ME 4101 Bachelor of Engineering Dissertation

NNaammee :: YYeeoo LLiiaannggyyii GGaabbrriieell

Supervisor : A/Prof Lim Kian Meng JSPL Supervisor : Mr Seow Tan Hong

Snr Vice President (Engineering) SembCorp Marine

AAAMMM 333000

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Summary

This project was proposed by Jurong Shipyard Pte Ltd (JSPL) to do a study of the

stresses involved and load distributions on a tanker when it is in the dry dock.

The model of the tanker was being modeled using plate elements and analyzed

using the finite element method with the MSC.FEA software. To simulate the

static condition of the tanker in the dry dock, the restrain forces were applied at

the keel block locations and the model loaded with pressure forces in lightship

condition. Results of the analysis reviewed acceptable global structural

displacement as well as stresses. Modification of keel block positions concluded

the importance of the keel block alignment at the keel of the tanker, supporting a

large amount of weight of the tanker.

i

Acknowledgements

The author would like to acknowledge and express earnest appreciation to

A/Professor Lim Kian Meng, Mr Seow Tan Hong (Snr Vice President, Sembcorp

Marine) and Mr Yap Chea Kim (Engineer, JSPL) for their advice, guidance and

helpful discussions throughout the project.

ii

Table of Contents

Page

Summary i

Acknowledgements ii

Table of contents iii

Lists of Figures v

1. Introduction 1

2. Ship Terminology 2

3. Objective 2

4. Literature Review 3

5. Finite Element Method Theory 5

6. Methodology 7

6.1 Structure Modeling 7

6.2 Material Properties 12

6.3 Coordinate System 13

6.4 Plate Element vs Solid Model 13

7. Loading the Model 16

7.1 Lightship Loading 16

7.2 Restrain Forces 17

8. Results and Discussion 19

8.1 Lightship Condition 19

8.2 Keel Blocks Concentrated at the Centre 24

8.3 Keel Blocks Concentrated at the Side 29

iii

9. Conclusion 31

10. Recommendations 32

11. References 33

iv

List of Figures

Figure 1 Photo taken of a vessel in dry

Figure 2 Picture of keel blocks

Figure 3 Schematic illustration of a vessel

Figure 4 Ship directional terminologies

Figure 5 Illustration of finite element shapes

Figure 6 Schematic illustration of degrees of freedom

Figure 7 Longitudinal curves of tanker

Figure 8 Transverse curves of tanker

Figure 9 Line model of tanker

Figure 10 Top view of line model modeled with plate elements

Figure 11 Bottom view of line model modeled with plate elements

Figure 12 Top view of model with double hull

Figure 13 Sectional view of model with stiffeners (yellow curves)

Figure 14 Cross sectional view of model with stiffeners

Figure 15 Sectional view of longitudinal plates between hulls

Figure 16 Box model using plate elements

Figure 17 Displacement fringe of plate element model

Figure 18 Solid model of box

Figure 19 Displacement fringe of solid model

Figure 20 Lightship loading

Figure 21 Diagram illustrating restrain forces

v

List of Figures (continued)

Figure 22 Displacement fringe with idealized restrain case

Figure 23 Displacement fringe of inner hull without plates

Figure 24 Stress fringe of inner hull without plates

Figure 25 Stress distribution of lightship condition

Figure 26 Stress distribution of stiffeners in lightship condition

Figure 27 Constrain forces in lightship condition

Figure 28 Arrangement of keel blocks concentrated at the keel

Figure 29 Displacement fringe when keel blocks were concentrated at the

keel

Figure 30 Stress fringe when keel blocks were concentrated at the keel

Figure 31 Stress distribution of stiffeners when keel blocks were

concentrated at the keel

Figure 32 Constrain forces when keel blocks were concentrated at the

keel

Figure 33 Arrangement of keel blocks concentrated at the side

Figure 34 Displacement fringe when keel blocks were concentrated at

the keel

vi

1. Introduction

This project was proposed by Jurong Shipyard

Pte Ltd (JSPL) in June’06 to do a study of the

stresses involved and load distributions on a

tanker when it is in the dry dock (Figure 1). At

present, only simple calculations are made prior

to a vessel’s dry docking in order to determine

the positions of the keel blocks (Figure 2) to

support the weight of the vessel. The actual

stresses acting on the vessel’s hull as well as on the dry dock are often unknown.

Due to time and sheer size of modeling a tanker, JSPL

has never attempted to analyze the global structure.

