DESIGN AND ANALYSIS OF HYBRID TITANIUM-COMPOSITE HULL
STRUCTURES UNDER EXTREME WAVE AND SLAMMING LOADS
by
Md Hafizur Rahman
A Thesis Submitted to the Faculty of
The College of Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
Master of Science
Florida Atlantic University
Boca Raton, Florida
December 2013
iii
ACKNOWLEDGEMENTS
I wish to express my sincere appreciation and gratitude to my Committee
Advisor, Dr. Hassan Mahfuz for his magnificent and intelligent supervision, constructive
guidance, boundless energy, inspiration, and patience. Without his direction and
dedication, this work would not have been possible. I am also grateful to my dissertation
committee members: Dr. Manhar Dhanak, and Dr. Palaniswamy Ananthakrishnan for
their help and advices on my research.
In addition, I would like to thank Office of Naval Research for funding this
research under the grant “Transformational Craft (T-craft) Tool Development” (No:
N00014-07-1-0965). Thanks are also given to all the members of Nano-composites
Laboratory.
Finally, I am most grateful for the support, affection, and encouragement from my
wife, parents, family and friends throughout the research.
iv
ABSTRACT
Author : Md Hafizur Rahman
Title: Design and Analysis of Hybrid Titanium-Composite Hull
Structures Under Extreme Wave and Slamming Loads
Institution: Florida Atlantic University
Thesis Advisor: Dr. Hassan Mahfuz
Degree: Master of Science
Year: 2013
A finite element tool has been developed to design and investigate a multi-hull
composite ship structure, and a hybrid hull of identical length and beam. Hybrid hull
structure is assembled by Titanium alloy (Ti-6Al-4V) frame and sandwich composite
panels. Wave loads and slamming loads acting on both hull structures have been
calculated according to ABS rules at sea state 5 with a ship velocity of 40 knots.
Comparisons of deformations and stresses between two sets of loadings demonstrate that
slamming loads have more detrimental effects on ship structure. Deformation under
slamming is almost one order higher than that caused by wave loads. Also, Titanium
frame in hybrid hull significantly reduces both deformation and stresses when compared
to composite hull due to enhancement of in plane strength and stiffness of the hull. A 73
m long hybrid hull has also been investigated under wave and slamming loads in time
domain for dynamic analysis.
v
DESIGN AND ANALYSIS OF HYBRID TITANIUM-COMPOSITE HULL
STRUCTURES UNDER EXTREME WAVE AND SLAMMING LOADS
LIST OF TABLES .............................................................................................................. x
LIST OF FIGURES ........................................................................................................... xi
1 . INTRODUCTION ......................................................................................................... 1
1.1 . Composite Materials ............................................................................................... 1
1.2 . Sandwich Structured Composites............................................................................ 2
1.3 . Applications of Composite Materials in Marine Structures .................................... 5
1.4 . Existing Problems ................................................................................................... 8
1.5 . Hybrid Hull Structures ............................................................................................ 9
1.6 . Thesis Objectives .................................................................................................. 11
1.7 . Thesis Outlines ...................................................................................................... 11
2 . THEORETICAL BACKGROUNDS........................................................................... 13
2.1 . Basic of Composite Mechanics ............................................................................. 13
2.1.1 . Composite Materials ....................................................................................... 13
2.1.2 . Effective Modulus Theory .............................................................................. 14
2.1.3 . First-Order Shear Deformation Theory .......................................................... 15
2.2 . Failure Theories..................................................................................................... 17
2.2.1 . Tsai Wu Failure Criterion ............................................................................... 17
vi
2.2.2 . Maximum Stress Criterion.............................................................................. 18
2.3 . American Bureau of Shipping (ABS) Rules ......................................................... 19
2.3.1 . Wave Loads by ABS ...................................................................................... 20
2.3.2 . Slamming Loads by ABS ............................................................................... 22
2.4 . Dynamic Loads on Ship Structure ........................................................................ 23
2.4.1 . Low-Frequency Loads .................................................................................... 24
2.4.2 . High-Frequency Loads ................................................................................... 25
2.5 . Dynamic Loads by ABS ........................................................................................ 25
2.5.1 . Dominant Load Parameters (DLPs) ............................................................... 26
2.5.2 . Wave Spectra .................................................................................................. 26
2.5.3 . Vessel Motion and Wave Load Response Amplitude Operators (RAO) ....... 27
2.5.4 . Extreme Values for DLA Analysis ................................................................. 28
2.5.5 . Short-Term Response ..................................................................................... 28
2.5.6 . Equivalent Wave ............................................................................................. 29
2.5.7 . DLPs as Time Function .................................................................................. 31
3 . NUMERICAL SIMULATION OF COMPOSITE HULL .......................................... 32
3.1 . Design of Sandwich Hull Structure ....................................................................... 32
3.2 . Numerical Simulation ........................................................................................... 35
3.2.1 . Finite Element Model ..................................................................................... 36
3.2.2 . Calculation of Section Modulus ..................................................................... 37
3.2.3 . Boundary Conditions ...................................................................................... 38
3.2.4 . Gravitational Force and Buoyancy Force ....................................................... 38
vii
3.3 . Load Estimations ................................................................................................... 38
3.3.1 . Wave Loads Calculation ................................................................................. 39
3.3.2 . Slamming Loads Calculation.......................................................................... 41
3.4 . Results and Discussion .......................................................................................... 44
3.4.1 . Wave loads analysis........................................................................................ 44
3.4.2 . Slamming load analysis .................................................................................. 48
3.5 . Comparisons Between Wave Loads and Slamming Loads ................................... 52
4 . DESIGN AND ANALYSIS OF SMALL HYBRID HULL ........................................ 53
4.1 . Finite Element Model ............................................................................................ 53
4.2 . Calculation of Section Modulus ............................................................................ 55
4.3 . Load Calculations .................................................................................................. 57
4.3.1 . Wave Loads Calculation ................................................................................. 57
4.3.2 . Slamming Loads Calculation.......................................................................... 58
4.4 . Results and Discussion .......................................................................................... 61
4.4.1 . Under Wave Loads ......................................................................................... 61
4.4.2 . Under Slamming Loads .................................................................................. 63
4.5 . Comparisons Between Composite Hull and Hybrid hull ...................................... 65
5 . DESIGN AND ANALYSIS OF A LARGE HYBRID HULL STRUCTURE ............ 68
5.1 . Finite Element Model ............................................................................................ 68
5.2 . Load Estimations ................................................................................................... 71
5.2.1 . Estimation of Wave Loads ............................................................................. 71
5.2.2 . Estimation of Slamming Loads ...................................................................... 72
viii
5.3 . Results and Discussion .......................................................................................... 73
5.3.1 . Wave Load Analysis ....................................................................................... 73
5.3.2 . Slamming Load Analysis................................................................................ 76
5.4 . Comparisons between Wave Loads and Slamming Loads ................................... 77
6 . DYNAMIC ANALYSIS OF A LARGE HYBRID HULL ......................................... 79
6.1 . Dynamic Approach for Wave Loads ..................................................................... 79
6.1.1 . Wave Bending Moment .................................................................................. 79
6.1.2 . Wave Shear Force ........................................................................................... 83
6.1.3 . Comparisons Between Static and Dynamic Conditions for Wave Loads ...... 84
6.2 . Dynamic Loading Approach under Slamming Loads ........................................... 85
6.2.1 . Slamming Induced Bending Moment ............................................................. 85
6.2.2 . Slamming Induced Shear Force ...................................................................... 86
6.2.3 . Bottom Slamming Pressure ............................................................................ 87
6.2.4 . Comparisons Between Static and Dynamic Under Slamming Loads ............ 88
7 . SUMMARY AND RECOMMENDATIONS FOR FUTURE WORKS ..................... 89
7.1 . Summary ............................................................................................................... 89
7.2 . Recommendations for Future Works .................................................................... 90
APPENDIXES .................................................................................................................. 92
A. Sample calculation to determine wave loads for large hybrid hull .......................... 92
B. Sample calculation to determine slamming loads for large hybrid hull ................... 93
C. MATLAB code for two parameter Bretschneider Spectrum.................................... 94
D. MATLAB code to obtain time to frequency domain for wave bending moment ... 95
ix
E. MATLAB code to calculate RAO for wave bending moment ................................. 96
REFERENCES ................................................................................................................. 97
x
LIST OF TABLES
Table 3.1. Properties of Carbon/Epoxy Composite (Unidirectional) and Foam (DIAB
KLegecell® R 260 Rigid, Closed Cell PVC Foam) [30, 43] ........................... 33
Table 3.2. Strength parameters for Carbon/Epoxy and PVC Foam [30, 43] .................... 34
Table 3.3. Comparisons between wave loads and slamming loads .................................. 52
Table 4.1. Properties of Ti-6Al-4V [48] ........................................................................... 55
Table 4.2. Comparisons between composite hull and hybrid hull for both wave loads
and slamming loads.......................................................................................... 66
Table 5.1. Properties of Glass Fiber Reinforced Polymer and Foam (DIAM
Klegecell® R260 Rigid, Closed Cell PVC Foam) [30, 43] ............................. 69
Table 5.2. Strength parameters of Glass Fiber Polymer and PVC Foam [30, 43] ............ 70
Table 5.3. Comparisons between wave loads and slamming loads for large hybrid hull
at sea state 5 with ship velocity of 40 knots..................................................... 78
Table 6.1. Comparisons between static and dynamic situation for wave loads ................ 84
Table 6.2. Comparisons between static loads and dynamic loads under slamming ......... 88
xi
LIST OF FIGURES
Figure 1.1. The structure of a sandwich composite ............................................................ 3
Figure 1.2. Visby Class Corvette (Royal Swedish Navy) ................................................... 6
Figure 1.3. "Hunt" class mine counter-measure vessel (MCMV) ...................................... 7
Figure 2.1. Flowchart to determinate laminate engineering constants [30] ...................... 14
Figure 2.2. Deformed beam for first-order shear theory ................................................... 15
Figure 2.3. Distribution Factor, F1 along ship length [35]................................................ 21
Figure 2.4. Distribution Factor, F2 along ship length [35]................................................ 21
Figure 2.5. Vertical acceleration distribution factor, Kv, along ship length [35] .............. 23
Figure 2.6. Equivalent wave amplitude and wave length [37] ......................................... 30
Figure 3.1. Sandwich plate system with girders [39-42] .................................................. 35
Figure 3.2. Cross-sectional view of a sandwich plate ....................................................... 35
Figure 3.3. Finite element model of sandwich composite multi-hull ship [39-42]........... 36
Figure 3.4. Mesh distribution of hull structure [51-54] .................................................... 37
Figure 3.5. Location of neutral axis for composite hull .................................................... 37
Figure 3.6. Variation of total bending moment (hogging) along ship length ................... 40
Figure 3.7. Variation of wave shear force (positive) along ship length ............................ 40
Figure 3.8. Variation of wave shear force (negative) along ship length ........................... 41
Figure 3.9. Slamming induced bending moment distribution along ship length .............. 42
xii
Figure 3.10. Variation of slamming induced shear force (positive) along ship length ..... 42
Figure 3.11. Distribution of bottom slamming pressure along ship length ...................... 43
Figure 3.12. Deformation distribution under wave loads (Ux = 0.514 mm, Uy = 2.57
mm, Uz = 12.57 mm) ..................................................................................... 44
Figure 3.13. Von Mises Stress distribution under wave loads (bottom View) ................. 45
Figure 3.14. Von Mises Stress distribution under wave loads (side view) ....................... 45
Figure 3.15. Distribution of shear stress along ship hull (bottom view)........................... 46
Figure 3.16. Distribution of shear stress along ship hull (side view) ............................... 46
Figure 3.17. Deformation under slamming loads (Ux = 6.71 mm, Uy = 35.77 mm, Uz =
211.53 mm) ................................................................................................... 48
Figure 3.18. Von Mises Stress distribution under slamming loads (bottom view)........... 49
Figure 3.19. Von Mises Stress distribution under slamming loads (side view) ............... 49
Figure 3.20. Distribution of shear stress along ship hull under slamming loads
(bottom view) ................................................................................................ 50
Figure 3.21. Distribution of shear stress along ship hull under slamming loads (side
view) ............................................................................................................. 51
Figure 4.1. 3D view of hybrid hull ................................................................................... 54
Figure 4.2. Sketch of Ti frame (wall thickness of 70 mm) ............................................... 55
Figure 4.3. Location of neutral axis for hybrid hull .......................................................... 56
Figure 4.4. Variation of total bending moment (hogging) along ship length ................... 58
Figure 4.5. Variation of wave shear force (positive) along ship length ............................ 58
Figure 4.6. Slamming induced bending moment distribution along ship length .............. 59
xiii
Figure 4.7. Distribution of slamming induced shear force (positive) along ship length .. 60
Figure 4.8. Variation of bottom slamming pressure along ship length ............................. 60
Figure 4.9. Deformation of hybrid hull under wave loads along ship length ................... 61
Figure 4.10. Von Mises Stress distribution along hybrid hull under wave loads ............. 62
Figure 4.11. Shear stress distribution along hybrid hull under wave loads ...................... 62
Figure 4.12. Deformation distribution of hybrid hull under slamming loads ................... 64
Figure 4.13. Von Mises Stress distribution along hybrid hull under slamming loads ...... 64
Figure 4.14. Shear stress distribution along hybrid hull under slamming loads ............... 65
Figure 5.1. 3D view of a large hybrid ship hull (length = 73m) model ............................ 68
Figure 5.2. Sketch of Ti frame (wall thickness is 75 mm) ................................................ 70
Figure 5.3. Total bending moment (hogging) distribution along ship length ................... 71
Figure 5.4. Wave shear force (positive) distribution along ship length ............................ 72
Figure 5.5. Distribution of slamming induced bending moment along ship length ......... 73
Figure 5.6. Distribution of bottom slamming pressure along ship length ........................ 73
Figure 5.7. Deformation under wave loads of large hybrid hull (Ux = 0.11 mm, Uy =
0.58 mm, Uz = 2.04 mm) ................................................................................. 74
Figure 5.8. Von Mises Stress distribution under wave loads for large hybrid hull .......... 75
Figure 5.9. Distribution of shear stress along ship hull under wave loads ....................... 75
Figure 5.10. Deformation under slamming loads (Ux = 1.51 mm, Uy = 9.19 mm, Uz =
41.85 mm) ..................................................................................................... 76
Figure 5.11. Von Mises Stress distribution under slamming loads .................................. 76
Figure 5.12. Distribution of shear stress of hybrid hull under slamming loads ................ 77
xiv
Figure 6.1. Maximum wave bending moment vs. time (wave period = 7.5 sec) .............. 80
Figure 6.2. Maximum wave bending moment in frequency domain ................................ 80
Figure 6.3. Wave power spectrum (Two parameter Bretschneider spectrum) ................. 81
Figure 6.4. Response amplitude operator (RAO) for wave bending moment .................. 81
Figure 6.5. Probability distribution function for wave bending moment ......................... 82
Figure 6.6. Probability distribution function for slamming induced shear force .............. 86
Figure 6.7. Response amplitude operator (RAO) for bottom slamming pressure ............ 87
1
1. INTRODUCTION
1.1. Composite Materials
The development of composite materials and their related design and
manufacturing technologies is one of the most important advances in the history of
materials. Composites are multifunctional materials having unprecedented mechanical
and physical properties that can be tailored to meet the requirements of a particular
application. Many composites also exhibit great resistance to high temperature corrosion
and oxidation and wear. These unique characteristics provide mechanical engineers with
design opportunities not possible with conventional monolithic (unreinforced) materials.
