontal shear force’s

adjustment’s error on the results needs to be checked. Criteria of comparison are the relative

deflexions of hatch diagonals. To do so, two 115-189 Cargo Hold models are created: One

with independent points at neutral axis, and another one with independent points at the shear

center (figure 34). Both use the standard boundary conditions as in HCSR. The second model

does not require a high accuracy of adjustments since the lever of the horizontal reaction force

is very low and does not create any additional torsion. The results showed that the

independent points’ position does not have high effect on the deflexions with the highest



difference of 1.4% (figure 35 and table 11) and that the adjustments’ script is accurate enough

for horizontal shear forces even when the shear centers are far from the independent points.

Figure 34: Cargo Hold model 115-189 with independent points at z = -9.94m

Figure 35: Relative deflexions of Cargo Hold models with different independent points

(L i - T j: Corresponding to Loadcase i – Truss j)

48 Ngoc-Do NGUYEN


Table 11: Relative deflexions of Cargo Hold models with different independent points


Independent Points at z = -9.94m Independent Points at z = 8.6m Difference (%)

L6-T1 70.1 70.8 1.1

L6-T2 -65.9 -66.6 1.1

L6-T3 59.8 60.6 1.2

L6-T4 -53.6 -54.3 1.4

L7-T1 -58.6 -58.0 -1.0

L7-T2 63.2 62.6 -0.9

L7-T3 -47.2 -46.6 -1.3

L7-T4 53.9 53.1 -1.4

4.7. Influence of longitudinal force

It exists always a longitudinal force in the ship structure mainly due to the static pressure of

the water. The resulting force acts at the position z = 1/3 of the draft. It means if the neutral

axis is located at different height from 1/3 of the draft, there will be additional vertical

bending moment due to the longitudinal force. However, this additional moment is normally

small compared to the total value of vertical bending moment and will be ignored in HCSR.

There are two effects of the longitudinal force: The existence of compression stress and the

influence on the deflexion. The first effect will change the longitudinal stress of about some

MPa. The second effect is analysed in this part by two Cargo Hold models 78-189 (figure 36):

One without the adjustment of the longitudinal force which means it comes only from the

local loads and another one with adjustment of Nx (figure 37). In the loadcase A9 –tor m, the

initial value of the longitudinal force Nx is -17200 KN at the aft part. The target value for this

force is -12870 KN. The difference between with and without adjustment of Nx is about 4330

KN. The results show that the effect of longitudinal force on deflexion (figure 38 and table

12) is very small and the adjustment of Nx is not important regarding the deflexion. But while

comparing with the longitudinal stress of the Entire and Cargo Hold model, careful attention

needs to be paid on the longitudinal force because the difference of Nx between the Entire

model and the Cargo Hold one without adjustment of Nx will create a difference of some MPa

in the longitudinal stress. For example, in the loadcase A9 –tor m, with the shear area in x-

direction of the midship of about 3.43 m2, the error of the longitudinal stress due to this effect

will be 4300/3.43 = 1253KN/m2 = 1.25 MPa.




Figure 37: Longitudinal force Nx in the load case A9 –tor m of the Cargo Hold Model

(Left: Without adjustment; Right: With adjustment)

50 Ngoc-Do NGUYEN


Figure 38: Relative deflexions of Cargo Hold models with and without adjustments of Nx

(L i - T j: Corresponding to Loadcase i – Truss j)

Table 12: Relative deflexions of Cargo Hold models with and without adjustments of Nx


Without Adjustment of Nx With Adjustment of Nx Difference (%)

L6-T1 50.6 50.6 0.0

L6-T2 -44.2 -44.2 0.0

L6-T3 53.1 53.1 0.0

L6-T4 -46.5 -46.5 0.0

L7-T1 -42.9 -42.8 -0.3

L7-T2 49.4 49.4 0.0

L7-T3 -48.8 -48.8 0.0

L7-T4 55.8 56.0 0.3



5. WORK WITH THE BIG CONTAINERSHIP OF 11000 TEU

5.1. FE model of 11000 TEU containership

The main characteristics of the ship are shown in table 13.

