Probability of Failure of Damaged Ship Structures Phase IV

Probability of Failure of Damaged Ship Structures Phase IV Report

Unyime Akpan Lloyd's Register Applied Technology Group Prepared By: Lloyd's Register Applied Technology Group Suite 400, 1888 Brunswick Street Halifax, NS B3J 3J8 Contractor's Document Number: TR-15-18 Rev 02 Contract Project Manager: Colin Clark, 902-425-5101 PWGSC Contract Number: W7707-145679, Task 6 Technical Authority: Malcolm Smith, 902-426-3100

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Contract Report DRDC-RDDC-2016-C347 March 2015

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Probability of Failure of Damaged Ship Structures Phase IV Report

Technical Report #TR-15-18 Rev 02

Job No: 14.28008.1106 March 2015

Prepared for:

Malcolm Smith DRDC Atlantic

Martec Limited is a member of the Lloyd’s Register Group, doing business as the Applied Technology Group

REVISION CONTROL

REVISION REVISION DATE Revision 1 Sept 15, 2015 Revision 2 October 7, 2015

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EXECUTIVE SUMMARY This is Phase IV of a multiyear effort to develop methodologies for computing failure probabilities of ships with gross damage. In Phase I literature review and assessment of available tools for damaged ship was undertaken. Estimate of gross damage to large surface ships was carried out using simulations of collisions. In Phase II a general methodology for assessing the failure probabilities of intact and damaged ship was formulated. In Phase III a case study was conducted with the methodology developed in Phase II on an intact and damaged Annapolis Class Destroyer HMCS NIPIGON, a former Canadian naval vessel, decommissioned in 1998. The present study is aimed at improving the load models used to compute the failure probabilities of ships with gross damage. Improvements are implemented in two areas, namely (i) the modelling of flooding due to damage and (ii) the incorporation of correlation between the vertical and the horizontal wave bending moments. In Phase III, the hydrodynamic loads on the damaged ship with flooding were not computed, instead the load on the intact vessel was used in the damaged ship case. In the present study flooding is now explicitly modeled to provide more realistic loading for the damaged ship case. Furthermore, in Phase III, as well as in other related studies that have been reviewed, correlation of the vertical and the horizontal wave loads were not considered. This is included in the current study. Two correlation models, Method 1 and Method 2 are proposed and used. Method 1 uses only the amplitudes of the RAOs. Method 2 utilises the real and the imaginary parts of the RAOs, and the sea spectrum. Based on literature review, Method 2 is selected as the best strategy that reflects the ‘actual’ correlation between the vertical and the horizontal wave loads. The Annapolis Class Destroyer HMCS NIPIGON, a former Canadian naval vessel, decommissioned in 1998, is used to demonstrate the methodology. An operational profile involving four ship speeds, 5 speed headings and three sea states is defined. The RAOs of the wave loads and the stillwater bending moments are computed for four flooding conditions: (i) 56% flooding of one damaged mid compartment; (ii) 74% flooding of one damaged mid compartment (iii) 60% flooding of two damaged mid compartments; and (iv) 100% flooding of two damaged bow compartments. The statistics of the extreme wave loads and the correlation between the vertical and the horizontal wave loads are determined for all the flooding conditions and operational profiles. The strength model is based on the midship section of the ship. Bottom and side damage scenarios are defined and the ultimate strengths are computed. Both the linear model that uses only the vertical components of the load and strength, and the interaction equation model, that uses both the vertical and the horizontal components of loads and strengths are employed in the reliability assessment. Failure probabilities of the damaged ship are computed with the improved models. Results indicate that the higher the vessel speed the higher the failure probabilities and vice versa. The failure of all the damaged ships is governed by the hogging mode. Failure probabilities of the bottom

damaged ship are always higher than the side damaged ship. When correlation between the wave loads is neglected, the interaction model always gives failures probabilities that are bigger than the linear model. Estimates of the correlation between the wave loads are positive for Method 1 and negative for Method 2. The largest failure probabilities are obtained with correlation from Method 1 and the smallest failure probabilities are obtained with correlation from Method 2. This underscores the need to include correlation in the reliability assessment of damaged ship. This feature is lacking in most of the published work. Failure probabilities are highest for the 74% flooded one bottom damaged mid compartment, followed by the 60% flooded two bottoms damaged mid compartments. Uncertainties in the wave load, vertical and horizontal wave bending moments, and uncertainties in the model used to represent the ultimate vertical bending moment capacity are the main parameters that govern the failure probabilities. The results underline the need to use strength and load models that best reflect the damage under consideration. Recommendations to gain further insight on the impact of load and strength models on the failure probabilities of damaged ships are provided, including the following: (i) Investigate the effect on failure probabilities of including or excluding operational

conditions with very small extreme wave loads and the selection and use of probability distributions that best fit the extreme wave loads based on the data;

(ii) Conduct study using more realistic operational conditions applicable to a damaged Canadian naval ship that has travel from a specified location to another for analyses. This could include the effect of alternate travel routes and combination of different flooding scenarios to reflect the impact of increasing hazards, starting with for example, small flooding water in a single compartment, followed by more water in a single compartment and more flooding in two compartments etc. can also be included;

(iii) Conduct study that incorporates the impact of slamming or dynamic loads; (iv) Additional study on the current ship for other damaged locations; (v) More studies with other intact and/or damage ships; and (vi) Improving the strength models through finite element so that other components such as

torsion can be incorporated into the assessment. .

TABLE OF CONTENTS

1.0 INTRODUCTION ......................................................................................................................................... 1 1.1 BACKGROUND .............................................................................................................................................. 1 1.2 OBJECTIVE AND SCOPE ................................................................................................................................. 1 1.3 ORGANIZATION OF THIS REPORT................................................................................................................... 2

2.0 SUMMARY OF OVERALL METHODOLOGY AND DEFINITION OF VESSEL CHARACTERISTICS ............................................................................................................................................ 3

2.1 SUMMARY OF OVERALL METHODOLOGY .................................................................................................... 3 2.2 DESCRIPTION OF SELECTED VESSEL ............................................................................................................ 4 2.3 BRIEF SUMMARY OF APPROACHES AND TOOLS USED IN THE STUDY........................................................... 6

2.3.1 Approach and Tools Used to Estimate the Damage Size ................................................................... 6 2.3.2 Approaches and Tools Used to Estimate the Loads........................................................................... 6 2.3.3 Approach and Tools Used to Estimate the Ultimate Strength ........................................................... 9 2.3.4 Approach and Tool Used to Estimate the Reliability....................................................................... 10

3.0 RELIABILITY ASSESSMENT OF THE INTACT VESSEL ................................................................. 13 3.1 INTRODUCTION .......................................................................................................................................... 13 3.2 LOADS ON THE INTACT VESSEL ................................................................................................................. 13

3.2.1 Operational Profile ......................................................................................................................... 13 3.2.2 Still Water Bending Moment ............................................................................................................ 13 3.2.3 Wave-Induced Bending Moment ...................................................................................................... 15 3.2.4 Extreme Value of the Wave Loads ................................................................................................... 15

3.3 ULTIMATE STRENGTHS OF THE INTACT VESSEL ........................................................................................ 17 3.4 RELIABILITY OF THE INTACT SHIP ............................................................................................................. 18

4.0 RELIABILITY ASSESSMENT OF THE DAMAGED SHIP ................................................................. 23 4.1 INTRODUCTION .......................................................................................................................................... 23 4.2 DAMAGE DEFINITION................................................................................................................................. 23 4.3 LOADS ON THE DAMAGED SHIP ................................................................................................................. 23

4.3.1 Operational Profile ......................................................................................................................... 23 4.3.2 Still Water Bending Moment ............................................................................................................ 24 4.3.3 Wave-Induced Bending Moment ...................................................................................................... 26 4.3.4 Extreme Value of the Wave Loads ................................................................................................... 32

4.4 ULTIMATE STRENGTHS OF THE DAMAGED SHIP ........................................................................................ 34 4.5 ESTIMATES OF THE RELIABILITY OF THE DAMAGED SHIP .......................................................................... 37

5.0 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .............................................................. 46 5.1 SUMMARY AND CONCLUSIONS .................................................................................................................. 46 5.2 RECOMMENDATIONS .................................................................................................................................. 47

6.0 REFERENCES ............................................................................................................................................ 52

LIST OF FIGURES FIGURE 2-1: OVERALL METHODOLOGY FOR ESTIMATING PROBABILITY OF FAILURE OF A DAMAGED SHIP .............. 3 FIGURE 2-2: THE ANNAPOLIS CLASS DESTROYER HMCS NIPIGON ........................................................................ 4 FIGURE 2-3: FINITE ELEMENT MODEL OF THE SHIP ................................................................................................... 5 FIGURE 3-1: THE VERTICAL STILL WATER BENDING MOMENT ON THE INTACT VESSEL ......................................... 14 FIGURE 3-2: THE VERTICAL STILLWATER SHEAR FORCE ON THE INTACT VESSEL ................................................... 14 FIGURE 3-3: RAOS OF THE WAVE BENDING MOMENT AT 12 KNOTS ON THE INTACT VESSEL (A) VERTICAL (B)

HORIZONTAL .................................................................................................................................................. 15 FIGURE 3-4: CDF OF THE EXTREME VERTICAL WAVE BENDING MOMENT ON THE INTACT SHIP............................. 16 FIGURE 3-5: CDF OF THE EXTREME HORIZONTAL WAVE BENDING MOMENT ON THE INTACT SHIP ........................ 16 FIGURE 3-6: ULTIMATE BENDING CAPACITY OF THE INTACT VESSEL ...................................................................... 17 FIGURE 3-7: IMPORTANCE FACTOR FOR THE RANDOM VARIABLES .......................................................................... 22 FIGURE 4-1: FLOODING SCENARIOS USED FOR THE DAMAGED SHIP (A) 1 MID COMPARTMENT (B) 2 MID

