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Structural DesignAssessment

Sloshing Loads andScantling Assessment

May 2004

ShipRightDesign and construction

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Lloyd’s Register,2004

Document History

Document Date: Notes:

October 1994 New document

November 2001 Intranet user review version

July 2002 General release.

Notice 1 October 2002

Revisions as identified in ‘ History of Development up to January 2004’.

May 2004 Revisions as identified in ‘Structural Design Assessment – Sloshing Loads and Scantling Assessment, Changes incorporated in May 2004 version’.

LLOYD’S REGISTER

Chapter Contents

Sloshing Loads and Scantling Assessment, May 2004

1 Introduction

2 Scope of Procedure

3 Fluid Sloshing Phenomena

4 Definitions

5 Units

6 Data Requirements

7 Levels of Assessment

8 Loading Conditions and Ship Motions Sloshing Analysis

9 Level 1 Sloshing Pressure Determination



12 Post-Processing of SDA Fluids Data

13 Strength Assessment

PROCEDURES MANUAL

Chapter Contents


PROCEDURES MANUAL

LLOYD’S REGISTER

14 Acceptance Criteria

15 Applications

References

Appendices

A Examples

B Natural Frequencies of Structural Components

C Determination of Equivalent UniformlyDistributed Loading

D SDA Fluids Data FileDYN_STAT Data File & Output

LLOYD’S REGISTER

Contents


CHAPTER 1 INTRODUCTION 1

Summary

Section 1 Introduction

CHAPTER 2 SCOPE OF PROCEDURE 3

Section 1 Scope of Procedure

CHAPTER 3 FLUID SLOSHING PHENOMENA 5

Section 1 Sloshing Waves

Section 2 Sloshing Induced Loads

CHAPTER 4 DEFINITIONS 9

Section 1 Filling Factor Coefficient

Section 2 Fluid Natural Period

Section 3 Tank Depth

Section 4 Maximum ‘Lifetime’ Ship Motions4.1 Ship’s Natural Rolling Period4.2 Ship’s Natural Pitching Period4.3 Maximum ‘Lifetime’ Roll Angle4.4 Maximum ‘Lifetime’ Pitch Angle4.5 Maximum ‘Lifetime’ Heave Amplitude4.6 Maximum ‘Lifetime’ Sway Amplitude4.7 Maximum ‘Lifetime’ Vertical Acceleration4.8 Centre of rotation

Section 5 Effect of Wash Bulkhead on Sloshing Pressures

Section 6 Pressure at Tank Corners

Section 7 Pressure in Tapered Tanks

CHAPTER 5 UNITS 17

Section 1 Units

CHAPTER 6 DATA REQUIREMENTS 19

Section 1 Data Requirements

PROCEDURES MANUAL

Contents


PROCEDURES MANUAL

LLOYD’S REGISTER

CHAPTER 7 LEVELS OF ASSESSMENT 21

Section 1 Pressure Determination

Section 2 Sloshing Criteria

Section 3 Critical Fill Range

Section 4 Level 1 Assessment



Section 7 Structural Capability

CHAPTER 8 LOADING CONDITIONS AND SHIP 23MOTIONS FOR SLOSHING ANALYSIS

Section 1 General Considerations

Section 2 Loading Conditions2.1 Unrestricted Filling Levels -

Unspecified Sea-Going Loading Conditions2.2 Restricted Filling Levels -

Unspecified Sea-Going Loading Conditions2.3 Unrestricted Filling Levels -

Specified Sea-Going Loading Conditions2.4 Restricted Filling Levels -

Sea-Going Loading Conditions

Section 3 Level 2 Sloshing Assessment Parameters3.1 Level 2 Ship Motions3.2 Level 2 Fill Range

Section 4 Level 3 Assessment Parameters4.1 Level 3 Ship Motions4.2 Level 3 Investigation Fill Range

CHAPTER 9 LEVEL 1 SLOSHING PRESSURE 27DETERMINATION

Section 1 Level 1 Sloshing Pressure Determination


Section 1 Smooth Rectangular Tanks

Section 2 Smooth Hopper Tanks

Contents

LLOYD’S REGISTER


PROCEDURES MANUAL


Section 1 General

Section 2 Limitations and Assumptions of SDA FluidsProgram

Section 3 Data Preparation3.1 Mesh Spacing3.2 Fitting the mesh to the tank3.3 Boundary Conditions3.4 Including Internal Tank Structure3.5 Properties of the Fluid3.6 Pressure Output Sampling Points3.7 Ullage Pressure3.8 Sloshing Excitation Spectrum3.9 Time Control