Many manufacturing processes for composites are well adapted to the fabrication of
large, complex structures, which allows consolidation of parts, reducing manufacturing
costs.
The applications for composite materials are extensive, covering all forms of end-
uses, markets, and applications: military, defense, aerospace, automotive, sporting goods
equipment, medical applications, electronics, conductivity, utility poles, household
appliances, storage tanks, beams, drive shafts, engine components, bearing, seals,
furniture etc. That list is endless. Following the trend, naval architects are also rapidly
accepting the latest construction techniques using composites to benefit from the
following advantages [1]:
2
Very low weight : enables increased speed; increases payload; reduces fuel
consumption
Stealth benefits: non-magnetic; absorbs radar energy rendering structure with
lower radar cross-sections (RCS); lower harmonic resonance; thermal properties
provide considerable lower thermal signatures
Fire performance : excellent fire resistance; interior panels prevent flame spread
and smoke emission
High stiffness : reduces supporting framework; carries fitting readily
Durability : excellent fatigue, impact and environmental resistance; fiber-
reinforced composites are non-corrosive
Improved appearance : panels can have smooth or textured finishes; integral
decorative facings can be incorporated
Rapid fitting: modular construction ensures panels are interchangeable; large
panels are easy to handle and install due to light weight
Versatile: wide range of design possibilities to suit circumstances
1.2. Sandwich Structured Composites
A structural sandwich is a special form of a laminated composite comprising a
combination of different materials that are bonded to each other so as to utilize the
properties of each separate component to the structural advantage of the whole assembly.
Typically a sandwich composite consists of three main parts; two thin, stiff and strong
faces separated by a thick, light and weaker core. The faces are adhesively bonded to the
3
core to obtain a load transfer between the components. The structure of sandwich
composites is shown in Figure 1.1.
Figure 1.1. The structure of a sandwich composite
Concept of sandwich structured composite materials can be traced back to as early
as the year 1849 AD [2] but potential of this construction could be realized only during
Second World War. Developments in aviation posed requirement of lightweight, high
strength and highly damage tolerant materials. Sandwich structured composites, fulfilling
these requirements became the first choice for many applications including structural
components. Now their structural applications spread even to the ground transport and
marine vessels.
The primary advantage of a sandwich composite is very high stiffness-to-weight
and high bending strength-to-weight ratio. The sandwich enhances the flexural rigidity of
the structure without adding substantial weight. Sandwich structures also have fatigue
strength, acoustical insulation and additional thermal insulation. The absorption of
4
mechanical energy can be multiplied compared with monocoque structures due to an
imposed shorter mode of buckling waves. The use of cellular cores obviates the need to
provide additional thermal insulation, ensuring low structural weight, since most cellular
cores have a low thermal conductivity.
GRP terrain vehicles use sandwich in parts of the structure to obtain higher
stiffness and strength and integrated thermal insulation. Low structural weight is a feature
of the vehicle in order to be able to operate in deep snow conditions. By reducing the
structural weight, the pay-load can be increased. A similar application to the truck
structure is the sandwich containers which posse low weight with high thermal insulation
for the transportation of cold goods, e.g. fruit or other types of food. Sandwich structures
are also used for transportation applications, including cars, subway cars and trains with
an aim of reducing weight, emissions, and to integrate details for reduced manufacturing
costs, acoustical and thermal insulation. Sandwich design is also included in flooring,
interior and exterior panels. There is a variety of pleasure boats and ships made in
sandwich design. In pleasure boats, decks and hull are commonly made in a sandwich
design. Even larger ships utilize GRP sandwich design to combine high energy
absorption capability and low structural weight. In civil engineering applications,
sandwich panels have been used for a long time in low weight and thermal insulation. In
aerospace, sandwich construction has been used for a long time and applications include
wings, doors, control surfaces, stabilizers, space structures, antennas and solar panels for
both military and civil aircraft [1, 3-4].
5
1.3. Applications of Composite Materials in Marine Structures
Due to the excellent characteristics mentioned in the previous sections, the uses of
composite materials in marine industry have increased dramatically in recent years. Now
a days, composite structure becomes the main attraction for ship builder to construct
small to medium size ship. After the Second World War, US Navy constructed small
composites personnel craft, which was the start of composites application in marine
industry. These boats were stiff, strong, durable and easy to repair, and these features led
to a rapid expansion of composites use in other types of craft between 40s and 60s [3]. By
the 1970s, mine-hunting ships, pilot boats and landing craft were being built of
composites, which marked the beginning of the application of composites to large ship
structures [3-6].
At present, all composites naval ships are 40-80 m long. The Royal Swedish Navy
has built the “Visby” class corvette illustrated in Figure 1.2. [7]. It is 72.7 m long and
10.4 m wide with a full load displacement of 640 tons. The hull is a carbon fiber
reinforced sandwich construction made of T700 carbon fiber skins and PVC foam core.
Vacuum injection process is used to manufacture the sandwich composite. The design of
Visby mainly focuses on low visibility or stealth technology. Good conductivity and
surface flatness mean a low radar signature, while good heat insulation lowers the
infrared signature and increases survivability in case of fire. The composite sandwich
used in Visby is also non-magnetic, which lowers the magnetic signature.
6
Figure 1.2. Visby Class Corvette (Royal Swedish Navy)
The “Skjold” class patrol boat has been constructed by the Royal Norwegian
Navy, is an air surface effect ship (SES) with a catamaran hull form that is 46.8 m long,
13.5 m wide and 270 tons of full load displacement. Water jets propel the patrol boat and
lift fans reduce the draft to 2.6 m to achieve a top speed of 57 knots in calm water and 44
knots in Sea State 3. This operational speed is really high compared with many other
same scale ships. It is completely built by a sandwich composite consisting of glass fiber
laminate skins with a polyvinyl chloride (PVC) foam core [8]. Skjold‟s boat builders used
the sandwich composite instead of conventional steel or aluminum alloy as it simplified
the construction of the hull as well as superstructure. The composites provide not only
high strength to weight ratio and good impact properties, but also some of stealth
characteristics. Extensive use of carbon laminates provides the necessary high stiffness in
structures like beam frames, mast and support base to the gun.
7
The 60 m long, 9.8 m beam, and 750 tons of full load displacement “Hunt” class
mine counter-measure vessel (MCMV), has developed by the Royal U.K. Navy is
illustrated in Figure 1.3. It has a framed single skin hull design. This design consists of
transverse frames and longitudinal composite girders that are adhesively bonded in the
transverse and longitudinal directions to a pre-laminated GRP hull [9].
Figure 1.3. "Hunt" class mine counter-measure vessel (MCMV)
The 55 m long and 350 tones full load displacement, called Standard Flex 300
corvette is built by Royal Danish Navy. The core of this sandwich composite is made by
PVC foam and the face-sheets by glass fiber reinforced polymer [10]. U.S. OSPREY
class coastal mine-hunters, 57.3 m long, is built of a specially developed spun woven
roving laminate, using fabric impregnators to wet the glass fiber into a sectional steel
mold [11].
Composites are also used in ship superstructures, mast systems, decks, bulkheads
etc. Composites combining fiber reinforced polymer face sheets with a light core material
8
offer an alternative to stiffened single skin construction for the shell, deck and bulkhead
structure of ships [4]. The French La Fayette class Frigates has been built by glass/balsa
core panels with polyester resin for both the deckhouse and the deck structure in order to
reduce the weight [12]. A 28 m tall and 10.7 m in diameter composite Advanced
Enclosed Mast/Sensor system were installed on a US Navy Spruance class destroyer [13].
1.4. Existing Problems
Despite the above successful benefits and applications of composites in marine
industry, most of the applications are found to be restricted in relatively small ships (e.g.
Patrol boats, corvettes), or in non-structural, non-critical components on large ships [3].
Several problems appear to prohibit the use of composites in large ships or critical
structural parts. One of the major problems is that the composites lack both the stiffness
and the in-plane strength required for the larger combatant ship hulls. The structures of
long Navy combatants carry loads by the alternating axial tension and compression of the
hull during hogging and sagging modes due to waves at sea. Therefore, for long hull
structures, stiffness and in-plane strength of the composite become the critical design
factor. Also, the increased hull deflection may cause problems such as fatigue cracking
around joints and connections, and may also cause mis-alignment in the propeller shaft-
line. Thus, the sandwich composite construction technology common in smaller ships or
boats, cannot provide the necessary stiffness and in-plane strength for sea loads in long
ship hulls.
9
1.5. Hybrid Hull Structures
The hybrid structure concept may be a feasible alternative. Composites have the
advantages of light weight, corrosion resistance especially low magnetic, acoustic, radar,
and thermal signatures [14]. On the other hand, some metals have high stiffness and in-
plane strength. Thus, a hybrid ship structure could possibly combine attributes of both
composites and metal. A part of hybrid hull can therefore be constructed by composites
which provide the lower signatures and some part by metal which can provide the
required stiffness and in-plane strength.
Several hybrid ship designs have been proposed already. One design is combining
a mid-section of steel advance double hull (ADH) construction, and bow and stern
sections made of fiber reinforced composites. The strength and stiffness required for
large ships is obtained by using steel ADH in the middle of the ship. The composites at
bow and stern side usually lead to reduced mass and moment of inertia. It also allows to
manufacture of complex shapes [15-16]. An alternative hybrid approach has been
investigated that consists of attaching composite panels to a steel framework, which is
known as “The Modular Advanced Composite Hull (MACH)" [17-20]. In the MACH
concept, a steel “rib cage” is also attached to some smaller composite panels.
Jun Cao et al. investigated a hybrid ship hull made of a steel truss and composite
sandwich skins [21]. It was a 142 m long ship which was designed, analyzed and
optimized using finite element analysis. Loads applied to the hull structure were
calculated according to the American Bureau of Shipping (ABS) rules. Based on the
optimized design of this full scale hybrid model, a six meter subscale steel-
10
truss/composite-skin hybrid hull model was developed [22]. This subscale model was
then manufactured and tested under sagging loads. It was found that there was no
indication of damage in any of the composite sandwich panels, nor in the bonds between
the panels and the steel truss under the applied loads which were 36% above of the design
load. Later, the same subscale model was numerically analyzed and also experimentally
tested under hogging loads [23]. All loads were introduced as shear through brackets
welded to bulkheads. Results from the numerical analyses were then compared with the
data obtained from both sagging and hogging tests and good correlation was found.
In another study, H. J. Garala [24] has shown that Titanium alloy (Ti-6Al-4V)
could be a good choice for ship hull structure due to high stiffness, high in-plane strength
and good corrosion resistant properties. Other researchers have also showed that Ti is
resistant to corrosion as well as fatigue [25-28]. But the idea to manufacture the whole
ship hull with Titanium (Ti) alloy is not a feasible concept due to its high cost. Thus, it
could be a better approach to construct a hull in part from composites which provide the
lower signatures and in part from Ti alloy to provide the required structural integrity. In
the present study, first, a 39 m small hybrid ship hull assembled by Titanium alloy (Ti-
6Al-4V) frame and sandwich composites skins is designed and investigated under wave
and slamming loads calculated by ABS rules. Comparisons of deformations and stresses
have been made between similar composite hull and hybrid hull under same
environmental conditions. Finally, a long 73 m hybrid hull has been designed and
analyzed at high sea state condition.
11
1.6. Thesis Objectives
The main objective of the current study is to develop a new hybrid hull model and
then to investigate the hull structure under wave and slamming loads at high sea state
using both static and dynamic analysis. The details are described below:
i) Develop a multi-hull composite ship hull and hybrid hull of identical length and
beam by a finite element tool.
ii) Analyze the structural responses of both hull configurations under wave and
slamming loads.
iii) Compare between the effects of wave loads and slamming loads on both hull.
iv) Compare between the composite hull and hybrid hull to find out the benefits of
hybrid hull.
v) Model a large hybrid ship hull and simulate under static load conditions.
vi) Perform time domain analyses on large hybrid model to investigate the variations
between static and dynamic loads on ship hull.
