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Influence of Trimaran Geometric Parameters on Intact and Damaged Ship Stability
William Scott Weidle
Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in
partial fulfillment of the requirements for the degree of
Master of Science
In
Ocean Engineering
Alan J. Brown, Committee Chair
Stefano Brizzolara
Christopher C. Bassler
September 22, 2017
Blacksburg, VA
Keywords: Trimaran, Ship, Stability, Intact, Damaged
© 2017 by William S. Weidle
Influence of Trimaran Geometric Parameters on Intact and Damaged Ship Stability
William Scott Weidle
ACADEMIC ABSTRACT
Multi-hull vessels have been considered for high-speed, military and commercial applications for
decades. More recently the trimaran vessel, with three hulls, has captured interest among naval
ship designers and stakeholders. A definition of multi-hulls is introduced as a continuum with
monohulls on one end, catamarans on the other, and trimarans in-between. A review of
methods to assess intact and damaged stability follows in addition to an overview of current
research in the area of dynamic stability for monohulls and trimarans. An investigation of intact
stability characteristics for multi-hulls along the continuum is presented and their trends are
examined. Next, a series of trimaran configurations are modeled in CAD with subdivision to
determine their allowable KG according to USN deterministic criteria and using quasi-static
methods. A response surface model was determined for allowable KG as a function of center
hull length to beam, side hull beam to draft, transverse position, and displacement for use as a
rule of thumb measure and potential optimization constraint.
Influence of Trimaran Geometric Parameters on Intact and Damaged Ship Stability
William Scott Weidle
GENERAL ABSTRACT
Ships which have more than one hull are called multi-hulls. In contrast to a monohull, which
comprises of a single hull form, catamarans, trimarans, quadramarans, and pentamarans consist
of two, three, four, and five hulls respectively and make up the multi-hull group of ships. Multi-
hulls have been considered for high-speed, military and commercial applications for decades.
More recently trimaran ships, with three hull forms, have captured interest among naval ship
designers and stakeholders. This thesis provides a definition of multi-hulls as a continuum with
monohulls on one end, catamarans on the other, and trimarans in-between.
An assessment of ship stability quantifies a ship’s risk of capsize in a particular loading condition
and environment. The intact ship condition is assessed as well as damaged ship conditions
where the certain compartments are subject to flooding from the sea. Initially, a quasi-static
method is first undertaken where the ship characteristics are computed at discrete motion
points of interest and integrated. This thesis includes a review of quasi-static methods
employed by United States and British navies to assess intact and damaged stability.
Additionally, an overview of current research in the area of dynamic stability for monohull and
trimaran. Assessing stability using dynamic methods involves a more detailed assessment of
extreme events a ship may encounter in its lifecycle. These assessments were beyond the scope
of this thesis.
The first investigation of this thesis assessed the intact stability characteristics for multi-hulls
along the continuum, including a monohull, five trimarans, and a catamaran. Trends were
examined and the effect of displacement distribution between hulls on stability characteristics
was analyzed. The results of the first investigation set the bounds for a more detailed
investigation of trimaran geometric parameters affecting intact and damaged stability.
The second investigation established a design of experiments to formulate a series of trimaran
configurations consisting of a center hull, two side hulls and cross deck structure joining the
hulls from above. This series of trimaran hull forms varied particular parameters that were
determined to be influential to intact and damaged stability. Each hull form configuration was
modeled using three-dimensional CAD software with subdivision defining compartments to be
flooded. The maximum center of gravity was determined which would satisfy all stability criteria
published by the United States Navy for each intact and damaged condition. After tabulating the
data, a response surface model was determined for maximum vertical center of gravity as a
function of center hull ratios, side hull ratios, and the transverse distance between the center
and side hulls. The response surface model is intended for use as a rule of thumb measure and
potential constraint for optimization.
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Dedication
The thesis is dedicated to my wife, Stephanie, and daughter, Virginia, for their incredible
patience and understanding as I spent countless hours away from them to finish.
Acknowledgements
I would like to thank my committee chair and advisor, Dr. Brown, for his willingness to work with
me remotely over the past two years to converge on this thesis. Also, many thanks to my
colleague and committee member, Dr. Bassler, who encouraged me to start this thesis and
whose door was always open to discuss aspects of this thesis and other material. Additional
thanks to Dr. Brizzolara for joining my committee and his readiness to support me.
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Table of Contents
1 Introduction ............................................................................................................................ 1 1.1 Trimaran Background ..................................................................................................... 1 1.2 Naval Trimaran ............................................................................................................... 3 1.3 Literature Survey ............................................................................................................ 4
1.3.1 Trimarans ................................................................................................................... 4 1.3.2 Trimaran Intact Stability and Seakeeping .................................................................. 4 1.3.3 Trimaran Intact and Damaged Stability ..................................................................... 6 1.3.4 Related Research: Dynamic Stability ......................................................................... 8
1.4 Problem and Thesis Motivation ..................................................................................... 9 1.5 Thesis Objective ............................................................................................................. 9 1.6 Thesis Outline................................................................................................................. 9
2 Intact and Damage Stability: Methods and Issues ................................................................ 11 2.1 Methods and Standards for Quasi-Static Stability Assessment of Naval Combatants in
Intact and Damage Conditions ..................................................................................... 13 2.2 Probabilistic Methods to Assess Stability of Damaged Naval Combatants ................. 16 2.3 Dynamic Intact and Damaged Stability and Trimaran Seakeeping .............................. 17
2.3.1 Transverse Stability of Trimarans in Longitudinal Waves ........................................ 18 2.3.2 Estimation of Roll Damping for Trimarans............................................................... 19 2.3.3 Sloshing Effect in Cross Deck: Case of RO/RO Passenger Ferries ............................ 20 2.3.4 Experimental and Numerical Modeling of Dynamic Stability for Damaged
Monohulls ................................................................................................................ 20 3 Multi-Hull Intact Stability Comparison ................................................................................. 22
3.1 Quasi-Static Intact Stability Assessment in Heel .......................................................... 23 3.2 Quasi-Static Intact Stability Assessment with Trim ..................................................... 28 3.3 Intact Stability Conclusions .......................................................................................... 32
4 Trimaran Quasi-Static Stability Study ................................................................................... 33 4.1 Study Overview ............................................................................................................ 33 4.2 Assumptions ................................................................................................................. 33
4.2.1 Ship Design Concepts ............................................................................................... 33 4.2.2 Design Variables and Parameters ............................................................................ 36
4.3 Intact and Damaged Stability Model ........................................................................... 37 4.4 Design of Experiments ................................................................................................. 40 4.5 Relationships ................................................................................................................ 46 4.6 Response Surface Model ............................................................................................. 52 4.7 Application of Stability RSM to Ship Design ................................................................. 54
5 Conclusions: Stability Criteria Adaption for Trimaran .......................................................... 55 References ..................................................................................................................................... 57 Figure References .......................................................................................................................... 60 Appendix A KGa/D RSM Visualized through Contour Plots ...................................................... 61
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Figures
Figure 1-1 Multi-Hull Continuum..................................................................................................... 1 Figure 1-2 Trimaran Waterplane View ............................................................................................ 2 Figure 1-3 Trimaran Body Plan View ............................................................................................... 2 Figure 1-4 RV Triton and USS Independence (LCS 2) ....................................................................... 4 Figure 1-5 Free-Body Diagram: Righting Arm, GZ ........................................................................... 5 Figure 1-6 Anatomy of Righting Arm, GZ Curve .............................................................................. 5 Figure 1-7 Metacentric Height calculation for the Trimaran .......................................................... 6 Figure 2-1 Stability of Intact Vessel: Effect of Adverse Rolling ...................................................... 13 Figure 2-2 Stability of Damaged Vessel: Effect of Adverse Rolling................................................ 14 Figure 2-3 Susceptibility Index to Longitudinal Waves of Different ship typologies compared to
the Trimaran Unina1 ........................................................................................................ 18 Figure 2-4 Stationary Parametric Roll Amplitude for Unina1 in configuration S1L4. .................... 19 Figure 3-1 Body Plan Comparison across Generated Multi-Hull Continuum ................................ 22 Figure 3-2 Change in WP Area Comparing Multi-Hulls at Heel Angle ........................................... 24 Figure 3-3 Changes in Righting Arm, GZ Comparing Multi-Hulls at Heel Angle ............................ 26 Figure 3-4 Correlation between Area A1 and Side Hull Displacement.......................................... 27 Figure 3-5 Correlation between Maximum Righting Arm and Side Hull Displacement ................ 27 Figure 3-6 Deck Edge Height above Waterline, Comparing Multi-Hulls at Heel Angle ................. 28 Figure 3-7 Isometric View of 2.5-95-2.5 Trimaran (Mid and Aft SH Position) and Monohull ....... 29 Figure 3-8 Percent Difference in Wetted Surface and WP Area at Trim Angle, Level Heel .......... 29 Figure 3-9 Change in LCB and LCF with Trim Angle, Level Heel .................................................... 30 Figure 3-10 Change in Longitudinal and Transverse Metacentric Height with Trim Angle, Level
Heel................................................................................................................................... 30 Figure 3-11Righting Arm, GZ, with Change in Heel and Trim Angle .............................................. 31 Figure 3-12 Change in WP Area with Heel Angle .......................................................................... 31 Figure 3-13 Change in Trim Angle with Change in Heel Angle ...................................................... 31 Figure 3-14 LCF and TCF with Change in Heel and Trim Angle ...................................................... 32 Figure 4-1 Isometric and Profile View of the Frigate Design Concepts ......................................... 35 Figure 4-2 Engine and VLS components in 85% Permeable Spaces .............................................. 36 Figure 4-3 Beam-Wind Calculation using 2m strip method for L1 and L2 side hull position ........ 37 Figure 4-4 Asymmetric, Transverse Damage Extents Subject to Flooding .................................... 38 Figure 4-5 Damaged Conditions Subject to Flooding .................................................................... 39 Figure 4-6 Damaged Stability Model: Hull 1 Equilibrium, KG/D = 0.51 ......................................... 39 Figure 4-7 DoE Hull Variants, Hull 1, 1a, 1b ................................................................................... 40 Figure 4-8 Side Hull Positions to Examine in DoE .......................................................................... 41 Figure 4-9 DoE Hull Variants with Vratio = 2% .............................................................................. 43 Figure 4-10 DoE Hull Variants with Vratio = 3% ............................................................................ 43 Figure 4-11 DoE Hull Variants with Vratio = 4% ............................................................................ 43 Figure 4-12 Effect of Side Hull Beam to Draft Ratio, (B/T)SH (Hulls 1, 1a, 1b) ............................... 46 Figure 4-13 Effect of Side Hull Longitudinal Position, SHX (Hulls 1-6) ........................................... 47 Figure 4-14 KGa / D Results at L2 Side Hull Position: Intact Condition ......................................... 48 Figure 4-15 KGa / D Results at L2 Side Hull Position: Intact Condition During OTD...................... 48 Figure 4-16 KGa / D Results at L2 Side Hull Position: Worst Damaged Condition ........................ 49 Figure 4-17 Worst Damaged Condition (Hulls 1-24) ..................................................................... 50 Figure 4-18 Stability Assessment, Feasible Region: KGa/D > 0.5(Hulls 1-3, 7-24)......................... 51 Figure 4-19 KGa/D Function Actual v. Predicted, Hulls 1 and 7 - 24 ............................................. 52
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Tables
Table 1-1 Characteristics of Triton and Independence Class Trimarans ......................................... 3 Table 1-2 Effect of Trimaran Geometry Configuration on Metacentric Height, Mt ........................ 6 Table 2-1 Methods for Intact and Damaged Stability Analysis ..................................................... 13 Table 2-2 Criteria for Intact Vessel, Normal Operations ............................................................... 15 Table 2-3 Criteria for Damaged Vessels ........................................................................................ 15 Table 2-4 Assumptions for Damaged Stability Assessments ......................................................... 16 Table 2-5 Relationship of RDLI with Quasi-Static Stability Criteria for Monohulls........................ 21 Table 3-1 Hydrostatics Comparison across Generated Multi-Hull Continuum ............................. 23 Table 3-2 Significant Multi-Hull Events and Corresponding Heel Angles ...................................... 25 Table 3-3 Stability Criteria, Comparing Multi-Hulls ....................................................................... 26 Table 4-1 Design Concept Principal Characteristics ...................................................................... 34 Table 4-2 Design Concept Hydrostatics ......................................................................................... 34 Table 4-3 Design Concept Non-Dimensional Parameters ............................................................. 34 Table 4-4 Design Concept Propulsion and Electrical Plant ............................................................ 34 Table 4-5 Design Concept Systems ................................................................................................ 35 Table 4-6 Design Variable Values used for Design of Experiments (DoE) ..................................... 36 Table 4-7 Design Parameters used for DoE – Constrained for All Designs.................................... 36 Table 4-8 Damaged Conditions Subject to Flooding ..................................................................... 38 Table 4-9 DoE varying (B/T)SH: Hydrostatic Characteristics ........................................................... 40 Table 4-10 DoE varying (B/T)SH: Waterplane Characteristics ........................................................ 40 Table 4-11 DoE varying (B/T)SH: Stability Characteristics .............................................................. 40 Table 4-12 DoE varying Side Hull Position, SHX and SHY: Hydrostatics Characteristics................ 41 Table 4-13 DoE varying Side Hull Position, SHX and SHY: Waterplane Characteristics ................ 41 Table 4-14 DoE varying Side Hull Position, SHX and SHY: Stability Characteristics ....................... 41 Table 4-15 DoE varying (L/B)CH, Vratio, and SHY: Hydrostatic Characteristics ............................... 42 Table 4-16 DoE varying (L/B)CH, Vratio, and SHY: Waterplane Characteristics .............................. 44 Table 4-17 DoE varying (L/B)CH, Vratio, and SHY: Stability Characteristics .................................... 45 Table 4-18 KGa/D functions of (B/T)SH, Hulls 1,1a, and 1b at Intact and Damaged Conditions .... 52 Table 4-19 KGa/D Function Statistics Summary, Hulls 1 and 7 - 24 .............................................. 52 Table 4-20 KGa/D Functions of Hulls 1 and 7 – 24 at Intact and Damaged Conditions ................ 52 Table 4-21 Valid Bounds for Variables in KGa/D RSM ................................................................... 53
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Acronymns and Nomenclature
ANEP ........................................................................................ Allied Naval Engineering Publication ASW ............................................................................................................ Anti-Submarine Warfare ASUW ............................................................................................................... Anti-Surface Warfare B/T ......................................................................................................................Beam to Draft Ratio Bratio ................................................................................................ Side to Center Hull Beam Ratio Bwl ........................................................................................................................... Beam Waterline CDH ....................................................................................................................... Cross Deck Height CH .................................................................................................................................... Center Hull CL ....................................................................................................................................... Centerline D ........................................................................................................................... Depth at Midships D/T ..................................................................................................................... Depth to Draft Ratio DERA ...................................................................... Defence Evaluation and Research Agency (U.K.) DoE ................................................................................................................ Design of Experiments FB ....................................................................................................................................... Freeboard GZ................................................................................................................................... Righting Arm IMO .......................................................................................... International Maritime Organization KG ............................................................................................. Vertical Center of Gravity from Keel KGa ............................................................................................................................... Allowable KG L/B.................................................................................................................... Length to Beam Ratio LCB ................................................................................................. Longitudinal Center of Bouyancy LCG ..................................................................................................... Longitudinal Center of Gravity LCS ..................................................................................................................... Littoral Combat Ship Lpp .................................................................................................. Length between perpendiculars Lratio ............................................................................................... Side to Center Hull Length Ratio Lwl .......................................................................................................................... Length Waterline MoD .......................................................................................................... Ministry of Defence (U.K.) MCM ............................................................................................................. Mine Countermeasures MS ........................................................................................................................................ Mid-Ship Mt ....................................................................................................................... Metacentric Height NATO .......................................................................................... North Atlantic Treaty Organization OTD .................................................................. Overhaul, Towing, and Decommissioning condition RDLI ........................................................................................................ Relative Damage Loss Index RV ............................................................................................................................. Research Vessel SOLAS ................................................................................................................. Safety Of Life At Sea SH ..........................................................................................................................................Side Hull SHX ............................................................... Side Hull Longitudinal Location, Mid-ship to Mid-span SHY ............................................................. Side Hull Transverse Location, Centerline to Centerline SWATH .......................................................................................... Small Waterplane Area Twin Hull T ................................................................................................................................................. Draft TCG ...................................................................................................... Transverse Center of Gravity Tratio ................................................................................................. Side to Center Hull Draft Ratio USN ...................................................................................................................... United States Navy Vratio ........................................................................................ 1x Side to Total Displacement Ratio VCB ........................................................................................................ Vertical Center of Bouyancy VCG ............................................................................................................ Vertical Center of Gravity WDC ................................................................................................................... Wet Deck Clearance
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WL ....................................................................................................................................... Waterline WP .................................................................................................................................. Waterplane
Symbols and Units
Δ ................................................................................................................................. Displaced Mass λ ...................................................................................................................................... Wavelength θ ........................................................................................................................................ Heel Angle
........................................................................................................................... Displaced Volume ft .................................................................................................................................................. foot kt ................................................................................................................................................ knots LT ......................................................................................................................................... long tons m ............................................................................................................................................. meters mt ....................................................................................................................... metric ton (tonnes) deg. ........................................................................................................................................ degrees
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1 Introduction
1.1 Trimaran Background A “trimaran” is a type of hull form within a larger set of hull forms called “multi-hulls.” In a
broad sense, the term “multi-hull” vessel is defined for this thesis as a continuum of vessels
between a monohull and catamaran. Using this definition, a monohull can be considered as a
vessel with center hull and infinitesimally small side hulls and a catamaran is considered as a
vessel with two side hulls and infinitesimally small center hull. Trimaran vessels are “in-
between” a monohull and catamaran on this multi-hull continuum where the center hull and
side hulls each contribute a portion of the total displacement. Figure 1-1 is a graphical
representation of the multi-hull vessel continuum with the monohull on the lower left,
catamaran on the upper right, and trimaran linearly in-between.