COMPARTMENTS (C) 2 BOW COMPARTMENTS ................................................................................................ 24 FIGURE 4-2: THE VERTICAL STILLWATER BENDING MOMENT ON THE DAMAGED VESSEL ...................................... 25 FIGURE 4-3: VERTICAL STILLWATER SHEAR FORCE ON THE DAMAGED VESSEL ..................................................... 25 FIGURE 4-4: RAOS OF THE VERTICAL WAVE BENDING MOMENTS AT 12 KNOTS, 0 HEADING.................................. 27 FIGURE 4-5: RAOS OF THE HORIZONTAL WAVE BENDING MOMENTS AT 12 KNOTS, 0 HEADING ............................ 27 FIGURE 4-6: RAOS OF THE VERTICAL WAVE BENDING MOMENTS AT 12 KNOTS, 45 HEADING ................................ 28 FIGURE 4-7: RAOS OF THE HORIZONTAL WAVE BENDING MOMENTS AT 12 KNOTS, 45 HEADING ........................... 28 FIGURE 4-8: RAOS OF THE VERTICAL WAVE BENDING MOMENTS AT 12 KNOTS, 90 HEADING ................................ 29 FIGURE 4-9: RAOS OF THE HORIZONTAL WAVE BENDING MOMENTS AT 12 KNOTS, 90 HEADING ........................... 29 FIGURE 4-10: RAOS OF THE VERTICAL WAVE BENDING MOMENTS AT 12 KNOTS, 135 HEADING............................ 30 FIGURE 4-11: RAOS OF THE HORIZONTAL WAVE BENDING MOMENTS AT 12 KNOTS, 135 HEADING ....................... 30 FIGURE 4-12: RAOS OF THE VERTICAL WAVE BENDING MOMENTS AT 12 KNOTS, 180 HEADING............................ 31 FIGURE 4-13: RAOS OF THE HORIZONTAL WAVE BENDING MOMENTS AT 12 KNOTS, 180 HEADING ....................... 31 FIGURE 4-14: CUMULATIVE DISTRIBUTION OF THE EXTREME VERTICAL WAVE BENDING MOMENTS .................... 33 FIGURE 4-15: CUMULATIVE DISTRIBUTION OF THE EXTREME HORIZONTAL WAVE BENDING MOMENTS ................ 33 FIGURE 4-16: DAMAGED LOCATIONS USED FOR ULTIMATE STRENGTH ANALYSIS (A) SIDE (B) BOTTOM ............... 35 FIGURE 4-17: ULTIMATE STRENGTH OF THE DAMAGED SHIP BASED ON FE CALIBRATED MODEL (A) BOTTOM (B)

SIDE ................................................................................................................................................................ 35 FIGURE 4-18: ULTIMATE STRENGTH OF THE DAMAGED SHIP ................................................................................... 36 FIGURE 4-19: IMPORTANCE FACTOR FOR THE SIDE DAMAGED 74% FLOODED SHIP RANDOM VARIABLES ............. 44 FIGURE 4-20: IMPORTANCE FACTOR FOR THE BOTTOM DAMAGED 74% FLOODED SHIP RANDOM VARIABLES ....... 45

LIST OF TABLES TABLE 2-1: SUMMARY OF THE MAIN PARTICULARS OF THE SHIP............................................................................... 5 TABLE 2-2: SUMMARY OF POSSIBLE VESSEL SECTIONS FOR ANALYSIS ..................................................................... 6 TABLE 2-3: DESTROYER DAMAGE STATISTICS........................................................................................................... 6 TABLE 2-4: COMPONENTS OF THE LOADS ON A DAMAGED SHIP ................................................................................ 7 TABLE 2-5: COMPONENTS OF THE ULTIMATE STRENGTHS OF A DAMAGED SHIP ....................................................... 9 TABLE 2-6: A SUMMARY OF THE RANDOM VARIABLES ........................................................................................... 11 TABLE 3-1: OPERATIONAL PROFILE USED ............................................................................................................... 13 TABLE 3-2: PROBABILISTIC CHARACTERISTIC OF THE STILL WATER LOADS ........................................................... 14 TABLE 3-3: PROBABILISTIC CHARACTERISTIC OF THE EXTREME WAVE LOADS ON THE INTACT SHIP ..................... 17 TABLE 3-4: ULTIMATE BENDING MOMENT CAPACITIES OF THE INTACT SHIP .......................................................... 18 TABLE 3-5: RANDOM VARIABLES USED FOR THE INTACT VESSEL ANALYSIS .......................................................... 18 TABLE 3-6: PROBABILITIES OF FAILURES OF THE INTACT SHIP (SAGGING MODE) ................................................... 20 TABLE 3-7: PROBABILITIES OF FAILURES OF THE INTACT SHIP (HOGGING MODE)................................................... 20 TABLE 4-1: STATISTICS OF THE DAMAGE ON THE MIDSHIP SECTION ....................................................................... 23 TABLE 4-2: PROBABILISTIC CHARACTERISTIC OF THE STILL WATER LOAD ON THE DAMAGED SHIP ....................... 26 TABLE 4-3: CORRELATION BETWEEN THE VERTICAL AND HORIZONTAL WAVE LOADS ........................................... 32 TABLE 4-4: STATISTICS OF THE EXTREME VERTICAL AND HORIZONTAL WAVE BENDING MOMENTS ..................... 33

TABLE 4-5: PROBABILISTIC CHARACTERISTIC OF THE DAMAGED SHIP ULTIMATE BENDING CAPACITIES ............... 36 TABLE 4-6: SUMMARY OF THE DAMAGED VESSEL RANDOM VARIABLES ................................................................ 38 TABLE 4-7: HOGGING FAILURES PROBABILITIES OF DAMAGED SHIP ....................................................................... 40 TABLE 4-8: SAGGING FAILURES PROBABILITIES OF DAMAGED SHIP ........................................................................ 41 TABLE 4-9: HOGGING FAILURES PROBABILITIES FOR CASE 3 (9 & 12 KNOTS) ........................................................ 42 TABLE 4-10: STATISTICS OF THE EXTREME WAVE BENDING MOMENTS CASE 3 (9 & 12 KNOTS) ............................ 42 TABLE 5-1: POSSIBLE STATISTICS OF THE EXTREME WAVE LOADS FOR A DAMAGED 2 MID COMPARTMENT WITH

60% FLOODED ................................................................................................................................................ 49

Probability of Failure of Damaged Ship Structures Phase IV Report 1

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1.0 INTRODUCTION

1.1 BACKGROUND

Both naval and commercial vessels may experience various forms of damage in normal service conditions or as a result of an accident. Typical kinds of damage incurred by commercial ships include fatigue cracking, corrosion, as well as indentation and rupture due to collision, grounding or heavy seas (e.g. slamming or green seas). In addition, naval vessels may also suffer combat-related blast, fragmentation, and ballistics damage. Assessing the survivability of ships in a damaged condition continues to be an active area of research that encompasses the loss of structural strength, changes to the loading and dynamic stability, damage control, and crew evacuation. Due to the large number of uncertainties associated with operating a ship with damage, a probabilistic framework is considered. The proposed project addresses the following question: for a ship operating in a given seaway how does the presence of damage affect the probability of structural failure? The present work is concerned with the effect of gross damage to structure, such as may be caused by collisions, groundings, and explosive effects. The ability to assess the probability of failure with gross damage would allow ship designers to improve damage tolerance of new designs through assessment of likely damage scenarios. It would also give naval architects and engineers a tool to assess the risk of operating ships with damage, and would ultimately provide ship owners with improved availability of their vessels without compromising safety. The present work is aimed at improving the modelling strategy used in the Phase III work. In Phase III, the hydrodynamic loads on the damaged vessel that incorporates flooding were not computed. Instead, the load on the intact vessel was used to analyse the damaged ship. Also, the correlation between the vertical and the horizontal wave loads was not included in the study.

1.2 OBJECTIVE AND SCOPE

The overall objective of the study is to develop methodologies for assessing the failure probabilities of ship structures with gross damage. The present study is aimed at refining the modelling strategy presented in Phase III. The scope includes:

Improved load models on the damage ship that include the effect of flooding. Improved load models that incorporates the effect of correlation between the horizontal

and the vertical wave bending moments.


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Computing the probabilities of failure of the damaged ship with the improve models for the following cases:

1. Intact ship; 2. Damaged ship with side or bottom damaged to a single midship compartment; 3. Damaged ship with side or bottom damaged to a multiple midship

compartments; and 4. Damaged ship with a multiple bow compartment damage.

1.3 ORGANIZATION OF THIS REPORT

The remainder of this document is organized as follows:

Chapter 2 summarises the overall methodology for the reliability assessment and provides the definition of the vessel characteristics. A very brief review is presented of the approaches and the tools used for the analyses.

Probabilities of failure estimates for ex-HMCS NIPIGON in the intact condition are presented in Chapter 3.

Chapter 4 provides the estimates of the probabilities of failure for ex-HMCS NIPIGON in the damaged condition.

Summaries, conclusions and recommendations are discussed in Chapter 5.

A list is provided in Chapter 6 of the references used in the study.


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2.0 SUMMARY OF OVERALL METHODOLOGY AND DEFINITION OF VESSEL CHARACTERISTICS

2.1 SUMMARY OF OVERALL METHODOLOGY

The methodology used for the study is shown in Figure 2-1. It was developed in the Phase II of the work.

Figure 2-1: Overall Methodology for Estimating Probability of Failure of a Damaged Ship

STEP 1: Define Ship Characteristic

STEP 2: Determine Damage Sizes and Damage Scenarios

Define Vessel Characteristics and Operation Profile

STEP 3: Estimate Load on Damage Ship

Investigate Damage Profile Using SIMCOL and LSDYNA Consider Other Damage Sizes and Locations In Literature Determine extent of Flooding

Still Water with/without Flooding Wave Load with/without FloodingShort Term DurationLoad Combination

TOOL/ INPUT

TOOLSLSDYNASIMCOLCOMPASS

INPUT Ship particulars Operational Profile Accident Scenario

STEP 4: Estimate Ultimate Strength of Damaged Ship

STEP 5: Estimate Deterministic Structural Integrity of Damaged Ship

STEP 6: Estimate Probabilistic Structural Integrity of Damaged Ship

1. Identify and code applicable limit state function2. Calibrate model parameters3. Perform reliability assessment4. Identify values of parameters at failure surface5. Repeat 3, 4 and 5 until convergence is achieved

OUTPUT/ NEED

OUTPUTDamage Sizes and Characteristics3-D FE Model of Damaged Ship2-D ULTMAT model of Damaged Ship Section

TOOLSPARAMARINETRIDENTFDWAVELOAD

INPUTShip particulars Operational Profile

TOOLSULTMATTRIDENT

INPUT Ship particulars Damage Scenario

Determine Suitability of ULTMATConfirm using TRIDENTDevelop Applicable Interaction Equation

INPUT Combined LoadExponents of Interaction EquationResidual Strength of Section

TOOL COMPASS

INPUTStructural Parameters Load Parameters Damage ParametersCombined Load and Coefficient of Interaction Equation

OUTPUTExtreme Value of Wave LoadsStill Water Load Combined Loads

NEEDTo Implement algorithm for extreme load estimation and load combination

OUTPUTValue or Curve Denoting Strength at Selected SectionsCoefficients of Applicable Interaction Equations at Selected Sections

NEEDModels for Interaction EquationCurve Fitting

OUTPUTSafety Factor

NEEDCode for interaction equations

OUTPUTProbability Characterization of Structural, Damage and Load ParametersProbability of FailureImportance of Uncertain Parameters

NEEDCode for interaction equations


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The methodology has six steps: 1. Definition of the ship characteristics and an operational profile. 2. Determination of the damage size and scenarios and development of suitable 3-D

and/or 2-D model representations of the damaged vessel. Determination of the extent of flooding resulting from damage and the change in ship attitude (heel, trim) resulting from flooding;

3. Estimation of the loads on the damaged ships; 4. Estimation of the ultimate strengths of the damaged ship; 5. Estimation of the deterministic structural integrity of the damaged vessel using

appropriate interaction equations; and 6. Estimation of the reliability of the damaged vessel.