Table 13: Main characteristics of 11000 TEU containership

Length btw. perpendiculars (m): 331 Scantling draught (full load cond.)(m): 15.5

Length of water line at T(m): 337.67 Block coefficient: 0.722

Breadth (m): 45.2 Dead weight at T(tons): 130700

Scantling length(m): 327.54 Depth (m): 29.7

Figure 39: Entire FE model of 11000 TEU containership

Load case A0 hogg m Load case A0 – hor m

52 Ngoc-Do NGUYEN


Load case A0 + tor a Load case A5 –tor a

Load case A5 +warp m Load case A9 + tor m

Load case A9 –tor m

Figure 40: Load diagrams of 7 considered loadcases of 11000 TEU containership

5.2. Cargo Hold 74-97

So far, the analyses have been carried out with the small containership and its Cargo Hold

models covering two and three holds. The end cross sections of these Cargo Hold models are

located at the bulkheads. The conclusions have been obtained from the analysis of the small

ship will be used to work with this big one. The Cargo Hold model of the 11000 TEU ship

covers two holds with two watertight bulkheads in the middle as can be seen in the figure 41

and figure 42. The end sections are located at one frame away from the support bulkheads. A

fine mesh is used with the mesh length of about 0.8m at the middle. As mentioned before, the

mesh at the midship of the Cargo Hold and Entire model must be the same. In this case, for

the Entire FE model, in order to have a compromise between accuracy and calculation time,



only the midship will have a fine mesh using beams and shell elements. The fore and aft part

still have a coarse mesh using truss and plane stress elements. In order to do so, the fine mesh

Cargo Hold and coarse mesh Entire models were first created (figure 39). The midship part of

the Entire model is then cut off and replaced by the Cargo Hold mesh. The effect of the rapid

change of mesh at the intersections is reduced by applying one transition area where the mesh

is at intermediate size. Also, the hold in the middle is supposed to be far enough to neglect the

border effect at the end cross sections. The loadcases considered and the work procedures are

as described in the work with the small ship and in the figure 40. Taking into account the

conclusions from the part 4, this Cargo Hold model has the following characteristics:

- Nodal loads exist at the end cross sections.

- There is no adjustment of the longitudinal force. This does not affect the deflexion but

the longitudinal stress in the scale of some MPa.

- Independent points are located near the neutral axis of the sections.

The following studies try to limit the variables of the constraint beams’ stiffness and to work

separately with torsion in order to better understand its effect on deflexion and warping stress.

Figure 41: Cargo Hold model 74-97 of 11000 TEU containership

54 Ngoc-Do NGUYEN



5.3. Influence of (Ax, Ay, Az) and (Ixx, Iyy, Izz) on Cargo Hold model’s

stiffness

Homogeneous constraint beams have been used in the study of part 4 which means the beams

have the same flexural stiffness and cross area in three directions. This is in fact a

simplification and not all of these six parameters (Ax, Ay, Az, Ixx, Iyy, Izz) are necessary. The

orientation of the constraint beams in one end cross sections is as in the figure 43. With this

type of orientation and with the rigid links in global coordinates in y-translation, z-translation

and x-rotation, only Ay, Ixx, Izz of the constraint beams should have effect on the stiffness. To

check this, we analyse two Cargo Hold models with the two boundary conditions as given in

table 14. The second has the same values of Ay, Ixx, Izz with the first one but all other

parameters are set to be zero.

The relative deflexions between the two Cargo Hold models are absolutely the same (table

15) showing that only three parameters Ay, Ixx, Izz are important for torsional and horizontal

bending effects. The other parameters of the constraint beams have no effect and will be

neglected in the calculations of the next parts.