Previous works by Andrews and Zhang (1995, 2004a, 2004b) have found that trimarans with side
hulls contributing 2-5% each of the total displacement to be a sweet spot, for resistance
performance, where benefits of the monohull are seen with high heel angle before deck
submergence as well as benefits of the catamaran with high righting arm values. This range is
plotted in red (Figure 1-1) where the two side hulls contribute 2 to 10% of the total
displacement and the center hull contributes 98 to 90% of the total displacement.
Figure 1-1 Multi-Hull Continuum
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In traditional naval architectural terms, a trimaran is defined as “a tri-hull ship with three
identical hulls of a traditional form.” (Dubrovsky and Lyakhovitsky, 2001) In modern times, the
term ‘trimaran’ has become more general to include all ships with three hulls, often with the
side hulls distinct in size and shape from the center hull. The trimaran hull form, when applied to
practical military naval applications, has typically taken the shape of what is classically termed a
“stabilized monohull” or “ship with outriggers”, where significantly smaller side hulls (typically 2
– 5% each of the overall displacement) are placed on the port and starboard side of a slender
monohull for increased transverse stability and increased overall beam over traditional
monohulls. The waterplane diagram (Figure 1-2) shows a typical arrangement of a trimaran for
port and starboard symmetry and defines the nomenclature used in this thesis. SHX denotes the
longitudinal position as the distance between mid-ships (MS) of center hull and mid-span of side
hull (SH); SHY denotes the transverse position as the distance between centerlines (CL) of center
and side hull. Figure 1-3 shows the body view of a trimaran and defines nomenclature including
the freeboard (FB), cross deck height (CDH), wet deck clearance (WDC), and waterline (WL).
(Brown, 2015)
Figure 1-2 Trimaran Waterplane View
Figure 1-3 Trimaran Body Plan View
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1.2 Naval Trimaran Two existing naval trimarans have been designed, built and are in use today, the ACV Triton and
the Littoral Combat Ship: LCS 2 USS Independence class. Triton was originally commissioned by
the United Kingdom’s Defence Research and Evaluation Agency (DERA) as a research vessel (RV)
prototype at two-thirds scale. At 98m long, RV Triton was launched in 2000 and began a series
of sea trials assessing its suitability as a potential design to replace the Type 23 frigate. The RV
Triton was sold to Gardline Marine Services in 2005 and was used for hydrographic survey,
before the vessel was chartered to the Austrialian Customs and Border Protection Service in
2006.
LCS was conceived by the USN in the early 2000s as the next generation small surface
combatant to fulfill mission requirements in the areas of anti-surface warfare (ASUW), anti-
submarine warfare (ASW), and mine countermeasures (MCM) through modular mission
packages contained within a ship platform. Two variants of the LCS ship were designed and built
by Lockheed Martin Marinette and Austal shipyards which produce the monohull and trimaran
variant respectively. The lead ship of the trimaran variant, USS Independence, was
commissioned in 2010. Principal particulars for both the RV Triton and USS Independence (LCS
2) are listed in the Table 1-1.
Additionally, the Indonesian Navy built a Trimaran Fast Missile Boat made of carbon fiber. At
63m long the trimaran was launched in 2012 only to catch fire a month later due to a short-
circuit in the engine room resulting in the loss of the vessel.
Table 1-1 Characteristics of Triton and Independence Class Trimarans
ACV Triton1 USS Independence (LCS 2)2
Length [m] 98 128.5
Beam [m] 22.5 31.6
Depth [m] 8.6 15.4
Draft [m] 3.0 4.6
Wet Deck Clearance (WDC) [m] 3.1 3.7
Displacement [mt] 1,000 3,200
Speed [kt] 20 40+
Propulsion Plant Diesel Electric 1x FP Propeller on CL 2x FP Propellers in SH
Combined Diesel and Gas Turbine
4x Waterjets
Accommodations 44 983
Mission Capability Customs and Border Protection
ASUW, ASW, or MCM Mission Packages
1 ACV Triton Northern Patrol Vessel Fact Sheet
http://www.border.gov.au/AustralianBorderForce/Documents/ACV%20Triton%20Northern%20Patrol%20Vessel.pdf#search=triton 2 US Navy Fact File - http://www.navy.mil/navydata/fact_display.asp?cid=4200&tid=1650&ct=4 3 USNI on LCS Manning: https://news.usni.org/2013/09/24/report-lcs-manning-permanently-increase-2015
4
Figure 1-4 RV Triton4 and USS Independence (LCS 2) 5
1.3 Literature Survey
1.3.1 Trimarans
Advantages for the trimaran over an equivalent monohull include a wide, convenient
mission/cargo bay above the waterline, satisfactory initial transverse stability, and lower wave
resistance of the main hull. Additionally, for a given displacement, the trimaran typically has less
probability of bow slamming, and a smaller transverse bending moment compared to
catamarans and Small Waterplane Area Twin Hull (SWATH) vessels. The trimaran also has
disadvantages including a wider overall beam than equivalent displacement monohull affecting
access to ports and canals and a larger relative wetted surface area at level trim and heel for the
same displacement. Additionally, trimarans see higher longitudinal bending moments with
increasing magnitude in high seas and poorer maneuverability/controllability with side hulls
positioned at the stern than equivalent monohulls. (Dubrovsky, 2004, p. 53) Additionally the
trimaran structural weight fraction typically must be higher compared to a monohull of the
same displacement due to the increase internal volume. (Brown, 2015)
1.3.2 Trimaran Intact Stability and Seakeeping
As the ship heels the center of gravity is assumed to remain fixed while the center of buoyancy
shifts transversely. Physically, the righting arm, GZ, is the restoring lever resulting from small
perturbations in heel. Mathematically, GZ is the horizontal distance between the center of
gravity and center of buoyancy. Figure 1-5 (Zubaly, 1996) shows a submerged body subject to
heel angle, θ and the resulting weight-buoyancy force couple which causes the restoring
moment from which GZ is derived.
One of the largest factors affecting ship stability and the righting arm, GZ, is the transverse
metacentric height, Mt. Mathematically, Mt is the moment of inertia of the waterplane area
divided by the displaced volume, which physically is the point above the waterplane about
which small heel angle changes occur, shown in Figure 1-5. At small heel angles, up to 10
4 www.naval-technology.com 5 www.navy.mil
5
degrees, there exists a linearized relation between GZ and Mt: GZ = GMt sin(θ). At larger heel
angles, greater than 10 deg., the Mt changes position and the GZ curve (Figure 1-6) exhibits non
- linear behavior with heel angle. (Zubaly, 1996)
Figure 1-5 Free-Body Diagram: Righting Arm, GZ
Figure 1-6 Anatomy of Righting Arm, GZ Curve
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For a trimaran, Mt is significantly higher above the waterplane than for an equivalent monohull
due to the presence of the side hulls. “It should be noted, however that a change in the overall
width has a much greater impact on lateral metacentric radius than the outrigger’s dimensions
have.” (Dubrovsky, 2004, p. 31) The impact of the transverse separation between the side and
center hull on Mt can be seen using a trimaran with hulls having a rectangular cross section
depicted in Figure 1-7. (Grafton, 2007) Using the parallel axis theorem, Equation 1 can be
derived for such a trimaran. (Grafton, 2007) For example, a trimaran with a total displaced
volume of 3,000 m3 and side hulls displacing 5% of the total within the study range in Figure 1-1
is used. This trimaran has a center hull length (LCH) of 100m, center hull beam (BCH) of 10m, side
hull length (LSH) of 40m, and side hull beam (BSH) of 1m. Table 1-2 shows the effect of two SHY
locations. A 50% increase in side hull beam results in a smaller BMT than for increasing the
transverse side hull position by 50%. It is also seen that the contribution of side hull (second and
third term) and center hull (first term) to the BMT is similar for both increases transversely.
233
122
12
1YSHSH
SHSHCHCHT SHBL
BLBLBM Equation 1
Table 1-2 Effect of Trimaran Geometry Configuration on Metacentric Height, Mt
Variable Initial Configuration Increased SH Beam Increased Overall Beam
BSH 1 1.5 1
SHY [m] 10 10 15
BMT 5.5 7.9 8.8
% BMT (CH) 51% 35% 32%
% BMT (SH) 49% 65% 68%
Figure 1-7 Metacentric Height calculation for the Trimaran
1.3.3 Trimaran Intact and Damaged Stability
The current state of the art regarding trimaran stability in an intact and damaged condition
includes investigations from Dr. Victor Dubrovsky, Krylov Shipbuilding Research Institute, and
Dr. David Andrews, University College London. These investigations are based on limited early-
stage concept design studies. Additional experiments conducted at Osaka Prefecture University
7
(Katayama et al, 2011) and University of Trieste (Francescutto, 2001) have led to developments
in roll damping models for multi-hulls and assessment of dynamic stability qualities for the
intact vessel. It appears, no comprehensive examination of geometry configuration has been
related to stability quantities for the intact and damaged trimaran vessel to date.
Studies under Professor David Andrews, University College London, have resulted in useful
insight regarding the configuration of trimarans related to their performance for early-stage to
preliminary ship design. In papers Andrews (2004a) and Andrews and Zhang (1995) the authors
discuss trimaran concepts and general rules of thumb for hull form configuration in early-stage
design.
In general, Andrews recommends increasing draft over increasing the waterplane area for
improved stability. This increases side hull displacement, to a reasonable minimum Vratio of 3 –
5% in the deep condition while maintaining resistance performance. Additionally, the side hull
will remain in the water for larger heel angles. Also, Andrews notes that critical stability
problems occur in the damaged condition, and thus should be a principal factor in choosing the
size and configuration of the side hulls.
Four transverse extent conditions are recommended for assessing damage stability: center hull
only; side hull plus associated cross-structure; center hull, cross-structure and 1 side hull, and
complete transverse extent of ship. (Andrews, 2004a) It is also noted that for flooding in center
hull only, improvements are made by increasing side hull displacement and lowering COG. Also,
longitudinal subdivision is necessary in the center hull to preserve transverse stability. Any
additional flooding in the side hull reduces buoyancy and waterplane area, with no COG change
causing adverse effects to stability. Conversely, increasing the length of the side hull and adding
bulkheads in the side hull minimizes this effect, meeting criteria
Also, conveniently, ballasting the side hulls provides an automatic way of meeting damage
criteria, the heel is controlled after damage because the sea cannot flood into tanks that are
already full, possibly allowing for heel in the opposite direction. Andrews and Zhang (1995)
concludes, “With this radical hull form [trimaran] the criteria for satisfactory stability must be
re-examined.” “Trimaran stability assessment is a much more complex task than that for
monohulls and catamarans. There are many variables which effect the final stability
characteristics, and a small change in a single variable may result in quite different stability
qualities.”