Only sections of vessels are often modeled for analysis.

This study on the stresses involved when the tanker is in

the dry dock will thus be able to affirm the engineer’s

estimation and prevent overloading on the vessel as well

as the dry dock.

Figure 1: Photo taken of a vessel in dry

Figure 2 Picture of keel blocks

1

2. Ship Terminology

Anchor

Navigation bridge

Bulbous bulb

Propeller Rudder Keel

Figure 3: Schematic illustration of a vessel

igures 3 and 4 illustrate the various parts of the vessel and its directional

rminologies which will be used in this report.

. Objective

his project aims to analyze the stresses and displacements experienced by a

nker when it is in the dry dock.

Port Side

Starboard Side

Forward

Stern / Aft

Figure 4: Ship directional terminologies

F

te

3

T

ta

2

4. Literature Review

“Guidelines for Tanker Structures” by Nippon Kaiji Kyokai[1] (ship classification

ociety) was reviewed as a reference for analysis of tanker structures. Nippon

nds that structural analysis be performed by the Finite

extent of analysis is decided such that the actual

stress conditions of the tanker can be reproduced by considering the arrangement

of cargo oil and ballast tanks, the loading pattern and the arrangement of members

near the bulkhead. Members in the model should consist of members to be

evaluated and primary members within the extent of the analysis. Longitudinal

stiffeners and watertight bulkhead stiffeners should also be included in the model

as load transmitting members.

The size of the mesh should be appropriately selected, considering the stress

condition in the model and the meshing of elements is to be performed rationally.

Care should also be taken to avoid meshes with large aspect ratios. The standard

size of the element in the stress evaluation part is decided by taking one side of

the element as approximately equal to the spacing of the nearby stiffeners.

s

Kaiji Kyokai recomme

Element Method (FEM) with the structural members replaced by structural

models using plate elements. The

3

The next literature that was reviewed was “Rules for Classification and

d stresses of the hull and the primary structural

components.

global

strength analysis. These simplifications are permissible, provided that the results

are only impaired to a negligible extend. A common simplification is to neglect

small secondary components or details such as the brackets at the frames and

small cut-outs.

Construction” by Germanischer Lloyd[2], section 5: Analysis Techniques –

Guidelines for Strength Analyses of Ship Structures with the Finite Element

Method. Lloyd categorized strength analysis in the following steps:

• Determination of the objective, type and extend of the analysis.

• Modeling of the structure and the boundary conditions.

• Determination and modeling of the loads.

• Execution of the analysis.

• Evaluation and assessment of the results.

In ship structures, the deformation and stresses can usually be subdivided into the

following categories, depending on the structural conditions:

• Global deformations an

• Local deformation and stresses of the primary and secondary components.

• Locally increased stresses at structural details and discontinuities.

Lloyd also mentioned that owing to the complexity of the ship structure,

simplifications are generally necessary in the modeling, especially for

4

5. Finite Element Method Theory

The finite element method is a way of analyzing a complex engineering problem

by breaking it up into many smaller, simpler problems. In the case of structural

analysis, the complex structure is broken up into many small pieces call finite

elem odes. The

ass b

The in which are relatively easy to formulate and

ana e he stress and strain within each

ents. The elements are connected to each other at grid points or n

em lage of elements is called a finite element model.

f ite elements have shapes

lyz : beams, plates and blocks (Figure 5). T

element is a function of the displacement of the grid points it is connected to.

Figure 5: Illustration of finite element shapes

he displacement of each grid point may be described by six independent degrees T

of freedom (Figure 6). A degree of freedom is defined as an independent

component of translation or rotation.

Figure 6: Schematic illustration of degrees of freedom

5

A continuous structure has theoretically an infinite number of degrees of freedom.

• Model bodies composed of several different materials.

• Handles unlimited number of boundary conditions.

• Elements can vary in size allowing use of small elements when necessary.

• Alter finite el

• Model many different types of physics.

The finite element method approximates the behavior of a continuous structure

with a finite number of grid points. Finer mesh will result in better the

approximation of the characteristics and behavior of the structure. The Finite

Element Method is thus capable of solving large, complex problems with general

geometry, loading and boundary conditions.

Advantages of the finite element method:

• Model irregularly shaped bodies easily.

• Handles general loading conditions.

ement model relatively easily and cheaply.

• Handle non-linear behavior.

6

6. Methodology

6.1 Structure Modeling

The tanker to be modeled for analysis is the 11-1056, 30 000 dead weight ton

product tanker.