1.7. Thesis Outlines
Chapter 2 discusses some relevant theories related to composite materials and
sandwich structures. Failure theories associated with sandwich materials are also
included. The brief procedures to estimate wave loads and slamming loads according to
American Bureau of Shipping (ABS) rules have been explained. Dynamic Load
Approach (DLA) to calculate time domain loads are also described.
12
Numerical simulation of a multi-hull sandwich composite ship structure is dealt in
Chapter 3. The composite hull is subjected to both wave loads and slamming loads and
structural responses under these two loads have been compared.
Chapter 4 presents finite element simulation of a small hybrid hull identical in
length and beam of the composite hull. This chapter includes simulation of hybrid hull
and also the comparisons between composite and hybrid hulls.
A large hybrid hull has been modeled and analyzed statically in Chapter 5. Both
wave loads and slamming loads are considered. Comparisons between wave loads and
slamming loads for large hybrid hull have been made.
Time domain dynamic analyses for wave loads and slamming loads acting on a
large hybrid hull (73 m) have been performed in Chapter 6. This chapter shows how the
ship structure behavior changes when dynamic conditions are taken into consideration.
Summary and recommendations for future works are presented in the final
Chapter 7. At the end of the thesis, references are attached.
13
2. THEORETICAL BACKGROUNDS
Chapter 2 represents some important theories related to composite materials and
sandwich composite structures. Tsai Wu Failure criterion and Maximum stress criterion
are also discussed. The procedures to calculate the static wave loads and static slamming
loads according to American Bureau of Shipping (ABS) rules have been explained.
Dynamic Load Approach (DLA) by ABS is also described to explain how to obtain time
domain loads.
2.1. Basic of Composite Mechanics
2.1.1. Composite Materials
A composite is a structural material that consists of two or more constituents that
are combined at a macroscopic level and are not soluble in each other. One constituent is
called the reinforcing phase and the other one in which it is embedded is called the
matrix. The reinforcing phase material may be in the form of fibers, particles, or flakes.
The matrix phase materials are generally continuous. The individual components remain
separate and distinct within the finished structure. The new materials may be preferred
for many reasons such as stronger, lighter or less expensive compared to traditional
materials. Examples of composite systems include concrete reinforced with steel and
epoxy reinforced with graphite fibers, etc.
14
2.1.2. Effective Modulus Theory
In a practical analysis of composite structure, if the laminate is composed of a
large number of layers, it becomes impractical to consider each individual lamina in a
three-dimensional stress analysis. Especially for ship structure using composite material,
the information on each layer will increase the time for calculation to a large extent.
However, most thick composite laminates possesses a periodic stacking sequence. Thus,
inhomogeneous properties over each typical layer can be neglected by solving the
laminate engineering constants [29]. A flowchart for the determination of engineering
properties of multi-directional laminates is given in Figure 2.1. [30].
Figure 2.1. Flowchart to determinate laminate engineering constants [30]
Some softwares have already included the above procedure for calculation of
elastic engineering constants of composite laminate, such as PROMAL. Experimental
methods can also be used to determine these constants.
15
In chapter 3, while analyzing the composite sandwich hull structure, the hull has
been modeled as a shell structure. Since computation time would be high if inner
structures (bulkheads etc.) are considered, the hull does not include these components.
Additionally using this effective modulus theory, the face laminate of sandwich structure
is assumed to be an isotropic single layer.
2.1.3. First-Order Shear Deformation Theory
The most popular composite shell theory is the first-order shear deformation
theory (FSDT). Most of beam and shell elements used in ANSYS are based on this shear
deformation theory. It is based on the following assumptions [31]:
1. As the Figure 2.2. shows, a straight line is drawn through the thickness of the
undeformed shell. The cross section may rotate but it will remain straight when the shell
deforms. The angles with the normal vector to the undeformed mid-surface are donated
by Φx and Φy when measured in the x-z and y-z planes, respectively.
2. The thickness of the shell remains same along the member as the shell deforms.
Figure 2.2. Deformed beam for first-order shear theory
16
Based on above assumptions, the displacement of a random point at one cross
section surface can be written in terms of the displacement and rotation at the mid-surface
as follows [31]:
( ) ( ) ( ) (2.1)
( ) ( ) ( ) (2.2)
( ) ( ) (2.3)
The mid-surface variables on the right-hand side are functions of only two
coordinates (x and y), meaning that it is in two-dimension. On the left-hand side, the
displacements are functions of three coordinates as it correspond to the three-dimensional
representation of the material. The 3D constitutive equations and the 3D strain-
displacement equations can be expressed in terms of 2D quantities as follows [31]:
( )
(2.4)
( )
(2.5)
(2.6)
( )
(
)
(2.7)
( )
(2.8)
( )
(2.9)
Where: o
x , o
y , and o
xy are membrane strains. They represent stretching and in-plane
shear of the mid-surface.
17
The composite structures used in composite and hybrid hull in this study, are
modeled as a 3D surface body using SHELL 181 elements of ANSYS. This shell element
is based on this first-order shear deformation theory [32].
2.2. Failure Theories
Failure means a component has separated into two or more pieces, has become
permanently distorted, and has had its reliability downgraded. Failure modes of
composite sandwich structure mainly include face or core yielding, face/core debonding,
shear crimping, face dimpling, local indentation, buckling etc. [33]. As sandwich
composite consists of core and skin material so two types of failure will be associated
with ultimate failure of the component if core and skin are perfectly bonded. Tsai Wu
failure criteria will be appropriate for defining failure of the skin of composite. On the
other hand, maximum stress criterion can be a good approximation for defining failure
theory of the core as core is considered as isotropic material. In this thesis, face sheet and
core are assumed perfectly bonded. Therefore, these two failure criteria have been used.
Tsai-Wu failure criterion on the face sheet, and the maximum stress criterion on the foam
core have been applied.
2.2.1. Tsai Wu Failure Criterion
For the failure study of composite face sheet, Tsai-Wu failure criterion has been
widely used. Tsai-Wu theory is a simplification of Gol‟denblat and Kapnov‟s generalized
failure theory for anisotropic materials [34]. It is capable of predicting strength under
18
general states of stress if there is no experimental data available. It uses the concept of
strength tensors, which allows for transformation from one coordinate system to another.
For Tsai-Wu failure criterion, we have,
√
√
√ (2.10)
(
) (
) (
) (2.11)
Where: σ1t, σ1c, σ2t, σ2c, σ3t, σ3c are failure strength in uni-axial tension and compression in
the three directions of anisotropy. τ23, τ13, τ12 are shear strengths in the three planes of
symmetry. σ1, σ2, σ3 are normal stresses and σ4, σ5, σ6 are shear stresses in three
directions. The constants C4, C5, and C6 are coupling coefficients.
Thus, The Tsai-Wu failure index, ,
√(
)
- (2.12)
Failure will occur if IF ≥ 1.0
2.2.2. Maximum Stress Criterion
Since sandwich core material is considered as an isotropic material, maximum
stress criterion can be used for its failure analysis. For maximum stress criterion, SF is
defined as follows [31]:
19
612
513
412
333333
222222
111111
/)(
/)(
/)(
0/0/
0/0/
0/0/
min
abs
abs
abs
iforif
iforif
iforif
Sct
ct
ct
F
(2.13)
Where: σ1t, σ1c, σ2t, σ2c, σ3t, σ3c are failure strength in uni-axial tension and compression in
the three directions of anisotropy. τ23, τ13, τ12 are shear strengths in the three planes of
symmetry. σ1, σ2, σ3 are normal stresses and σ4, σ5, σ6 are shear stresses in three
directions.
Failure will occur if SF ≤ 1.0
2.3. American Bureau of Shipping (ABS) Rules
The American Bureau of Shipping (ABS) is a classification society, with a
mission to promote the security of life, property and the natural environment, primarily
through the development and verification of standards for the design, construction and
operational maintenance of marine-related facilities. Rules are derived from principles of
naval architecture, marine engineering and associated disciplines and developed over
many years through extensive research and development and service experience. Rules
and regulations are subjected to constant refinement based upon additional research or
practical experience.
The ABS guide [35] describes the requirements for direct load assessment for
vessels. The guide contains a detailed description and sequential procedures for
20
calculation of wave loads and slamming loads under the specified design and
environmental conditions. ABS rules have been extensively used in this thesis to
calculate loads on the hull structures.
2.3.1. Wave Loads by ABS
Wave loads are basically induced by waves. When the ship is on the top of a wave
crest in head sea condition, it causes a “hogging” bending moment and a shear force.
When in a wave trough a “sagging” bending moment and shear force are experienced.
These loads act alternately on the hull girder as the wave progresses along the ship. These
wave loads can be assessed by following equations according to ABS rules [35]:
Wave Bending Moment (Hogging) at amidships,
(kN-m) (2.14)
Wave Bending Moment (Sagging) at amidships,
( )
(kN-m) (2.15)
Still Water Bending Moment (Hogging) at amidships,
( ) (kN-m) (2.16)
Still Water Bending Moment (Sagging) at amidships,
(2.17)
Wave Shear Force (Positive),
( ) (kN) (2.18)
21
Wave Shear Force (Negative),
( ) (kN) (2.19)
Where: k2 = 190, k1 = 110, C2 = 0.01, k = 30, C1 = 0.044L + 3.75, fp = 17.5 kN/cm2
L = Length of the craft, B = Beam of the craft, Cb = Block Coefficient,
F1, F2 = Distribution Factor.
The values of F1 and F2 can be obtained from the following Figure 2.3. and Figure 2.4.:
Figure 2.3. Distribution Factor, F1 along ship length [35]
Figure 2.4. Distribution Factor, F2 along ship length [35]
22
2.3.2. Slamming Loads by ABS
In rough seas, the vessel‟s bow and stern may occasionally emerge from a wave
and re-enter the wave with a heavy impact or slam as the hull structure comes in contact
with the water. A vessel with such excessive motions is subject to very rapidly developed
hydrodynamic loads. The vessel experiences impulse loads with high-pressure peaks
during the impact between the vessel‟s hull and water. Usually, slamming loads are much
larger than other wave loads. Sometimes ships suffer local damage from the impact load
or large-scale buckling on the deck. For high speed ships, even if each impact load is
small, frequent impact loads accelerate fatigue failures of hulls. Thus, slamming loads
may threaten the safety of ships. Hence, assessment of slamming loads or pressures are
important criteria for structural integrity of the ship.
The formulas to attain the slamming load according to ABS rules are briefly
described below [35]:
Maximum Vertical Acceleration of craft,
[
] , -
(g‟s) (2.20)
Slamming Induced Bending Moment at amidships,
( )( ) (kN-m) (2.21)
Slamming Induced Shear Force (Positive),
( ) (kN) (2.22)
Slamming Induced Shear Force (Negative),
( ) (kN) (2.23)
23
Bottom Slamming Pressure,
( ) (kN/m
2) (2.24)
Where: N1 = 0.1, N2 = 0.0078, C3 = 1.25, C4 = 4.9, h1/3 = Significant Wave Height,
τ = Running Trim, βcg = Deadrise Angle, Δ1 = Displacement in kg, d = Draft,
FD = Design Area Factor, Δ2 = Full load displacement in metric tons,
AR = Reference area = (0.697Δ2) / d (m2)
ls = Length of slam load = AR / B (m), Kv = Vertical Acceleration Distribution Factor
The vertical acceleration distribution factor, Kv can be obtained from Figure 2.5. as
shown below:
Figure 2.5. Vertical acceleration distribution factor, Kv, along ship length [35]
2.4. Dynamic Loads on Ship Structure
Dynamic loads are loads that vary in time with periods ranging from a few
seconds to several minutes. A vessel in motion will also experience a number of dynamic
loads as the result of machinery vibrations and seaway interactions. These dynamic loads
24
on ship structures can be divided into two categories. One is low frequency loads and
other is high frequency loads.
2.4.1. Low-Frequency Loads
Waves
Waves induce dynamic loadings both by induced variances in buoyancy along the
length of the hull and water impacts from the resultant heaving, rolling, and pitching.
Waves may vary in frequency, duration, height, and direction. As a ship moves along the
waves, the wave induced stress can result in the center of the ship keel bending upwards
and downwards, known as hogging and sagging, respectively, mentioned in earlier
section. The dynamic hogging is caused due to the fact that the crest of the wave is
amidships. Otherwise, when the trough of the wave is amidships sagging will occur.
When the vessel is operated in the ocean, the interlaced hogging and sagging will lead to
vibrations of the ship structure. In such situations, the stresses acting on the ship
structural details vary in time (dynamic stress), which may cause fatigue problems.
Inertial Accelerations of Equipment & Cargoes
Dynamic loads occur due to
Cargo shifting or transmitting forces through lashing points
Free surface effect
25
2.4.2. High-Frequency Loads
High frequency loads, or common vibration, can be amplified and transmitted by
structure; structures exposed to vibrations near their natural frequency can 'pick up' the
vibration (resonance). This may be felt on many vessels, for example, as a shudder in a
vessel as it accelerates & the engines rpm passes through the natural frequencies of
structural elements. High frequency dynamic loads increase displacement & deflection of
structural members, potentially leading to or accelerating fatigue failures. The main
reasons of high-frequency loads are:
Propeller Interactions (too close to hull, insufficient propeller immersion,
unbalanced or damaged blades, pressure gradient across propeller disc)
Machinery Vibration
Flow around appendages
High-frequency waves (if resonant with some portion of the structure or
hull, springing or whipping may occur)
2.5. Dynamic Loads by ABS
The ABS „Dynamic Loading Approach (DLA)‟ represents a first principles
systematic dynamic loads and strength assessment procedure to evaluate ship structural
strength under realistic dynamic load conditions. The concept was comprehensively
described in Liu, et al [36] for tanker design and has been successfully used for other ship
types such as bulk carriers, LNG carriers and FPSOs. In 2003, ABS has published the
guide to apply this „Dynamic Load Approach (DLA)‟ for high speed craft [37]. This
26
guide provides the procedures of enhanced structural analyses to assess the capabilities
and sufficiency of a structural design. The enhanced realism provided by the DLA
analysis has benefits that are of added value to structural safety. Additionally, the more
specific knowledge of expected structural behavior and performance is very useful in
more realistically evaluating and developing inspection and maintenance plans especially
for aluminum and FRP hulls. A potentially valuable benefit that can arise from the DLA
analysis is that it provides access to a comprehensive structural evaluation model, which
may be readily employed in the event of emergency situations that might arise during the
service life of the craft, such as structural damage, repairs or modifications; ocean transit
to a repair facility or redeployment to another operating route.