“Specifics of damage buoyancy and stability of multi-hull ships, as compared with monohulls,
are due to: the presence and configuration of their cross-structure; subdivision of both hulls and
cross-structure into watertight compartments and their ratios; specifics of principal dimensions
of individual hulls.” (Multi-Hull Ships, Dubrovsky, p. 89) Dubrovsky and Lyakhovitsky propose
multi-hull properties that influence stability under damage including SHY, beam to draft ratio
(B/T), block coefficient (CB), vertical center of gravity to depth ratio (KG/D), depth to draft ratio
(D/T), forecastle length to ship length ratio (LF/L), length of damage extent to overall length
8
ratio, and permeability coefficient. They present analysis by Fishkis (1976) of catamarans with
symmetric hulls with V-shaped frames. Also damage extents and characteristics of SWATH
vessels are discussed. “Due to lack of experience, there are no recommendations on the length
of flooded compartment (used to assess stability) of tri-hull ships. A conservative estimate is
30% of the overall length [for trimaran].” (Dubrovsky et al, 2001, p. 89) This recommendation is
very conservative; the USN uses 15% of overall length for the assumed length of damage extent
for naval monohulls. (NAVSEA, 2016)
“In an emergency, the flooding of multi-hull ships is usually considerably more non-symmetric
than in monohull ships. And therefore, the watertight volume above WL is usually subdivided by
longitudinal bulkheads as well. It is desirable to have the lower deck of the cross-structure
watertight as well as hatchway trunks.” (Dubrovsky et al, 2001, p. 89) This suggests that the
lowest cross deck of the trimaran is best as bulkhead/DC deck. “In analyzing the damage
stability and unsinkability, the outrigger of a small displacement should be considered as fully
flooded.” (Dubrovsky et al, 2001, p. 89) The transverse extent of damage will be a key
parameter to study for this thesis.
1.3.4 Related Research: Dynamic Stability
Four modes of stability failure are commonly examined for the intact vessel using dynamic
methods: pure-loss of stability, parametric roll, surf-riding, and dead-ship condition. (Belenky et
al, 2009) Only within the past decade have hydrodynamicists studied these modes of stability
failure in detail for intact trimaran forms, in addition to monohull forms. A documented area of
transverse instability exists in longitudinal waves for the head and following seas heading. A
series of experiments with an intact trimaran model has confirmed the potential for parametric
roll in longitudinal waves. For large wavelengths, near to or greater than the length of the ship,
the side hull exits the water in the trough indicating a potential for danger and parametric roll.
(Bulian et al, 2011)
Additionally, the capability to model the dynamic stability of a damaged monohull vessel has
only recently been investigated. This data coupled with experiments of monohull models has
begun to link the quasi-static stability criteria developed with historic data to dynamic
performance of the vessel in waves. A recent study used time domain results from the
numerical simulation program, FREDYN, to determine a relative damage loss index (RDLI) by
evaluating dynamic damage stability performance of a monohull vessel in a particular damage
scenario and load condition. (Peters and Wing, 2009) Relationships between RDLI and quasi-
static stability criterion were found and the strongest correlations were highlighted.
Another potential area of instability for the trimaran is sloshing effects on the cross deck which
could lead to capsize with a significant sea state. The effect of entrained water on the decks of
RO/RO passenger ferries is a well-documented danger posing to these vessels where the
entrained water accelerates transversely amplifying any perturbation of waves beyond the
remaining restoring force of the vessel leading to capsize. (Ross et al, 2000) (Pawlowski, 1999)
(Egan et al, 1998) Similarly, given significant sinkage, the damaged and flooded trimaran with
9
reduced restoring force could be at risk for capsize if any entrained water in the cross deck
creates a sloshing effect.
Additional work has sought to determine the characteristics of roll motion for trimarans; with
emphasis placed on calculating the roll damping terms. Methods for determining the roll
damping and dynamic characteristics for trimarans have been suggested. (Zhang and Andrews,
1999) (Grafton, 2007) (Katayama et al, 2008) (Katayama et al, 2011)
1.4 Problem and Thesis Motivation Compared to monohulls there is a lack of knowledge concerning the assessment of both intact
and damaged stability of trimarans. The trimaran vessel is known to possess significantly
different stability qualities than an equivalent monohull; however, the effect of design changes
on the trimaran’s stability qualities is in it’s infancy for the intact vessel and non-existent for the
damaged vessel subject to flooding. The general perspective for many monohull forms has been
that a lower center of gravity and large metacentric height benefit ship stability, however
motions are often unfavorable for crew. For monohulls, in general, an increased beam increases
the transverse metacentric height. For a trimaran, the transverse separation of hulls is a more
important factor but the beam of each hull and distribution of displacement are important as
well. How do these different geometries affect the stability and what behavior does a damaged
trimaran, subject to flooding, exhibit?
1.5 Thesis Objective Trimaran vessels have only recently been examined for issues related to stability. The intact
trimaran vessel has been investigated for instability in longitudinal waves and for general
stability qualities encountered during early-stage design. Single naval trimaran ship
configurations have been evaluated for intact and damage stability. (Andrews, 1995) (Ordonez,
1995) However, to date no other research appears to have examined the stability qualities for a
series of damaged trimaran vessels subject to flooding to understand the influence of the
geometric configuration.
This thesis examines trimaran stability qualities in both intact and damaged conditions using a
series of trimaran hull forms with varying geometric configurations in a Design of Experiments
(DoE) approach. The relationships between geometry changes in the trimaran hull-form to intact
and damaged stability qualities are examined.
1.6 Thesis Outline In Chapter 2, current criteria and methods to assess intact and damaged stability are surveyed
and examined to clearly define the problem. A brief history of stability criteria is explained,
followed by a description of current deterministic stability criteria. Additionally, IMO regulations
which use probabilistic methods to assess stability are described for commercial vessels
including their recent application to naval combatants. Issues related to trimaran stability are
discussed, including transverse stability in longitudinal waves, the sloshing effect in the cross
deck and roll damping estimation methods.
10
In Chapter 3 an initial series of multi-hulls are generated from common center and side hull lines
along the continuum with different distributions of displacement between hulls. These multi-
hull variants are assessed for intact stability qualities including GMt, righting arm (GZ),
waterplane area, and heel of deck immersion. These multi-hull variants have the same overall
displacement, center of gravity, overall depth, and transverse separation between hulls to focus
on the effect of displacement distribution. The intact stability qualities are assessed and
compared at different heel and trim angles.
In Chapter 4 a series of 26 trimaran hulls is set up for analysis of stability qualities using a DoE
approach. A baseline frigate concept displacing 3,780 mt is considered, utilizing its hull form
configuration to investigate the influence of center hull length to beam (L/B)CH, vertical center of
gravity to depth (KG/D), 1x side hull to total displacement (Vratio), side hull longitudinal location
(SHX), and side hull transverse location (SHY) on ship stability for intact and damaged conditions.
Certain parameters like length at waterline (Lwl), depth to draft (D/T), longitudinal subdivision,
and damage extents remained constant for consistency.
A subdivided model is generated for each trimaran hull form variation and stability analysis is
run for various KG values to determine the allowable KG (KGa) for each intact and damaged
condition. This was done using the Rhinoceros plug-in Orca3D. The limiting KGa is determined
for each hull and the influences of (L/B)CH, Vratio, SHx, and SHy are examined. Using the main
effects, a response equation is developed to predict the KGa for non-dimensional hull form
parameters.
Chapter 5 summarizes observations made from quasi-static examination made in chapters 3 and
quantitate relationships formulated in chapter 4. Additionally, a way ahead for future work is
described including expanding the DoE of trimaran hulls and using probabilistic methods to
capture the effects of trimaran geometric parameters on intact and damaged transverse
stability.
11
2 Intact and Damage Stability: Methods and Issues In basic terms, the stability of a body is defined by its ability to return to a previously established
steady motion after being perturbed. For the case of a ship transiting through a seaway, ocean
and wind waves perturb the vessel. Often due to the complexity of ship motions in waves,
solving the fully dynamic problem in six degrees of freedom is a difficult and resource intensive
process. To simplify the problem a quasi-static approach is often utilized whereby particular
motions are examined for known areas of instability.
Monohulls have been thoroughly studied for several centuries. In that time, methods to assess
and evaluate the stability of monohulls have been developed in response to catastrophic events
such as storms, collisions, and other flooding events. Criteria measures used by the US Navy and
other organizations to assess surface ship stability are based on observations at sea and
numerical simulations for dynamic motions of primarily intact and damaged monohull vessels.
Naval trimaran ships, on the other hand, have only been built in the past two decades and in
drastically fewer numbers compared to monohulls. Thus, criteria specifically crafted to address
the stability issues for the trimaran hull form does not currently exist due to lack of experience
and knowledge.
Brown and Deybach (1998) provide a useful history of the evolution of monohull stability criteria
and methods of assessment. “Before World War II criteria were based primarily on GM, range of
stability and maximum righting arm [GZ].” Hazards surrounding World War II including battle
damage reports and natural disasters such as tropical typhoon Calhoun provided a basis for
extensive analysis. Ship characteristics were correlated with survival history to determine
effects. Additionally, insight into the evolution of damaged stability criteria and a survey of
criteria used by navies across the globe is gained from Surko (1994).
The first refined, significant work that formed the basis for standards assessing the stability of
the intact, flooded and damaged ships was performed shortly after World War II by Sarchin and
Goldberg. (1962) “The criteria established as guides for U.S. Naval ships are essentially empirical
in nature and result from war-damage experience, model and full-scale caisson explosion tests
and general operating experience.” External influences and hazards identified for the intact ship
include beam-winds combined with rolling, lifting of heavy weights, crowding of passengers to
one side, high speed turning, and topside icing. To the damaged ship, influences and hazards
include stranding involving moderate flooding, bow collision, and extensive flooding caused by
collision or enemy explosive action. Once flooded, external influences and hazards include beam
winds combined with rolling and progressive flooding. General practice for early-stage ship
design is focus on the beam wind combined with rolling hazard for both the intact and damaged,
flooded ship as it is typically the most severe and then assess the other hazards depending on
the ship’s mission and operating environment. This thesis will focus on the beam wind combined
with rolling hazard for the intact and damaged ship.
12
Stability criteria for commercial ships developed alongside their military counterparts using
similar qualities but with additional criteria on the ship subdivision. Stability criteria are
regulated by the International Maritime Organization (IMO) through the International
Convention for the Safety of Life at Sea (SOLAS). The first SOLAS was issued in 1914 in response
to the Titanic disaster. The 1948 SOLAS convention first established damaged stability
requirements and residual stability standards were included in the 1960 SOLAS convention.
(Vassalos et al, 2007) In the late 1960s Kurt Wendel (1968) introduced the probabilistic concept
for ship subdivision which quantified ship survivability in relation to collision scenarios and
provided an alternative to deterministic requirements. “Challenging the prescriptive practice, he
put on the table a method for optimizing subdivision without compromising the specification of
a potential damage.” (Spyrou, Roupas, 2007) Probability of damage (pi) is coupled to
consequence after damage (si) for each compartment to achieve the risk of collision, capsize and
other causes of instability expressed by the attained subdivision index, A. The attained
subdivision index, A, must be greater than the required index, R, typically determined by ship
size, shown in Equation 2. IMO regulations in the SOLAS using the probabilistic concept for ship
subdivision were first developed for passenger ships and then for cargo ships. More recently a
collaboration effort has harmonized the different regulations for passenger ships and cargo
ships.
RspA
iii Equation 2
Since probabilistic concepts for commercial ships have been standardized, researchers have
begun applying these concepts to naval combatants. Additionally, with the perspective of
numerical evaluations of intact and damaged dynamic stability, “there are serious concerns
about the limitation of the current semi-empirical deterministic criteria in which a combatant’s
damage stability is assessed upon.” (Boulougouris, 2016) Shortfalls of the quasi-static, semi-
empirical criteria include: (Surko, 1994)
• Capability of modern warships to survive damage from current threats, in demanding environmental conditions, is not known
• Modern hull forms and construction techniques differ greatly from the ships used to determine the criteria, and
• Assumption of moderate wind and sea conditions at the time of damage
These shortfalls don’t negate the importance of quasi-static methods, but warn of their
limitations and the necessity to use intact and damaged dynamic methods in concert. Table 2-1
summarizes different analysis methods, dynamic and quasi-static, criteria employed, and
information/insight gained; where quasi-static methods are less complex and have lower fidelity
compared to time-domain dynamic methods.
13
Table 2-1 Methods for Intact and Damaged Stability Analysis
Analysis Quasi-Static Dynamic
Criteria Semi-empirical, Deterministic Probabilistic
Methods Added Weight Time Domain (e.g. FREDYN,
TEMPEST, etc) Lost Buoyancy
Results
Hydrostatic equilibrium at each heel angle Righting arm, GZ compared to Heeling arm Righting energy approximated by area under righting arm curve
Ship motions in up to six degrees of freedom in regular and irregular waves
2.1 Methods and Standards for Quasi-Static Stability Assessment of
Naval Combatants in Intact and Damage Conditions The current NAVSEA technical publication (2016) outlines deterministic criteria to assess
stability for USN surface ships using a quasi-static approach which evaluates the righting arm,
GZ, as a function of heel angle. Figure 2-1 and Figure 2-2 (NAVSEA, 2016, p. 8-13) show example
righting arm analyses for an intact vessel and one under damage respectively, where the
stability when subject to an adverse roll heeling arm is assessed. Additionally, stability criteria
published by the UK MoD (2014), which comply with NATO Allied Naval Engineering Publication
(ANEP) 77 (2017), are examined for comparison with USN criteria.
Figure 2-1 Stability of Intact Vessel: Effect of Adverse Rolling
14
Figure 2-2 Stability of Damaged Vessel: Effect of Adverse Rolling
Table 2-2 lists the deterministic criteria measures currently published by the US Navy (NAVSEA,
2016, p. 8-1) and UK MoD (2014, p. B-3 - B-6) to assess the stability of intact vessels. The last
column summarizes recommendations for trimarans from Dubrovsky and Lyankhovitsky.
Criterion (a) specifies that the ratio of the GZ at equilibrium (Point C) to the GZ at maximum be
less than 0.6. Criterion (b) specifies that the ratio A1/A2 be greater than 1.4. The bounds of A1, a
measure of righting energy, start at equilibrium (point C) and end at the angle of unrestricted
downflooding or 70 deg., whichever is less. The bounds of A2 end at the equilibrium (point C)
and start 25 deg to the left of equilibrium, estimating rollback energy. Criterion (c) specifies that
the equilibrium heel angle (Point C) be less than 15 deg for the USN and less than 30 deg for the
UK. Criterion (d) specifies that the transverse metacentric height from the center of gravity be
greater than 1 ft (0.3 m). It should be noted that the USN requires criteria (c) and (d) solely for
overhaul, towing, and decommissioning. Also, the UK MoD uses a lower 90 kt nominal beam-
wind instead of 100 kt beam-wind specified by the USN.