2.2 DESCRIPTION OF SELECTED VESSEL

Figure 2-2 shows the ship that is used for the study. A finite element model of the vessel is shown in Figure 2-3. Its structural and hydrodynamics particulars are summarised in Table 2-1. The midship section of the Annapolis Class Destroyer HMCS NIPIGON, a former Canadian naval vessel, decommissioned in 1998, is used in the study. Table 2-2 provides a list of the vessel sections that can be used for structural integrity assessment. The current study uses the midship section of the vessel to assess the ultimate strength.

Figure 2-2: The Annapolis Class Destroyer HMCS NIPIGON (Wikipedia [1])


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Figure 2-3: Finite Element Model of the Ship

Table 2-1: Summary of the Main Particulars of the Ship

Ship Particulars

Ship type Destroyer Length between perpendiculars (m)

108.4

Breadth (m) 12.75 Depth (m) 20.04 Draft (m) 5.18 Displacement (kg) 3.265X106

Number of transverse bulkheads 9 Locations of transverse bulkheads, measured from FP (m)

2.67, 6.93, 12.95, 24.38, 43.51, 68.20, 78.10, 81.76, 94.87

Number of longitudinal bulkheads

0

Smeared shell thickness (mm) 7.62 Smeared deck thickness (mm) 8.59 Smeared bottom thickness (mm) 9.52 Material grade for shell plating Mild Steel


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Table 2-2: Summary of Possible Vessel Sections for Analysis

Location Distance from FP Forward Quarter 2.74 m (1080 in) Forward Midship 4.27 m (1680 in) Midship 5.43 m (2136 in) Aft Mid 6.78 m (2670 in) Aft Quarter 8.14 m (3204 in)

2.3 BRIEF SUMMARY OF APPROACHES AND TOOLS USED IN THE STUDY

Detailed approaches and the tools for estimating the probabilities of failure of a damaged vessel were presented in Akpan et al., 2010, 2011, 2013. For completeness, these approaches and tools are very briefly summarised. Also, the models used for computing the correlation

2.3.1 Approach and Tools Used to Estimate the Damage Size

Damaged to the vessel is assumed to be due to collisions with a Handyman Dry Cargo Carrier. Realistic estimates of the damage sizes are obtained with two tools, SIMCOL and COMPASS. SIMCOL is used to calculate the maximum damaged length and the maximum damaged penetration depth during a ship collision event. It performs a two-fold analysis: external ship dynamics, to estimate collision forces and velocities, and internal ship deformations, to calculate the extent of the damage in the struck ship. The outputs from SIMCOL are fed into COMPASS to estimate the statistics and the probability characteristics of the damaged sizes. Table 2-3 gives the damage statistics.

Table 2-3: Destroyer Damage Statistics

Location Parameter Mean Value

Standard Deviation

40 m from midship Damaged Length (m) 4.0 3.6 Maximum Penetration (m) 1.2 1.1

Midship Damaged Length (m) 4.6 4.0 Maximum Penetration (m) 1.6 1.5

-40 m from midship Damaged Length (m) 4.0 3.6 Maximum Penetration (m) 1.1 1.2

2.3.2 Approaches and Tools Used to Estimate the Loads

The two global load components on the damaged ship considered are the stillwater and the wave loads. The stillwater load is induced by the difference between the weight and the buoyancy distribution along the hull. The wave load is generated by the interaction between the waves and the hull and is represented by accelerations, pressure distributions, and body forces and moments. The magnitudes of these loads depend on many factors, including ship geometry, sea state, and ship speed and heading. In order to evaluate the ship structural


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response with respect to environmental and operational conditions, linear response theory is used. The wave load response amplitude operators, RAO, are determined from hydrodynamic analyses using a 3-D frequency domain boundary approach based on velocity potential theory. In case of flooding, the internal fluid motion in flooded spaces needs to be coupled with the external sea motion (Lin et al., 2009). The capabilities for computing the stillwater and the RAOs of vertical and the horizontal wave bending moments with flooding are encapsulated in WAVELOAD. The tool is used in the current study to compute the loads. The load parameters required for the reliability assessment of an intact and a damaged vessel are summarised in Table 2-4.

Table 2-4: Components of the Loads on a Damaged Ship

Load Component Symbol Still Water Vertical Bending Moment VSWM

Horizontal Bending Moment HSWM Wave Induced Extreme Vertical Bending Moment VWM

Extreme Horizontal Bending Moment HWM Correlation between the vertical and the horizontal wave bending moments

vh

The steps used to compute the loads are summarised below: 1. Estimate the components of the still water loads using WAVELOAD; 2. For a selected operational profile defined by the vessel’s speed, heading and location,

estimate the response amplitude operators, RAOs, of the components of the wave loads on the vessel using WAVELOAD.

3. Apply the methodology outlined in Chapter 5 of Phase II work, (Akpan et. el. (2011)), to estimate an extreme load for an operational condition. The method uses spectral moments of the load and a one hour time frame for each operational condition.

4. Obtain the probability of occurrence of an extreme load in an operational condition by multiplying the probability of the speed, heading and the sea state associated with that operational condition.

5. Use the probability of occurrence of the extreme loads to estimate the extreme statistics, mean values and standard deviations, of the loads.

6. Obtain the correlation coefficient associated with the vertical and horizontal wave loads.

The following two correlation approaches, called Method 1 and Method 2, are proposed:

Method 1: Method 1 uses Equation (1) to calculate the correlation coefficient between the two loads.


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n

ihmih

n

ivmiv

n

ihmihvmiv

vhi

RAORAORAORAO

RAORAORAORAO

1

2,

1

2,

1,,

)()(

))(( (1)

where ivRAO , is a realization of the amplitude derived from the real and imaginary parts of the

vertical response operator at a frequency with a specified operational condition. ihRAO , is a realization of the amplitude derived from the real and imaginary parts of the

horizontal response amplitude operator at a frequency with a specified operational condition. vmRAO is the mean value of the vertical response amplitude operator for a specified

operational condition. hmRAO is the mean value of the horizontal response amplitude operator for a specified

operational condition. It is noted that the response amplitude operators, RAOs are the resultant amplitude from the real and imaginary parts of the response amplitude operator. They have only real values. The numerator of Equation (1) is the covariance between the amplitudes of the vertical and the horizontal response amplitude operators while the denominator is the square root of the variance (standard deviation), of both processes. Because there are many operational conditions in the operational profile, Equation (2) uses the probability of occurrence of each operational condition to compute the effective correlation coefficient between the wave loads;

vhi

n

iivh P

1

(2)

where Pi is the probability of occurrence of operational condition i. Method 2: Equation (3) is the second method used for obtaining the correlation between the vertical and the horizontal bending moment. It is based on suggestions in the literature (Deco et al., 2013, Mansour and Thayamballi, 1994)

dSRAORAO wihivihiv

vhi )Re(1

0

*,,

,,

(3)

where


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dSRAO wiv iv

0

2, )(Am

, is the standard deviation of the vertical wave bending moment of

operational condition i

dSRAO wih ih0

2, )(Am

, is the standard deviation of the horizontal wave bending moment

of operational condition i Re (*) is the real part of the bending moment RAO and Am () is the amplitude of the bending moment RAO. Sw is the spectrum used to capture the sea state. Bretschneider wave spectrum defined in Equation (4) is used in the study.

(4)

Where and p are the wave frequency and the modal or peak energy frequency

respectively and Hs is the significant wave height.

7. Combine the components of the stillwater and the wave loads using appropriate load combination factors.

2.3.3 Approach and Tools Used to Estimate the Ultimate Strength

The components of the ultimate strength used in the reliability assessment are summarised in Table 2-5. ULTMAT, the 2-D progressive collapse tool, is used to compute the sectional ultimate bending moment capacities.

Table 2-5: Components of the Ultimate Strengths of a Damaged Ship

Strength Component Symbol Ultimate Ultimate Vertical Bending Moment VUM

Ultimate Horizontal Bending Moment HUM For a selected cross section of the ship, the interaction curve analysis capability of ULTMAT is used to generate the envelope of the vertical bending moment and the horizontal bending moment structural strengths. A regression and an optimization technique are then used to calibrate the interaction parameters, m and n, defined in Equation (5), for the various quadrants of the interaction curves.

4

5

42

25.1exp16

5 ppsw

HS


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1m

HU

n

VU Mx

My (5)

2.3.4 Approach and Tool Used to Estimate the Reliability

Probabilistic structural reliability methods attempt to estimate the failure probability of a structure. The methods account for the uncertainties with the models and the parameters. It involves defining the performance or limit state function. Equations (6) and Equation (7) are the two limits state function used. Equation (6) is referred to as the linear model and Equation (7) is called the interaction equation model. The parameters in Equation (6) and (7) are defined in Table 2-6.