2.5.1. Dominant Load Parameters (DLPs)
Dominant Load Parameter (DLP) refer to load effects, arising from ship motions
and wave loads, that yield the maximum structural response for all critical structure.
Typical DLPs include vertical bending moment, vertical shear force, vertical
acceleration, torsional moment at various stations, etc. DLPs may vary from ship type to
ship type. These parameters are to be maximized to establish Load Cases for FE
structural analysis.
2.5.2. Wave Spectra
The shape of a spectrum supplies useful information about the characteristics of
the ocean wave system to which it corresponds. There exist many wave spectral
27
formulations (e.g., Bretschneider spectrum, Pierson-Moskowitz spectrum, ISSC
spectrum, ITTC spectrum, JONSWAP spectrum, Ochi-Hubble 6-parameterspectrum,
etc.).
The Bretschneider spectrum or two-parameter Pierson-Moskowitz spectrum is the
spectrum recommended for open-ocean wave conditions (e.g., the Atlantic Ocean).
( )
,
(
) -
m2/(rad/s) (2.25)
Or, ( )
(
)
,
(
) -
m2/(rad/s) (2.26)
Where: ωp = modal (peak) frequency (rad/s)
Hs = significant wave height (m)
ω = circular frequency of wave (rad/s)
Ts= average zero up-crossing period of the wave (s)
2.5.3. Vessel Motion and Wave Load Response Amplitude Operators (RAO)
RAOs are calculated for the DLPs for each load case. Only these DLPs need to be
considered for the calculation of extreme values. The RAOs represent the pertinent range
of wave headings (β), in increments not exceeding 15 degrees. It is important that a
sufficiently broad range of wave frequencies are considered based on the site-specific
wave conditions. The recommended range is 0.5 (rad/s) to 2.5 (rad/s) in increments of
0.05 rad/s. The worst wave frequency-heading (ω, β) combination is determined from an
examination of the RAOs for each DLP. Only the heading (βmax) and the wave frequency
ωe at which the RAO of the DLP is a maximum need to be used in analysis [37].
28
2.5.4. Extreme Values for DLA Analysis
Extreme value analysis is performed for each DLP to determine the maximum
values to be used in the DLA Analysis. The long-term extreme value refers to the long-
term most probable value at the exceedance probability level of 10−8
corresponding to an
approximately 25-year service life of vessel [37]. Preference is given to an Extreme
Value method that follows the so-called long-term approach commonly used for ship
structures. However, the use of a validated short-term extreme value approach, which is
appropriate to the vessel type and installation site‟s environmental data, should also be
considered. The supplementary use of such a short-term approach to confirm or test the
sensitivity of the long-term based design values is required.
2.5.5. Short-Term Response
The spectral density function, Sy(ω) of the wave-induced response is calculated
from the following equation for a particular wave spectrum [38]:
( ) ( ) ( ) (2.27)
In the above equation, Sζ(ω) represents a wave spectral density function and H(ω)
represents the response amplitude operator (RAO).The zero-th and second moments of
Sy, denoted by mo and m2, are defined by:
∫ ( )
(2.28)
∫ ( )
(2.29)
where, ω is the wave frequency. For a vessel operating at a forward speed U in waves of
heading angle θ, the moments of the response spectrum are given by:
29
∫ ( )
(2.30)
Where, ωe is the frequency of encounter defined by:
(2.31)
Assuming the wave-induced response is a Gaussian stochastic process with a zero
mean and the spectral density function Sy(ω) is narrow banded, the probability density
function of the maxima (peak values) can be represented by a Rayleigh distribution. The
probability of the response exceeding xo, Pr{xo} in the short-term prediction is calculated
by:
* + (
) (2.32)
While calculating the RAO for each of DLP, Equation (2.27) would be utilized. It
is noted here that we considered only the wave frequency (ω) neglecting the encounter
frequency (ωe). It could be included in Equation (2.27) to obtain a precision analysis.
2.5.6. Equivalent Wave
An equivalent wave in deep water is a sinusoidal wave characterized by its:
amplitude, length (or frequency), heading, and crest position (or phase angle) relative to
the longitudinal center of gravity (LCG) of the hull. For each load case, an equivalent
wave is determined which simulates the magnitude and location of the extreme value of
the dominant load component of the load case.
Equivalent Wave Amplitude
The wave amplitude of the equivalent wave illustrated in Figure 2.6., is
determined by dividing the extreme value of a DLP under consideration by the RAO
30
value of that DLP occurring at the wave frequency and wave heading corresponding to
the maximum amplitude of the RAO.
Figure 2.6. Equivalent wave amplitude and wave length [37]
The amplitude of the equivalent wave is given by,
(2.33)
Where: awj = wave amplitude as shown in Figure 2.6.
MPEVj = Most Probable Extreme Value of the jth
DLP at a probability level
equivalent to the design criterion.
Max. RAOj = maximum amplitude of the jth
DLP‟s RAO.
Wave Frequency and Length
The frequency and length of the equivalent wave for each DLP are determined
from the peak value of the DLP‟s RAO for each considered heading angle. When the
RAO is maximum, the corresponding peak frequency is denoted, ωe. The wavelength of
the equivalent wave system sketched in Figure 2.6. is calculated by:
λ = (2πg)/ωe2 (2.34)
Where: λ = wave length,
g = acceleration due to gravity,
ωe = frequency of the equivalent wave
31
2.5.7. DLPs as Time Function
For the equivalent wave, the DLPs as a function of time value, can be calculated
using the following equation [37]:
Mi = (Ai) (aw) sin (ωet+ ∈i) (2.35)
Where: Mi = i-th (other) load effect being considered (i.e., vertical bending moment and
shear force, external and internal pressures, or acceleration at selected points) at a
particular time
ωe = frequency of the equivalent wave when the RAO of the dominant load
component of the load case reaches its maximum
Ai = amplitude of the other load component‟s RAO,
aw = equivalent wave amplitude
∈I = phase angle of the (other) load component‟s RAO,
t = time under consideration
The above equation is applied to motions, accelerations, hydrodynamic pressures,
and the bending moments and shear forces at the selected stations at a particular time.
32
3. NUMERICAL SIMULATION OF COMPOSITE HULL
3.1. Design of Sandwich Hull Structure
Major parameters for designing sandwich hull structure include: material
properties for face and core, thickness of the face and core, ratio of face/core thickness,
fiber orientation angle, and layer stacking sequence. These parameters are directly related
with the performance of the sandwich structure. In the current study, the stiffness and
strength of the structure is the primary concern for the sandwich design. For the face
material, the sandwich face should have high stiffness and high tensile and compressive
strength to withstand wave and slamming loads. The common fiber-reinforced
composites used as face materials are carbon/epoxy, glass/epoxy, carbon/vinyl ester and
glass/vinyl ester. For the resin, both vinyl ester and epoxy have been widely used. Epoxy
resin normally provides higher stiffness and strength and for this reason epoxy has been
chosen as the resin of the face laminate. For the fiber reinforcement, E-glass fiber has
been used more often than the carbon fiber in marine application due to its lower cost.
However carbon fiber can be very competent in weight-critical structure, like surface
effect ship; plus its modulus and strength are higher than that of glass fiber. Considering
these superior characteristics of carbon/epoxy composites, this has been chosen as the
face sheet material of sandwich composite hull [39-42].
33
For the choice of core material of sandwich structure, the density should be low
so that the overall weight of sandwich structure can be largely reduced. At the same time
the compressive modulus should be high enough to prevent large deformation. In
addition, since the core material is mainly subject to shear deformation, the shear
modulus and strength should be fairly high enough to meet the structural requirement.
Polyvinyl chloride (PVC) foam has been used widely in marine sandwich construction
and it is chosen as the core material in this investigation. In this study, R260 closed cell
PVC foam has been chosen as the core material of sandwich structure [39-42]. Material
properties and strength parameters for both Carbon/Epoxy face sheet and PVC core are
listed in the Table 3.1. and Table 3.2. Thickness for both top and bottom laminates is 5.2
mm while that for core is 60 mm.
Table 3.1. Properties of Carbon/Epoxy Composite (Unidirectional) and Foam (DIAB
KLegecell® R 260 Rigid, Closed Cell PVC Foam) [30, 43]
Properties (GPa) E1 E2 E3 G12 G23 G13 ν12 ν23 ν13
Carbon/Epoxy 147 10.3 10.3 7 3.7 7 0.27 0.54 0.27
Properties (MPa) E G
PVC Foam 290 115
Where:
E = Modulus of Elasticity, G = Modulus of Rigidity, and ν = Poisson‟s Ratio
34
Table 3.2. Strength parameters for Carbon/Epoxy and PVC Foam [30, 43]
Properties (MPa) σ1t σ2t σ3t σ1c σ2c σ3c τ12
Carbon/Epoxy 2280 57 57 1725 228 228 76
Properties (MPa) Through-thickness
compressive strength
Shear
Strength
PVC Foam 6.6 4
Where: σ1t = Longitudinal tensile strength, σ2t = Transverse tensile strength,
σ3t = Out-of-plane tensile strength, σ1c= Longitudinal compressive strength,
σ2c = Transverse compressive strength, σ3c= Out-of-plane compressive strength,
τ12 = In-plane shear strength
Hull girders and stiffened panels are the main components of a typical ship hull
structure. The sandwich plate forms a much stiffer and stronger system than a single
stiffened metal plate, so it eliminates the need for closely spaced discrete stiffeners [44].
The conventional SPS [Sandwich plate system] structure and the finite element model of
a sandwich panel using current design scheme is shown in Figure 3.1. and Figure 3.2.,
respectively. In this study steel web frame and girders are the main supports for the panel
and they are modeled implicitly as an isotropic layer with a constant thickness based on
orthotropic plate theory. In applying this theory to panels having discrete stiffeners and
girders, the structure is idealized by assuming that girders are distributed evenly and the
structural properties of the stiffeners may be approximated by their average values [45-
35
46]. Based on the design scheme, the equivalent thickness for web frames/girders in the
FE model is set as 50 mm.
Figure 3.1. Sandwich plate system with girders [39-42]
Figure 3.2. Cross-sectional view of a sandwich plate
system with idealized girders [39-42]
3.2. Numerical Simulation
The numerical simulation in this study consists of finite element structural model
of composite hull in ANSYS, calculation of wave loads and slamming loads according to
ABS rules and then applying these loads to the structural model and then solving the
model with appropriate boundary condition. Finally, the stresses are extracted for failure
analysis
36
3.2.1. Finite Element Model
The finite element model used in this study consists of a ship hull structure as
shown in Figure 3.3. The hull is 39 m long and the beam is 12 m. The draft is assumed as
2.5 m. The structure is modeled as a 3D surface body using SHELL181 element [32].
This particular element is suitable for analyzing thin to moderately-thick shell structures.
It is a 4-noded element with six degrees of freedom at each node: translations in the x, y,
and z directions, and rotations about the x, y, and z-axes. This layered shell element is
capable of accounting for first order shear deformation across the thickness according to
Mindlin-Reissner shell theory [47].
Figure 3.3. Finite element model of sandwich composite multi-hull ship [39-42]
The model is simplified for the following aspects in order to reduce the simulation
complexity: a) Inner structures and bulkheads are neglected; b) thickness and material
properties are assumed constant through the entire structure. The sandwich composite
material properties were imported into ANSYS using section setup for shell element.
Mesh distribution of hull structure is shown in Figure 3.4.
37
Figure 3.4. Mesh distribution of hull structure [51-54]
3.2.2. Calculation of Section Modulus
The section modulus for composite hull is calculated as depicted in Figure 3.5. It
is found that the neutral axis is 2.72 m above from the bottom surface or bottom deck of
ship structure. The moment of inertia about neutral axis is 52.255 m4. The section
modulus for top surface or top deck has a value of 29.35 m3 whereas that for bottom deck
is 19.21 m3.
Figure 3.5. Location of neutral axis for composite hull
(3.1)
( )
(3.2)
( )
(3.3)
38
3.2.3. Boundary Conditions
Since, static wave loads and slamming loads are applied on the ship hull, “Inertia
Relief” boundary conditions were used. The concept of inertia relief is that the applied
loads are balanced by a set of translational and rotational accelerations. The acceleration
provides body forces distributed over the structure in such a way that the sum total of the
applied forces on the structure is zero. This provides steady-state stress and deformed
shape in the structure as if it were freely accelerating under the applied loads.
3.2.4. Gravitational Force and Buoyancy Force
Materials are defined in finite element software with all material properties and
strength parameters. The depth of the ship is chosen along z-axis during modeling of hull
structure. Thus, The gravity force is applied on the hull by choosing gravitational
acceleration as 9.81 m/s2 along –z direction using ANSYS. The draft is selected as 2.5 m.
Therefore, the buoyancy force acting on the hull was calculated using the area of the
water plane and the selected draft.