Table 14: Homogeneous and unhomogeneous constraint beams' characteristics of the Cargo Hold

model 74-97

Homogeneous Constraint Beams Unhomogeneous Constraint Beams

Ax,y,z (m2) Ixx,yy,zz (m

4) Ax,y,z (m

2) Ixx,yy,zz (m

4)

Aft part

0.069 39.159 0 39.159

0.069 39.159 0.069 0

0.069 39.159 0 39.159

Fore part

0.048 37.411 0 37.411

0.048 37.411 0.048 0

0.048 37.411 0 37.411

Figure 43: Local coordinates of constraint beams

56 Ngoc-Do NGUYEN


Table 15: Relative deflexions of Cargo Hold models using standard boundary conditions

(Li-Tj: Loadcase i – Truss Element j as defined in figure 42)


Entire FE model Cargo Hold Model Difference (%)

L6-T1 94.0 106.9 14

L6-T2 -87.0 -99.8 15

L6-T3 75.1 94.2 25

L6-T4 -67.4 -86.5 28

L7-T1 -80.3 -83.6 4

L7-T2 87.4 90.7 4

L7-T3 -63.1 -71.8 14

L7-T4 70.1 79.1 13

5.4. Influence of the constraint beams’ stiffness in case of pure torsion

and real-load case

The last two loadcases (A9 +tor m and A9 –tor m) which have a high absolute value of the

torsional moment in the midship always give a high deflexion of hatch diagonals in this area.

This part works separately with a pure torsion similar to the one of A9 +tor m loadcase to find

an appropriate set of boundary conditions. The deflexion of the hatch diagonals are due to the

warping and the horizontal bending of the hull. By analysing the effect of the torsion

separately, it helps to understand how the hull is deformed under different torsional moments.

The pure torsion is created by a Python script. Starting from the given values of torsional

moments at some sections, the script interpolate them to obtain the distribution of torsional

moment along the ship and calculate the nodal loads which need to be applied on the nodes of

one section to reach the required torsional moment. These are the nodal loads which act

vertically.

The real-load case means it exists in the FE models all the shear forces and moments.

5.4.1. Pure torsion whose maximum value exists in the midship area

The distribution of torsional moment generated looks like the one in the loadcase A9 +tor m

(figure 44). Almost pure torsion has been successfully implemented in the model. The other

forces are very small compared to the torsional moment and are negligible.



Figure 44: Pure torsional moment similar to A9 +tor m loadcase applied in 11000 TEU ship

Part 5.3 concluded that the stiffness of the constraint beams depends only on the Ay, Ixx and Izz

with the local coordinates as defined in figure 43. This part tries to evaluate the importance

level of each parameter. In order to do so, many models have been tested with two fixed

parameters and the left one changing. The initial values of this set of parameters for both end

cross sections were taken from the midship section (table 16). The criterion of comparison is

the relative deflexion of the hatch diagonals compared to the one in the Entire model (figure

45 and 46). Figures 45 and 46 show that the stiffness of the model depends highly and almost

equally on the parameters Ay and Ixx of the constraint beams. Because in case of pure torsion,

the hull is deformed equally at the starboard and at the port side, we only show here one

relative deflexion per one hold. The parameter Izz seems to be much less important than the

two others. Learning from this, the adjustments of the stiffness of the constraint beams will

only focus on Ay and Ixx.

The best suited constraint beams found for this pure torsion case is with Ay and Ixx 50% stiffer

than the original one in table 16. The relative deflexion of the hold 1 and 2 has only 1.2% and

7.9% of difference compared to the one in the Entire model.