VWVWVSWVVSWVUVU MxMxMxXg )( (6)

m

HUHU

HWHWHSWHHSW

n

VUVU

VWVWVSWVVSW

MxMxMx

MxMxMxXg 1)( (7)

Once the limit state function is defined, the reliability of the vessel is the likelihood of it functioning according to its designed purpose. The failure probability is one minus the reliability. The reliability of the vessel can be computed using Equation (6) or Equation (7). The failure domain ( ) is the negative performance function (that is, ]0)([ Xg ), while its compliment ( ]0)([' Xg ) defines the safe region. The failure probability is computed using Equation (8) dXXfPf )( (8)

where )(Xf denotes the joint probability density function of the basic random variables, X. COMPASS is used for the reliability assessment of the intact and the damaged ship. Several practical approaches for computing failure probabilities are available in COMPASS, including the first order reliability methods (FORM) and the Monte Carlo Simulation (MCS). These two methods are used in the study. Identification of the main sources of uncertainty that impact the failure probability is carried out by computing the probability sensitivity measures. COMPASS has capabilities for computing several parameter sensitivity measures based on FORM results. The importance factors, i, are used to determine the sources of uncertainties that drive failure. i

2 are computed. i is given by

*)(*)(

UgUg

(9)


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where g is the limit state function, U* is the most probable point in the standard normal space (u-space) and is the gradient operator. The u-space is defined by the mean value of the random variable divided by the standard deviation. Importance factors express the relative importance of the different sources of uncertainty associated with the basic random variables that define a problem.

Table 2-6: A Summary of the Random Variables

Name Symbol Mean Value COV Probability Distribution

Ultimate Vertical Bending Moment Capacity (kNm),

VUM Depends on vessel condition,

cross section etc.

Depends on vessel

condition, cross section etc.

Weibull

Modelling Uncertainty Factor for Ultimate Vertical Bending Moment Capacity,

VUx 1 0.10 Normal

Vertical Wave Bending Moment (kNm),

VWM Depends on operational profile, hull form, weight

distribution etc.

Depends on operational profile, hull form, weight

distribution etc.

Gumbel

Modelling Uncertainty Factor for Vertical Wave Bending Moment,

VWx 1 0.10 Normal

Vertical Still Water Bending Moment (kNm),

VSWM Depends on vessel hull form,

weight distribution etc.

0.10 Normal

Modelling Uncertainty Factor for Vertical Still Water Bending Moment

VSWx 1 0.10 Normal

Vertical Load Combination Factor,

V 1 Fixed

Ultimate Horizontal Bending Moment Capacity (kNm),

HUM Depends on vessel condition,

cross section etc.

Depends on vessel

condition, cross section etc.

Weibull

Modelling Uncertainty Factor for Ultimate Horizontal Bending Moment Capacity,

HUx 1 0.10 Normal

Horizontal Wave Bending Moment (kNm),

HWM Depends on operational profile, hull form weight

distribution etc.

Depends on operational profile, hull form, weight

distribution etc.

Gumbel


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Name Symbol Mean Value COV Probability Distribution

Modelling Uncertainty Factor for Horizontal Wave Bending Moment,

HWx 1 0.10 Normal

Horizontal Still Water Bending Moment (kNm),

HSWM Depends on vessel hull form,

weight distribution etc.

Depends on Vessel

Normal

Modelling Uncertainty Factor Horizontal Still Water Bending Moment,

HSWx 1 0.10 Normal

Horizontal Load Combination Factor,

H 1 Fixed

Vertical Bending Moment Interaction Coefficient

m Depends on operational profile, hull form weight

distribution etc.

Depends on operational profile, hull form weight

distribution etc.

Fixed

Horizontal Bending Moment Interaction Coefficient

n Depends on operational profile, hull form weight

distribution etc.

Depends on operational profile, hull form weight

distribution etc.

Fixed


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3.0 RELIABILITY ASSESSMENT OF THE INTACT VESSEL

3.1 INTRODUCTION

In Phase III, the failure probability of the intact ship was computed using linear limit state, which involves only the vertical ultimate capacity and loads, and the interaction equation limit state that uses the vertical and the horizontal ultimate capacities and loads. Analyses were based on the mid-ship section. The effect of correlation between the vertical and the horizontal loads, neglected in Phase III, are included in the current study.

3.2 LOADS ON THE INTACT VESSEL

3.2.1 Operational Profile

The operation profile defined in Table 3-1 is used.

Table 3-1: Operational Profile Used Vessel Speed (knots) % Time Spent

Case 1 3 50 6 50

Case 2 6 50 9 50

Case 3 9 50 12 50

Headings(Degrees) % Time Spent 0 5 45 30 90 20 135 30 180 15

Location: North Atlantic; Spectrum: Bretschneider (Hs, Tz)

Hs(m) Tz(sec) % Time Spent in Each State 6.5 9.00 20 4.5 7.75 40 2.5 5.50 40

Vessel Loading Condition - 3,474,900 kg

3.2.2 Still Water Bending Moment

Figure 3-1and Figure 3-2 show the still water loads. Since the structural integrity assessment for the intact case is based on the midship section, only the still water loads at this location are relevant to the current study. Table 3-2 summarises the probabilistic characteristic of the still water loads used.


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Figure 3-1: The Vertical Still Water Bending Moment on the Intact Vessel

Figure 3-2: The Vertical Stillwater Shear Force on the Intact Vessel

Table 3-2: Probabilistic Characteristic of the Still Water Loads

Name Mean Value COV Probability Distribution Vertical Still Water Bending Moment (kNm)

37087.20 0.1 Normal

Horizontal Still Water Bending Moment (kNm)

0.00 Fixed

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

3.0E+04

3.5E+04

4.0E+04

0 20 40 60 80 100 120

Ben

ding

Mom

ent,

Vert

ical

(kN

m)

Longitudinal Location (m)

-2.0E+03

-1.5E+03

-1.0E+03

-5.0E+02

0.0E+00

5.0E+02

1.0E+03

1.5E+03

2.0E+03

0 20 40 60 80 100 120

Shea

r For

ce, V

ertic

al (k

N)

Longitudinal Location (m)


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3.2.3 Wave-Induced Bending Moment

3.2.3.1 RAO of Wave Loads on the Midship Section

Figure 3-3 shows typical value of the RAOs of the wave loads.

(a)

(b)

Figure 3-3: RAOs of the Wave Bending Moment at 12 knots on the Intact Vessel (a) Vertical (b) Horizontal

3.2.4 Extreme Value of the Wave Loads

The cumulative distribution function, CDF, of the extreme wave loads are shown in Figure 3-4

and Figure 3-5.


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Table 3-3 summarises their probabilistic characteristics.

Figure 3-4: CDF of the Extreme Vertical Wave Bending Moment on the Intact Ship

Figure 3-5: CDF of the Extreme Horizontal Wave Bending Moment on the Intact Ship


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Table 3-3: Probabilistic Characteristic of the Extreme Wave Loads on the Intact Ship

Load Case

Name Mean Value (kNm)

COV Correlation Coefficient

Probability Distribution

Method 1 Method 2 3 and 6 knots

VWM 47729.97

0.60

0.90 -0.47 Gumbel

HWM 18887.04

0.74 Gumbel

6 and 9 knots

VWM 49926.87

0.63 0.85 -0.49 Gumbel

HWM 19368.84

0.76 Gumbel

9 and 12 knots

VWM 50460.98 0.64 0.82 -0.50 Gumbel

HWM 18849.83

0.78 Gumbel

3.3 ULTIMATE STRENGTHS OF THE INTACT VESSEL

The ultimate strength of the intact ship mid section is shown in Figure 3-6. The +y quadrants (1 and 2) represent hogging mode and the –y quadrants represent the sagging modes. Figure 3-6 and Table 3-4 summarise the ultimate bending moment capacities for the intact vessel.

Figure 3-6: Ultimate Bending Capacity of the Intact Vessel

-4.E+05

-3.E+05

-2.E+05

-1.E+05

0.E+00

1.E+05

2.E+05

3.E+05

-4.E+05 -3.E+05 -2.E+05 -1.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05

Vert

ical

BM

Cap

acity

(kN

m)

Horizontal BM Capacity (kNm)


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Table 3-4: Ultimate Bending Moment Capacities of the Intact Ship

Load Case Name Mean Value COV Probability Distribution Quadrant -1: Hogging

VUM (kNm) 216622.80 0.04 Weibull

HUM (kNm) 293815.80 0.04 Weibull n 2.50 Fixed m 2.50 Fixed

Quadrant -2: Hogging



Quadrant -3: Sagging



Quadrant -4: Sagging



3.4 RELIABILITY OF THE INTACT SHIP

The probabilistic characteristics of the random variables are summarised in Table 3-5.

Table 3-5: Random Variables Used for the Intact Vessel Analysis

Name Mean Value COV Probability Distribution

Ultimate Vertical Bending Moment Capacities (kNm), VUM

Depends on Quadrant

Table 3-4

0.04 Weibull

Modelling Uncertainty Factor for Ultimate Vertical Bending Moment Capacities, VUx

1 0.10 Normal

Vertical Wave Bending Moment (kNm), VWM

Depends on Speed Case Table 3-3


Gumbel

Modelling Uncertainty Factor for Vertical Wave Bending Moment, VWx

1 0.10 Normal

Vertical Still Water Bending Moment (kNm), VSWM

33534.22 0.10 Normal


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Modelling Uncertainty Factor for Vertical Still Water Bending Moment VSWx

1 0.10 Normal

Vertical Load Combination Factor, V 1 Fixed Ultimate Horizontal Bending Moment Capacity (kNm), HUM

Depends on Quadrant

Table 3-4

0.04 Weibull

Modelling Uncertainty Factor for Ultimate Horizontal Bending Moment Capacities,

HUx

1 0.10 Normal

Horizontal Wave Bending Moment (kNm), HWM



Gumbel

Modelling Uncertainty Factor for Horizontal Wave Bending Moment, HWx

1 0.10 Normal

Horizontal Still Water Bending Moment (kNm), HSWM

1643.32 0.10 Normal

Modelling Uncertainty Factor Horizontal Still Water Bending Moment, HSWx

1 0.10 Normal

Horizontal Load Combination Factor, H 1 Fixed Vertical Interaction Coefficient m Depends on

Quadrant Table 3-4

Fixed

Horizontal Interaction Coefficient n Depends on Quadrant

Table 3-4

Fixed

Correlation Coefficients Between Vertical and Horizontal Wave Bending Moments

Depends on Load Case Table 3-3

The results of the structural integrity assessment for the sagging and the hogging failure modes are summarised in Table 3-6 and Table 3-7 respectively. Both FORM and MCS results are presented. As Monte Carlo Simulation method is generally considered the most robust approach for reliability assessment, conclusions on failure probabilities are based on it. FORM results are included for comparison purposes and also because sensitivity analyses results are based on FORM. FORM and Monte Carlo Simulation estimates of reliabilities are very close to each other.