Table 2-3 lists the deterministic criteria measures currently published by the US Navy (NAVSEA,
2016, p. 8-13 – 8-15) and UK MoD (2014, p. B-15 – B-18) to assess the stability of damaged
vessels. The last column summarizes recommendations for trimarans from Dubrovsky and
Lyankhovitsky (2001, 2004). Criterion (a) measures the angle of static heel after damage. USN
criterion specifies this angle be less than 15 degrees while UK MoD specifies the angle be less
than 20 deg. “Heel angle of a damaged multi-hull ship shall not exceed 10 deg. However, the
initial heel angle can be allowed up to 15 deg; provided that there are provisions for decreasing
it automatically to 10 deg.” (Multi-Hull Ships, Dubrovsky, p. 87)
As with intact criteria, criterion (b) dictates that the ratio, A1/A2, be greater than 1.4, shown in
Figure 2-2. However, for damaged conditions the bounds of area A1 start at the equilibrium heel
15
angle (point C) and end at the angle of unrestricted downflooding or 45 degrees whichever is
less. The bounds of area A2 end at the equilibrium heel angle (point C) and start at point C
minus the rollback angle determined by Equation 3. (NAVSEA, 2016, p. 8-27) Additionally,
criterion (e) specifies that area A1 must be greater than the polynomial function based on
displacement.
ΘR = C0Δ7 + C1Δ6 + C2Δ5 + C3Δ4 + C4Δ3 + C5Δ2 + C6Δ + C7. Equation 3 Criterion (c) states that the first GZ maxima or minima minus the HA (at the same angle) be
greater than 0.076 m (0.25 ft), and is used by the USN to identify periods of instability where GZ
is negative at small heel angles, known as loll. Criterion (d) defines a margin line below deck
edge which is not to be submerged under any damage condition. USN criterion specifies this line
be 0.076m (0.25ft) below downflood points or the deck edge. Others recommend, “The damage
waterline is to be at least 300mm below any opening in the hull, which can cause flooding.”
(Dubrovsky, 2001, p. 87) Also, UK MoD adds intact criterion (a) to apply to damaged vessels,
listed as damaged criterion (f).
Table 2-2 Criteria for Intact Vessel, Normal Operations
Measure US Navy
T9070-AF-DPC-010/079-1 UK MoD STAN 02-900 Part 4
Dubrovsky et al recommendations
(a) GZ (Point C) / GZ (max) < 0.6 < 0.6 < 0.6
(b) A1/A2 (25 deg rollback) > 1.4 > 1.4 > 1.4
(c) Equil. Heel Angle (Point C) < 15 deg* < 30 deg
(d) GMT > 0.3 m* > 0.3m
* applies to overhaul, towing, and decommissioning conditions only
Table 2-3 Criteria for Damaged Vessels
Measure US Navy
T9070-AF-DPC-010/079-1 UK MoD STAN 02-900 Part 4
Dubrovsky et al recommendations
(a) Static Heel Angle (Point D) a.k.a. Angle of List or Loll
< 15 deg < 20 deg < 10 deg or < 15 deg with remedy
(b) A1/A2 (rollback angle) > 1.4 > 1.4
(c) GZ(max) – HA(max) > 0.076 m
(d) Margin line unsubmerged (after 15 minutes of flooding at static heel angle)
0.076 m below deck edge or downflood points
0.3 m below any deck openings
(downflood points)
(e) A1 > aΔ7+bΔ6+cΔ5+dΔ4+eΔ3+fΔ2+gΔ+h 1000 < Δ < 5000 mt:
> aΔ+b
(f) GZ (Point C) / GZ (max) < 0.6
Assumptions for subdivision, extent of damage and weight conditions can vary from one
assessment of ship stability to the other. However, as long as consistency is maintained with a
level of realism for the assessment, most assumptions are valid. Table 2-4 lists these
assumptions published by USN, UK MoD, and those recommended by Dubrovsky. As noted
16
previously, a damaged length equal to 30% of the ship’s length is very conservative; the USN and
UK MoD published standard (15% of the ship’s length) will be used. Three transverse extents will
be examined including side hull only, up to center hull centerline, and full extent. Vertical
damage extents, weight conditions and permeability for each compartment will be assigned
based on the design concept under evaluation.
Table 2-4 Assumptions for Damaged Stability Assessments
Damage Condition US Navy
T9070-AF-DPC-010/079-1 UK MoD STAN 02-900
Part 4 Dubrovsky et al
recommendations
Beam-Wind Speed Δ < 40000 LT:
VWind = C0 + C1Δ + C2Δ2 + C3Δ3 1000 < Δ < 5000 mt:
VWind = a*ln(Δ) - b
Length of Damage Extent (for combatants)
LBP > 300’: 15% of LBP Lwl > 92m: greater of
15% of Lwl or 21m 30% of LOA
Vertical Damage Extent
Up to DC/Bulkhead Deck (with and without inner
bottom)
Transverse Damage Extent
(1) Full extent (2) Up to but not including
any CL bkhd
(3) Side hull fully
flooded
Weight Conditions (1) Pre-ballast
(2) Ballast (3) De-ballast
Permeability: Watertight Voids Accommodation
Machinery Stores, etc.
0.95 0.95 0.85
0.6 – 0.95
0.97 0.95 0.85
0.8 – 0.95
0.6 – 1.0
2.2 Probabilistic Methods to Assess Stability of Damaged Naval
Combatants The sole criterion used for probabilistic methods resides in the utilization of Equation 2 as the
summation of probable catastrophic events leading to ship loss taking into account possible
damage conditions in the wave environments specified in the operational profile. Formulations
presented herein for probabilistic assessment of naval combatants stems from research and
case studies by Boulougouris et al. (2013, 2016) Survivability is defined for a naval combatant as,
“the capability of a ship and its ship board system to avoid and withstand a weapons effects
environment without sustaining impairment of their ability to accomplish designated missions”
(Said, 1995) Survivability can be expressed as the product of susceptibility, the inability to avoid
being hit (PH), vulnerability, the inability to withstand damage when hit by threats(PK|H), and
recoverability, the ability to prevent loss and restore mission function after damage.
Susceptibility depends on the ship signature, countermeasures, and self-protection armament.
Vulnerability depends on ship design features including size, compartmentation, structural
detailing, and shock hardening as well as separation and redundancy of vital equipment.
17
Recovery depends on damage control protocols and equipment on board as well as crew
training and readiness. (Said, 1995) Ignoring recoverability, survivability can be quantified thus:
PS = 1 - ( PH - PK|H ). (Ball and Calvano, 1994) This definition of survivability forms the basis of the
formulations for the probability index, pi, and survival index, si, whose sum of products
determines the attained subdivision index, A.
The probability index, pi, is the probability that a compartment or group of compartments may
be flooded from a damage event such as collision, grounding, or weapon impact. For early-stage
design a simple mathematical distribution longitudinally can be assumed where pi is a joint
distribution considering the weapon impact location and damage length. A piecewise linear
impact distribution has been proposed for air-to-surface missiles with maximum probability at
mid-ships (Boulougouris and Papnikolaou, 2004). Similar linear impact distributions have been
proposed for contact mines (Harmsen and Krikke, 2000). The damage length distribution uses
the Damage Function concept used in theory from Defense Analysis. (Przemieniecki, 1994) A
log-normal distribution is considered most appropriate as a function of assumed damage and
flooded lengths and spacing between compartments.
Two approaches have been used to determine the survival index, si, which is the probability of
non-exceedance of roll events with respect to a critical roll angle determined to capsize the
vessel in a particular damage condition. The first approach adjusts quasi-static estimates by the
established semi-empirical, deterministic criteria for damaged vessels. Another approach uses a
time domain code, such as FREDYN, with flooding capability to capture capsize events and
determine their probability for each given damaged condition. (MARIN, 2017)
2.3 Dynamic Intact and Damaged Stability and Trimaran Seakeeping In recent years, the ability to solve the fully dynamic problem has increased in efficiency due to
advances in computing and theories of motion making these assessments more common. This
has led to the development of a next generation of intact stability criteria and assessment
where the areas of instability including a pure loss of stability, parametric roll and surf-
riding/broaching are identified and the probability of these events occurring in a particular sea
state and region is determined using a wave scatter diagram.
Dynamic stability for ships in the damaged condition was not addressed until the early 1990s
with simplified models. “The subject of [damaged] dynamic ship stability in waves with the hull
breached received much attention following the tragic accident of Estonia, to the extent that
lead to a step change in the way damage stability is being addressed, namely by assessing the
performance of a vessel in a given environment and loading condition on the basis of first
principles.” (Vassalos et al, 2009) Known modes in which a damaged ship may capsize include:
high freeboard ships with damage leeward, low freeboard Ro-Ro and conventional ships, and
multi free-surface effect where flooded water spreads through complex non-watertight
subdivision. More recently models, both numerically and experimentally, have been developed
to analyze dynamic performance of damaged monohulls with increasing fidelity. Results have
18
been linked to that of quasi-static stability criteria to determine their ability to assess dynamic
events for the damaged ship. (Peters, Wing, 2009)
2.3.1 Transverse Stability of Trimarans in Longitudinal Waves
Quasi-static assessments of an intact trimaran with the side hull positioned in the trough of the
wave reveal a potentially dangerous condition due to parametric roll. Typically, GZ fluctuates
around 0 (unstable) up to 5-10 deg of heel then goes positive. Further analysis using potential
flow theory, higher fidelity RANS CFD codes, and model tests would likely provide further
clarification if the parametric roll phenomenon is present. Analysis for parametric roll in
longitudinal waves, beyond quasi-static, will not be included for this thesis but it is an important
consideration for stability of trimaran vessels.
Experiments at University of Trieste, Italy have evaluated some trimaran configurations in
several regular wave types and speeds in head and following seas. They used a susceptibility
index measuring the change in metacentric height, normalized by the wave amplitude (δGM/ζa).
“The experimental results indicate extreme sensitivity of this type of trimaran to the danger
connected with the onset of parametric rolling." (Bulian et al, 2011). The side hull Unina1 was
the focus of the paper with a Vratio of 1% and has a higher susceptibility index (δGM/ζa) than
side hull Dinav1 with a Vratio of 10%. This effect is seen in Figure 2-3 and indicates that the
danger of parametric roll is more likely in smaller side hulls. Additionally, Figure 2-4 shows roll
amplitudes are highest at speed in following seas and lowest at speed in head seas with steeper
waves causing increased roll amplitude. The waves used had λ/Lpp = 0.5 and different
steepnesses, sW. Negative speed values indicate the following waves.
Figure 2-3 Susceptibility Index to Longitudinal Waves of Different ship typologies compared to
the Trimaran Unina1
19
Figure 2-4 Stationary Parametric Roll Amplitude for Unina1 in configuration S1L4.
2.3.2 Estimation of Roll Damping for Trimarans
Component based models have been employed since the 1970s to estimate roll damping, in
general, of ships. Most modern potential flow codes use a formulation derived from Ikeda et al
(1978) for ships at forward speed. These methods focused primarily on displacement and semi-
displacement monohulls and only in recent decades have changes have been made to Ikeda’s
method to include high-speed planing craft (Ikeda & Katayama, 2000) and high speed multi-hulls
vessels. (Katayama et al, 2008) Most have examined wave-making, eddy, and lift components of
roll damping. More recently, the zero-speed roll damping formulation is developed and
contrasted with the forward speed formulation for multi-hulls. (Katayama et al, 2011)
A PhD thesis by Grafton [2007] from UCL has substantial coverage and overview of roll damping
for trimarans. "Doctors and Scrace [2004] allowed for the extra roll damping not considered by
the potential theory to comprise of two components, skin friction damping and appendage lift
roll damping." (Pt 1, pg 187) "Comparison of this theory with model experiment results for RV
Triton showed that the frictional damping component was very small and could almost be
ignored. Hence all of the additional damping (not computed by potential theory) was attributed
to the appendages(in side hull) ." (Pt 1, pg 188)
"To date, the most comprehensive study of trimaran roll motion has been by Zhang and
Andrews (1999). Following the well-established monohull investigative approach, they predicted
the motions in all six degrees of freedom using a frequency domain Potential Flow method using
the Green Function approach and either augmented the roll damping term with a suitable range
of existing empirically based theoretical formulae, or they used roll damping coefficients
measured in roll decay experiments." (Pt 1, Pg 189)
20
Additionally, the roll motion in regular beam waves were studied for models of two catamarans
and three-Wigley trimarans at University of Trieste, Italy. (Francescutto, 2001) Simplified
mathematical models for roll motion description were developed in both cases and
experimental data was used to validate numerical results.
2.3.3 Sloshing Effect in Cross Deck: Case of RO/RO Passenger Ferries
Another concern for a trimaran is flooding in the cross decks and how the trimaran will behave if
subjected to motions, most notably rolling. Any water that is entrained in the cross decks, unless
there is sufficient watertight subdivision will likely increase motions in rough seas due to
sloshing effects. This effect is seen and has been thoroughly examined in RO/RO ferry designs
where car decks, often below the waterline, span the entire breadth of the ship with minimal
subdivision. In such cases, any water inflow will cause the vessel to capsize due to sinkage and
increased motions from sloshing. Investigations were performed on a RO/RO ferry model of the
Herald of Free Enterprise in which different subdivisions were tested to improve damage
stability at the expense of some efficiency. The capsize times of models with modified
subdivision were increased by 3 to 5 times that of the original vessel. (Ross et al, 2000)
In addition to the tradegy of the Herald of Free Enterprise, the subsequent tragedies of RO/RO
ferries Scandenavian Star, European Gateway, and Estonia stimulated additional SOLAS
regulation. This regulation influenced designers to provide guidance to achieve efficient, yet
compliant RO/RO passenger vessels. (Pawlowski, 1999) These efforts have enabled shipbuilders
to construct safer RO/RO passenger vessels like the MV Kennicott employed by the State of
Alaska. (Egan et al, 1998)
2.3.4 Experimental and Numerical Modeling of Dynamic Stability for Damaged
Monohulls
In recent years, an increase in fidelity of non-linear, time-domain solutions coupled with the
utilization of high performance computing (HPC) resources have enabled sophisticated
numerical analysis to be employed for modeling of dynamic stability events. With this numerical
capability comes a need to validate each method using experimental models with
compartments open to the sea and waves. A recent paper (Riola et al, 2017) highlights monohull
model testing, subject to compartment flooding, used to verify case studies performed by the
Spanish Navy to assess dynamic rolling, parametric excitation, resonant excitation, impact
excitation, transient flooding, broaching, and survivability tests for the future F-110 frigate class.