58 Ngoc-Do NGUYEN


Table 16: Boundary conditions with the same constraint beams for both end cross sections

Ax,y,z (m2) Ixx,yy,zz (m

4)

Fore part

0 39.159

0.069 0

0 39.159

Aft part

0 39.159

0.069 0

0 39.159

Figure 45: Deflexions of Cargo Hold models (hold 1) (in % of Entire Model’s deflexion) having

different values of Ay, Ixx and Izz of constraint beams



Figure 46: Deflexions of Cargo Hold models (hold 2) (in % of Entire Model’s deflexion) having

different values of Ay, Ixx and Izz of constraint beams

5.4.2. In real-load case and pure torsion whose minimum value exists in the midship area

When we use the optimised set of boundary conditions found in the last part for the real-load

Cargo Hold model, the corresponding difference reaches -6.5% and 1% in the loadcase A9

+tor m. It means that in this loadcase, the horizontal bending moment is responsible for about

7% of difference. That is why Ay and Ixx are proposed to be increased only 30% (table 17).

The results of the relative deflexion are then 7.8% and 15% in case of pure torsion A9 +tor m;

-1% and 7.5% in case of real-load model. We also observe from the table 18 that the initial

boundary condition gives the best results regarding the deflexion in the loadcase 7 (A9 –tor

m). Meanwhile in the loadcase 6 (A9 +tor m), the best suited boundary condition is the one

with Ay and Ixx 30% stiffer. This second one will be used for the next calculations.

A first verification is done for the case of pure torsion similar to A9 –tor m where the

minimum value of torsional moment locates in the midship area (figure 47). Using this third

boundary condition of the last part (Ay and Ixx 30% stiffer), the difference of deflexion

between Entire and Cargo Hold model is 2.7% and 6.2% compared to 7.8% and 15% in the

pure torsion loadcase A9 +tor m (table 19).

60 Ngoc-Do NGUYEN


Table 17:Constraint beams’ characteristics of the Cargo Hold modell 74-97

Initial BC (Ay,Ixx) 50% stiffer (Ay,Ixx) 30% stiffer

Aft part

Ay(m2) 0.0689 0.1033 0.0895

Ixx(m4) 39.1591 58.7386 50.9068

Izz(m4) 39.1591 39.1591 50.9068

Fore part

Ay(m2) 0.0689 0.1033 0.0895

Ixx(m4) 39.1591 58.7386 50.9068

Izz(m4) 39.1591 39.1591 50.9068

Table 18: Relative deflexions of Cargo Hold models using different sets of boundary conditions in

real-load cases

(T i: Truss element i)

Relative Deflexion


Initial Boundary Condition (Ay,Ixx) 50% stiffer (Ay,Ixx) 30% stiffer

(mm) (mm) (%) (mm) (%) (mm) (%)

L6-T1 94.0 102.7 9 88.0 -6 93.0 -1

L6-T2 -87.0 -95.7 10 -81.1 -7 -86.1 -1

L6-T3 75.1 90.1 20 75.7 1 80.7 7

L6-T4 -67.4 -82.6 22 -68.3 1 -73.0 8

L7-T1 -80.3 -80.3 0 -67.9 -16 -72.0 -10

L7-T2 87.4 87.2 0 74.7 -14 78.9 -10

L7-T3 -63.1 -68.5 9 -56.4 -11 -60.6 -4

L7-T4 70.1 75.7 8 63.7 -9 67.7 -4

In a real-load model, this optimised one has a difference of -10% and -4% in loadcase A9 –tor

m compared to -1% and 7.5% in loadcase A9 +tor m (table 19). It means that the difference of

deflexion between the Entire and Cargo Hold models does not only depend on the stiffness of

the constraint beams but also on the distribution of the torsional moment. It is then not

possible by only modifying the stiffness of the constraint beams to find an optimised one

working with all the loadcases regarding the deflexion.

So far, for this big containership, a stiffer boundary condition with the constraint beams

having Ay and Ixx 30% bigger than the original one based the midship’s dimensions gave an

improvement for the deflexion in the loadcase A9 +tor m.