Figure 7: Longitudinal curves of tanker

Figure 8: Transverse curves of tanker

The curves of the tanker were first obtained from the ship’s drawing. Figures 7

and 8 show the longitudinal as well as the transverse curves. These lines were

used to construct the many frames and bulkheads that are essential in the tanker.

The frames were then connected to obtain a gross line model of the tanker as

shown in Figure 9.

7

Figure 9: Line model of tanker

As recommended by Kaiji Kyokai, the structural analysis of the tanker will be

analyzed using FEM. The line model was thus imported into the MSC.FEA

software. With MSC.FEA, plate elem the line mode

to simulate the tanker’s structure. Comparison between modeling in plate

elements and solid modeling will be discussed in section 5.4.

Figures 10 and 11 show the line model of the tanker being modeled using plate

elements.

ents were constructed from l

Figure 10: Top view of line model modeled with plate elements

8

Figure 11: Bottom view of line model modeled with plate elements

March 1989, the oil tanker Exxon Valdez owned by the former Exxon

er to be realistic, all

ain structures have to be modeled and the double hull is essential in the global

ith a double hull as well as the top deck.

In

Corporation hit Prince William Sound's Bligh Reef and spilled an estimated 11 to

30 million U.S. gallons (50,000 m³ to 150,000 m³) of crude oil. This accident

resulted in the U.S congress passing the Oil Polution Act in 1990 resulting in a

mandate for tankers to have double hull design, providing an additional layer

between the oil tanks and the ocean. For the model of the tank

m

structural strength of the model. Therefore, as shown in Figure 12, the model was

modelled w

Figure 12: Top view of model with double hull

9

The hull, double hull and the frames were meshed using the isomesh mesher with

quad shaped elements for 90% of the model. However, due to the curvature of the

model, several parts particularly at the forward and stern required meshing using

tria shaped elements. The prevalent mesh size of the model measures 2 m by 2 m.

Method o areas of

adings and restrains with a mesh size of 1m by 1m. “Mesh on mesh” proved

esence of stiffeners.

f “mesh on mesh” was employed to vital areas, such as the

lo

advantageous as it enables the global structure to maintain its mesh size while

having a specified mesh size based on requirements at selected areas. These

meshings are considered to be relatively fine as the model’s length is

approximately 300 m. This will result in at least 150 elements through the

model’s length.

Another important feature that is essential in the model is the stiffeners on the

tanker. As the tanker is produced from 2 cm thick metal plates, it can easily

deform along its longitudinal direction without the pr

Figure 13: Sectional view of model with stiffeners (yellow curves)

10

Figure 13 shows a cross section of the model with yellow curves modeled on the

hulls. These curves were modeled to be defined as stiffeners for the model. The

curves were meshed using “bar 2” topology and were defined as 1D beams.

Figure 14: Cross sectional view of model with stiffeners

es to the analysis of the model as

the outer hull will be seated on the keel blocks but the loads will be applied on the

inner hull. Therefore, the plates connecting the inner hull and the outer hull are

essential structures to be modeled to prevent excessive deformation of the inner

hull as well as to effectively transmit the loads to the outer hull.

Figure 14 shows a cross sectional view of the model with a 3D view of the curves

being modeled as stiffeners.

The double hull posed as a difficulty when it com

11

Figure 15: Sectional view of longitudinal plates between hulls

Figure 15 shows a cross sectional view of the model with the plates connecting

the inner hull to plate elements

nd meshed using isomesh with quad shaped element in order to match the

the outer hull. The plates were also modeled using

a

meshing of the hulls to ensure that the forces will be effectively transmitted

throughout the model.

6.2 Material Properties

A single material of steel was assumed for all components in the model. The

linear elastic material model was employed. The material parameters are,

Young’s modulus = 200GPa, Poisson Ration = 0.3 and Density = 7.83×103 . kg/m3

12

6.3 Co-ordinate System

Cartesian coordinate system was adopted in the Finite Element Modeling with

configurations as follows.

X-axis: Longitudinal, positive FWD

Y-axis: Vertical, positive upwards

Z-axis: Transverse, positive towards PORT side

6.4 Plate element vs Solid model

As mentioned, the model of the tanker was modeled using plate element instead of

el of accuracy.

esh size and result of the two different methods of

a solid model. This was basically due to the scale of the model and the thickness

of the steel plates used in the construction of the tanker. As the steel plates are 2

cm in thickness, modeling the tanker in solid model will result in a very fine mesh

as at least two elements must be meshed through the thickness of the plate to

obtain an acceptable lev

In order to illustrate the m

modeling, a box model was modeled as a test platform due to its geometric

simplicity instead of modeling the tanker in both plate elements as well as solid

model.