Another recent study used the numerical simulation program FREDYN to estimate the
probability of loss for a monohull frigate under specific damaged scenarios and wave conditions
using a relative damage loss index (RDLI). These results were correlated with quasi-static
stability criteria under damage. (Peters, Wing, 2009) Table 2-5 shows their correlation results.
They followed stability criteria published in UK MoD [2014] which is very similar to USN
published standards. Their data suggests that A1/A2 is the best measure of stability under
damage, followed by GZ (Point C) / GZ (max), then area A1. The damage list angle criterion
showed poor correlation and GMt showed none. It was noted that all but one of the scenarios
21
evaluated had a GZ (Point C) / GZ (max) < 0.6, “this suggests that this criterion is not particularly
good in these cases if used in isolation.” (Peters, Wing, 2009)
Table 2-5 Relationship of RDLI with Quasi-Static Stability Criteria for Monohulls
RDLI v. Relationship Fit R2 RDLI at Criteria value
Damage List Angle Power 0.71 28% at 20 deg
Area A1 Logarithmic 0.85 28% at 0.02 m-rads
GZ (Point C) / GZ (max) Linear 0.93 62% at 0.6
A1/A2 (15 deg rollback) Exponential 0.99 12% at 1.4
GMt None
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3 Multi-Hull Intact Stability Comparison In order to examine the stability advantages of trimarans compared to other multi-hull vessels
from a quasi-static viewpoint in heel, a series of intact hull forms were generated along the
multi-hull continuum. Starting from a parent center and side hull, geometrical similar scaling
(geosim) of the principal dimensions length, beam, and draft at the waterline was used to vary
the distribution of displacement between the center and side hulls while maintaining a total
displacement in saltwater of 3,500 mt and an initial waterplane (WP) area of 1,100 m2.
Additionally, all multi-hulls have the same WDC at 3.0 m above the waterline, FB at 6.6 m, and
SHY distance of 13.5 m. The floating plane hydrostatic characteristics were calculated at the
design displacement and center of gravity where LCG = LCB for level trim, TCG = 0 on centerline,
and VCG = 0 at the design draft line. Figure 3-1 shows the body view of the hull forms generated
along the multi-hull continuum. Table 3-1 lists the principal characteristics and hydrostatic
properties for each multi-hull. Additionally, parameters including side to center hull length ratio,
Lratio, length to beam ratio, L/B, and beam to draft ratio, B/T are listed in Table 3-1 to inform
the length beam and draft changes made to vary the displacement distribution. The origin is
placed at the forward perpendicular, on centerline, and at the design waterline.
Figure 3-1 Body Plan Comparison across Generated Multi-Hull Continuum
23
Table 3-1 Hydrostatics Comparison across Generated Multi-Hull Continuum
Monohull Trimaran Catamaran
SH-CH-SH Displacement Distribution (% of total)
0-100-0 2.5-95-2.5 5-90-5 12.5-75-12.5 25-50-25 37.5-25-37.5 50-0-50
Lwl [m] 120.1 118.1 116.0 109.2 95.2 116.6 134.6
Bwl [m] 11.4 28.8 29.2 30.1 30.8 31.6 33.1
T [m] 5.2 5.5 5.5 5.3 6.0 6.1 5.3
(L/B)CH 10.5 11.4 11.7 11.9 11.8 12.0
(B/T)CH 2.2 1.9 1.8 1.7 1.6 1.7
Lratio 0.40 0.52 0.66 1.00 1.54
(L/B)SH 26.9 28.2 23.9 25.5 25.6 22.3
(B/T)SH 0.68 0.64 0.61 0.62 0.75 1.14
Disp. Vol [m3] 3,412 3,412 3,412 3,412 3,412 3,412 3,412
Disp. Mass [mt] 3,500 3,500 3,500 3,500 3,500 3,500 3,500
Swet [m2] 1,680 2,136 2,402 2,858 3,427 3,631 3,105
SHX - 0 0 0 0 0 -
SHY - 13.5 13.5 13.5 13.5 13.5 13.5
LCB [m] 64.4 64.2 64.1 63.6 63.0 62.4 62.0
VCB [m] -1.9 -2.0 -2.0 -1.9 -1.9 -2.0 -1.9
WP Area [m2] 1,100 1,100 1,100 1,100 1,100 1,100 1,100
LCF [m] 71.0 69.9 69.3 68.0 66.0 64.0 62.0
BMt [m] 2.8 8.1 11.1 17.1 26.6 38.7 59.5
BMl [m] 308.3 273.7 254.6 213.0 177.3 187.8 292.3
KG [m] 5.2 5.5 5.5 5.3 6.0 6.1 5.3
GMt [m] 0.9 6.1 9.1 15.2 24.7 36.8 57.6
3.1 Quasi-Static Intact Stability Assessment in Heel The change in WP area from level (1100m2 all designs) to 55 degrees heel for each multi-hull is
plotted in Figure 3-2. The WP area plots show how the trimaran vessels transition from the
monohull trends to the catamaran trends for increasing heel angles. These results and trends
are similar to those found by Dubrovsky and Lyakhovitsky for monohulls, catamarans, and Tri-
SWATH vessels. (Multi-hull ships, 2001, Fig. 2.8) Table 3-2 shows the progression of motion
phenomena as different multi-hulls heel. The key developments are pictured including when the
side hull exits, the cross deck enters and the deck edge submerges. Note that superstructure
was assumed non-watertight and thus downflooding would occur at the deck edge, any
watertight superstructure would positively impact the waterplane area right arm at large heel
angles.
Similarly, the righting arm, GZ, is plotted in Figure 3-3 showing the positive range of stability for
each multi-hull. The monohull has the greatest range of positive stability and the smallest
maximum GZ; while the catamaran has the smallest range of positive stability and the largest
24
maximum GZ. The trimaran falls in-between where larger side hulls exhibit behavior like the
catamaran and smaller side hulls exhibit behavior like the monohull. It is prudent to more
closely examine the GZ range up to the angle of downflooding. This angle typically has an upper
limit, US published criteria specifies 70 deg. (NAVSEA, 2016) For this limit drawn in Figure 3-3. At
70 degrees of heel trimarans with relatively small sides hulls, at 2.5% and 5% each, have
advantages of larger GZ and A1 area over a monohull with a higher angle of vanishing stability.
Stability qualities including deck edge submersion, area A1, maximum righting arm, and heel
angle of maximum righting arm are listed for each multi-hull in Table 3-3. A strong trend is
exhibited between side hull displacement and area, A1, bounded by the deck edge submersion
and angle of vanishing stability. Additionally, a strong trend is found between side hull
displacement and maximum righting arm, GZ, as shown in Figure 3-4 and Figure 3-5. For the
monohull, the deck edge submerged at large heel angles around 50 degrees; in contrast to the
trimarans which submerge at 30 degrees of heel. This effect is shown in Figure 3-6.
“Emergence of outrigger from the water constitutes a dramatic drop in stability due to the total
loss of waterplane area at a side.” (Ships with Outriggers, Dubrovsky, p. 31) This behavior is
most significant in larger side hulls where the side hull opposite from the angle of heel exits the
water around 10 degrees, at small heel angles. For large heel angles, larger than 10 degrees of
heel until 15-20 degrees, the cross deck enters the water on the heeled side increasing the WP
area. After 20 degrees of heel, the trimarans with a larger center hull follow the monohull
behavior with increasing WP area, while the trimarans with a larger side hull follow the
catamaran behavior with decreasing WP area. This effect is best seen in Figure 3-2 and Figure
3-3.
Figure 3-2 Change in WP Area Comparing Multi-Hulls at Heel Angle
25
Table 3-2 Significant Multi-Hull Events and Corresponding Heel Angles
Level Heel (0 deg) Side Hull Exits Cross Deck Enters Deck Edge Submergence
Monohull
2.5-95-2.5 Trimaran
12.5-75-12.5
Trimaran
Catamaran
10 deg 15 deg 29 deg
47 deg
17 deg
50 deg
15 deg 19 deg 29 deg
26
Figure 3-3 Changes in Righting Arm, GZ Comparing Multi-Hulls at Heel Angle
Table 3-3 Stability Criteria, Comparing Multi-Hulls Monohull Trimaran Catamaran
SH-CH-SH Displacement Distribution (% of total)
0-100-0 2.5-95-2.5 5-90-5 12.5-75-12.5 25-50-25 37.5-25-37.5 50-0-50
Vratio 0 2.5% 5% 12.5% 25% 37.5% 50%
Deck Edge Submersion [deg.]
45 25 25 25 25 30 50
Range of Positive Stability [deg]
155 122 116 108 100 96 98
A1 Area [m-deg.] 30 35 50 86 142 258 524
A1 Area (Range of Positive Stability) [m-deg]
200 353 420 526 629 671 719
Maximum GZ [m] 2.4 4.6 5.8 7.7 10.3 11.6 12.8
Heel Angle of Max RA [deg.]
80 55 52.5 50 45 30 20
Upper Bound -
Downflooding
27
Figure 3-4 Correlation between Area A1 and Side Hull Displacement
Figure 3-5 Correlation between Maximum Righting Arm and Side Hull Displacement
28
Figure 3-6 Deck Edge Height above Waterline, Comparing Multi-Hulls at Heel Angle
3.2 Quasi-Static Intact Stability Assessment with Trim The stability and dynamic motion of an intact ship can be further assessed in a quasi-static
manner by a thorough examination of the hydrostatic properties at trim and heel angles. Using
basic static methods some results found in more advanced hydrodynamic simulations can be
achieved. This method is practical for use in early-stage ship design projects (Belenky and
Bassler, 2010) and is employed here to analyze the monohull and 2.5% - 95% - 2.5% trimaran
discussed previously, shown in Figure 3-7. Additionally, the 2.5% - 95% - 2.5% trimaran with side
hull in an aft longitudinal position was examined in contrast to the mid-ship position examined
previously. The hydrostatic properties of interest are: WP and wetted surface area, center of
buoyancy, center of floatation, transverse metacentric height, and righting arm. Heel angles up
to 55 degrees are tracked at level trim and at 2.5 and 5 degrees of trim in both directions. With
the origin at the forward perpendicular, x is positive aft, y is positive to starboard, and z is
positive up. A trim angle with bow down is negative and bow up is positive.
29
Figure 3-7 Isometric View of 2.5-95-2.5 Trimaran (Mid and Aft SH Position) and Monohull
First, the effect of trim angle on the WP and wetted surface area is examined at level heel. The
monohull and 2.5% - 95% - 2.5% trimaran with mid SH positions exhibit similar WP area changes
with trim angle as shown in Figure 3-8; however, the trimaran with aft SH position gains WP
area in 2.5 deg bow up trim. The wetted surface area trend is similar for monohull and both
trimaran SH positions. Changes in longitudinal center of buoyancy and floatation are consistent
between the trimaran and monohull as shown in Figure 3-9. Additionally, the longitudinal and
transverse metacentric height (GML and GMT) at level heel are examined, shown in Figure 3-10.
The monohull and trimarans show a consistent trend with trim angle for GML; where the
trimaran with aft SH position has the largest GML and the monohull has the least overall. The
monohull and trimaran with mid SH position have a little change in GMT with trim angle
compared to the trimaran with aft SH position which exhibits a substantial, non-linear trend
with trim angle. This is because the side hull’s waterplane area is substantially reduced for bow
down trim angle equal to that of the monohull at 5 degrees bow down trim. This suggests that
the trim angle will have a greater effect on the transverse stability of the trimaran with aft SH
position.
Monohull Mid SH Position 2.5-95-2.5 Trimaran Aft SH Position
Figure 3-8 Percent Difference in Wetted Surface and WP Area at Trim Angle, Level Heel
x
y
z
30
Monohull Mid SH Position 2.5-95-2.5 Trimaran Aft SH Position
Figure 3-9 Change in LCB and LCF with Trim Angle, Level Heel
Figure 3-10 Change in Longitudinal and Transverse Metacentric Height with Trim Angle, Level
Heel
Next, the effect of heel angle on the righting arm, GZ, and WP area is examined at different trim
angles. The GZ curve is plotted against heel angle up to point of vanishing stability in Figure
3-11. These plots further show how the trim angle effects the trimaran with aft SH position
more than the monohull and trimaran with mid SH position. Also note that both trimaran
positions have a substantially higher maximum GZ then compared to the monohull. For both the
monohull and the trimarans, the WP area changes are consistent in trend where an increase at
small heel angles is followed by a decrease and then another increase a heel angle up to the
angle of vanishing stability. Some trim angle had some additional oscillations but followed the
trend overall as shown in Figure 3-12. The change in trim angle at heel angles up to 55 degrees is
plotted in Figure 3-13. These results confirm the possibility that the side hull longitudinal
position, SHX, has an impact on transverse stability.
31
Monohull Mid SH Position 2.5-95-2.5 Trimaran Aft SH Position
Figure 3-11Righting Arm, GZ, with Change in Heel and Trim Angle
Monohull Mid SH Position 2.5-95-2.5 Trimaran Aft SH Position
Figure 3-12 Change in WP Area with Heel Angle
Monohull Mid SH Position 2.5-95-2.5 Trimaran Aft SH Position
Figure 3-13 Change in Trim Angle with Change in Heel Angle
32
In addition to change in WP area, the change in center of floatation for the trimaran was
different than for the monohull. Figure 3-14 shows the longitudinal center of floatation (LCF)
and transverse center of floatation (TCF). Firstly, the trimarans exhibit greater change in TCF
than the monohull. Secondly, the LCF moves towards midships for all trim angles on the
monohull and for the trimaran with aft SH position, where the LCF moves forward for bow down
trim angles and aft for bow up trim angles for the trimaran with mid SH position. This indicates
that the stability may be more of a concern for the trimaran with a trim at large heel angle
greater than 30 degrees more so than for a monohull.
Figure 3-14 LCF and TCF with Change in Heel and Trim Angle (Monohull top, Trimaran Mid SH Pos., Aft SH Pos. bottom)
3.3 Intact Stability Conclusions The quasi-static examinations and intact stability analysis for multi-hulls along the continuum
provided useful insights that will guide the next chapters of this thesis. First the side hull
displacement range, Vratio from 2% to 5%, identified by Andrews (2004a) is confirmed by Figure
3-3 where the angle of vanishing stability is high exhibiting monohull behavior and the
maximum righting arm, GZ, is relatively high exhibiting catamaran behavior. Secondly, the effect
of side hull longitudinal position is observed to have an effect on intact transverse stability at all
trim angles and should be considered.