Figure 47: Pure torsional moment similar to A9 -tor m loadcase applied in 11000 TEU ship

Table 19: Relative deflexions of Cargo Hold models using different sets of boundary conditions in 2

pure-torsion loadcases


Cargo Hold model

Entire FE model Initial BC (Ay,Ixx) 50% stiffer (Ay,Ixx) 30% stiffer

(mm) (mm) (%) (mm) (%) (mm) (%)

A9 +tor m-T1 104.2 125.3 20.3 105.4 1.2 112.3 7.8

A9 +tor m-T2 -104.2 -125.3 20.3 -105.4 1.2 -112.3 7.8

A9 +tor m-T3 94.2 121.2 28.6 101.7 7.9 108.3 15.0

A9 +tor m-T4 -94.2 -121.2 28.6 -101.7 7.9 -108.3 15.0

A9 -tor m-T1 -108.7 -124.7 14.7 -105.0 -3.4 -111.6 2.7

A9 -tor m-T2 108.7 124.7 14.7 105.0 -3.4 111.6 2.7

A9 -tor m-T3 -103.1 -122.4 18.7 -102.9 -0.2 -109.6 6.2

A9 -tor m-T4 103.1 122.4 18.7 102.9 -0.2 109.6 6.2

It is now important to check with this optimised one how the warping stress has been

changed. To do so, four points at the center of four shell elements are considered: two on the

hatch coaming and at the middle of the second hold (x = 164.8m) and two in the bilge area

62 Ngoc-Do NGUYEN


with the same x coordinate and z = 2.43m (figure 48). Those shell elements are located in the

middle of the hold to better neglect the local effect due to the local bending of the transversal

members because of torsion. The difference of longitudinal stress between the two models

only come from the warping since the adjustment of longitudinal and horizontal bending

moments are considered accurate enough baring in mind that it exists always a difference in

the scale of some MPa due to the non-adjustment of the longitudinal force Nx.

Figure 48: Considered points for the comparison of longitudinal stress

We will look at longitudinal stress at the centre of those shell elements (table 20) in the Entire

FE model, the Cargo Hold models using the initial boundary condition and the one with Ay

and Ixx 30% stiffer (table 19). Two loadcases are considered: A9 +tor m which have a good

deflexion using the second boundary condition and A0 +tor a which has a zero-crossing of the

torsional moment in the midship area (figure 40). It is observed that even though by using the

new constraint beams, deflexions have been improved, the longitudinal stress and

consequently the warping stress did not change correspondingly and is still very different

from the one in the Entire model in the scale of 30 to 40 MPa.

Table 20: Longitudinal stress in the Cargo Hold model using different boundary conditions in real-

load cases

Longitudinal stress σx (MPa)

Entire model Cargo Hold model

Initial BC (Ay,Ixx) 30% stiffer

A0 +tor a

Element 1 97 59 59

Element 2 60 97 97

Element 3 -138 -88 -87



Element 4 25 -3 -3

A9 +tor m

Element 1 108 138 137

Element 2 159 124 125

Element 3 -134 -90 -90

Element 4 -34 -53 -53

5.5. Influence of torsional moment in the fore and aft part on the midship

area in case of pure torsion

So far, the modification of the boundary condition by changing the stiffness of the constraint

beams does not seem to work with the warping stress. These parts 5.5 and 5.6 try to find the

reason. In this part 5.5, we analyse the effect of torsional moment of the fore and aft part on

the midship area regarding the deflexion and warping stress.

The loadcases considered are the two pure torsion ones having the torsional distribution

similar to A0 +tor a and A9 +tor m which were used to compare the longitudinal stress in the

midship area. In each pure torsion loadcase, there are two torsional moments which have the

same value in the midship but differ in the aft and fore parts (figure 49 and 50). To have a

clear difference, high values of torsional moments were created. The purpose is not to use the

results of this pure torsional loadcase to superimpose the effects in real-load cases but to

conclude whether the torsional moment at the ends of the ship will affect the deflexion of the

structures in the midship.