13

Figure 16: Box model using plate element

Figure 16 shows the model of the box using plate

and quad elements. Each

element. A relatively coarse mesh was used to mesh

this model using isomesh

element measures 20 mm by 20 mm.

Figure 17: Displacement fringe of plate element model

The maximum displacement when the plate element model was loaded (Figure

17) yielded 14.3 mm.

14

For comparison, a solid model of the box

imensions was modeled as

.

The same loading and restrain conditions were

employed to the solid model (Figure 18). In

order to yield acceptable results, the mesh size

on the solid model has to be very fine as the

thickness of the box is very small.

The solid model was meshed using paver mesh. The dimension of the mesh was 2

mm by 2 mm. With this mesh size, three elements were meshed through the

thickness of the box. The result of the analysis is as shown in Figure 19.

Figure 18: Solid model of box

having the same d

shown in Figure 18

Figure 19: Displacement fringe of solid model

15

The maximum displacement for th

mm of the plate element model. O

experienced by the solid model as ent model. This

comparison thus prove that the plate element model is a more effective model as it

can yield approximately the same r

thus saving a significant amount of

only apply to thin plate structures.

7. Loading the Model

odel is

perative in this condition. In addition, assignment of restrain forces is

paramount in a realistic stress state simulation of the tanker when in dry dock.

7.1 Lightship Loadings

e solid model is 14.5 mm as compared to 14.3

nly a slight variance in the displacements was

compared to the plate elem

esults as the solid model with a coarser mesh,

time needed for analysis. However, this may

Loadings on the model are critical elements in the determination of result

accuracy in the analysis. Vessels in lightship condition are a majority when dry

docking is due. As such, effective static lightship loading of the m

im

Lightship weight distribution of the tanker was obtained through the referencing

of the tanker’s specifications. This distribution primarily illustrates the weight in

metric tons through a specified longitudinal distance of the tanker. Application of

weight distribution is achieved through pressure (kg/m2) loads converted from

weight (kg).

16

Figure 20: Lightship loading

Figure 20 illustrates the lightship pressure loads in accordance to the lightship

weight distribution. Observation of the model in Figure 20 reveals the

concentration of pressure loads at the stern towards the mid ship. This is

evitable with onboard machineries such as the main engine located at the stern.

bsence of pressure loads in the mid ship section to the forward are justified by

in

A

the tanker’s empty ballast and cargo tanks.

7.2 Restrain forces

As mentioned, restrain forces play a vital role in the result of the static load

analysis. Placement of the restrain forces are warranted on the basis of the keel

locks’ positions when the tanker is in the dry dock. This justification is deemed

na s are the only structures supporting the tanker’s static

iciently large to only exert only a downward force. Rotations about

e X-axis, Y-axis and Z-axis were also assumed to be zero.

b

reaso ble as the keel block

weight. The translation of the restrain forces in the Y-axis was resolved to be zero

as the keel blocks are directly supporting the tanker and no movement in the Y-

axis is assumed to be present in the keel blocks. The translations in the X-axis

and Z-axis were also resolved to be zero based on the assumption that the tanker’s

weight is suff

th

17

Assumptions made with regards to the restrain forces were in attempt for a more

simplified global analysis with time constrains in view. Nevertheless, results were

not unacceptably compromised. On the contrary, results obtained on the

eformation and stresses on the tanker as

oor, resulting in a global deformation of the dock floor.

d well as the forces needed to be exerted

by the keel blocks were well within acceptable limits. More precise analysis can

be obtained with the modeling of the keel blocks as well as the dock floor with

foundations modeled as springs. Such meticulous details will ensure that the

forces exerted by the tanker will be transmitted via the keel blocks to the dock

fl

Figure 21: Diagram illustrating restrain forces

Figure 21 shows the restrain forces applied at the positions of the keel blocks at

the hull of the model.

18

8. Results and Discussion

8.1 Lightship Condition

The tanker was simulated in the dry dock with lightship condition in accordance

to the loading conditions as mentioned earlier. Idealized keel block positions

were initially simulated as a benchmark for other variance in loading conditions.