33
4 Trimaran Quasi-Static Stability Study As concluded in Chapters 2 and 3, other geometric factors affecting the intact and damaged
transverse stability of the trimaran vessel in addition to the side hull displacement ratio, Vratio,
include the length to beam ratio, L/B, and side hull longitudinal and transverse position, SHX and
SHY. These effects are more comprehensively examined for trimarans with side hulls between
2% and 5% of the total displacement which have higher vanishing angles of stability than a
catamaran and higher maximum righting arms than monohulls as concluded in Chapter 3.
4.1 Study Overview A design of experiments (DoE) was constructed, based on a parent design concept, varying the
trimaran hull form configuration. The bounds and variables of the DoE was informed by
conclusions derived from the literature review and results of intact stability investigations in
Chapter 3. While the focus of this study is to examine the transverse stability of the damaged
vessel, the intact vessel is also examined for additional understanding. The objective of the DoE
is to observe trends among variables examined and use the data to develop a response surface
model (RSM) to quantify the stability relationships observed. The application of the stability
RSM includes use as rule-of thumb and constraint for early-stage design of trimaran ships.
4.2 Assumptions A quasi-static, analysis method was employed as the least resource intensive for adequate
results to inform a rule-of-thumb method. Probabilistic methods are not standard practice yet
for naval ships and are more resource-intensive requiring many more assumptions which are
subject for debate. As such, probabilistic methods for stability analysis were not considered. For
the quasi-static method, the hydrostatic properties were determined for discrete heel angles
and the stability qualities were assessed numerically. For damaged conditions, the ship was
balanced to determine hydrostatic equilibrium. This can be done by treating the flooded water
as an added-weight or the compartment volume as lost buoyancy. A lost buoyancy approach,
free to sink and trim as a function of heel angle was employed as specified by NAVSEA for
flooding that is in free communication with the sea. (NAVSEA, 2016) Additionally, this approach
allows the hydrostatic equilibrium and righting arm at heel angles to be computed in a single
evaluation for the specified weight and center of gravity. The subdivided, trimaran hull form was
modeled as closed polysurface compartments in Rhinoceros. Then using the Orca3D plug-in,
stability qualities and criteria were evaluated. “Orca3D is a suite of tools, written as a plug-in for
Rhinoceros, providing powerful naval architectural design and analysis capabilities that are easy
to learn and run in a powerful 3D CAD environment.” (DRS, 2017)
4.2.1 Ship Design Concepts
A multi-mission frigate concept previously developed by Virginia Tech students under the
direction of Dr. Alan Brown, was used as baseline for the series of trimaran hulls. (Manzitti et al,
2016) The principal characteristics are listed in Table 4-1, hydrostatics in Table 4-2, non-
34
dimensional parameters in Table 4-3 and propulsion and mission systems in Table 4-4 and Table
4-5 respectively. The design concept is pictured in Figure 4-1. (Manzitti et al, 2016)
Table 4-1 Design Concept Principal Characteristics
Length Overall, LOA [m] 150 Structures Weight [mt] 1,311
Beam Overall, BOA [m] 24.9 Light Ship Weight [mt] 2,979
Depth Overall, D [m] 11.6 MinOp Weight [mt] 3,374
Wet Deck Clearance [m] 3.7 Vertical Center of Gravity, KG [m] 6.3
Cross Deck Height [m] 3.0 Full Load Weight [mt] 3,780
Hull Internal Volume Avl. [m3] 18,576 Vertical Center of Gravity, KG [m] 5.9
Deckhouse Volume Avl. [m3] 2,500 Design Displacement [mt] 3,780
Total Internal Volume Req. [m3] 13,087
Total Internal Volume Avl. [m3] 21,076
Table 4-2 Design Concept Hydrostatics
Center Hull 1x Side Hull
Length Waterline, Lwl [m] 143.0 64
Beam Waterline, Bwl [m] 10.1 1.9
Draft, T [m] 4.9 3.0
Displaced Volume [m3] 3,390 147
WP Area, Aw [m2] 1,208 81
SHX [m] 10.7
SHY [m] 11.5
Table 4-3 Design Concept Non-Dimensional Parameters
CH LtoB 14.2
CH BtoT 2.1
Lratio 0.45
Bratio 0.19
Tratio 0.61
Vratio (1x SH / Total) 0.04
SHX / Lwl 0.073
SHY / Lwl 0.080
KG / D 0.51
DtoT 2.4
Table 4-4 Design Concept Propulsion and Electrical Plant
Propulsion Plant Hybrid Electric:1 x LM 2500+ GT, 2x CAT C280-16
DE, 2x Perm Mag Motors, 3x propellers
Total Propulsion Power [kW] 36,660
Electrical Plant Gas Turbines:2x Allison 501K
Maximum Functional Load 10,850
Endurance Range [nm] 4,066 @ 21 kt
Sustained Speed [kt] 30
No. of MMRs 1
No. of AMRs 2
35
Table 4-5 Design Concept Systems
Frigate
Hull & Deckhouse Material Steel
Personnel 110
AAW Systems Sea Giraffe 3D radar, ICMS, 1x CIWS/SEARAM, ES-3601ESM, 2XSKWS
DECOY LAUNCHER, IFF
ASUW Systems AN/SPS-73 RADAR, 57 mm gun, SeaStar EO/IR, 4x50 cal Machine
Guns, SMALL ARMS and Pyro Locker, 2x7m RHIB, 8 Hellfire ASM VLS
ASW Systems 2xMK 32 SVTT, AN/SLQ-25 NIXIE, AN/SWQQ-28 LAMPS MK LLL
Sonobuoy Process System, NDS 3070 Vanguard Mine Avoidance Sonar
CCC Systems ExComm Level B, Cooperative Engagement Capability (CEC) and Link
11, Navigation System
AIR Complement & Facilities Embarked 2xLAMPS w/ Hangar
Figure 4-1 Isometric and Profile View of the Frigate Design Concepts
Recent capability developed to analyze righting arm, GZ curves in Orca3D was used. The USN
published criteria listed in Table 2-2 and Table 2-3 for intact and damaged vessels was evaluated
against the righting arm GZ calculated by Orca3D at specified heel angles. Box volumes
representing engines (equivalent to 15% of each MMR’s volume at the centroid) were modeled
to account for assumed 85% permeability in machinery rooms at bulkheads 32, 62, and 92.
Similarly, missile like components (equivalent to 15% of launcher’s volume at the centroid) were
added to account for assumed 85% permeability at bulkhead 20. 100% permeability was
assumed for all other spaces. These components are shown in Figure 4-2.
The LOA and Lwl for each design were kept constant to maintain transverse bulkheads and
longitudinal subdivision. Center hull Bwl was determined by the specified L/B ratio. The center
hull draft, T, was adjusted to achieve the center hull displacement distribution, 1-2*Vratio.
Similarly, each side hull maintained constant Lwl, while Bwl and T were adjusted uniformly to
achieve displacement distribution, Vratio. Each hull variation had the same subdivision,
arrangement, and downflood points (X, Y) located on the main/weather deck. These constant
values are listed in Table 4-7. The total displacement was changed proportionally with an
increase in cross structure volume. The percent increase of cross structure was calculated from
36
Hull 1 and applied to structures weight (1,311 mt). This weight was added to the design
displacement and used for stability analysis.
138 130 122 112 102 92 78 62 48 32 20 10
Figure 4-2 Engine and VLS components in 85% Permeable Spaces
4.2.2 Design Variables and Parameters
Certain ratios between hull dimensions and properties have a noticeable effect on stability
qualities, as noted by Dubrovsky and Andrews. Most notably the KG/D, D/T, L/B, Vratio, and the
position of the side hull transversely, SHY is examined. Additionally, the longitudinal position
(SHX) was examined for its effect on trim and list angle under damage. The design variables (DVs)
and values used in the design of experiments (DoE) are listed in Table 4-6. To maintain
consistency, the D/T ratio was fixed for all designs. All fixed values, design parameters (DPs), are
listed in Table 4-7.The KGa/D ratio was determined iteratively by determining the maximum KG
which fulfills all stability criteria.
Table 4-6 Design Variable Values used for Design of Experiments (DoE)
Design Variable, DV Values
Center Hull Length to Beam Ratio, (L/B)CH 12.7 14.2 15.6
SH Vratio (1x SH displacement, relative to total) 0.02 0.03 0.04
SHX / Lwl (from Midships) 0.073 (L2) 0.276 (L1)
SHY / Lwl (from Centerline) 0.063 (T1) 0.080 (T2) 0.098 (T3)
Table 4-7 Design Parameters used for DoE – Constrained for All Designs
Design Parameter, DP Value
Center Hull Length Waterline, Lwl 143 m
Side Hull Length Waterline, Lwl 64 m
Depth to Draft Ratio, D/T 2.4
Downflood Points (X, Y) @ Depth
Bow Fwd Edge
Fwd Cross Deck Edge Mid Cross Deck Edge Aft Cross Deck Edge
Aft Deck Edge Transom
[-7, 0] [20, 4.2] [48, 8.5] [72, 8.5]
[112, 8.5] [130, 5.8] [143, 0]
Longitudinal Damage Extents 21.5 m (15% Lwl)
Vertical Damage Extents Keel to Bottom of Cross Deck Structure
Transverse Damage Extents (1) Side Hull to Center Hull Centerline
(2) SH Only
37
4.3 Intact and Damaged Stability Model The method of analysis was to create a DoE and evaluate a subdivided model using Rhinoceros,
and Orca3D. Using this method trends were observed and relationships between DVs and
stability metrics were developed, noting key designs with excellent stability. The deckhouse was
assumed to be a non-watertight space and was not modeled for intact or damaged stability
analyses.
Significant effort was spent to automate the DoE process using ModelCenter to process tasks
and data between a macro-based excel sheet VTPAM, Rhinoceros, and Orca3D plug-in.
Unfortunately, the commands necessary to generate the subdivided hull model in an automated
fashion for damaged stability analysis was deemed infeasible to accomplish this thesis research
in a reasonable time-frame. Additional effort beyond the scope of this thesis research will be
required to have an automated trimaran damaged stability analysis capability.
In order to complete the research, a manual DoE was undertaken utilizing the same process to
examine the principal variable changes with some coupled hull form variations. Various criteria,
listed in Table 2-2 and Table 2-3, were used to assess the resulting stability. The KGa / D ratio
was calculated for adequate intact and damaged stability of the trimaran vessel at the
displacement examined. Beam-wind heeling arm amplitude was determined using a 2m strip
method to integrate the wind pressure on the hull and deckhouse above the waterline as shown
in Figure 4-3. The longitudinal location of the side hull has minimal affect the beam-wind heeling
arm amplitude. A 15% Lwl longitudinal damaged length of 21.5 m is used to determine the
damaged conditions listed in Table 4-8. The two asymmetric, transverse extents examined are
pictured in Figure 4-4. The ten longitudinal extents examined are pictured in Figure 4-5. The
worst of these conditions is determined for each hull. For compartments with side hull
components below the bulkhead deck two transverse asymmetric extents are evaluated: ½ CH
+SH and SH only. Damage is considered vertically from the keel up to the bulkhead deck for all
conditions. The equilibrium state of the baseline hull configuration after damage at KG/D = 0.51
is pictured in Figure 4-6.
Figure 4-3 Beam-Wind Calculation using 2m strip method for L1 and L2 side hull position
2100 cos577.0ktHA
2100 cos578.0ktHA
38
Table 4-8 Damaged Conditions Subject to Flooding
Condition Number
Flooded Compartments
L1: SHX / Lwl = 0.276 L2: SHX / Lwl = 0.073
Transverse Extents Transverse Extents
1 0-10-20 ½ CH ½ CH
2 10-20-32 ½ CH ½ CH
3 20-32-48 ½ CH ½ CH +SH
4 32-48-62 ½ CH ½ CH +SH
5 48-62-78 ½ CH (1) ½ CH +SH (2) SH Only
6 62-78-92 ½ CH (1) ½ CH +SH (2) SH Only
7 78-92-102-112 (1) ½ CH +SH (2) SH Only
(1) ½ CH +SH (2) SH Only
8 92-102-112-122 (1) ½ CH +SH (2) SH Only
½ CH
9 102-112-122-130 (1) ½ CH +SH (2) SH Only
½ CH
10 112-122-130-138 (1) ½ CH +SH (2) SH Only
½ CH
(1) ½ CH + SH (2) SH Only
Figure 4-4 Asymmetric, Transverse Damage Extents Subject to Flooding
39
0-10-20 Condition
10-20-32 Condition
20-32-48 Condition
32-48-62 Condition
48-62-78 Condition
62-78-92 Condition
78-92-102-112 Condition
92-102-112-122 Condition
102-112-122-130 Condition
112-122-130-138 Condition
Figure 4-5 Damaged Conditions Subject to Flooding
Figure 4-6 Damaged Stability Model: Hull 1 Equilibrium, KG/D = 0.51
40
4.4 Design of Experiments The generation of a subdivided trimaran geometry and stability analysis for intact and damaged
conditions was performed manually. As such, the design of experiments (DoE) was completed in
stages where the principal effects of Vratio, (L/B)CH, SHX, SHY, and (B/T)SH were assessed
individually first and then those principal effects showing the most variation were coupled to
further examine their effect on the stability response. (B/T)SH and SHX showed little change in
damaged stability response, thus emphasis was placed on Vratio, (L/B)CH, and SHY for coupled
effect on stability response.
The matrix of design variables assessed for (B/T)SH variations, hulls 1, 1a, and 1b are listed in
Table 4-9 and pictured in Figure 4-7. The WP area and other characteristics are listed in Table
4-10 for hulls 1, 1a, and 1b. The stability characteristics and results are listed in Table 4-11 for
hulls 1, 1a, and 1b. The matrix of design variables for side hull position, SHY and SHX, is listed in
Table 4-12 and pictured in Figure 4-8 for hulls 1-6. Table 4-13 lists WP area and hydrostatics for
hulls 1-6. Table 4-14 lists the stability characteristics.