3.3. Load Estimations
According to ABS rules, the sea state 5 corresponds to a significant wave height
of 4.0 m. This value of wave height was considered throughout the current analysis. The
ship velocity is assumed as 40 knots. By using these environmental conditions along with
the ship geometric dimensions, the wave loads and slamming loads were calculated
according to ABS rules [35] from Equations (2.14) to (2.24).
39
3.3.1. Wave Loads Calculation
The procedures and rules to calculate the wave loads have been described briefly
in chapter 2. Wave bending moment (hogging and sagging), still water bending moment
(hogging and sagging), and wave shear force (positive and negative) can be estimated
directly from Equations (2.14) to (2.19). By using these equations, the corresponding
wave loads acting on ship hull structure are as follows:
Wave Bending Moment (Hogging) at amidships, Mwh = 8910 kN-m
Wave Bending Moment (Sagging) at amidships, Mws = − 12840 kN-m
Still Water Bending Moment (Hogging) at amidships, Mswh = 7660 kN-m
Still Water Bending Moment (Sagging) at amidships, Msws = 0
Wave Shear Force (Positive), Fwp = 898 kN (maximum)
Wave Shear Force (Negative), Fwn = − 826 kN (maximum)
The envelopes for wave bending moment, wave shear forces (positive and
negative) were obtained by using the distribution factor provided by ABS rules [35]. The
distribution factors for positive wave shear force and also for negative wave shear force
are shown in Figure 2.3. and Figure 2.4., respectively in chapter 2. Figure 3.6. shows how
the total bending moment (hogging) varies along the length of ship hull. It is seen that the
maximum bending moment occurs at amidships and gradually decreases to both fore
body and after body of hull structure.
40
Figure 3.6. Variation of total bending moment (hogging) along ship length
The variation of wave shear force (positive), and wave shear force (negative)
along the ship hull length are illustrated in Figure 3.7. and Figure 3.8. For positive wave
shear force, the peak value occurs at fore body section whereas for negative wave shear
force, the highest value is occurs at after body section of hull structure.
Figure 3.7. Variation of wave shear force (positive) along ship length
02000400060008000
1000012000140001600018000
0 10 20 30 40
To
tal
Ben
din
g M
om
ent
(Ho
gg
ing
), M
(k
N-m
)
Ship Length, L (m)
Total Bending Moment (Hogging) vs. Ship Length
0
200
400
600
800
1000
0 10 20 30 40
Wa
ve
Sh
ear
Fo
rce
(Po
siti
ve)
,
Fw
p (
kN
)
Ship Length, L (m)
Wave Shear Force (Positive) vs. Ship Length
41
Figure 3.8. Variation of wave shear force (negative) along ship length
3.3.2. Slamming Loads Calculation
The formulas to calculate the slamming load parameters such as maximum
vertical acceleration, slamming induced bending moment, slamming induced shear force,
bottom slamming pressure etc. have been explained in chapter 2. By using the Equations
(2.20) to (2.24), slamming loads were calculated as follows:
Maximum Vertical Acceleration, ηcg = 2.70 (g‟s)
Slamming Induced Bending Moment at amidships, Msl = 16943 kN-m
Slamming Induced Shear Force (Positive), Fsl = 1810 kN (maximum)
Slamming Induced Shear Force (Negative), Fsl = 1662 kN (maximum)
Bottom Slamming Pressure, Pbxx = 8.55 (1+2.70 Kv) kN/m2
Here, Kv is the vertical acceleration distribution factor which varies along the ship
length and can be calculated from Figure 2.5. Similar to wave loads, the envelopes for
slamming induced bending moment, slamming induced shear forces (positive and
0
100
200
300
400
500
600
700
800
900
0 10 20 30 40
Wa
ve
Sh
ear
Fo
rce
(Neg
ati
ve)
,
Fw
n (
kN
)
Ship Length, L (m)
Wave Shear Force (Negative) vs. Ship Length
42
negative) and bottom slamming pressure are obtained by using the distribution factor
provided by ABS rules [35].
Variation of slamming induced bending moment is shown in Figure 3.9.
Maximum bending moment due to slamming occurs at the amidships and gradually
decreases to both side of bow and stern.
Figure 3.9. Slamming induced bending moment distribution along ship length
Figure 3.10. Variation of slamming induced shear force (positive) along ship length
02000400060008000
1000012000140001600018000
0 10 20 30 40
Sla
mm
ing
In
du
ced
Ben
din
g
Mo
men
t, M
sl (
kN
-m)
Ship Length, L (m)
Slamming Induced Bending Moment vs. Ship Length
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 10 20 30 40Sla
mm
ing
In
du
ced
Sh
ear
Fo
rce
(Po
siti
ve)
, F
sl (
kN
)
Ship Length, L (m)
Slamming Induced Shear Force (Positive) vs. Ship Length
43
Positive slamming induced shear force also changes along the length. Maximum value
for this shear force is 1810 kN which occurs at the bow side as shown in Figure 3.10.
The bottom slamming pressure remains constant at the stern side of the hull
structure and it then gradually increases towards the bow. The maximum value occurs
at the very front of the hull structure. The envelope is obtained from Figure 2.5. as
bottom slamming pressure is a function of vertical acceleration distribution factor. In our
case, the maximum slamming pressure generated was equal to a value of 55 kN/m2. The
distribution is shown in Figure 3.11.
Figure 3.11. Distribution of bottom slamming pressure along ship length
After applying these wave loads and slamming loads separately into the
composite hull model with inertia relief boundary conditions, the model was solved in
ANSYS workbench. Deformations and stress components for both loading conditions are
extracted for failure analyses.
0
10
20
30
40
50
60
0 10 20 30 40Bo
tto
m S
lam
min
g P
ress
ure
,
Pb
xx (
kN
/m2)
Ship Length, L (m)
Bottom Slamming Pressure vs. Ship Length
44
3.4. Results and Discussion
3.4.1. Wave loads analysis
Deformation
Figure 3.12. illustrates the distribution of deflection of hull under wave loads. It is
observed that the deflection does not show any particular pattern; rather it varies
irregularly along the length. Maximum deflection occurs at the middle part as well as on
the side of the hull. The maximum value of the deflection was 12.85 mm.
It is interesting that since we are considering the wave shear force and wave
bending moment corresponding to hogging and sagging conditions, maximum
deformation occurs mainly is in the vertical direction (z-direction). This vertical
deflection is 12.57 mm whereas in x and y directions those are only 0.514 mm and 2.57
mm, respectively.
Figure 3.12. Deformation distribution under wave loads (Ux = 0.514 mm, Uy = 2.57
mm, Uz = 12.57 mm)
45
Stresses
Variation of Von Mises Stress along the bottom and side of the hull structure
under wave loads are shown in Figures 3.13. and 3.14., respectively. By analyzing these
two figures, it is observed that the maximum Von Mises stress generated in the hull is
around 133.82 MPa. This stress varies over the bottom part of the ship, and middle
portion has the maximum value.
Figure 3.13. Von Mises Stress distribution under wave loads (bottom View)
Figure 3.14. Von Mises Stress distribution under wave loads (side view)
46
For shear stress, the distribution behavior is random along hull length similar to
Von Mises Stress. Maximum shear stress occurs at the middle part of the upper deck, and
is about 68.19 MPa under the wave loading. Figures 3.15. and 3.16. display the
distribution of shear stress along the bottom and side of the hull structure, respectively.
Figure 3.15. Distribution of shear stress along ship hull (bottom view)
Figure 3.16. Distribution of shear stress along ship hull (side view)
47
Failure Analysis
Failure criterion has been applied on the model to verify the integrity of the
sandwich structure under the wave loads. As mentioned earlier, failure modes of
composite sandwich structure mainly include face or core yielding, face/core debonding,
buckling, etc. In this model, face sheet and core are assumed perfectly bonded. Therefore,
we primarily consider face sheet and core yielding by employing Tsai-Wu failure
criterion on the face sheet, and maximum stress criterion on the foam core. Tsai-Wu
failure criterion and maximum stress criterion have been briefly explained in chapter 2.
For Tsai-Wu failure criterion, by using Equations ( 2.10), ( 2.11) and (2.12), we have,
Thus, Tsai-Wu failure index,
It is known that, failure will occur if IF ≥ 1.0. But in this case, IF is less than 1.0, which
ensures that there was no failure in composite skin.
Also, from Equation (2.13), by applying maximum stress criterion for isotropic sandwich
core, we have calculated the Safety Factor, SF = 2.40
Failure is predicted when SF < 1.0 indicating that no failure occurred in core either for
this case.
Thus, the hull structure is verified to survive without any indication of failure at assumed
wave load conditions.
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48
3.4.2. Slamming load analysis
Deformations
Figure 3.17. shows the distribution of deformation along ship hull under
slamming loads. Under slamming loads, the maximum deformation of the hull was
214.64 mm, whereas the maximum deformation for wave load was only 12.85 mm. This
large deformation takes place at middle of the top deck. Similar to wave loads, maximum
deformation occurs primarily in the vertical i.e. in z-direction. The deformation in the z-
direction was 211.53 mm while the deformations in x and y directions were 6.71 mm and
35.77 mm, respectively.
Figure 3.17. Deformation under slamming loads (Ux = 6.71 mm, Uy = 35.77 mm, Uz =
211.53 mm)
Stresses
The Von Mises Stress under slamming loads varied haphazardly along the length
of the ship similar to that of deformation. Variation of Von Mises Stress along the bottom
49
hull and as well as along the side of the hull are depicted in Figures 3.18. and 3.19.,
respectively.
Figure 3.18. Von Mises Stress distribution under slamming loads (bottom view)
Figure 3.19. Von Mises Stress distribution under slamming loads (side view)
Middle part of the bottom deck experiences the maximum value of Von Mises
Stress equivalent to 229 MPa. This value is much higher compared to Von Mises Stress
under wave loads (133.82 MPa).
50
The distribution of shear stress along the hull length is uneven in nature as
indicated by Figures 3.20. and 3.21., respectively. Maximum shear stress under slamming
loads is also seen at the middle portion of the bottom deck as observed with wave loads.
For slamming loads, maximum shear stress was around 118 MPa which for wave loads
was 68.19 MPa.
Figure 3.20. Distribution of shear stress along ship hull under slamming loads
(bottom view)
Failure Analysis
Similar to wave load condition, Tsai-Wu failure criterion was applied on the face
sheet, and maximum stress criterion was applied on the foam core of sandwich composite
hull structure.
51
Figure 3.21. Distribution of shear stress along ship hull under slamming loads (side view)
For Tsai-Wu failure criterion, from Equations (2.10), (2.11), and (2.12), we obtain:
So, Tsai-Wu failure index,
Failure is predicted when IF ≥ 1.0. Since, IF has a value of lower than 1.0, it confirms that
sandwich composite skin would not fail.
Similarly, by using Equation (2.13) described earlier for maximum stress
criterion, the safety factor, SF for core was calculated as 1.25, which is more than 1.0
suggesting that no failure would occur in the composite core.
Thus, the hull structure is also verified without any indication of failure under
slamming loads.
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52
3.5. Comparisons Between Wave Loads and Slamming Loads
Typically, slamming load has more disastrous and detrimental effects on ship hull
than wave loading. The results from both cases also verify this general phenomenon. By
comparing these two conditions, it is clear that slamming load will induce larger
deformation and stresses than wave loading. The maximum deformation under wave load
is 12.85 mm while for slamming load it is 214.64 mm. Similarly, maximum Von Mises
stress under wave loads is 133.82 MPa which is lower than 228.49 MPa generated under
slamming loads. Shear stress also follows the same trend.
Although the failure analyses confirm that no failure would occur at the existing
loading conditions, it is very likely that if ship velocity or significant wave height
increases, the failure will occur first due to slamming load. Table 3.3. shows a
comparison of deformation and stresses under wave and slamming loads.
Table 3.3. Comparisons between wave loads and slamming loads
Parameters Wave Loads (location) Slamming Loads (location)
Maximum
Deformation
0.012 m
(Middle of upper deck and side
hull)
0.214 m
(Middle of upper deck and side
hull)
Maximum Von
Mises Stress
133.82 MPa
(Middle of upper deck)
228.49 MPa
(Middle of bottom deck and side
hull)
Maximum
Shear Stress
68.19 MPa
(Middle of upper deck)
118.43 MPa
(Middle of bottom deck)
Tsai-Wu Failure
Index 0.3095 0.8093
Safety Factor
for Maximum
stress criterion
2.40 1.25
53
4. DESIGN AND ANALYSIS OF SMALL HYBRID HULL
As mentioned earlier, in spite of superior characteristics and advantages of
sandwich composites, their uses are only limited to smaller boats or ships, as they lack
both required stiffness and the in-plane strength for large ships exceeding 60 m in length.
In that sense, hybrid hull could be an alternative and appropriate choice. The basic goal
of hybrid ship hull concept is to combine metal and composite such that the advantages
of both materials can be utilized. In present study, first, a 39 m small and simplified
model of hybrid ship hull consisting Titanium alloy (Ti-6Al-4V) frame and sandwich
composites skins is designed and then investigated under wave and slamming loads
calculated by ABS rules. Comparisons of deformation and stresses have been made
between this hybrid hull and composite hull (analyzed in chapter 3) under same loading
conditions.
4.1. Finite Element Model
A simplified model of a small hybrid hull consisting of metal frames with
composite panels has been designed by utilizing Design Modular of ANSYS workbench
shown in Figure 4.1. The ship hull is 39 m long and has 12 m of beam. Total depth and
draft are 6 m and 2.2 m, respectively.
54
Figure 4.1. 3D view of hybrid hull
For composite panels construction, similar materials and parameters were chosen
as the previous composite ship. The composite skin is made of Carbon/Epoxy laminate
(thickness 5 mm) whereas core was by PVC foam (thickness 60 mm). Material properties
and strength parameters for both Carbon/Epoxy face sheet and PVC core were listed in
the Table 3.1. and Table 3.2.