Figure 22: Displacement fringe with idealized restrain case

Figure 22 shows the displacement fringe of the model with idealized restrain case

in lightship condition. The maximum displacement is approximately 6 cm acting

on the port side of the stern. This displacement is deemed to be reasonable

considering the 60 m width of the tanker. The occurrence of the maximum

displacement at the stern of the tanker can be accorded to the fewer number of

19

keel blocks at the stern despite the maximum load due to the engine room at the

n. Additional keel blocks are unable to be

ositioned at the stern due to the curvature of the stern and the placement (or

ell as the rudder.

As for the mid ship section, the maximum displacement is approximately 4cm

acting at the centre of the tanker’s width. It flexes as a beam restrained at both

ends with load at the centre. This displacement is mainly due to the loadings on

the inner hull and restrains on the outer hull. The loads were being transferred

from the inner hull to the outer hull via longitudinal plates as shown in Figure 15.

As the longitudinal plates are uniformly spaced, the inner hull tends to deform in

between the longitudinal plates with the largest deformation at the centre. In

general, the inner hull experienced approximately 1.5 cm of displacement.

stern during lightship conditio

p

position) of the propeller as w

However, the maximum displacement is only over a relatively small area of the

hull. Generally, the stern’s displacement is approximately 1.5 cm.

Figure 23: Displacement fringe of inner hull without plates

20

As mentioned, the longitudinal plates between the inner and outer hulls are vital in

the amount of deformation experienced by the inner hull.

Figure 23 illustrates the displacement of the inner hull without the longitudinal

plates. The maximum displacement is two orders of magnitude more than in

Figure 22.

Figure 24: Stress fringe of inner hull without plates

As shown in Figure 24, the stress experienced by the inner hull at the position of

maximum displacement has far exceeded the yield stress of its material of

approximately 250 MPa. This illustrates the importance of modeling the

longitudinal plates.

21

Next is the analysis of the stress distribution of the tanker in lightship condition.

Figure 25: Stress distribution of lightship condition

The analysis result (Figure 25) indicates that the maximum Von Mises stress at

162 MPa for lightship condition is within the yield strength of steel of about

250MPa. This occurs only over a very minute area at the stern. The global stress

experienced by th

that is independent of the element coordinate system used.

As such, it is always a positive number.

e model is approximately 43 MPa.

The maximum stress will result in a Factor of Safety of 1.54 for the model.

The Von Mises stress combines three dimensional stresses to a single value and is

an invariant quantity

22

The Von Mises stress is calculated as follows:

)(32

)()()(222

222

zxyzxy

xzzyyx τττσσσσσσ

σ +++−+−+−

=′

Figure 26: Stress distribution of stiffeners in lightship condition

Stresses on the stiffeners had to be analyzed in order to determine whether

yielding has occurred. Figure 26 shows a maximu

m stress of 15 MPa on the

feners. This value is also below the yielding stress of the material, giving a

3 MPa. Only the stiffeners at the stern experienced higher stresses due to the

stif

safety factor of 17. The stresses experienced by the stiffeners are generally about

loadings and lesser restrains at the stern.

23

Figure 27: Constrain forces in lightship condition

Figure 27 shows the constrain forces supporting the tanker in lightship condition.

The maximum constrain force is 1×106 N. It can be calculated that each keel

block is supporting a weight of 100 tons. At present, the keel block’s maximu

load capacity is determined to be 200 tons. This will then result in a safety factor

of 2 for the keel blocks which can affirm the under loading of the keel blocks,

preventing damage.

8.2 Keel blocks concentrated at the centre

m

The next case is to determine the important keel blocks’ positions to prevent

damage to the hull of the tanker. It is known that the structural keel is a large

beam which the hull of a ship is built around. The keel runs in the middle of the

ship, from the bow to the stern, and s on or spine of the erves as the foundati

24

structure, providing the major source of structural strength of the hull. Therefore,

the restrain forces were concentrated at the keel and the alternate rows of restrain

forces were removed to minimise the amount of keel blocks. This arrangement is

shown in Figure 28.

Figure 28: Arrangement of keel blocks concentrated at the keel

The results of concentrating the keel blocks at the keel and minimizing the rest of

the keel blocks are shown in Figures 29, 30 and 31.