Figure 4-7 DoE Hull Variants, Hull 1, 1a, 1b
Table 4-9 DoE varying (B/T)SH: Hydrostatic Characteristics
Hull Bratio Tratio Vratio (LtoB)CH SHx SHy Disp. [mt]
LCB [m]
VCB [m]
Bwl [m]
Depth [m]
1 0.19 0.61 0.04 14.16 L2 T2 3,780 73.9 3.09 24.9 11.6
1a 0.25 0.45 0.04 14.16 L2 T2 3,780 73.9 3.11 25.6 11.6
1b 0.15 0.75 0.04 14.16 L2 T2 3,780 73.9 3.07 24.5 11.6
Table 4-10 DoE varying (B/T)SH: Waterplane Characteristics
Hull WP Area
[m2] LCF [m]
BMt [m]
FB - WDC [m]
WDC [m]
SHX/ Lwl
SHY/ Lwl
Awp Ratio
(B/T)CH (L/B)SH (B/T)SH
1 1,289.0 80.7 7.9 3.0 3.7 0.073 0.080 0.063 2.07 33.7 0.64
1a 1,345.7 80.7 10.0 3.0 3.7 0.073 0.080 0.082 2.07 25.0 1.16
1b 1,257.5 80.8 6.8 3.0 3.7 0.073 0.080 0.053 2.07 41.7 0.42
Table 4-11 DoE varying (B/T)SH: Stability Characteristics
Hull B-W HA Amp. [m] Down flood
CD Y Extent
KGa/D Worst Condition
Intact Damaged Intact Intact-OTD Damaged
1 0.578 0.062 -8.5 0.69 0.76 0.51 48-62-78 1/2CH+SH
1a 0.578 0.062 -8.5 0.64 0.77 0.56 48-62-78 1/2CH+SH
1b 0.578 0.062 -8.5 0.73 0.62 0.47 48-62-78 1/2CH+SH
41
Figure 4-8 Side Hull Positions to Examine in DoE
Table 4-12 DoE varying Side Hull Position, SHX and SHY: Hydrostatics Characteristics
Hull Vratio (LtoB)CH SHx SHy CD %
Increase Disp. [mt]
LCB [m]
VCB [m]
Bwl [m]
Depth [m]
1 0.04 14.16 L2 T2 0% 3,780 73.9 3.1 24.9 11.6
2 0.04 14.16 L2 T1 0% 3,780 73.9 3.1 19.9 11.6
3 0.04 14.16 L2 T3 4.1% 3,833 74.0 3.1 29.9 11.6
4 0.04 14.16 L1 T2 0% 3,780 76.4 3.1 24.9 11.6
5 0.04 14.16 L1 T1 0% 3,780 76.4 3.1 19.9 11.6
6 0.04 14.16 L1 T3 4.1% 3,833 76.5 3.1 29.9 11.6
Table 4-13 DoE varying Side Hull Position, SHX and SHY: Waterplane Characteristics
Hull WP Area
[m2] LCF [m]
BMt [m]
FB - WDC [m]
WDC [m]
SHX/ Lwl
SHY/ Lwl
Awp Ratio
(B/T)CH (L/B)SH (B/T)SH
1 1,289.0 80.7 7.9 3.0 3.7 0.073 0.080 0.063 2.07 33.7 0.64
2 1,289.0 80.7 5.7 3.0 3.7 0.073 0.063 0.063 2.07 33.7 0.64
3 1,294.2 80.8 10.7 3.0 3.7 0.073 0.098 0.064 2.06 33.6 0.64
4 1,289.0 84.7 7.9 3.0 3.7 0.276 0.080 0.063 2.07 33.7 0.64
5 1,289.0 84.7 5.7 3.0 3.7 0.276 0.063 0.063 2.07 33.7 0.64
6 1,293.9 84.8 10.7 3.0 3.7 0.276 0.098 0.064 2.06 33.5 0.64
Table 4-14 DoE varying Side Hull Position, SHX and SHY: Stability Characteristics
Hull B-W HA Amp. [m] Down flood
CD Y Extent
KGa/D Worst Condition
Intact Damaged Intact Intact-OTD Damaged
1 0.578 0.062 -8.5 0.69 0.76 0.51 48-62-78 1/2CH+SH
2 0.578 0.062 -8.5 0.49 0.58 0.41 48-62-78 1/2CH+SH
3 0.555 0.060 -8.5 0.98 0.8 0.73 48-62-78 1/2CH+SH
4 0.577 0.062 -8.5 0.59 0.67 0.53 78-92-102-112 1/2CH+SH
5 0.577 0.062 -8.5 0.43 0.53 0.41 92-102-112-122 1/2CH+SH
6 0.567 0.061 -8.5 0.89 0.83 0.75 102-112-122-130 1/2CH+SH
T3 T2 T1
L2 L1
CH MS CH CL
42
The coupled matrix of design variables for SHY, (L/B)CH, and Vratio, hulls 1-3 and 7-24, are listed
in Table 4-15 and pictured in Figure 4-9, Figure 4-10, and Figure 4-11.Table 4-16 lists WP area
and hydrostatics for hulls 1-3 and 7-24. Table 4-17 lists the stability characteristics and results
for hulls 1-3 and 7-24. It was found for all hulls that the A1/A2 ratio criterion (b) limited the KGa
for the normal, intact condition; however, for the intact vessel undergoing overhaul, towing, or
decommissioning (OTD) the limiting criteria was the equilibrium heel angle criterion (c). The
KGa/D for the worst damage condition was limited by the list angle criterion (a). KGa/D values
less than the concept design KG/D=0.5 are colored red.
Table 4-15 DoE varying (L/B)CH, Vratio, and SHY: Hydrostatic Characteristics
Hull Vratio (LtoB)CH SHx SHy CD %
Increase Disp. [mt]
LCB [m]
VCB [m]
Bwl [m]
Depth [m]
1 0.04 14.16 L2 T2 0% 3,780 73.9 3.1 24.9 11.6
2 0.04 14.16 L2 T1 0% 3,780 73.9 3.1 19.9 11.6
3 0.04 14.16 L2 T3 4.1% 3,833 74.0 3.1 29.9 11.6
7 0.04 15.6 L2 T2 0% 3,780 73.8 3.4 24.9 12.8
8 0.04 15.6 L2 T1 0% 3,780 73.8 3.4 19.9 12.8
9 0.04 15.6 L2 T3 4.8% 3,843 73.9 3.4 29.9 12.8
10 0.03 15.6 L2 T2 0% 3,780 73.7 3.4 24.7 13.0
11 0.03 15.6 L2 T1 0% 3,780 73.7 3.5 19.7 13.0
12 0.03 15.6 L2 T3 5.0% 3,846 73.8 3.5 29.7 13.0
13 0.02 14.16 L2 T2 0% 3,780 73.6 3.2 24.3 12.1
14 0.02 14.16 L2 T1 0% 3,780 73.6 3.2 19.3 12.1
15 0.02 14.16 L2 T3 4.3% 3,836 73.7 3.2 29.4 12.1
16 0.03 14.16 L2 T2 0% 3,780 73.8 3.2 24.7 11.8
17 0.03 14.16 L2 T1 0% 3,780 73.8 3.2 19.7 11.8
18 0.03 14.16 L2 T3 4.1% 3,834 73.9 3.2 29.7 11.8
19 0.02 12.7 L2 T2 0% 3,780 73.6 2.9 24.3 10.8
20 0.02 12.7 L2 T1 0% 3,780 73.6 2.9 19.3 10.8
21 0.02 12.7 L2 T3 4.1% 3,834 73.7 2.9 29.4 10.8
22 0.03 12.7 L2 T2 0% 3,780 73.8 2.9 24.7 10.6
23 0.03 12.7 L2 T1 0% 3,780 73.8 2.8 19.7 10.6
24 0.03 12.7 L2 T3 3.9% 3,831 73.8 2.8 29.7 10.6
43
SHY /
Lw
l
0.098 (T3)
0.080 (T2)
0.063 (T1)
12.7 14.16 15.6 (L/B)CH
Figure 4-9 DoE Hull Variants with Vratio = 2%
SHY /
Lw
l
0.098 (T3)
0.080 (T2)
0.063 (T1)
12.7 14.16 15.6 (L/B)CH
Figure 4-10 DoE Hull Variants with Vratio = 3%
SHY /
Lw
l
0.098 (T3)
0.080 (T2)
0.063 (T1)
12.7 14.16 15.6 (L/B)CH
Figure 4-11 DoE Hull Variants with Vratio = 4%
Hull 22
Hull 23
Hull 24
Hull 16 Hull 10
Hull 17
Hull 18
Hull 11
Hull 12
Hull 19
Hull 20
Hull 21
Hull 13
Hull 14
Hull 15
Hull 1 Hull 7
Hull 2
Hull 3
Hull 8
Hull 9
44
Table 4-16 DoE varying (L/B)CH, Vratio, and SHY: Waterplane Characteristics
Hull WP Area
[m2] LCF [m]
BMt [m]
FB - WDC [m]
WDC [m]
SHX/ Lwl
SHY/ Lwl
Awp Ratio
(B/T)CH (L/B)SH (B/T)SH
1 1,289.0 80.7 7.9 3.0 3.7 0.073 0.080 0.063 2.07 33.7 0.64
2 1,289.0 80.7 5.7 3.0 3.7 0.073 0.063 0.063 2.07 33.7 0.64
3 1,294.2 80.8 10.7 3.0 3.7 0.073 0.098 0.064 2.06 33.6 0.64
7 1,184.5 80.7 7.4 3.7 3.7 0.073 0.080 0.069 1.71 33.8 0.64
8 1,184.5 80.7 5.1 3.7 3.7 0.073 0.063 0.069 1.71 33.8 0.64
9 1,190.1 80.7 10.1 3.7 3.7 0.073 0.098 0.069 1.69 33.6 0.64
10 1,163.2 80.7 6.6 3.8 3.7 0.073 0.080 0.061 1.67 38.8 0.65
11 1,163.2 80.7 4.7 3.8 3.7 0.073 0.063 0.061 1.67 38.8 0.65
12 1,169.0 80.7 9.0 3.8 3.7 0.073 0.098 0.061 1.66 38.5 0.64
13 1,241.1 80.8 6.2 3.2 3.8 0.073 0.080 0.047 1.99 47.7 0.64
14 1,241.1 80.8 4.6 3.2 3.8 0.073 0.063 0.047 1.99 47.7 0.64
15 1,246.5 80.8 8.2 3.2 3.8 0.073 0.098 0.047 1.97 47.3 0.63
16 1,267.6 80.7 7.2 3.1 3.8 0.073 0.080 0.056 2.03 38.7 0.64
17 1,267.6 80.7 5.2 3.1 3.8 0.073 0.063 0.056 2.03 38.7 0.64
18 1,272.7 80.8 9.6 3.1 3.8 0.073 0.098 0.056 2.02 38.6 0.64
19 1,370.1 80.8 7.0 3.0 3.3 0.073 0.080 0.042 2.47 47.7 0.64
20 1,370.1 80.8 5.4 3.0 3.3 0.073 0.063 0.042 2.47 47.7 0.64
21 1,375.7 80.8 8.9 3.0 3.2 0.073 0.098 0.042 2.45 47.3 0.64
22 1,396.4 80.8 7.9 3.0 3.2 0.073 0.080 0.051 2.52 38.8 0.64
23 1,396.4 80.8 6.0 3.0 3.2 0.073 0.063 0.051 2.52 38.8 0.64
24 1,401.7 80.8 10.3 3.0 3.1 0.073 0.098 0.051 2.50 38.7 0.64
45
Table 4-17 DoE varying (L/B)CH, Vratio, and SHY: Stability Characteristics
Hull B-W HA Amp. [m] Down flood
CD Y Extent
KGa/D Worst Condition
Intact Damaged Intact Intact-OTD Damaged
1 0.578 0.062 -8.5 0.69 0.76 0.51 48-62-78 1/2CH+SH
2 0.578 0.062 -8.5 0.49 0.58 0.41 48-62-78 1/2CH+SH
3 0.555 0.060 -8.5 0.98 0.8 0.73 48-62-78 1/2CH+SH
7 0.670 0.072 -8.38 0.69 0.58 0.47 48-62-78 1/2CH+SH
8 0.670 0.072 -8.38 0.46 0.45 0.35 48-62-78 1/2CH+SH
9 0.658 0.071 -8.38 0.91 0.74 0.53 62-78-92 SH Only
10 0.696 0.075 -8.38 0.61 0.51 0.46 48-62-78 1/2CH+SH
11 0.696 0.075 -8.38 0.47 0.4 0.35 48-62-78 1/2CH+SH
12 0.68 0.073 -8.38 0.84 0.65 0.50 62-78-92 SH Only
13 0.617 0.066 -8.46 0.64 0.51 0.45 48-62-78 1/2CH+SH
14 0.617 0.066 -8.46 0.46 0.42 0.37 48-62-78 1/2CH+SH
15 0.609 0.066 -8.46 0.83 0.63 0.53 62-78-92 SH Only
16 0.600 0.065 -8.46 0.69 0.6 0.47 48-62-78 1/2CH+SH
17 0.600 0.065 -8.46 0.41 0.49 0.38 48-62-78 1/2CH+SH
18 0.529 0.057 -8.46 0.76 0.53 0.56 62-78-92 SH Only
19 0.522 0.056 -8.5 0.74 0.64 0.58 48-62-78 1/2CH+SH
20 0.522 0.056 -8.5 0.46 0.54 0.48 48-62-78 1/2CH+SH
21 0.513 0.055 -8.5 0.96 0.79 0.69 48-62-78 SH Only
22 0.504 0.054 -8.5 0.82 0.73 0.63 78-92-102 SH only
23 0.504 0.054 -8.5 0.48 0.61 0.47 48-62-78 1/2CH+SH
24 0.496 0.053 -8.5 1.05 0.9 0.71 62-78-92 SH Only
46
4.5 Relationships First, the effect of side hull beam and draft on KGa/D is investigated for Hulls 1, 1a, and 1b,
shown in Figure 4-12. An increase in side hull beam to draft ratio, (B/T)SH, decreases the KGa for
the intact condition. This is because the limiting criterion (b) is A1/A2, in which the shallower
draft at high (B/T)SH adversely effects KGa. In contrast, the KGa increases with an increase in side
hull B/T for intact-OTD, and damaged conditions. This is because the limiting criterion is the
equilibrium and static heel angle respectively, which is most effected by beam.
Figure 4-12 Effect of Side Hull Beam to Draft Ratio, (B/T)SH (Hulls 1, 1a, 1b)
Next, the effect of side hull longitudinal position, SHX, on KGa/D is investigated for Hulls 1-6. As
shown in Figure 4-13, the KGa increases from aft L1 to mid L2 SH position for intact conditions,
however for damaged conditions minimal effect is present. Upon this understanding, SHX was
dropped from the remainder of the DoE. Despite a minimal effect seen from quasi-static
methods evaluating the transverse stability, the effect SHX should be included for dynamic
methods to capture its effect on longitudinal stability in both intact and damaged conditions.
Because a manual approach was taken to accomplish the stability analysis a reduced number of
points are used for the DoE. Given this sparse DoE, the data and results should only be used for
rule-of-thumb and initial guidance early-stage design.