Figure 49: Two pure torsional moments similar to A0 +tor a loadcase

(Left: A and Right: B)

64 Ngoc-Do NGUYEN


Figure 50: Two pure torsional moments similar to A9 +tor m loadcase

(Left: A and Right: B)

The points to consider the warping stress are the same with the two first points in the figure

48 which are located in the coamings. The results in table 21 show the dependence of the

deflexion and warping stress of the structures in the midship on the torsion at the fore and aft

parts of the model. However, one Cargo Hold model can not modelise the torsional moment

outside of its limit. And because the torsion at the aft and fore part of the ship differs from one

loadcase to another, it does not exist one set of constraint beams that works for all the

loadcases regarding the deflexion and warping stress. This is the same conclusion with the

one obtained in the part 5.4. There could be however one set of boundary condition that works

for one specific loadcase or some similar ones. But then, it is not assured that it will give

relevant results compared the Entire model regarding both deflexion and warping stress. For

example, the optimised set of constraint beams found above works quite well for the

deflexion in the loadcase A9 +tor m but not for warping stress.

Table 21: Comparison of the deflexion and warping stress in Entire FE models analysing the effect of

the torsion at the fore and aft part

(Trusses are used to calculate the deflexion between two hatch diagonals)

Load cases

Deflexion (mm) Warping stress (MPa)

Truss 1 ( in Hold 1) Truss 3 (in Hold 2) Element 1 Element 2

A0 +tor a (A) -6.0 8.5 -68.7 68.7

A0 +tor a (B) -18.3 0.6 -95.4 95.4

A9 +tor m (A) 94.6 84.5 54.4 -54.4

A9 +tor m (B) 89.4 78.7 56.7 -56.7



5.6. Independence of constraint beams’ stiffness on warping stress in

Cargo Hold model

So far, for this big containership, the optimised boundary condition found works only with the

deflexion in the loadcase A9 +tor m and A9 –tor m and does not improve the inaccuracy

regarding the warping stress of the Cargo Hold model. This work tries to find out whether it

exists a relation between the stiffness of the constraint beams and the warping stress of the

midship structure. We work with the big 11000 TEU containership and the bulkcarrier 48000

DWT (see more in part 1.2) where good results of deflexion and warping stress have been

obtained using the standard boundary conditions of the future HCSR. To verify the

dependence of the constraint beams’ stiffness on warping stress, for each Cargo Hold model,

two sets of constraint beams are used: one very soft and another one very stiff.

For the 11000 TEU ship, the difference of stiffness of the two models is 50 times for the shear

area (A) and 1000 times for the moment of inertia (I) (table 22). The difference of longitudinal

stress and so warping stress is about 5 MPa for the loadcase A0 +tor a and 8 MPa for the

loadcase A9 +tor m (table 23). Meanwhile, the difference of warping stress that needs to be

corrected is in the order of 40 MPa (table 20). That means it is not feasible to correct the

stiffness of the constraint beams to have a good correlation regarding the warping stress

between the Cargo Hold and Entire models.

Table 22: Two boundary conditions (very soft and very stiff) applied to the 11000 TEU ship

A (m2) I (m

4)

Very soft Boundary Condition 0.01 0.1

Very stiff Boundary Condition 0.5 100

Table 23: Longitudinal stress in the two Cargo Hold models of 11000 TEU ship corresponding to two

different boundary conditions


Very soft BC Very stiff BC

A0 +tor a Element 1 56 51

Element 2 82 86

A9 +tor m Element 1 145 137

Element 2 93 101

66 Ngoc-Do NGUYEN


For the 48000 DWT bulkcarrier, a similar calculation has been done using two boundary

conditions as in table 24. The shear area (A) and moment of inertia (I) are four times different

between the two models. Meanwhile the difference of warping stress of the two shell

elements located on the coaming of the midship area in the loadcase LC1 Tor1 (figure 51) is

only 3 MPa (Table 25).

To conclude, this set of boundary conditions has a small effect on the warping stress. The

change of boundary conditions’ stiffness could be effective to correct the difference of

warping stress between the Entire and Cargo Hold models only when this difference of stress

is small such as in bulkcarriers. In containerships, the difference of warping stress is normally

much higher and modifying the stiffness of the constraint beams is not effective to

compensate the difference.