The hogging mode governs the failure of the intact ship. The estimate of the failure probability is dependent on the model used and the correlation between the horizontal and the vertical wave bending moment. When the correlation between the wave loads is neglected, the interaction model always gives estimates of failure probabilities that are bigger than the linear


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model. Estimates of the correlation between the wave loads are positive for Method 1 and negative for Method 2. A positive correlation implies the horizontal and vertical wave loads are in sync (phase) that is the value of the two load types increase or decrease at the same time, and a negative correlation means they are out of sync (out of phase), that is when one load type increases the other one decreases and vice versa. For all operational scenarios, the largest failure probabilities are obtained with correlation from Method 1 and the smallest failure probabilities are obtained with correlation from Method 2. These results underscore the need to use accurate load models that include the correlation between the load components. Since the nature of the correlation, positive or negative, is not known without computation, a neglect of correlation could result in non-conservative estimates of failure probabilities. The higher the vessel speed the higher the probability of failure and vice versa. FORM and Monte Carlo Simulation estimates of reliabilities are very close to each other. The results underline the need to use strength and load models that best reflect the damage under consideration.

Table 3-6: Probabilities of Failures of the Intact Ship (Sagging Mode) Speed Case

Failure Probability (Linear Model)

Failure Probability (Interaction Model)

No Correlation Method 1: Correlation Method 2: Correlation

FORM MCS FORM MCS FORM MCS FORM MCS 3 and 6 5.92E-06 5.27E-06 6.87E-06 6.36E-06 1.66E-05 1.76E-05 2.34E-06 2.40E-06 6 and 9 1.20E-05 1.15E-05 1.39E-05 1.32E-05 2.91E-05 2.91E-05 4.89E-06 4.79E-06 9 and 12 2.67E-05 2.56E-05 3.03E-05 2.97E-05 5.76E-05 5.74E-05 9.45E-06 9.54E-06

Table 3-7: Probabilities of Failures of the Intact Ship (Hogging Mode)

Speed Case




FORM MCS FORM MCS FORM MCS FORM MCS 3 and 6 3.52E-04 3.10E-04 3.55E-04 3.35E-04 3.83E-04 3.72E-04 1.72E-04 1.61E-04 6 and 9 5.99E-04 5.24E-04 6.03E-04 5.68E-04 6.53E-04 6.40E-04 2.65E-04 2.60E-04 9 and 12 1.00E-03 8.94E-04 1.02E-03 9.67E-04 1.12E-03 1.08E-03 4.88E-04 4.78E-04

The importance factors for the random variables involved in the reliability analyses are computed for all load and strength cases. The results for the importance factor are similar for all the cases considered. Typical results are summarised in Figure 3-7 for Load Case 3 and hogging failure mode. The importance factors express the relative importance of the different sources of uncertainty associated with the basic random variables that define a problem. For the three limit state models under consideration, linear, uncorrelated interactive and correlated interactive, uncertainties in four parameters governs the failure of the intact vessel. The four parameters are the vertical wave bending moment/horizontal wave bending moment, the modelling uncertainty of the ultimate vertical bending moment, the modelling uncertainty of


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the vertical wave bending moment and the ultimate vertical bending moment govern the failure of the intact vessel. Of the four, the vertical wave bending moment/ horizontal wave bending moment are the main parameter that governs the failure of the intact vessel. This is followed by the uncertainties in the model used to estimate the ultimate vertical bending moment. The high importance attributed to the model used to estimate the ultimate vertical bending moment underscores the need to ensure that an accurate structural model that reflects the failure under consideration is used for the assessment. It is also noted that the higher the vessel speed, as represented by the higher values and uncertainties in the vertical bending moment, the higher the importance factor of the vertical bending moment. This demonstrates the importance of using a model that accurately estimates the RAOs on the vessel for the speed under consideration.


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(a) Linear Model

(b) Interactive Model –Correlated (Method 2)

(c) Interactive Model –Uncorrelated

Figure 3-7: Importance Factor for the Random Variables

68%

3%

9%

20%

Importance Factor for the Random Variables

Vertical Wave Bending Moment

Ultimate Vertical Bending Moment Capacity

Modelling Uncertainty Vertical Wave BendingMoment

Modelling Uncertainty Ultimate VerticalBending Moment Capacity

69% 2%

9%

20%


Vertical & Horizontal Wave BendingMomentUltimate Vertical Bending Moment Capacity

Modelling Uncertainty Vertical WaveBending MomentModelling Uncertainty Ultimate VerticalBending Moment Capacity

68%

3%

9%

20%



Ultimate Vertical Bending Moment Capacity

Modelling Uncertainty Vertical Wave BendingMomentModelling Uncertainty Ultimate VerticalBending Moment Capacity


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4.0 RELIABILITY ASSESSMENT OF THE DAMAGED SHIP

4.1 INTRODUCTION

In Phase III, the loads on the intact ship were used to compute the failure probabilities of the damaged ship. The loads on the damaged ship were not estimated or used in the analyses. In this chapter flooding is explicitly modeled in the hydrodynamic analyses to provide more realistic loading on the damaged ship. Furthermore, correlation between the vertical and the horizontal is included in the current study. It was not considered in Phase III. The structural integrity analyses are performed on the mid-section of the damaged ship. The following damaged scenarios are investigated:

Side or bottom damaged to a single mid compartment with 75% and 60% flooding; Side or bottom damaged to two mid compartments with 60% flooding; and 100% flooding of two bow compartments with no structural damage.

4.2 DAMAGE DEFINITION

For completeness, the statistics of the damage lengths and penetrations on the mid-section, which were estimated in the previous phase, are summarised in Table 4-1. Although the damage sizes are presented in the form of damage lengths and penetrations, the strength of the damaged ship is executed in ULTMAT with spherical damage sizes of 2 m, 3 m, 4 m, 5 m, and 6 m diameters.

Table 4-1: Statistics of the Damage on the Midship Section

Location Parameter Mean Value

Standard Deviation

Midship Damaged Length (m) 4.6 4.0 Maximum Penetration (m) 1.6 1.5

4.3 LOADS ON THE DAMAGED SHIP

4.3.1 Operational Profile

The operational profile in Table 3-1 is used to study the structural integrity of the damaged ship. Figure 4-1 shows the flooding scenarios that are used.


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(a)

(b)

(c)

Figure 4-1: Flooding Scenarios Used for the Damaged Ship (a) 1 Mid Compartment (b) 2 Mid Compartments (c) 2 Bow Compartments

4.3.2 Still Water Bending Moment

Figure 4-2and Figure 4-3 show the stillwater loads. The values at the mid-section used for analyses are summarised in Table 4-2.


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Figure 4-2: The Vertical Stillwater Bending Moment on the Damaged Vessel

Figure 4-3: Vertical Stillwater Shear Force on the Damaged Vessel


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Table 4-2: Probabilistic Characteristic of the Still Water Load on the Damaged Ship

Name Mean Value COV Probability Distribution Intact

VSWM (kNm) 37087.20 0.10 Normal

HSWM (kNm) 0.00 Fixed 1 Mid Compartment 74% Flooding

VSWM (kNm) -4729.79 0.10 Normal




VSWM (kNm) -4750 0.10 Normal

HSWM (kNm) 0.00 Fixed 2 Bow Compartment 100 Flooding


HSWM (kNm) 0.00 Fixed

4.3.3 Wave-Induced Bending Moment

4.3.3.1 RAO of Wave Loads on the Midship Section

The RAOs for the wave bending moments, torsional moments and shear forces for the operational speeds (3, 6, 9 and 12 knots), and the headings (0, 45, 90, 135 and 180 degrees) in Table 3-1 are computed. For brevity, only the RAOs of the wave bending moments at 12 knot speed are shown in Figure 4-4 to Figure 4-13. In general, the RAOs of the vertical wave bending moments are not very sensitive to the flooding scenarios, while the RAOs of the horizontal wave bending moments are sensitive. This is due to the sloshing of the fluid in the damage vessel. Furthermore, the RAOs of the horizontal wave bending moments are significantly smaller than the vertical wave bending moments. For all cases, except bow flooding, the flooding models are symmetric, that is, the heel angle is zero resulting in zero horizontal wave bending moment RAO values. The flooding model for bow flooding is however asymmetric, with non-zero heel angle resulting in non-zero values for the horizontal wave bending moments (See Figure 4-5 and Figure 4-13). The correlation between the vertical and the horizontal wave bending moments are summarised in Table 4-3. Estimates of the correlation between the loads are positive for Method 1 and negative for Method 2. A positive correlation means the two component of wave load, horizontal and vertical, are in sync (phase), that is, the value of the two load types increase at the same time and vice versa. A negative correlation means they are out of sync (out of phase), that is, when one load type increases the other decreases and vice versa. Estimates from the two models are out of sync,


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both methods cannot be correct. Method 2 is backed with literature, and also includes phase information (imaginary and real parts of the RAOs) and the sea spectrum. Method 1 does not consider the phase information or the spectrum. It is based solely on the amplitude of the RAOs. Therefore, Method 2 is taken as the best representative of the ‘actual’ correlation between the applied wave loads. Method 1 is used to gain insight on the effect of positive correlation and is included in the report for completeness

Figure 4-4: RAOs of the Vertical Wave Bending Moments at 12 knots, 0 heading

Figure 4-5: RAOs of the Horizontal Wave Bending Moments at 12 knots, 0 Heading


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Figure 4-7: RAOs of the Horizontal Wave Bending Moments at 12 knots, 45 heading


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Table 4-3: Correlation between the Vertical and Horizontal Wave Loads