For the metal part, Titanium alloy (Ti-6Al-4V) was chosen. Table 4.1. lists all
material properties and strength parameters for this alloy. All Ti frames were hollow with
a wall thickness of 70 mm. The cross-sections of different hollow Ti frames were (0.5m ×
0.5m), (0.5m × 0.866m), (1m× 0.532m), (0.5m × 1m), and (0.866 × 1m). Figure 4.2.
shows the Ti frame.
Figure 1: Deadrise and Flare Angles
CHAPTER 2: SECTION 2: DESIGN PRESSURES
“ABS GUIDE FOR BUILDING AND
CLASSING HIGH SPEED CRAFT, (FEBRUARY 2012); PART 3: HULL
CONSTRUCTION AND EQUIPMENT”
55
Table 4.1. Properties of Ti-6Al-4V [48]
Properties Ti-6Al-4V
Density, ρ
4420 kg/m3
Modulus of Elasticity, E 114 GPa
Modulus of Rigidity, G 44 GPa
Poisson‟s Ratio, ν 0.31
Ultimate Strength, σut 1000 MPa
Yield Strength, σy 910 MPa
Shear Strength, τ 550 MPa
Figure 4.2. Sketch of Ti frame (wall thickness of 70 mm)
4.2. Calculation of Section Modulus
The section modulus for hybrid hull is calculated as illustrated in Figure 4.3. It is
observed that the neutral axis is 2.75 m above from the bottom deck of ship structure. The
56
moment of inertia about neutral axis is 20.9177 m4. The section modulus for top surface
has a value of 6.44 m3 while that for bottom surface is 7.61 m
3.
Figure 4.3. Location of neutral axis for hybrid hull
(4.1)
( )
(4.2)
( )
(4.3)
By comparing, it is noted that, the hybrid hull has less moment of inertia with
respect to neutral axis comparing to that of composite hull (20.91 m4 vs. 52.25 m
4).
Similarly, hybrid hull has less section modulus for both top deck (6.44 m3 vs. 29.35 m
3)
and bottom deck (7.61 m3 vs. 19.21 m
3) than composite hull.
From strength of materials, we know that, bending stress is inversely proportional
to section modulus. It means that, lower value of section modulus of hybrid hull should
give higher values of stress when compared to composite hull.
57
4.3. Load Calculations
For load conditions, we chose the same parameters as we did for composite hull.
The wave loads and slamming loads were calculated according to ABS rules [35] by
utilizing the Equations from (2.14) to (2.24) at sea state 5 (Significant wave height of 4.0
m) with a ship forward velocity of 40 knots.
4.3.1. Wave Loads Calculation
Various wave loads such as wave bending moment (hogging and sagging), still
water bending moment (hogging and sagging), and wave shear forces (positive and
negative) have been calculated directly from Equations (2.14) to (2.19) as:
Wave Bending Moment (Hogging) at amidships, Mwh = 5679 kN-m
Wave Bending Moment (Sagging) at amidships, Mws = − 8153 kN-m
Still Water Bending Moment (Hogging) at amidships, Mswh = 4864 kN-m
Still Water Bending Moment (Sagging) at amidships, Msws = 0
Wave Shear Force (Positive), Fwp = 571 kN (maximum)
Wave Shear Force (Negative), Fwn = − 525 kN (maximum)
Figure 4.4. and Figure 4.5. show how the total bending moment (hogging) and
wave shear force (positive) vary along the length of ship hull. It has been found that the
maximum bending moment occurs at the amidships has a value of 10543 kN-m and
gradually decreases to both fore body and after body of hull structure while for positive
wave shear force, the peak value of 571 kN occurs at fore body part of hull structure.
58
Figure 4.4. Variation of total bending moment (hogging) along ship length
Figure 4.5. Variation of wave shear force (positive) along ship length
4.3.2. Slamming Loads Calculation
The different slamming load parameters such as maximum vertical acceleration,
slamming induced bending moment, slamming induced shear force, bottom slamming
pressure etc. were computed using the Equations (2.20) to (2.24). The computed
slamming loads are as follows:
0
2000
4000
6000
8000
10000
12000
0 10 20 30 40
To
tal
Ben
din
g M
om
ent
(Ho
gg
ing
),
Mw
h (
Kn
-m)
Ship Length, L (m)
Total Bending Moment (Hogging) vs. Ship Length
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35 40
Wa
ve
Sh
ear
Fo
rce
(Po
siti
ve)
, F
wp
(kN
)
Ship Length, L (m)
Wave Shear Force (Positive) vs. Ship Length
59
Maximum Vertical Acceleration, ηcg = 1.406 (g‟s)
Slamming Induced Bending Moment at amidships, Msl = 23908kN-m
Slamming Induced Shear Force (Positive), Fsl = 3538kN (maximum)
Slamming Induced Shear Force (Negative), Fsl = 3255kN (maximum)
Bottom Slamming Design Pressure, Pbxx = 58.7 (1+1.40628Kv) kN/m2
The bending moment, shear force and bottom pressure due to slamming
developed on hybrid hull were higher when compared to composite hull. The slamming
loads are proportional to the displacement (weight) of the hull structure. Since, hybrid
hull has higher weight than compared to composite hull; slamming loads were also higher
for hybrid hull.
Variation of slamming induced bending moment is depicted in Figure 4.6.
Maximum slamming bending moment generated in the ship hull is 23908 kN-m and it
occurs at the amidships and gradually decreases to both side of fore body and after body
of ship hull.
Figure 4.6. Slamming induced bending moment distribution along ship length
0
5000
10000
15000
20000
25000
30000
0 10 20 30 40
Sla
mm
ing
In
du
ced
Ben
din
g
Mo
men
t, M
sl (
KN
-m)
Ship Length, L (m)
Slamming Induced Bending Moment vs. Ship Length
60
Positive slamming induced shear force also varies along the length and has a
maximum value of 1810 kN occurring at the fore body part of hull structure shown in
Figure 4.7. The distribution of bottom slamming pressure is shown in Figure 4.8.
Figure 4.7. Distribution of slamming induced shear force (positive) along ship length
Bottom slamming pressure remains same at the after body part of ship structure
and then linearly increases to the fore body part. The maximum value of 224 kN/m2
occurs at the very front part of hull structure.
Figure 4.8. Variation of bottom slamming pressure along ship length
0
500
1000
1500
2000
2500
3000
3500
4000
0 10 20 30 40
Sla
m I
nd
uce
d S
hea
r F
orc
e
(Po
siti
ve)
, F
sl (
kN
)
Ship Length, L (m)
Slam Induced Shear Force (Positive) vs. Ship Length
0
50
100
150
200
250
0 10 20 30 40Bo
tto
m S
lam
min
g P
ress
ure
,
Pb
xx (
kN
/m2
)
Ship Length, L (m)
Bottom Slamming Pressure vs. Ship Length
61
For numerical simulation, we use the same boundary condition "inertia relief" as
before as composite hull. Both the gravity force (g = 9.81 m/s2) and buoyancy force were
taken into consideration during simulation. After applying the wave and slamming loads,
deformations and stress values were extracted for the hybrid hull.
4.4. Results and Discussion
4.4.1. Under Wave Loads
Deformation
Figure 4.9. shows the deformation of hybrid model under wave loads. Maximum
deflection occurs at the front part of the side hull has a value of about 1.63 mm.
Figure 4.9. Deformation of hybrid hull under wave loads along ship length
Stresses
Distributions of Von Mises Stress and shear stress have been plotted in Figures
4.10. and 4.11., respectively. Both stresses vary arbitrarily along the ship hull. But
62
it is noted that the sandwich panels have usually higher values of stresses than that to Ti
alloy frames. Maximum Von Mises Stress produced in the hull is 13.07 MPa whereas
maximum shear stress is 7.25 MPa. Both maximum values occurred in the sandwich
structure.
Figure 4.10. Von Mises Stress distribution along hybrid hull under wave loads
Figure 4.11. Shear stress distribution along hybrid hull under wave loads
63
Failure Analysis
Similar as sandwich composite ship structure, Tsai-Wu failure criterion and
maximum stress criterion were applied on the face sheet, and foam core respectively. It
was assumed that the sandwich panels and Ti frames were perfectly bonded. Also, we
assumed that face sheet and core were perfectly bonded.
While considering Tsai-Wu failure criterion, utilizing Equations (2.10), (2.11), and
(2.12), we found,
A = 0.0187, B = 0.2593 and IF = 0.0106
Since, IF is very small compared to 1, it means that there would be no failure in face
sheet.
And, also from Equation (2.13), by applying maximum stress criterion for isotropic
sandwich core, we found, Safety Factor, SF = 4.28 which is much larger than 1.
Thus, there would be no failure in the core either.
4.4.2. Under Slamming Loads
Deformation
Deformation of hybrid hull under slamming loads is shown in Figure 4.12.
Maximum deformation in hybrid hull is 20.67 mm and it was at the front part of the side
hull.
64
Figure 4.12. Deformation distribution of hybrid hull under slamming loads
Stresses
Figures 4.13. and 4.14. show the variation of Von Mises Stress and Shear stress
respectively. Similar to wave loads conditions, the sandwich panels have usually
higher values of stresses than compared to Ti alloy frames. With slamming loads, the
Figure 4.13. Von Mises Stress distribution along hybrid hull under slamming loads
65
values are much higher. Maximum Von Mises Stress generated in the hull is 76.06 MPa
whereas maximum shear stress is 44.70 MPa.
Figure 4.14. Shear stress distribution along hybrid hull under slamming loads
Failure Analysis
For Tsai-Wu failure criterion, from Equations (2.10), (2.11), and (2.12), we
found, A = 0.2697, B = 1.0274 and IF = 0.1821
Since, IF is less than 1, there would be no failure in face sheet.
Also from Equation (2.13), we found, Safety Factor, SF = 3.19 which is larger than 1.
4.5. Comparisons Between Composite Hull and Hybrid hull
Table 4.2. shows the comparisons between composite hull and hybrid hull for
both wave and slamming loads under identical environmental conditions. For both
composite sandwich hull and hybrid hull, we have considered sea state 5 (Significant
wave height, Hs = 4m) with a ship forward velocity of 40 knots. By observing the Table
66
4.2., it is obvious that the deformations and stresses generated in the hybrid hull are much
lower than that of composite hull.
Table 4.2. Comparisons between composite hull and hybrid hull for both wave loads and
slamming loads
Parameters
Composite Hull
(L = 39m, B = 12m,
D = 4m, d = 2.5m)
Hybrid Hull
(L = 39m, B = 12m,
D = 6m, d = 2.2m)
Wave Loads Slamming
Loads Wave Loads
Slamming
Loads
Deformation (max) 12.85 mm 214.64 mm 1.63 mm 20.67 mm
Von-Mises Stress(max) 133.82 MPa 228.49 MPa 13.07 MPa 76.06 MPa
Shear Stress(max) 68.19 MPa 118.43 MPa 7.25 MPa 44.70 MPa
Tsai-Wu Failure Index 0.3095 0.8093 0.0106 0.1821
Safety Factor for
Maximum stress criterion 2.40 1.25 4.28 3.19
The maximum deformation of hybrid hull under wave loads is 1.63 mm whereas
for composite hull is 12.85 mm. Similar phenomena is seen for slamming loads also.
Thus, inclusion of Ti alloy frames causes almost one order reduction in hull deformation.
Both Von Mises stress and Shear Stress generated in the hybrid hull are much
lower than compared to composite hull. The maximum Von Mises Stresses under wave
loads for both composite hull and hybrid hull are 133.82 MPa and 13.07 MPa
respectively showing that hybrid hull has almost one order lower magnitude of stresses. It
is significant to note that, the maximum Von Mises stress in hybrid hull under slamming
67
loads is 76.06 MPa which is surprisingly lower than maximum Von Mises stress in
sandwich composite hull under wave loads (133.82 MPa). Similar feature is also
observed for shear stress as well.
From strength of materials, it is known that the bending stress is inversely
proportional to section modulus. It means that, lower values of section modulus of hybrid
hull should give higher values of stresses when comparing with composite hull. But we
see that, the situation is opposite; as hybrid hull has lower values of deformation and
stresses than those of composite hull. The reason for this is clear that introduction of Ti
frames in between the sandwich composite panels enhances the in plane strength and
stiffness of the hull structure.
Since, deformation and stresses are lower with hybrid hull, corresponding failure
index (IF) and safety factor (SF) suggest safer structure. Tsai-Wu Failure Index for
composite hull under slamming loads is 0.8093 while that for hybrid hull under same
environmental conditions is 0.1821. The failure occurs when Tsai-Wu Failure Index will
reach to 1.0. Thus, hybrid hull gives more safe value of Tsai-Wu Failure Index than
composite hull. Also, safety factor for Maximum stress criterion applied to sandwich core
of hybrid hull provide much secured value comparing to composite sandwich hull. The
safety factor for composite hull under wave loads is 2.40 whereas that for hybrid hull is
4.28. Similar trend is also found for slamming loads.
68
5. DESIGN AND ANALYSIS OF A LARGE HYBRID HULL STRUCTURE
In chapter 4, the comparisons between composite hull and hybrid hull for various
parameters clearly proved that the addition of Ti alloy frames into the composite panels
would provide superior performance. In this chapter 5, a large scale hybrid hull model is
developed. This large hull is then analyzed under static load as well as under dynamic
load. Chapter 5 deals with the static analysis of the long hybrid hull.
5.1. Finite Element Model
A large scale hybrid hull model is shown in Figure 5.1. This ship hull is designed
by using the Design Modular of ANSYS Workbench. The small hull which we
Figure 5.1. 3D view of a large hybrid ship hull (length = 73m) model
69
considered in chapter 4 were 39 m long and 12 m of beam. The large ship model is 73 m
long and has 12 m of beam.