Table 24: Two boundary conditions (very soft and very stiff) applied to the 48000 DWT ship

A (m2) I (m

4)

Very soft Boundary Condition Aft part 0.026 3.01

Fore part 0.026 2.67

Very stiff Boundary Condition Aft part 0.104 12.05

Fore part 0.104 10.68

Figure 51: Loadcase LC1 Tor1 of the 48000 DWT bulkcarrier



Table 25: Longitudinal stress in the two Cargo Hold models of 48000 DWT ship corresponding to two

different boundary conditions


Very soft BC Very stiff BC

LC1 Tor1 Element 1 -57 -54

Element 2 -35 -38

5.7. Application of the suitable boundary condition of the big ship (11000

TEU) on the small one (2700 TEU)

The suitable boundary condition for the big ship requires the increase from 0% to 30% of Ay

and Ixx of the constraint beams starting from the midship parameters depending on the

loadcase considered (table 18). In the previous work with the small ship, the initial values of

constraint beams stiffness were taken from the end cross sections and not from the midship.

That is why the constraint beams’ stiffness at the aft and the fore parts are not the same. In

this part, they will all be based on the midship with the increase factor for Ay and Ixx of 30%,

50% and 100%. Izz remains constant (Table 26). The best suited boundary condition is the one

with 100% increase of stiffness (Table 27). This ratio is higher than in the big ship where the

required increase of constraint beams’ stiffness is only from 0% to 30%. The results is also

similar to the ones in the part 4.4.1 concluding that the stiffness of the constraint beams

should increase twice for the small ship in order to obtain a good correlation of deflexion in

the high-torsional loadcase (loadcases 6 and 7).

Table 26: Constraint beams' characteristics of the Cargo Hold model 115-189 (2700 TEU)

Initial BC (Ay,Ixx) 30% stiffer (Ay,Ixx) 50% stiffer (Ay,Ixx) 100% stiffer

Aft part

Ay (m2) 0.023 0.030 0.035 0.046

Ixx (m4) 7.794 10.132 11.691 15.588

Izz (m4) 7.794 7.794 7.794 7.794

Fore part

Ay (m2) 0.023 0.030 0.035 0.046

Ixx (m4) 7.794 10.132 11.691 15.588

Izz (m4) 7.794 7.794 7.794 7.794

68 Ngoc-Do NGUYEN


Table 27: Relative deflexions of Cargo Hold models with different boundary conditions (2700 TEU)

Relative Deflexion


Initial BC (Ay,Ixx) 30% stiffer (Ay,Ixx) 50% stiffer (Ay,Ixx) 100% stiffer

(mm) (mm) (%) (mm) (%) (mm) (%) (mm) (%)

L6-T1 54.8 68.7 25 65.5 20 63.7 16 60.0 9

L6-T2 -50.6 -64.6 28 -61.5 21 -59.5 18 -55.8 10

L6-T3 40.5 58.6 45 55.7 37 53.9 33 50.3 24

L6-T4 -34.3 -52.5 53 -49.4 44 -47.6 39 -44.2 29

L7-T1 -48.8 -56.3 15 -53.4 9 -51.8 6 -48.4 -1

L7-T2 53.3 60.9 14 57.9 9 56.3 6 52.8 -1

L7-T3 -34.3 -45.0 31 -42.2 23 -40.7 19 -37.6 10

L7-T4 41.1 51.5 25 48.8 19 47.2 15 44.1 7



6. CONCLUSIONS

The primary aim of this study has been to investigate the feasibility of using the Cargo Hold

model to analyse the torsional effects of containerships. The Cargo Hold model in the future

HCSR which gives good results for the torsional analysis of bulkcarriers was used for the

analyses of containerships. The warping rigidity of the cut-off parts plays an important role in

the deflexion and warping stress of the containerships’ Cargo Hold model. It is represented in

the model by the constraint beams’ stiffness. Even though the current set of boundary

conditions works with tankers and bulkcarriers, it did not give good results for all the

loadcases of containerships regarding the deflexions and warping stress.