Vessel Condition Speed (knots) Correlation between VWM and HWM Method 1 Method 2

Intact 3 and 6 0.90 -0.47 6 and 9 0.85 -0.49 9 and 12 0.82 -0.50

1 Mid Compartment 56% Flooding

3 and 6 0.78 -0.23 6 and 9 0.76 -0.19 9 and 12 0.74 -0.17

1 Mid Compartment 74% Flooding

3 and 6 0.76 -0.22 6 and 9 0.74 -0.18 9 and 12 0.72 -0.18

2 Mid Compartments 56% Flooding

3 and 6 0.71 -0.25 6 and 9 0.65 -0.24 9 and 12 0.61 -0.24

2 Bow Compartments 100% Flooding

3 and 6 0.91 -0.48 6 and 9 0.82 -0.50 9 and 12 0.73 -0.51

4.3.4 Extreme Value of the Wave Loads

The CDF of the extreme vertical and horizontal wave loads are presented in Figure 4-14 and Figure 4-15 respectively. The magnitude of an extreme load depends on its operational condition. There is a large scatter in the extreme loads. For example, the magnitudes of the extreme horizontal bending moments varies from 5.0E-02 to 6.0E+04, a difference of about six orders of magnitude, and the extreme vertical bending moments varies from 5.0E+03 to 1.4.0E+05, a difference of almost two orders of magnitude. The scatter can be characterised

by the coefficient of variation, COV, averagedeviationstandard . Table 4-4 summarises

the statistics (average or mean value, COV) of the extreme loads. The minimum COV is 0.54 and the maximum is about 0.78. These are large values. Most reported studies typically use COVs of about 0.20 for wave bending moments. One practical approach that can be used to narrow the COV is to eliminate from consideration the operational conditions with very small extreme values. However, there is no guidance in the literature. The effect of removing the small extreme loads needs to be investigated. It will alter the operational profile and impact estimates of both the correlation and the failure probabilities. The current study uses all the operational conditions in Table 3-1 regardless of the size of their extreme values. The statistics in Table 4-4 are used in the assessments. It is assumed that the extreme loads are characterised by Gumbel probability distributions. This was not confirmed with the extreme data from the operational profile, it might not be the best distribution for the current data. Since the raw data is available, the most suitable distribution can be determined and used in future study. Algorithms for determining the best distribution are available in COMPASS.


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Figure 4-14: Cumulative Distribution of the Extreme Vertical Wave Bending Moments

Figure 4-15: Cumulative Distribution of the Extreme Horizontal Wave Bending

Moments

Table 4-4: Statistics of the Extreme Vertical and Horizontal Wave Bending Moments


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Speed Vertical Wave Bending Moment

Horizontal Wave Bending Moment

Mean (kNm) COV Mean (kNm) COV 2 Mid Compartment 60% Flooding

9 and 12 54860.35 0.59 25676.63 0.69 6 and 9 54805.69 0.57 25752.54 0.68 3 and 6 55044.80 0.54 26158.26 0.67

1 Mid Compartment 74 % Flooding 9 and 12 54198.30 0.63 26394.37 0.69 6 and 9 53680.04 0.60 26549.34 0.67 3 and 6 53519.37 0.59 27061.77 0.65

1 Mid Compartment 56 % Flooding 9 and 12 52760.42 0.64 25219.69 0.69 6 and 9 52301.75 0.62 25338.63 0.67 3 and 6 52197.84 0.60 25537.19 0.66

Bow Flooding 9 and 12 50922.51 0.63 21865.19 0.74 6 and 9 50693.96 0.61 22284.96 0.73 3 and 6 50832.11 0.59 23347.37 0.71

Intact 9 and 12 50460.98 0.64 18849.83 0.78 6 and 9 49926.87 0.63 19368.84 0.76 3 and 6 47729.97 0.60 18887.04 0.74

4.4 ULTIMATE STRENGTHS OF THE DAMAGED SHIP

Figure 4-16 shows the two damaged locations, side and bottom, used to determine the structural strength. The statistics of the damaged strength, mean values and standard deviation, are computed using strength estimates for spherical damaged sizes of 2 m, 3 m, 4 m, 5 m, and 6 m diameters with equal probability of occurrence. The interaction curves for the various damage sizes are shown in Figure 4-17. The +y quadrants (1 and 2) represent hogging mode and the –y quadrants represent the sagging modes. The interaction coefficients are estimated using the mean values of the horizontal and the vertical bending moment capacities. Figure 4-18 shows the average ultimate strength of the intact, and the bottom and side damaged ship. The bottom damage results in larger reduction in the strength. The statistics of the ultimate bending moment capacities, for the sagging and hogging damage modes, are summarised in Table 4-5.


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(a) (b)

Figure 4-16: Damaged Locations Used for Ultimate Strength Analysis (a) Side (b) Bottom

(a)

(b)

Figure 4-17: Ultimate Strength of the Damaged Ship Based on FE Calibrated Model (a) Bottom (b) Side

-3.E+05

-2.E+05

-1.E+05

0.E+00

1.E+05

2.E+05

3.E+05

-4.E+05 -3.E+05 -2.E+05 -1.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05

Vert

ical

BM

Cap

acity

(kN

m)


No Damage 2-m Damage 4-m Damage 6-m Damage Average Damage

-3.E+05

-2.E+05

-1.E+05

0.E+00

1.E+05

2.E+05

3.E+05

-4.E+05 -3.E+05 -2.E+05 -1.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05

Vert

ical

BM

Cap

acity

Horizontal BM Capacity No Damage 2-m Damage 4-m Damage 6-m Damage Average Damage


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Figure 4-18: Ultimate Strength of the Damaged Ship

Table 4-5: Probabilistic Characteristic of the Damaged Ship Ultimate Bending Capacities

Analysis Case


Side Bottom Side Bottom Quadrant 1: Hogging

VUM(kNm)

201379.88 125751.30 0.07 0.20 Weibull

HUM(kNm)

262686.62 265379.4 0.10 0.08 Weibull

n 2.50 2.30 Fixed m 2.20 2.20 Fixed

Quadrant 2: Sagging

VUM(kNm)

206310.40 125080.11 0.06 0.20 Weibull

HUM(kNm)

253467.94 234143.13 0.11 0.11 Weibull

n 2.70 1.80 Fixed m 2.70 2.60 Fixed

Quadrant 3: Hogging

VUM(kNm)

274566.52 225072.62 0.05 0.15 Weibull

HUM(kNm)

237392.79 254087.35 0.10 0.06 Weibull

n 1.70 2.10 Fixed m 1.70 2.30 Fixed

-3.E+05

-2.E+05

-1.E+05

0.E+00

1.E+05

2.E+05

3.E+05

-4.E+05 -3.E+05 -2.E+05 -1.E+05 0.E+00 1.E+05 2.E+05 3.E+05 4.E+05

Vert

ical

BM

Cap

acity

(kN

m)


No Damage Average Bottom Damage Average Side Damage


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Analysis Case


Side Bottom Side Bottom Quadrant 4: Sagging

VUM(kNm)

259317.54 217359.23 0.07 0.13 Weibull

HUM(kNm)

238062.99 261852.87 0.10 0.06 Weibull

n 1.70 1.60 Fixed m 1.70 1.70 Fixed

4.5 ESTIMATES OF THE RELIABILITY OF THE DAMAGED SHIP

The probabilistic characteristics of the random variables used to estimate the failure probabilities is summarised in Table 4-6. The results of the structural integrity assessment are summarised in Table 4-7 for the hogging mode and Table 4-8 for the sagging mode. Both FORM and MCS results are presented. Conclusions are based on the Monte Carlo Simulation method. FORM results are included for comparison purposes and also to draw conclusions on parametric sensitivity analyses. In general, FORM and MCS estimates of failure probabilities are very close to each other. The magnitude of the failure probabilities depend on several factors including, the failure mode, the strength model, the damaged scenarios (bottom or side), the load models (flooding locations and magnitudes, the correlation between the horizontal and the vertical wave loads), and the vessel speeds. Several general conclusions are observed from the tables. Higher vessel speeds results in the higher failure probabilities and vice versa. The hogging mode governs the failure of the damaged ships because estimates of the hogging failure probabilities are always higher than sagging. When the correlation between the horizontal and vertical loads is neglected the interaction equation model always gives estimates of failure probabilities that are bigger than the linear model, for both the hogging and the sagging failure modes.

For all the damaged and loading scenarios, the smallest failure probabilities (MCS results, last column Table 4-7 and Table 4-8 ) are obtained when the correlation estimates from Method 2 (negative correlations) are included in the structural integrity reliability assessment. The largest failure probabilities are obtained with correlations from Method 1 (MCS results, third to the last column Table 4-7 and Table 4-8). Since Method 1 results in positive correlations between the vertical and the horizontal wave bending moment, and Method 2 gives negative correlation values, it can be concluded that a positive correlation between the loads results in higher estimates of failures probabilities and vice versa. Because it is not possible to determine the nature and magnitude of the correlation (positive or negative) without


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Table 4-6: Summary of the Damaged Vessel Random Variables

Name Mean Value

COV Probability Distribution

Ultimate Vertical Bending Moment Capacities (kNm), VUM

Depends on Quadrant Table 4-5

0.04 Weibull

Modelling Uncertainty Factor for Ultimate Vertical Bending Moment Capacities, VUx

1 0.10 Normal

Vertical Wave Bending Moment (kNm), VWM Depends on Speed Case Table 4-4


Gumbel

Modelling Uncertainty Factor for Vertical Wave Bending Moment, VWx

1 0.10 Normal

Vertical Still Water Bending Moment (kNm), VSWM

Table 4-2 0.10 Normal

Modelling Uncertainty Factor for Vertical Still Water Bending Moment VSWx

1 0.10 Normal

Vertical Load Combination Factor, V 1 Fixed Ultimate Horizontal Bending Moment Capacity (kNm), HUM

Depends on Quadrant Table 4-5

Intact 0.04 Weibull

Modelling Uncertainty Factor for Ultimate Horizontal Bending Moment Capacities, HUx

1 0.10 Normal

Horizontal Wave Bending Moment (kNm), HWM

Depends on Speed Case

Gumbel

Modelling Uncertainty Factor for Horizontal Wave Bending Moment, HWx

1 0.10 Normal

Horizontal Still Water Bending Moment (kNm), HSWM

0.0 0.0 Fixed

Modelling Uncertainty Factor Horizontal Still Water Bending Moment, HSWx

1 0.10 Normal

Horizontal Load Combination Factor, H 1 Fixed Vertical Interaction Coefficient m Depends on

Quadrant Table 4-5

Fixed

Horizontal Interaction Coefficient n Depends on Quadrant Table 4-5

Fixed

Correlation Coefficient Between Vertical and Horizontal Wave Bending Moments

Depends on Load Case Table 4-3


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computing it, correlation needs to be included in the reliability assessment of damaged ships. This is an issue that is lacking in most of the reported work on damaged ship. These results underscore the need to use accurate load models.