For the face material of sandwich composite panels, previously carbon/epoxy has
been chosen instead of glass fiber. The basic reasons were carbon epoxy has higher
modulus and strength comparing to glass fiber though its cost is higher than glass fiber.
From the comparison between composite hull and hybrid hull, it is clear that introduction
of Ti frames in hybrid hull will enhance the strength and stiffness of the hull structure.
For core material, Polyvinyl chloride (PVC) foam was chosen as before. Material
properties and strength parameters for both Glass Fiber Reinforced Polymer and PVC
core are listed in the Table 5.1. and Table 5.2.. Thickness for both top and bottom
laminates is 5.0 mm whereas that for core is 65 mm.
Table 5.1. Properties of Glass Fiber Reinforced Polymer and Foam (DIAM Klegecell®
R260 Rigid, Closed Cell PVC Foam) [30, 43]
Properties (GPa) E1 E2 E3 G12 G23 G13 ν12 ν23 ν13
Glass Fiber Polymer 41 10.4 10.4 4.3 3.5 4.3 0.28 0.50 0.28
Properties (MPa) E G
PVC Foam 290 115
Where: E = Modulus of Elasticity, G = Modulus of Rigidity, and ν = Poisson‟s Ratio
70
Table 5.2. Strength parameters of Glass Fiber Polymer and PVC Foam [30, 43]
Properties (MPa) σ1t σ2t σ3t σ1c σ2c σ3c τ12
Glass Fiber Polymer 1140 39 39 620 128 128 89
Properties (MPa) Through-thickness
compressive strength
Shear
Strength
PVC Foam 6.6 4
Where: σ1t = Longitudinal tensile strength, σ2t = Transverse tensile strength,
σ3t = Out-of-plane tensile strength, σ1c= Longitudinal compressive strength,
σ2c = Transverse compressive strength, σ3c = Out-of-plane compressive strength,
τ12 = In-plane shear strength
Titanium alloy (Ti-6Al-4V) has been chosen as a metal part of the hybrid hull as
before. The Ti frames have been placed in between composite sandwich panels. Material
properties and strength parameters for this alloy have already listed in Table 4.1. All Ti
frames are hollow with a wall thickness of 75 mm. Ti frames are shown in Figure 5.2.
The cross-sections of different hollow Ti frames were (0.5m×0.5m), (0.5m × 0.639m),
(1m× 0.532m), (0.5m × 1m), and (0.639 × 1m).
Figure 5.2. Sketch of Ti frame (wall thickness is 75 mm)
71
5.2. Load Estimations
5.2.1. Estimation of Wave Loads
Wave loads calculation according to Equations (2.14) to (2.19) for the 73 m
hybrid hull were carried out and shown below:
Wave Bending Moment (Hogging) at amidships, Mwh = 29607 kN-m
Wave Bending Moment (Sagging) at amidships, Mws = − 37138 kN-m
Still Water Bending Moment (Hogging) at amidships, Mswh = 22156 kN-m
Still Water Bending Moment (Sagging) at amidships, Msws = 0
Wave Shear Force (Positive), Fwp = 1388 kN (maximum)
Wave Shear Force (Negative), Fwn = − 1277 kN (maximum)
Variations of bending moment and shear forces along the ship length were similar
as before and are shown in Figures 5.3. and 5.4.
Figure 5.3. Total bending moment (hogging) distribution along ship length
0
10000
20000
30000
40000
50000
60000
0 20 40 60
To
tal
Ben
din
g M
om
ent
(Ho
gg
ing
), M
wh
(K
n-m
)
Ship Length, L (m)
Total Bending Moment (Hogging) vs. Ship Length
72
Figure 5.4. Wave shear force (positive) distribution along ship length
5.2.2. Estimation of Slamming Loads
Slamming loads calculated according to Equations (2.20) to (2.24) are as follows:
Maximum Vertical Acceleration, ηcg = 0.288288 (g‟s)
Slamming Induced Bending Moment at amidships, Msl = 48333kN-m
Slamming Induced Shear Force (Positive), Fsl = 3788kN (maximum)
Slamming Induced Shear Force (Negative), Fsl = 3485kN (maximum)
Bottom Slamming Pressure, Pbxx = 47.0 (1+0.288288Kv) kN/m2
Variations of slamming induced bending moment and bottom slamming pressure
were obtained as before and shown in Figures 5.5 and 5.6, respectively.
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 Wa
ve
Sh
ear
Fo
rce
(Po
siti
ve)
,
Fw
p (
kN
)
Ship Length, L (m)
Wave Shear Force (Positive) vs. Ship Length
73
0
10
20
30
40
50
60
70
80
0 20 40 60
Bo
tto
m S
lam
min
g P
ress
ure
,
Pb
xx (
kN
/m2
)
Ship Length, L (m)
Bottom Slamming Pressure vs. Ship Length
Figure 5.5. Distribution of slamming induced bending moment along ship length
Figure 5.6. Distribution of bottom slamming pressure along ship length
5.3. Results and Discussion
5.3.1. Wave Load Analysis
Deformation
Figure 5.7. displays the deformation of large hybrid hull structure under wave
loads. The deformation varies randomly along the hull length. Maximum deformation
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
0 20 40 60
Sla
mm
ing
In
du
ced
Ben
din
g
Mo
men
t, M
sl (
kN
-m)
Ship Length, L (m)
Slamming Induced Bending Moment vs. Ship Length
74
occurs in the hull is 2.12 mm. It may be noted that the maximum deformation that occurs
primarily is in the vertical direction (z-direction). The deformation in vertical direction
i.e. in z-direction is 2.04 mm. The deformations in other two directions i.e., longitudinal
and transverse directions have very small values comparing to vertical direction. Since
we are considering the wave shear force and wave bending moment corresponding to
hogging and sagging, thus the vertical deformation is the maximum one.
Figure 5.7. Deformation under wave loads of large hybrid hull (Ux = 0.11 mm, Uy = 0.58
mm, Uz = 2.04 mm)
Stresses
Figures 5.8. and 5.9. show the distribution of Von Mises stress and shear stress
along the hull length. Variations of both stresses were identical with that of 39m hull but
the magnitudes were higher. For 73 m hull, maximum Von Mises Stress and shear stress
were 33.10MPa and 19 MPa.
75
Figure 5.8. Von Mises Stress distribution under wave loads for large hybrid hull
Figure 5.9. Distribution of shear stress along ship hull under wave loads
Failure Analysis
By applying Tsai-Wu failure criterion, we found:
A = 0.32, B = 0.8671, and IF = 0.28
IF is less than 1; confirms that there would be no failure in composite skins.
Similarly, safety factor, SF is 3.73, above than 1; indicates no failure in sandwich core.
76
5.3.2. Slamming Load Analysis
Deformation and Stresses
The variations of deformation and stresses for large hybrid hull under slamming
loads were similar to those of small hull but the magnitudes were higher. Maximum
deformation was 42.87 mm as shown in Figure 5.10. Figures 5.11. and 5.12. illustrate
how the Von Mises Stress and shear stress varied under slamming loads.
Figure 5.10. Deformation under slamming loads (Ux = 1.51 mm, Uy = 9.19 mm, Uz =
41.85 mm)
Figure 5.11. Von Mises Stress distribution under slamming loads
77
Figure 5.12. Distribution of shear stress of hybrid hull under slamming loads
Failure Analysis
By using Tsai-Wu failure criterion for sandwich skin, we found:
A = 0.6733, B = 1.3479, and IF = 0.67293
Since IF is less than 1, there would be no failure in composite skins.
Safety factor, SF for sandwich core is 2.31; indicates no failure in sandwich core.
5.4. Comparisons between Wave Loads and Slamming Loads
The comparisons between wave loads and slamming loads for large hybrid hull
shown in Table 5.3., substantiate that slamming loads induce larger deformation and
stresses than wave loading.
The maximum deformation under wave loads for large hybrid hull was only 2.12
mm while that for slamming load was 42.87 mm. Similarly, failure index and safety
factor under wave loads provide more safe values.
78
Table 5.3. Comparisons between wave loads and slamming loads for large hybrid hull at
sea state 5 with ship velocity of 40 knots
Parameters Wave Loads Slamming Loads
Maximum Deformation 2.12 mm 42.87 mm
Maximum Von Mises Stress 33.10 MPa 125.38 MPa
Maximum Shear Stress 19 MPa 70.80 MPa
Tsai-Wu Failure Index 0.28 0.67293
Safety Factor for Maximum
stress criterion 3.73 2.31
79
6. DYNAMIC ANALYSIS OF A LARGE HYBRID HULL
The large (73 m) hybrid hull has been investigated under dynamic load in this
chapter. The procedures to perform dynamic analysis according to ABS DLA guide, have
been described in chapter 2.
6.1. Dynamic Approach for Wave Loads
For wave loads, two dominant load parameters (DLP) were considered. One was
wave bending moment and other was wave shear force. These two loads were expressed
as a function of time.
6.1.1. Wave Bending Moment
From the static analysis of the large hybrid hull, it was found that maximum wave
bending moment occurring at sea state 5 (Hs = 4 m) with ship velocity of 40 knots was
51763 kN-m. Also, for this wave height, average time period of sea wave is 7.5 sec [49].
It is assumed that this maximum value is changing as a sinusoidal wave [50]. Considering
that it is occurring as hogging and sagging conditions, the wave bending moment as a
function of time can be expressed as shown in Figure 6.1.:
80
Figure 6.1. Maximum wave bending moment vs. time (wave period = 7.5 sec)
By using the MATLAB function Fast Fourier Transform (FFT) this time domain function
can be converted into frequency domain function as shown below:
Figure 6.2. Maximum wave bending moment in frequency domain
Figure 6.2. actually represents the response curve for maximum wave bending
moment. To obtain an input wave density diagram, we have chosen the two-parameter
Bretschneider spectrum as specified in Equation (2.25). By using the time period of 7.5
sec with a significant height of 4 m, MATLAB was used to compute the input wave
spectrum diagram as follows:
81
Figure 6.3. Wave power spectrum (Two parameter Bretschneider spectrum)
If we recall the Equation (2.27), we have,
Load Response PSD (power spectrum density) = RAO × Wave PSD
In this case, we have already obtained the load response curve for wave bending
moment as well as wave power spectral density. Thus, by utilizing MATLAB code again,
we found out the transfer function (Frequency response function) or Response amplitude
operator (RAO) for wave bending moment as shown in Figure 6.4. It is noted that we
have considered the wave frequency, ω up to 2 rad/s as the limit specified by ABS [37,
50]. Maximum value in RAO curve is 6.5×104
kN-m/m.
Figure 6.4. Response amplitude operator (RAO) for wave bending moment
82
The next step was to determine the most probable extreme value (MPEV) for
wave bending moment. Using Equation (2.32), the probability distribution function is
obtained as shown in Figure 6.5. By taking a probability level of 10-8
according to ABS
DLA [37], the most probable extreme value for wave bending moment was 118,000 kN-
m.
Figure 6.5. Probability distribution function for wave bending moment
From Equation (2.33), we have,
So, Equivalent wave amplitude (wave bending moment), aw = 118,000 / 6.5×104
= 1.8 m
Also, from Equation (2.34), Wavelength of equivalent wave, λ = (2πg)/ωe2
λ = (2×π×9.8) / (0.6)2 = 170 m
Therefore, the amplitude and wavelength of equivalent wave for wave bending moment
is 1.8 m and 170 m, respectively.
The final time dependent wave bending moment along the longitudinal direction can be
obtained from Equation (2.35), Mi = (Ai) (aw) sin (ωet + εi)
0
20000
40000
60000
80000
100000
120000
140000
1E-091E-081E-071E-061E-050.00010.0010.010.11
Wa
ve
Ben
din
g M
om
ent,
Mw
(k
N-m
)
Probability of Exceedance
Most Probable Extreme Value (MPEV) for WBM
83
Mi = 6.5×104×1.8×sin(0.6×t + εi) = 117000sin(0.6×t + εi) (6.1)
For example,
at t = 2 sec, εi = 1.8 rad; Mi = 16512 kN-m
at t = 2.8 sec, εi = 1.2 rad; Mi = 30259 kN-m
at t = 7 sec, εi = 2.4 rad; Mi = 34451 kN-m
These values are lower than maximum wave bending moment (51763 kN-m) at
static condition.
6.1.2. Wave Shear Force
Similar procedures were followed to obtain the time dependent wave shear force
expression. Maximum wave shear force obtained from static analysis was 1388 kN. With
a wave period of 7.5 sec, this wave load was first expressed as a sinusoidal curve in time
domain. Then by using FFT of MATLAB, it was converted into frequency domain. Same
Bretschneider spectrum was used as a input wave spectrum to finally obtain the RAO
curve for the wave shear force. Maximum wave shear force from the RAO diagram was
1730 kN/m at 0.6 rad/s. From the probability distribution function, most probable
extreme value (MPEV) for wave shear force at a probability level 10-8
was 3900 kN.
So, Equivalent wave amplitude (wave shear force), aw = 3900 / 1730 = 2.3 m
Also, Wavelength of equivalent wave, λ = (2πg)/ωe2
= (2×π×9.8) / (0.6)2 = 170 m
Therefore, the amplitude and wavelength of equivalent wave for wave shear force are 2.3
m and 170 m, respectively.
Time domain wave shear force along the longitudinal direction is,
84
Mi = 1730×2.3×sin(0.6×t + εi) = 3979 sin(0.6×t + εi) (6.2)
Similarly,
at t = 1 sec, εi = 2.5 rad; Mi = 166 kN
at t = 4 sec, εi = 0.5 rad; Mi = 952 kN
at t = 8 sec, εi = 1.8 rad; Mi = 1240 kN
These values are lower than maximum wave shear force (1388 kN) at static
condition.