Figure 29: Displacement fringe when keel blocks were concentrated at the keel

25

The maximum displacement as shown in Figure 29 is approximately 6.5 cm as

compared to 6 cm in Figure 22. The difference of 0.5 cm is deemed to be

relatively small as the number of keel blocks was significantly reduced. However,

more areas especially at the mid ship section experienced slightly more

isplacement. In general, these increases in displacement are still well within

acceptable limits and can be considered to be very minute.

d

Figure 30: Stress fringe when keel blocks were concentrated at the keel

The stress analysis result (Figure 30) indicates that the maximum Von Mises

stress is 170 MPa as compared to 162 MPa in Figure 25. This is also within the

yield strength of steel at about 250MPa. The maximum stress is only an increase

of approximately 5% resulting in a new safety factor of 1.47 instead of 1.52.

26

Figure 31: Stress distribution of stiffeners when keel blocks were concentrated at the keel

The stress distribution of the stiffeners in Figure 31 is considerably larger than in

Figure 26, yielding 49.7 MPa as compared to 15 MPa. This can be accorded to

the reduction of keel blocks. The stiffeners have to take up the stress to resist the

deformation of the hull as a result of the reduction of keel blocks. However, the

maximum stress obtained by the stiffeners is still well below the yield stress of the

material.

Figure 32: Constrain forces when keel blocks were concentrated at the keel

27

The maximum constrain force in Figure 32 is 1.12×106 N as compared to 1×106N

in Figure 27. Each keel block will thus support a weight of 112 tons instead of

100 tons, yielding a safety factor of 1.78. This incremental load that each keel

block has to withstand is due to the reduction of the number of keel blocks and is

still well within the maximum load capacity of 200 tons.

28

8.3 Keel blocks concentrated at the side

For effective comparison, the keel blocks were rearranged to be concentrated at

the side instead of the keel as shown in Figure 33.

Figure 33: Arrangement of keel blocks concentrated at the side

Figure 34: Displacement fringe when keel blocks were concentrated at the side

29

As shown in Figure 34, the maximum displacement when the keel blocks were

oncentrated at the sides is approximately 8 cm as compared to 6 cm with the

ealized case.

Through the stress, displacement, constrain forces analysis and comparison; it can

thus be deduced that the keel of the tanker supports a considerable amount of the

total weight. Therefore, alignment of keel blocks under the keel is of utmost

importance. Positions of other keel blocks are not as critical but they must be

evenly spaced under the stiffeners. No doubt the keel blocks have not exceeded

their maximum capacity, it should be noted that the model is based on an even

distribution of load throughout. In reality, the tanker’s loading even in lightship

condition

c

idealized case and 6.5 cm when the keel blocks were concentrated at the keel.

This displacement is considered far-off from the id

are unlikely to be evenly distributed.

30

9. Conclusion

In conclusion, a successful global model structure of the tanker was modeled with

all the essential structures, mainly the double hull, frames, longitudinal plates and

stiffeners. This model was able to be loaded in its lightship condition to analyze

e stress distribution as well as the displacement throughout the tanker when it is

th

in the dry dock. The results of the stress distribution verified that the tanker will

not be damaged under idealized keel block positions and that the displacements of

the tanker due to the lightship loadings are well within acceptable limits.

Furthermore, the rearrangement of the keel blocks concluded that the alignment of

the keel blocks under the keel is paramount in the safe docking of the tanker.

Misalignment will result in structural damage of the tanker.

31

10. Recommendations

ting in a

lobal deformation of the dock floor.

ker is safe to dock at

uch situations.

As mentioned in the paper, the restrain forces were positioned in placement of the

keel blocks with the translations in the X-axis, Y-axis and Z-axis as zero. An

enhancement to the accuracy is to model the keel blocks as well as the dock floor.

The keel blocks can be modeled with actual properties to ensure that deformation

of the keel blocks occurs instead of zero displacement in the Y-axis as modeled by

a restrain force. The dock floor can also be modeled with foundations supporting

it to ensure that the load of the tanker is supported by the pilings resul

g

In addition, variations types of loading conditions such as filled ballast tanks or

minimal cargo load can be simulated to ensure that the tan

s

32

11. References

1. Rules for Classification and Construction, Germaischer Lloyd

2. Guidelines for Tanker Structures, Nippon Kaiji Kyokai

3. Mechanics of Materials, A.C. Ugural, McGraw Hill, 1991.

4. Finite Element Procedures in Engineering Analysis, K.J. Bathe, Prentice-Hall,

1982.

5. Concepts and Applications of Finite Element Analysis, R.D. Cook, John Wiley

& Sons, 1989.

6. A First Course in the Finite Element Method, Daryl I. Logan, PWS-Kent

Publishing Company, 1986.

7. The Finite Element Method, O.C. Zienkiewicz, McGraw Hill, 1994.

33