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Table 4-7: Hogging Failures Probabilities of Damaged Ship Speed Case



No Correlation Method 1: Correlation Method 2: Correlation FORM MCS FORM MCS FORM MCS FORM MCS

One Mid Compartment with Bottom Damage and 56% Flooding

3 and 6 5.67E-02 5.56E-02 5.79E-02 5.69E-02 5.98E-02 6.07E-02 5.42E-02 5.08E-02 6 and 9 6.10E-02 5.92E-02 6.20E-02 6.09E-02 6.38E-02 6.47E-02 6.02E-02 5.47E-02 9 and 12 6.69E-02 6.45E-02 6.81E-02 6.67E-02 6.96E-02 7.07E-02 6.39E-02 5.99E-02

One Mid Compartment with Bottom Damage and 74% Flooding


One Mid Compartment With Side Damage and 56% Flooding


One Mid Compartment with Side Damage and 74% Flooding


Two Mid Compartments with Bottom Damage and 60% Flooding


Two Mid Compartments with Side Damage and 60% Flooded


Two Damaged Bow Compartment 100% Flooded


Intact



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Table 4-8: Sagging Failures Probabilities of Damaged Ship Speed Case




FORM MCS FORM MCS FORM MCS FORM MCS

One Mid Compartment with Bottom Damage and 56% Flooded 3 and 6 4.20E-04 4.20E-04 4.37E-04 4.27E-04 6.16E-04 6.53E-04 3.81E-04 3.61E-04 6 and 9 5.15E-04 5.19E-04 5.39E-04 5.29E-04 7.54E-04 8.01E-04 4.69E-04 4.39E-04 9 and 12 6.67E-04 6.68E-04 6.97E-04 6.87E-04 9.87E-04 1.04E-03 5.99E-04 5.79E-04

One Mid Compartment with Bottom Damage and 74% Flooded


One Mid Compartment With Side Damage and 56% Flooded


One Mid Compartment with Side Damage and 74% Flooded


Two Mid Compartments with Bottom Damage and 60% Flooded


Two Mid Compartments with Side Damage and 60% Flooded


Two Damaged Bow Compartment 100% Flooded


Intact



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To gain insight on the impact of damage and loading scenarios, the Monte Carlo Simulation results for the most critical cases (Hogging mode, Speed of 9 and 12 knot) are summarised in Table 4-9. The statistics of the applicable extreme wave load are presented in Table 4-10.

Table 4-9: Hogging Failures Probabilities for Case 3 (9 & 12 Knots)

Failure Probability (Linear Model

Failure Probability (Interaction Equation Model)

No Correlation Method 2 (Correlation)

1 Mid Compartment, Bottom Damage, 56% Flooding 6.45E-02 6.67E-02 5.99E-02

1 Mid Compartment, Bottom Damage 74% Flooding 6.88E-02 6.87E-02 6.16E-02

2 Mid Compartments, Bottom Damage 60% Flooding 6.83E-02 6.47E-02 6.14E-02

1 Mid Compartment, Side Damage, 56% Flooding 4.79E-04 4.89E-04 4.27E-04

1 Mid Compartment, Side Damage, 74% Flooding 5.39E-04 5.48E-04 4.77E-04

2 Mid Compartments, Side Damage 60% Flooding 5.11E-04 6.04E-04 5.00E-04

2 Bow Compartment 100% Flooding 3.32E-03 3.12E-03 1.97E-03

Intact 8.94E-04 9.67E-04 4.78E-04

Table 4-10: Statistics of the Extreme Wave Bending Moments Case 3 (9 & 12 Knots)

Loading Condition Vertical Wave Bending Moment

Horizontal Wave Bending Moment

Method 2 Correlation

Mean (kNm) COV Mean (kNm) COV 1 Mid Compartment 74 % Flooding 54198.30 0.63 26394.37 0.69 -0.18

2 Mid Compartment 60% Flooding 54860.35 0.59 25676.63 0.69 -0.24

The failure probabilities of the bottom damaged ship are always higher than the side damaged ship. For the bottom damaged flooding scenarios, the 1 mid compartment with 74% flooding has the highest failure probabilities, followed by the 2 mid compartments with 60 % flooding. The statistics of the associated extreme loads (Table 4-10) indicate that, the mean values of the two loads differ by roughly 1.2%, with the 2 mid having the bigger value, while the COV, a measure of scatter, differ by 6.8 % with the 1 mid having the bigger value. In addition, the wave load components of the 2 mid have higher negative correlation than the 1 mid (-0.24 versus -0.18). Therefore, it can be inferred that the failure probabilities of the 1-mid are higher


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because of the wider scatter in its wave load and the reduced influence of the negative correlation between its wave loads. These results again underscore the need for accurate load models and statistics.

Typical results of the importance factors, which express the relative importance of the different sources of uncertainty associated with the basic random variables that define a problem, are presented in Figure 4-19 and Figure 4-20. The results are for the most critical damage scenarios, hogging, Load Case 3 (9 and 12 knots).

For the bottom damage scenario, and the three limit state models under consideration, linear, uncorrelated interactive and correlated interactive (Method 2), uncertainties in four parameters governs the failure of the vessel. The four parameters are the vertical wave bending moment/horizontal wave bending moment, the ultimate vertical bending moment capacity, the modelling uncertainties of the ultimate vertical bending moment, and the modelling uncertainty of the vertical wave bending moment. Of the four, the uncertainties in the vertical wave bending moment/ horizontal wave bending moment is the main parameter that governs the failure of the vessel. This is followed by the uncertainties in the ultimate vertical bending moment capacity.

For the side damage scenario, again uncertainties in the same four parameters govern the failure of the vessel. Again of the four, the uncertainties in the vertical wave bending moment/ horizontal wave bending moment is the main parameter that governs the failure of the vessel. However this time, it is followed by the uncertainties in the model used to represent the ultimate vertical bending moment capacity. The high importance attributed to the model used to estimate the ultimate vertical bending moment capacity underscores the need to ensure that an accurate structural model that reflects the structural failure and damage under consideration is used for the assessment. It is also noted that the higher the vessel speed, as represented by the higher values and uncertainties in the vertical bending moment, the higher the importance factor of the vertical bending moment/horizontal wave bending moment. This demonstrates the importance of using a model that accurately estimates the RAOs on the vessel for the speed under consideration.


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(a) Linear Model

(b) Interactive Model (Method 2 Correlation)

(c) Interactive Model Uncorrelated

Figure 4-19: Importance Factor for the Side Damaged 74% Flooded Ship Random Variables

66%

7%

10% 17%



Ultimate Vertical Bending MomentCapacity

Modelling Uncertainty Vertical WaveBending Moment

Modelling Uncertainty Ultimate VerticalBending Moment Capacity

68% 7%

9% 16%


Vertical & Horizontal Wave BendingMomentUltimate Vertical Bending MomentCapacityModelling Uncertainty Vertical WaveBending MomentModelling Uncertainty UltimateVertical Bending Moment Capacity

67% 7%

9% 17%



Ultimate Vertical Bending MomentCapacityModelling Uncertainty Vertical WaveBending MomentModelling Uncertainty UltimateVertical Bending Moment Capacity


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(a) Linear Model

(b) Interactive Model (Correlated Method 2)

(c) Interactive Model Uncorrelated

Figure 4-20: Importance Factor for the Bottom Damaged 74% Flooded Ship Random

Variables

61%

29%

4% 6%



Ultimate Vertical Bending MomentCapacityModelling Uncertainty Vertical WaveBending MomentModelling Uncertainty UltimateVertical Bending Moment Capacity

62% 29%

4% 5%


Vertical & Horizontal Wave BendingMomentUltimate Vertical Bending MomentCapacityModelling Uncertainty Vertical WaveBending MomentModelling Uncertainty UltimateVertical Bending Moment Capacity

61% 30%

4% 5%



Ultimate Vertical Bending MomentCapacityModelling Uncertainty Vertical WaveBending MomentModelling Uncertainty Ultimate VerticalBending Moment Capacity


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5.0 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.1 SUMMARY AND CONCLUSIONS

The present study was aimed at refining the modelling strategy presented in Phase III by improving the load models used to compute the failure probabilities of ships with gross damage. Improvements are implemented in two areas, namely the modelling of flooding due to damage and the incorporation of correlation between the vertical and the horizontal wave bending

moments. In Phase III, the hydrodynamic loads on the damaged ship with flooding were not computed. Instead, the load on the intact vessel was used in the damaged ship case. In the present study flooding is now explicitly modeled to provide more realistic loading for the damaged ship case. Furthermore, in Phase III, as well as in other related studies that have been reviewed, correlation of the vertical and the horizontal wave loads were not considered. This is included in the current study. Two correlation models, Method 1 and Method 2 are proposed and used. Method 1 uses only the amplitudes of the RAOs. Method 2 utilises the real and the imaginary parts of the RAOs, and the sea spectrum. Based on literature review, Method 2 is selected as the best strategy that reflects the ‘actual’ correlation between the vertical and the horizontal wave loads. The Annapolis Class Destroyer HMCS NIPIGON, a former Canadian naval vessel, decommissioned in 1998, is used to demonstrate the methodology. An operational profile involving four ship speeds, 5 speed headings and three sea states is defined. The RAOs of the wave loads and the stillwater bending moments are computed for four flooding conditions: 56% flooding of one damaged mid compartment; 74% flooding of one damaged mid compartment 60% flooding of two damaged mid compartments; and 100% flooding of two damaged bow compartments.

The statistics of the extreme wave loads and the correlation between the vertical and the horizontal wave loads are determined for all the flooding conditions and operational profiles. The strength model is based on the midship section of the ship. Bottom and side damage scenarios are defined and the ultimate strengths are computed. Both the linear model that uses only the vertical components of the load and strength, and the interaction equation model, that uses both the vertical and the horizontal components of loads and strengths are employed in


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the reliability assessment. Failure probabilities of the damaged ship are computed with the improved models. Results indicate that the higher the vessel speed the higher the failure probabilities and vice versa. The failure of all the damaged ships is governed by the hogging mode. Failure probabilities of the bottom damaged ship are always higher than the side damaged ship. When correlation between the wave loads is neglected, the interaction model always gives failures probabilities that are bigger than the linear model. Estimates of the correlation between the wave loads are positive for Method 1 and negative for Method 2. The largest failure probabilities are obtained with correlation from Method 1 and the smallest failure probabilities with correlation from Method 2. This underscores the need to include correlation in the reliability assessment of damaged ship. This feature is lacking in most of the published work. Failure probabilities are highest for the 74% flooded one bottom damaged mid compartment, followed by the 60% flooded two bottoms damaged mid compartments. Uncertainties in the wave load, vertical and horizontal wave bending moments, and uncertainties in the model used to represent the ultimate vertical bending moment capacity are the main parameters that govern the failure probabilities. The results highlights the need to use strength and load models that best reflect the damage under consideration.