6.1.3. Comparisons Between Static and Dynamic Conditions for Wave Loads
Wave bending moment and wave shear force were applied on the large hybrid
hull as before. But this time, these values were expressed as a function of time. All other
loads such as gravity force, buoyancy force were also applied. After solving, the stresses
are extracted and listed in Table 6.1.
Table 6.1. Comparisons between static and dynamic situation for wave loads
Von-Mises Stress Shear Stress
Static 33.10 MPa
(maximum value)
19 MPa
(maximum value)
Dynamic
After 2 sec (max) 31.48 MPa 17.717 MPa
After 4 sec (max) 31.445 MPa 17.696 MPa
After 6 sec (max) 31.392 MPa 17.668 MPa
After 8 sec (max) 31.325 MPa 17.631 MPa
After 10 sec (max) 31.244 MPa 17.586 MPa
85
It is observed that, for both Von-Mises and shear stresses, the static case provides
the higher values. We ran the simulation for 10 sec and it was found that due to time
dependent wave loads, stresses developed were usually below the static case. With
change of time, the stress variation was not significant.
6.2. Dynamic Loading Approach under Slamming Loads
The DLPs considered for slamming load analysis were slamming induced bending
moment, slamming induced shear force and bottom slamming pressure.
6.2.1. Slamming Induced Bending Moment
Maximum slamming induced bending moment generated in large hybrid hull
under static condition is 48333 kN-m. RAO for slamming induced bending moment is
obtained from MATLAB showing a maximum value of 60000 kN-m/m. Most probable
extreme value (MPEV) for slamming bending moment is 108000kN-m at a probability
level of 10-8
.
Equivalent wave amplitude (slamming bending moment), aw = 108000 / 60000 = 1.8 m
Also, Wavelength of equivalent wave, λ = (2×π×9.8) / (0.6)2 = 170 m
Slamming induced bending moment therefore has an equivalent wave of 1.8 m amplitude
and 170 m of wave length.
Time varying slamming induced bending moment is,
Mi = 60000×1.8×sin(0.6×t + εi) = 108000sin(0.6×t + εi) (6.3)
at t = 1 sec, εi = 2.1 rad; Mi = 46158 kN-m
at t = 3.5 sec, εi = 0.9 rad; Mi = 15241 kN-m
86
at t = 7 sec, εi = 2.4 rad; Mi = 33647 kN-m
These values are lower than maximum slamming induced bending moment (48333 kN-
m) at static condition.
6.2.2. Slamming Induced Shear Force
From static analysis, Maximum slamming induced shear force at sea state 5 is
3788 kN. The highest value from RAO diagram is 4720 kN/m at 0.6 rad/s. Most probable
extreme value (MPEV) for slamming induced shear force obtained from the probability
distribution function is 10600 kN.
Figure 6.6. Probability distribution function for slamming induced shear force
Equivalent wave amplitude for slamming shear force, aw = 10600 / 4720 = 2.2 m
Wavelength of equivalent wave is 170 m as before.
Slamming induced shear force as a function of time is:
Mi = 4720×2.2×sin(0.6×t + εi) = 10384sin(0.6×t + εi) (6.4)
0
2000
4000
6000
8000
10000
12000
1E-091E-081E-071E-061E-050.00010.0010.010.11
Sla
mm
ing
Sh
ear
Fo
rce,
Fw
(k
N)
Probability of Exceedance
Most Probable Extreme Value (MPEV) for SSF
87
6.2.3. Bottom Slamming Pressure
Maximum slamming pressure developed at the bottom surface of large hybrid hull
is 74 kPa as obtained from static slamming analysis. This slamming pressure is first
converted into time domain and then into frequency domain. The frequency response
function i.e. the RAO for bottom slamming pressure illustrated in Figure 6.7. shows that
maximum value of RAO curve is 265 kPa/m. This maximum value occurs when wave
frequency is equal to 0.62 rad/s.
Figure 6.7. Response amplitude operator (RAO) for bottom slamming pressure
Again, from the probability distribution function of slamming pressure, most
probable extreme value (MPEV) is 660 kPa.
Equivalent wave amplitude, aw = 660 / 265 = 2.5 m
Wavelength of equivalent wave, λ = (2×π×9.8) / (0.62)2 = 160 m
Bottom slamming pressure is considered as an equivalent wave having amplitude of 2.5
m and wave length of 160 m.
Finally, the time varying bottom slamming pressure is obtained from following equation,
Mi = 265×2.5×sin(0.62×t + εi) = 662.5sin(0.62×t + εi) (6.5)
88
6.2.4. Comparisons Between Static and Dynamic Under Slamming Loads
Equations (6.3) to (6.5) were used to compute slamming induced bending
moment, shear force and bottom pressure as a function of time along longitudinal
direction. After applying the loads on the large hybrid, stress values were extracted and
listed in Table 6.2.
Table 6.2. Comparisons between static loads and dynamic loads under slamming
Von-Mises Stress Shear Stress
Static 125.38 MPa
(max)
70.80 MPa
(max)
Dynamic
After 2 sec (max) 108.03 MPa 62.01 MPa
After 4 sec (max) 107.68 MPa 61.807 MPa
After 6 sec (max) 107.28 MPa 61.580 MPa
After 8 sec (max) 106.83 MPa 61.326 MPa
After 10 sec (max) 106.33 MPa 61.044 MPa
Similar trend is observed as we did with wave loads. Maximum Von-Mises stress
under static load is 125.38 MPa which is always higher than that of the dynamic loading.
Same is the case with shear stress. It is clear therefore that static loads provide a
conservative value while comparing with dynamic loads. Also, with respect to time, there
is not much of a change in stress magnitudes.
89
7. SUMMARY AND RECOMMENDATIONS FOR FUTURE WORKS
7.1. Summary
This study presents finite element analysis of a multi-hull composite ship
structure, and a hybrid hull of identical length and beam and also a large hybrid hull
model under wave and slamming loads. The followings are the summary of the research
work:
1. A multi-hull composite ship structure, and a hybrid hull of same length and
beam, have been designed and analyzed. The hybrid hull structure is made by Ti alloy
frame along with sandwich composite panels.
2. Wave loads and slamming loads acting on both hull structures have been
calculated according to ABS rules at sea state 5 with a ship velocity of 40 knots.
3. Tsai-Wu failure criterion along with Maximum Stress criterion have been
applied for both wave load and slamming load conditions on multi-hull and hybrid ship
structure. Both the hull structures are verified to survive without any indication of failure
at assumed sea conditions.
4. A comparison of deformation and stresses between the two sets of loadings,
reiterates the fact that slamming loads are more detrimental to ship structures.
Deformation under slamming is almost one order higher than that caused by wave loads.
However, stresses under slamming are 2-3 times larger.
90
5. Introduction of Ti alloy frames in between sandwich composite panels in
hybrid hull significantly reduce both deformation and stresses compared to identical
composite hull although hybrid hull has lower value of moment of inertia and section
modulus. The reason is that attachment of Ti frames with composite panels enhances the
in plane strength and stiffness of the hull which consequently lessen the deformation and
stresses.
6. A large hybrid hull have also been modeled and investigated for static as well
as time domain dynamic loads. Under static condition, it is observed that in case of 73m
long hybrid hull, the maximum deformation and stress values are lower than that of 39m
composite multi-hull under both wave and slamming loads.
7. Dynamic analysis of the large hybrid hull (73 m) also shows that Von-Mises
stresses and shear stresses are close but lower than those obtained from static analysis.
With change of time, the stresses variation is not significant.
7.2. Recommendations for Future Works
This research work can be extended on following facts:
1. Current analyses are based on sea state 5 which corresponds to a significant
wave height of 4 m. Further analysis can be performed by increasing the significant wave
height. Similarly, analyses can be also done by changing the ship velocity.
2. General failure theories for isotropic and composite materials have been used to
predict the failure. But for sandwich composites it is recommended that failure theories
associated with face-sheet/core delamination, that is strain energy release rate (GI) may
also be used.
91
3. The dynamic analysis of large hybrid hull can be extended to perform a
comprehensive fatigue analysis.
4. The RAO for each of the loading parameters can be calculated with different
heading angles. The heading angle which gives the maximum value of RAO should be
used for further analysis.
5. In the current study, to predict most probable extreme value (MPEV), short
term analysis was used. Long term analysis can be included utilizing the wave scatter
diagram provided by ABS.
92
APPENDIXES
A. Sample calculation to determine wave loads for large hybrid hull
We consider, Sea State 5 (Significant wave height, h1/3 = 4 m)
Ship velocity, V = 40 knots
L = 73 m, B = 7 m, d = 2.6 m, k2 = 190, k1 = 110, fp = 17.5 kN/cm2, C2 = 0.01, k = 30
C1 = 0.044L+3.75 = 6.962
Cb = (Δ2) / (1.025 L B d) = (1.025) / (1.025 × 73 × 7 × 2.6) = 0.4405
but for L ≥ 61 m , Cb should be at least 0.6 . So, we choose, Cb = 0.6
Recalling Equation (2.14) to Equation (2.19), we have:
Wave Bending Moment (Hogging) at amidships,
kN-m
Mwh = 190 × 6.962 × (73)2 × 7 × 0.6 × 10
-3 = 29607 kN-m
Wave Bending Moment (Sagging) at amidships, ( )
Mws = − { 110 × 6.962 × (73)2 × 7 × (0.6+0.7) × 10
-3 } = − 37138 kN-m
Still Water Bending Moment (Hogging) at amidships, ( )
Mswh = 0.375 × 17.5 × 6.962 × 0.01 × (73)2 × 7 × (0.6+0.7) = 22156 kN-m
Still Water Bending Moment (Sagging) at amidships, Msws = 0
Wave Shear Force (Positive), ( ) kN
Fwp = 30 × 1.0 × 6.962 × 73 × 7 × (0.6+0.7) × 10-2
= 1388 kN (maximum)
Wave Shear Force (Negative), ( ) kN
Fwn = − { 30 × 0.92 × 6.962 × 73 × 7 × (0.6+0.7) × 10-2
} = − 1277 kN (maximum)
93
B. Sample calculation to determine slamming loads for large hybrid hull
N2 = 0.0078, τ = 3°, βcg = 38°, Δ1 = 6 × 105 kg, C3 = 1.25, Δ2 = 600 metric tons, C4 = 4.9,
N1 = 0.1, FD = 0.4
AR = (0.697 Δ2) / d (m2) = (0.697 × 600) / 2.6 = 160.846 m
2
ls = AR / B = (160.846 / 7) = 22.978 m
Recalling Equation (2.20) to Equation (2.24), we have:
Maximum Vertical Acceleration, [
] , -
g's
ηcg = 0.0078 × [{(12×4)/7}+1] × 3 × [50 − 38] × [{(40)2 × (7)
2}/6 × 10
5] = 0.288288 g's
Slamming Induced Bending Moment at amidships, ( )( ) kN-m
Msl = 1.25 × 600 × (1+0.288288) × (73 − 22.978) = 48333 kN-m
Slamming Induced Shear Force (Positive), ( ) kN
Fsl = 4.9 × 1.0 × 600 × (1+0.288288) = 3788 kN (maximum)
Slamming Induced Shear Force (Negative), ( ) kN
Fsl = 4.9 × 0.92 × 600 × (1+0.288288) = 3485 kN (maximum)
Bottom Slamming Pressure,
( ) (kN/m
2)
Pbxx = {(0.1×6×105) / (73×7)} × {1+0.288288KV) × 0.4 = 47.0 (1+0.288288Kv) kN/m
2
94
C. MATLAB code for two parameter Bretschneider Spectrum
************************************************************************
% %% Two Parameter Bretschneider Spectrum
clc;
clear all;
W=0.20944:0.01265:2.094395;
for counter=1:length(W)
S(counter) =( (2.37/((W(counter))^5)) * (exp(-1.25*((0.83/(W(counter)))^4))) );
end
figure(1);
plot(W,S);
title('Power Spectrum (Bretschneider Spectrum)');
xlabel('Wave Frequency,w (rad/s)');
ylabel('Wave Spectral Density, S(w) (m^2-s)');
grid on
W=W(2:150);
S=S(2:150);
save('bretschneider.mat','W','S');
95
D. MATLAB code to obtain time to frequency domain for wave bending moment
************************************************************************
% Maximum Wave Bending Moment: Time to Frequency Domain
clc; clear all;
Fs = 10; % Sampling frequency
t = 0.10:1/Fs:15; % Time vector of 30 second
f = 0.133; % Create a sine wave of f Hz.
x = (51763*sin(2*pi*f*t)); % Moment unit: kN-m
figure(1);plot(t,x);
title('Maximum Wave Bending Moment vs. Time');
xlabel('Time,t (sec)'); ylabel('Maximum Wave Bending Moment, Mw (kN-m)');
grid on
nfft = 300; % Length of FFT
X = fft(x,nfft); % Take fft, padding with zeros so that length(X) is equal to nfft
X = X(1:nfft/2);
mx = abs(X);
f = (Fs/30)*linspace(0.1,1,150); % Frequency vector
figure(2); plot(f(1:60),mx(1:60));
title('Power Spectrum of MWBM (frequency domain)');
96
xlabel('Wave Frequency,f (Hz)'); ylabel('Spectral density function (Wave induced
response)');
grid on
figure(3);
w=(2*pi*f);
plot(w(1:25),mx(1:25));
title('Power Spectrum of MWBM (frequency domain)');
xlabel('Wave Frequency,w (rad/s)');
ylabel('Spectral density function (Wave induced response)');
grid on
w=w(2:150);
mx=mx(2:150);
save('bendingmoment.mat','w','mx');
E. MATLAB code to calculate RAO for wave bending moment
************************************************************************
clc; clear all;
load('jonswap.mat');
load('bendingmoment.mat');
RAO=mx./S;
plot(W(30:140),RAO(30:140));
97
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