47
SHY /
Lw
l 0.098 (T3)
0.080 (T2)
0.063 (T1)
Mid L1 → SH moving aft → Aft L2
Figure 4-13 Effect of Side Hull Longitudinal Position, SHX (Hulls 1-6)
Given the previous results, the major effects of center hull L/B, side hull displacement and
transverse position on KGa/D are investigated for Hulls 1 and 7-24. As expected the side hull
transverse location, SHY, has the most effect on the KGa/D ratio followed by (L/B)CH and Vratio
for intact and damaged conditions. Figure 4-14, Figure 4-15, and Figure 4-16 show the KGa/D
ratio colored by red-yellow-green gradient for Hulls 1 and 7-24 in SHX/Lwl, CH L/B, and Vratio
space under intact, intact during overhaul, towing, and decommissioning (OTD) and damaged
conditions.
48
1.05
D
KGa
0.89
0.73
0.57
0.41
Figure 4-14 KGa / D Results at L2 Side Hull Position: Intact Condition
0.90
D
KGa
0.78
0.66
0.53
0.40
Figure 4-15 KGa / D Results at L2 Side Hull Position: Intact Condition During OTD
49
0.75
D
KGa
0.65
0.55
0.45
0.35
Figure 4-16 KGa / D Results at L2 Side Hull Position: Worst Damaged Condition
For most of the hulls, a trend is observed in Figure 4-17 where the worst damaged condition is
48-62-78, ½ CH + SH for the two side hull transverse positions closer to the center hull, SHY /Lwl
= 0.063 and 0.08. However, for the side hull transverse position furthest from the center hull,
SHY /Lwl = 0.098, the 62-78-92, SH only damaged condition is the worst case. The 48-62-78
damaged condition removes the maximum volume from the center hull, while the 62-78-92
condition removes the maximum volume for the side hull. This shows, in general, that trimarans
with side hulls further from the center hull transversely will have more of an effect on the list
angle criterion (a) than those closer to the center hull.
Given that the baseline design concept had a KG/D ratio of 0.5, this value was set as a threshold
with which to determine hulls that passed or failed the stability criteria. 10 of the 19 hulls
examined for the mid L2 SH position failed in the intact and/or damaged conditions (KG > KGa).
These hulls are colored red in Table 4-17 and have an X marker in Figure 4-18. Except for Hull 20,
the limiting condition with the least KGa/D was the damaged criterion (a): list angle < 15 deg.
The passing trimaran hull form configurations have an O marker and the interpolation region is
approximated with an orange to blue gradient contour in Figure 4-18. It is worth mentioning
that all of the variations with side hulls at the inner T1 SH position failed as well as those at mid
T2 SH position except for those with smaller CH L/B or larger Vratio.
50
Figure 4-17 Worst Damaged Condition (Hulls 1-24)
51
Figure 4-18 Stability Assessment, Feasible Region: KGa/D > 0.5(Hulls 1-3, 7-24)
52
4.6 Response Surface Model Using the KGa/D results from executing the DoE through the stability model, statistical
relationships are developed to guide rule-of thumb processes and early-stage design efforts.
First, functions for intact and damaged conditions listed in Table 4-18 are fit to the Hull 1, 1a,
and 1b data pictured in Figure 4-12. The effect of longitudinal side hull position, SHX, was not
significant enough as shown in Figure 4-13, especially in damaged condition, and a function for
KGa/D could not be determined. Next using hulls 1 and 7-24, KGa/D as a function of SHY/Lwl,
(L/B)CH, and Vratio for intact and damaged conditions were determined. Summary statistics are
listed in Table 4-19, actual v. predicted values are plotted in Figure 4-19, and the formulas are
listed in Table 4-20. Examining these functions will guide additional work to determine areas of
the DoE to increase point density and further improve the functions.
Table 4-18 KGa/D functions of (B/T)SH, Hulls 1,1a, and 1b at Intact and Damaged Conditions
Intact (A1/A2): KGa/D ( (B/T)SH) = 0.83477 - 0.29637*(B/T)SH + 0.11088*(B/T)SH2
Intact-OTD (Equil. HA): KGa/D ( (B/T)SH) = 0.78786 - 0.07077*Log( (B/T)SH) – 0.30359*Log( (B/T)SH)2
Damaged (Static HA): KGa/D ( (B/T)SH) = 0.36523 + 0.29637*(B/T)SH - 0.11088*(B/T)SH2
Table 4-19 KGa/D Function Statistics Summary, Hulls 1 and 7 - 24
Condition R Sq RMSE Residual Range
Intact (A1/A2) 0.97 0.04 -0.08 / +0.06
Intact-OTD (Equil. HA) 0.90 0.05 -0.12 / +0.06
Damaged (Static HA) 0.99 0.01 -0.02 / +0.02
Intact (A1/A2) Intact-OTD (Equil. HA) Damaged (Static HA)
Figure 4-19 KGa/D Function Actual v. Predicted, Hulls 1 and 7 - 24
Table 4-20 KGa/D Functions of Hulls 1 and 7 – 24 at Intact and Damaged Conditions
Intact (A1/A2): KGa/D ( SHY/Lwl, (L/B)CH, Vratio ) = 5.70683 + 15.15576 *Vratio -
0.983797*(L/B)CH - 4.35111*Vratio*(L/B)CH +
34.42221*SHY/Lwl + 218.40014*Vratio* SHY/Lwl -
1.98964*CH L/B*SHy/Lwl) +546.57089*Vratio2 +
0.04347*(L/B)CH2
53
Intact-OTD (Equil. HA): KGa/D ( SHY/Lwl, (L/B)CH, Vratio ) = 7.61851 + 53.311*Vratio -
1.12158*(L/B)CH - 6.85626*Vratio*(L/B)CH +
6.34427*SHY/Lwl + 895.58703*Vratio2 + 0.04389*(L/B)CH2
Damaged (Static HA): KGa/D ( SHY/Lwl, (L/B)CH, Vratio ) = 4.06319 - 5.0862*Vratio -
0.61341*(L/B)CH + 24.8936*SHY/Lwl +
88.23142*Vratio*SHY/Lwl - 0.89659*(L/B)CH* SHY/Lwl +
0.02215*(L/B)CH2 - 58.22415* (SHY/Lwl)2
The damaged condition is the limiting KGa for all hulls except for hull 20 which had the smallest
SH, largest L/B, and SH position T1 closest to the CH. In this case the intact KGa/D for hull 20 was
0.02 less than the damage KGa/D. Since this discrepancy for hull 20 is small, we can use the
damage KGa/D function as an overall function to determine KGa/D for trimaran configurations
similar to the ones used in the DoE. Further by normalizing the KGa/D as function of (B/T)SH by
Hull 1, it can be added to the KGa/D as a function of SHY/Lwl, (L/B)CH, and Vratio for the
damaged condition. Equation 4 is the resulting response surface model (RSM). This RSM for
KGa/D can be visualized using a series of (L/B)CH v. SHY/Lwl contour plots shown in Appendix A
where discrete values of (B/T)SH and Vratio are used. This RSM has a very good R Squared and
RMSE values; however, to maintain valid results given the sparse DoE, care should be taken not
to extrapolate the variable values beyond the range examined. The bounds with which this RSM
maintains applicability are detailed in Table 4-21. Additionally, these bounds and DoE points are
plotted on the contours in Appendix A.
KGa/D ( (B/T)SH, SHY/Lwl, (L/B)CH, Vratio ) = 3.91842 + 0.29637*(B/T)SH -
0.11088*(B/T)SH2- 5.0862*Vratio - 0.61341*(L/B)CH + 24.8936*SHY/Lwl +
88.23142*Vratio*SHY/Lwl - 0.89659*(L/B)CH* SHY/Lwl + 0.02215*(L/B)CH2 -
58.22415* (SHY/Lwl)2
Equation 4
Table 4-21 Valid Bounds for Variables in KGa/D RSM
Variable Min Max
(B/T)SH 0.40 1.16
SHY/Lwl 0.063 0.098
(L/B)CH 12.7 15.6
Vratio 0.02 0.04
54
4.7 Application of Stability RSM to Ship Design With one small exception, the worst damaged condition determined the allowable KG, KGa, for
each hull variation and an overall function is established in Equation 4. For these damage
conditions, list angle criterion (a) determined the KGa. This is somewhat concerning given the
results Peters and Wing (2009) found where the list angle for a damaged monohull shows poor
correlation with dynamic stability results summarized in Table 2-5. Performing a similar analysis
for a trimaran hull form would enable further insight regarding the usefulness of the list angle
criterion (a) as a measure of stability.
Surface combatants in navies across the globe are employing more capable topside weapons,
radars, and other sensors on surface ships for increased coverage. These systems are often
heavier than the systems they’re replacing and continue to be mounted up high in the top-level
superstructure and on the mast. Placing too many of these systems in these areas will increase
the KG. Having knowledge at the concept stage regarding the maximum, allowable KG will guide
the design towards a stable ship from the start. The derived function in Equation 4 provides such
a rule of thumb, first approach when determining the KGa for a trimaran ship, displacing close to
3,780 mt, during the early-stage design stage. Coupled with displacement, the KG is an
important parameter to be considered not only for adequate stability but for seakeeping
performance as well.
Additionally, Equation 4 can be utilized as a constraint for design space exploration used in the
early-stage. In such a formulation, trimaran hull form variations could be examined and
optimized for resistance, ship motions, or both. The KGa/D function could be used to determine
the infeasible region which are likely to fail an assessment of damaged stability. Similarly, set
based methods using ship synthesis models to evaluate the ship systems could benefit from this
KGa/D constraint for trimaran damaged stability.
To examine the usefulness of Equation 4 and the contours plotted in Appendix A, let’s consider a
situation where the designer would like to modify the propulsion for the baseline hull. The
effect of twin propellers in the center hull is examined rather than one larger propeller in the
center hull and two smaller propellers in each side hull. The L/B ratio of the center hull may
need to decrease from 14.16 to accommodate the twin shafts. The baseline design had a 5.5m
propeller in the center hull and two 1.05m propellers in each side hull. If the total propeller area
is kept constant, two 4.03m propellers would provide similar thrust to the original design. Given
a half diameter propeller separation distance, the Bwl of the center hull needs be at least
10.1m. Thus, no change is required from the baseline hull to accommodate two shafts.
However, if the designer chose to change the displacement distribution to a smaller side hull,
say Vratio = 3%, for improved low-speed resistance then the (L/B)CH would need to decrease to
13.85 or the SHY/Lwl would need to increase to 0.084, 0.57m out from its current position. This
would maintain the same KGa/D = 0.5 assuming the (B/T)SH remains constant at 0.64.
55
5 Conclusions: Stability Criteria Adaption for Trimaran After defining a trimaran and giving an overview of its advantages and potential stability
concerns under damage, a review of different methods to assess ship stability were examined.
These methods include semi-empirical, deterministic criteria relying on quasi-static estimates
and probabilistic methods relying mostly on time domain simulations. Additionally, current
research was reviewed for total loss of stability (capsize) events of damaged monohulls and
parametric roll events of intact trimarans in longitudinal waves.
In Chapter 3 intact vessels along the multi-hull continuum with the same displacement and
waterplane area were examined. The trimaran hull forms were observed to possess stability
qualities between the monohull and catamaran with largely linear trends as displacement was
distributed between hulls. A strong dependency is exhibited between side hull displacement and
area, A1 as well as side hull displacement and maximum righting arm, GZ. For trimarans, the
deck edge submerges at a heel angle around 30 degrees, whereas the monohull and catamaran
deck edge submerges at around 50 degrees of heel. Next, a thorough examination of the
hydrostatic properties at trim and heel angles for the monohull and 2.5% - 95% - 2.5% trimaran
were assessed. Hydrostatic properties were examined for heel angles up to the angle of
vanishing stability for bow down and up trim angles. One key observation was the LCF moves
towards midships for all trim angles on the monohull and trimaran with side hulls in the aft
position, where the LCF moves forward for bow down trim angles and aft for bow up trim angles
on the trimaran with mid SH position. This indicates that the stability may be more of a concern
for the trimaran with a trim at large heel angle greater than 30 degrees more so than for a
monohull. Additionally, it confirms the possibility that the side hull longitudinal position, SHX,
may have an impact on transverse stability.
Insights gained from the quasi-static estimates determined in Chapter 3 along the multi-hull
continuum are used to develop a series of trimarans in Chapter 4. In which further geometric
configurations were examined for transverse stability under intact and damaged conditions.
Starting from a design concept with multi-mission frigate capability displacing 3,780 mt, 26
subdivided trimaran models were generated and analyzed for published USN transverse stability
criteria. The principal and coupled effects of (B/T)SH, SHX, SHY, (L/B)CH, and Vratio on the intact
and damaged stability response were examined through a multi-stage DoE approach.
An increase in side hull beam to draft ratio decreases the KGa for the intact condition; however,
for intact-OTD, and damaged conditions the KGa increases. The KGa increases from aft, L1, to
forward, L2, SH position for intact conditions, however for damaged conditions minimal effect is
present. As expected, the side hull transverse location SHY has the most effect KGa followed by
(L/B)CH and Vratio for intact and damaged conditions. Using these relationships, a response
surface model (RSM) for KGa/D as a function of (B/T)SH, SHY, (L/B)CH, and Vratio was determined,
Within the bounds of applicability, the KGa/D RSM can be used as a rule-of-thumb and
optimization constraint for future studies.
56
Several tasks for additional work to expand and enhance this thesis are recommended. Adding
more trimaran hull form configurations to the DoE would increase the applicability of the RSM
and its use in early-stage design. The density of points should be increased in areas where
contours aren’t well defined in the RSM. Applying probabilistic methods to assess damaged
stability of trimarans and compare to deterministic criteria would further necessitate the need
to use these methods for naval combatants. Additionally, using probabilistic methods to
compare damaged monohull and trimaran vessels would aid in the understanding of their
differences in terms of next generation criteria and dynamic stability.
57
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61
Appendix A KGa/D RSM Visualized through Contour Plots (B/T)SH = 0.4, Vratio = 0.02 (B/T)SH = 0.4, Vratio = 0.03 (B/T)SH = 0.4, Vratio = 0.04
(B/T)SH = 0.64, Vratio = 0.02 (B/T)SH = 0.64, Vratio = 0.03 (B/T)SH = 0.64, Vratio = 0.04
62
(B/T)SH = 0.88, Vratio = 0.02 (B/T)SH = 0.88, Vratio = 0.03 (B/T)SH = 0.88, Vratio = 0.04
(B/T)SH = 1.12, Vratio = 0.02 (B/T)SH = 1.12, Vratio = 0.03 (B/T)SH = 1.12, Vratio = 0.04