For both small and big containerships, the boundary conditions need to be stiffened to better

represent the deflexion of the hatch diagonals in the loadcases where high torsional moments

exist in the midship area. The current constraint beams have homogeneous characteristics and

their stiffness are based on end cross sections’ parameters with the shear area equal to 1/10 of

the outer bottom plating area. The moment of inertia equals to 1/25 of the section’s moment

of inertia. However, with the local coordinates as in figure 40 and the rigid links in y-

translation, z-translation and x-rotation, only Ay, Ixx and Izz will affect the warping rigidity and

need to be modeled. The last parameter is much less important than the two first ones. For

small containerships, starting from the midship’s parameters (A and I), the most relevant

deflexions in these high-torsion loadcases are obtained when the shear area and moment of

inertia are increased by 100%. For big containerships, this increase should reach 30% to have

a suitable result regarding the deflexion in the loadcase A9 +tor m. For the deflexion in the

loadcase A9 –tor m, the initial parameters Ay, Ixx, Izz of the constraint beams calculated from

the midship’ vertical bending moment of inertia and shear area of the outer bottom plate seem

to work well and no modification needs to be done.

It is also shown that using different sets of constraint beams corresponding to different

thicknesses of the plates does not give better results regarding the deflexion. Both linear and

exponential relations between the plate thickness and constraint beams’ stiffness have been

tried before having this conclusion.

Regarding the warping stress, the current Cargo Hold model using the boundary condition in

HCSR does not give a correct result. Besides, the stiffness of the constraint beams does not

have a necessary effect on the warping stress to correct this error. Changing the stiffness of

the constraint beams induces at maximum a change of some MPa in the warping stress in the

coaming area meanwhile the difference between the Entire and Cargo Hold models is in the

70 Ngoc-Do NGUYEN


order of 40 MPa for the big 11000 TEU containership. This is different from the bulkcarriers

where the difference of warping stress between the Entire and Cargo Hold models is in the

order of some MPa and changing the stiffness of the constraint beams could be a good

approach to correct the error.



ACKNOWLEDGEMENTS

This thesis was developed in the frame of the European Master Course in “Integrated

Advanced Ship Design” named “EMSHIP” for “European Education in Advanced Ship

Design”, Ref.: 159652-1-2009-1-BE-ERA MUNDUS-EMMC.

I would like to express my deep gratitude towards Prof. Robert Bronsart for giving me the

opportunity to follow my last semester at the University of Rostock and for his kind

assistance during my study there. His supports have favorized greatly my stay in Germany.

Furthermore, I would also like to thank Mr. Helge Rathje for accepting me to do my master

thesis in the department of Rules Development (NB-RA) at Germanischer Lloyd (GL) group.

I received the best conditions to focus on my thesis during the working time there.

Additionally, I would like to extend my deepest gratitude towards my direct supervisor Dr.

Jörg Rörup for his guidance and unlimited patience. His wisdom and advice from the initial to

the final level enabled me to develop an understanding of the subject.

Moreover, I would like to especially thank Prof. Patrick Kaeding whose advice and guidance

have greatly helped me to learn new concepts which are vital to my work.

I am also indebted to many of my colleagues in the NB-RA department to support me,

especially, Dr. Thomas Schellin, Samuel Fils Iloga Balep, Philipp Ulrich, Dr. Hubertus von

Selle. The countless discussions with them provided me with excellent opportunities to learn

new things and also to strengthen my foundations.

Last but not least, I would like to thank the professors and teaching assistants of the EMSHIP

program, especially, Prof. Philippe Rigo from the University of Liege, Prof. Lionel Gentaz

and Prof. Pierre Ferrant from Ecole Centrale de Nantes for the knowledge they have provided

me during the study of this master. That has helped me greatly mature as a capable engineer.

72 Ngoc-Do NGUYEN


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