5.2 RECOMMENDATIONS

The operational profile stipulated in Table 3-1 is used to determine the wave loads on the intact and the damaged vessel conditions in the study. For a specified damaged ship condition, extensive hydrodynamics analyses that involve combining a speed and a heading were used to determine various components of RAOs at various locations along the length of the vessel. For an operational condition, defined by a vessel speed, a heading, a stipulated time frame (1 hour, 3 hours etc.), a vessel longitudinal location, and a sea state (spectrum, Hs, Tz), the RAOs are used to computed the extreme load. The statistics of the extreme load is determined by the probability of the various operational conditions within the operational profile. Although, three sea states conditions [(Hs, Tz): (6.5 m, 7.5 s), (4.5 m, 7.5 s), (2.5 m, 7.5 s)] were employed in the current study, the number of sea states can be expanded and studied to reflect more realistic routes. The associated extreme values can be calculated using the existing RAOs without additional hydrodynamics analyses. Also, even though the computed RAOs include various load components such as torsion, all the analyses executed in the study are limited to the vertical and horizontal wave bending moments at the midship. Therefore, there is a lot of untapped data that can be used for other studies to gain further insight on the impact of load and strength modelling on the probability of failure. Some of the potential studies are discussed in this section.


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1. Impact of the Extreme Load Statistics and Choice of Probability Distribution on Vessel Integrity.

The current study was undertaken using the statistics of all the extreme loads for the operational profile. Some of the operational conditions had very small values of extreme loads and their inclusion in the extreme load statistics, resulted in ‘large” coefficient of variations (spread) and ‘small’ mean values. The question that needs to be resolved is what impact does inclusion/exclusion of operational conditions with low extreme values have on the reliability of the damaged ship? Because reliability assessment of ships, especially damaged ships is new there are no guidelines in the literature on this subject.

This will require a ranking of the various operational conditions and selecting subsets to determine the statistic of loads for assessment. Table 5-1 shows possible statistics of the extreme wave loads for the 2 Mid Compartments damage case with 60% flooding. Depending on weather all, top 50%, or top 25% of the extreme loads are considered, the mean values can be doubled and the COV can be tripled. In addition, the correlation coefficients between the loads will also depend on the subset of the load used. The result of any reliability assessment is dependent on the statistic of the extreme loads used and the question of the appropriate statistics to use for integrity assessment is important.

Also, although the extreme loads were generated based on the operational profile, the most appropriate probability distribution that suit the data set for the operational profile was never investigated and used. The Gumbel distribution was used for all cases. The choice of probability distribution can affect the probability of failure. There is provision for testing and selecting the best probability distribution within COMPASS which will be used to enhance the quality of the result. In addition, the impacts of other alternative distributions need to be investigated.

These issues need to be studied to gain further insight and to assist in the development of guidance for reliability analysis of damaged ships.


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Table 5-1: Possible Statistics of the Extreme Wave Loads for a Damaged 2 Mid Compartment with 60% Flooded

Speed (knots)

All Data Top 50% Top 25%

Mean (kNm)

COV Mean (kNm)

COV Mean (kNm)

COV

Vertical Wave Bending Moments 9 and 12 54860.35 0.59 79251.49 0.32 98277.74 0.20 6 and 9 54805.69 0.57 79389.00 0.29 96425.12 0.19 3 and 6 55044.80 0.54 79557.59 0.25 94948.12 0.16

Horizontal Wave Bending Moments 9 and 12 25676.63 0.69 39420.90 0.29 48680.71 0.18 6 and 9 25752.54 0.68 39765.62 0.27 48231.59 0.17 3 and 6 26158.26 0.67 40328.64 0.25 47995.79 0.16

2. Use of More Realistic Operational Conditions and Routing. The operational conditions (Hs, Tz) used in the current study were selected to demonstrate the concept. More realistic operational conditions that are applicable to a naval ship that is damaged at a specific location and has to get back to another specified location can be used for further study. The effect of using different alternate routes, essentially defined by different combination of sea states, can also be explored. Also, the possibility of combining the various flooding scenarios, starting with for example, less water in a single compartment, followed by more water in a single compartment, followed by flooding in two compartments, to reflect the impact of increasing hazards can be investigated as well. The RAOs that have already been computed can be used in the assessment. No new hydrodynamic analyses may be required because the changes will only affect the extreme values.

3. Impact of Slamming or Dynamic Load on Reliability Both damaged and undamaged ships are subject to slamming or dynamic loads during service. These loads, which could impact the survivability of a damaged vessel, were not included in the previous assessments. Deco and Frangopol (2013) and Sikora and Brady (1989) proposed the following formula for computing the dynamic bending moment on fine bow ships induced by whipping

62

622

10x5for 4.510x5for0022.0

bLBPbLBPbLBPbLBP

MWIP

Where WIPM is the peak-to-peak whipping bending moment (ft-ton), LBP is the length between perpendiculars of the ship (ft) and b is the ship breadth (ft). Inclusion of the slamming load leads to the modification of the limit state functions as given below. The linear limit state function becomes


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VWDWIPVWVWVSWVVSWVUVU MkxMxMxMxXg )(

And the interaction equation limit state function, which involves both vertical and horizontal ultimate capacities and loads, becomes

m

HUHU

HWHWHSWHHSW

n

VUVU

VWDWIPVWVWVSWVVSW

MxMxMx

MxMkxMxMxXg 1)(

WIPx is the modelling uncertainty factor for the slamming loads

Dk is the correlation between the slamming load and the wave load 4. Reliability of Other Ship Sections The RAOs of other sections of the ship exist and their reliability can be investigated.

5. Reliability of Another Intact and Damaged Ships The reliability of another ship can also be studied. 6. Improvement of Reliability Assessment by Improving Strength Models The tasks performed for DRDC on intact and damaged surface ship and submarines structures involve modelling the global strength of the structure. So far, the global strengths are determined using 2-D models, namely ULTMAT tool that has built-in simplifying assumptions. For example, in the reliability assessment of damaged ship, although the global torsional moment wave loads are computed, analyses are restricted to models that involve only the vertical and horizontal wave bending moments. These assumptions need to be relaxed by using 3-D finite element modelling tool, namely TRIDENT with its solver engine VAST. However, the direct application of finite element analysis tool in reliability assessment is very computational expensive. Simplified models that indirectly use the finite element results have to be employed. One such method involves the response surface approach (Decò and Frangopol, 2013; Koko et al., 2013).

To position LR Applied Technology Group to satisfy DRDC and other clients’ needs in the area of finite-element based reliability assessment, lR Applied Technology Group embarked on a pilot Lloyd’s Register’s Marine Product Development, titled “Pilot Integrated FE-Based Reliability Analysis Capability for Stiffened Plate Structures” (Koko et al., 2013). The work was aimed at developing a finite element based structural reliability analysis (SRA) capability that integrates LR Applied Technology Group COMPASS reliability tool with LR Applied Technology Group finite element analysis software, TRIDENT/VAST. The study was motivated by the desire to have an integrated system for performing structural reliability analysis that usually involves manipulation of data between independent FEA and reliability


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analysis software packages. The capability developed in the pilot study is incomplete and very limited. It needs to be completed and expanded so that DRDC and other projects in this area can be successfully executed. The features available in the pilot tools include the following: three types of FE elements, namely, the 4-node quadrilateral shell; 3-node triangular shell and 2-node general beam element; random variables description for material properties (Young’s modulus, Poisson’s ratio, yield stress, density); geometric properties (plate/ shell thickness, beam section properties); and loading (element and concentrated loads); limit state functions for nodal displacements, element/ nodal stress/ strain; and buckling loads, based on linear elastic theory; linear and quadratic response surfaces to define the limit states based on finite element analysis; and display of reliability analysis results through text files. The capability was tested using three cases studies including a simply supported square plate, orthogonally stiffened panel, and a three-hold model of a general cargo ship.

To ensure that the capability can be used to meet the client’s needs, several features have to be added and tested including the following:

Testing of all the limit states that are implemented Inclusion of nonlinear analysis for the three selected element types. Beam sections to be defined by scantling dimensions, rather than the overall beam

cross sections only Inclusion of more FE element types Further testing and validation Additional case studies for different types of ships Limit states (performance types) based on nonlinear FE analysis; Response surfaces for the limit states based on nonlinear FE, including ultimate

strength This task can be executed in collaboration with LR.


6.0 REFERENCES

Q. Liu, I.R. Orisamolu and M.W. Chernuka, COMPASS Version 1.4 User’s Manual, Martec Limited, Halifax, Nova Scotia, February 1995.

Decò A and Frangopol, D. M (2013), Risk-informed optimal routing of ships considering different damage scenarios and operational conditions, Reliability Engineering and System Safety vol. 119, pp. 126–140

Koko, T. S., Akpan, U.O., Norwood, M., Yuen, B., (2013) Pilot Integrated FE-Based Reliability Analysis Capability for Stiffened Plate Structures, Martec Technical Report #TR-13-34, RCN 1987, August 2013.

Akpan U.O, B. Yuen, T. S. Koko, F. Lin, and J. Wallace, (2013), Probability of Failure of Damaged Ship Structures: Phase III, TR-13-15, Martec Limited, Halifax, NS.

Akpan, U.O, B.K. Yuen, K.O. Shahin, T.S. Koko, and F. Lin, (2012), Probability of Failure of Damaged Ship Structures: Phase II, CR 2012-010, DRDC Atlantic, Dartmouth, NS.

Akpan, U.O, T.S. Koko, B.K. Yuen and K.O. Shahin, (2010), Probability of Failure of Damaged Ship Structures: Phase 1, TR-10-16, Martec Limited, Halifax, NS.

Wikipedia [1] https://en.wikipedia.org/wiki/HMCS_Nipigon_(DDH_266)

Documents

Probability of Failure of Damaged Ship Structures Phase IV