CHAPTER 12 POST-PROCESSING OF SDA FLUIDS DATA 41

Section 1 Sloshing Simulation Quality Assurance Procedure1.1 General1.2 Minimum Quality Assurance Post Processing

Requirements1.3 Inconsistencies and Applied Results

Section 2 Pressure Pulse Time Averaging Scheme

Section 3 Dynamic and Static Pressures3.1 Conversion of Dynamic Pressure to Static

Pressure3.2 Response Calculation3.3 Pressure Conversion Procedure

Section 4 Structure Natural Frequency Calculation

Section 5 Force and Couple

Section 6 Pressure Applied to Internal Structural Members

CHAPTER 13 STRENGTH ASSESSMENT 47

Section 1 Pressure and Stresses

Section 2 Collapse Analysis Procedures for ClampedStiffened Panels

2.1 Description2.2 Assumptions and Limitations2.3 Applied Loads2.4 Output

Section 3 Minimum Factors of Safety

Contents


PROCEDURES MANUAL

LLOYD’S REGISTER

Section 4 Girder Structural Analysis Procedure4.1 Finite Element Analysis4.2 Analytical Structural Analysis4.3 Applied Loads

CHAPTER 14 ACCEPTANCE CRITERIA 55

Section 1 Strength based acceptance criteria

Section 2 Service based acceptance criteria

CHAPTER 15 APPLICATIONS 57

REFERENCES 59

APPENDICES

APPENDIX A EXAMPLES 61

Section 1 Level 1 Investigation



APPENDIX B NATURAL FREQUENCIES OF 85STRUCTURAL COMPONENTS

Section 1 Natural Frequency of Plate

Section 2 Natural Frequency of Plate Stiffener

Section 3 Effect of Submergence

Section 4 Dynamic Load Factor Charts4.1 Gradually Applied Load4.2 Triangular Pulse Load

APPENDIX C DETERMINATION OF EQUIVALENT 93UNIFORMLY DISTRIBUTED LOADING

Section 1 General

Section 2 Determination of Equivalent UniformlyDistributed Loading

2.1 Trapezoidal Distributed Loading2.2 Arbitrary Distributed Loading

APPENDIX D SDA FLUIDS DATA FILE 97DYN_STAT FILE & OUTPUT

1LLOYD’S REGISTER 1

Introduction


Chapter 1SUMMARY

SummarySection 1: Introduction

■■ Summary

This document describes the ShipRight SDA Sloshingprocedure for the assessment of boundary structures ofpartially filled tanks and liquid carrying holds. Three levelsof assessment are defined, each requiring a differentapproach to the estimation of likely maximum sloshingpressures.

Level 1 assessment is based on equivalent static loadsresulting from lifetime angular motions.

Level 2 assessment uses the SDA Tank Assessment program(10603) to determine the pressure on the tank boundaries.

Level 3 assessment uses the SDA Fluids finite differenceprogram to determine sloshing pressures on the tankboundaries and internal structural elements.

The strength assessment is based on safety against collapse.The calculations may be carried out using the SDA UltimateStrength program (10604).

■■ Section 1: Introduction

1.1 Where partial filling of tanks and liquid carryingholds is required, the likelihood of sloshing from both theship rolling and pitching is to be investigated. Sloshing isdefined as a dynamic magnification of internal pressuresacting on the boundaries of the tank to a level greater thanobtained from static considerations alone.

1.2 For any tank design, dimensions, internal stiffeningand filling level, a resonant period (or frequency) of thefluid exists, which, if excited by ship motions, can result invery high pressure magnifications.

1.3 The purpose of this procedure is to enable thedetermination of lifetime maximum design sloshingpressures for anticipated filling levels, tank position withinthe ship and ship’s loading conditions.

1.4 The estimated dynamic pressures may then be usedto determine the scantlings necessary to prevent structuralcollapse using appropriate structural collapse theory inassociation with defined criteria.

1.5 A typical sloshing investigation is illustrated in theflow chart in Figure 1.1.

1.6 The procedure is divided in two parts :

1. Assessment of pressures on tank boundaries2. Scantling determination and acceptance criteria

Note:LNG tanks in partial filling conditions exhibit a complexbehaviour of the fluid as a result of the possible change ofphase of the fluid under high velocity impacting with aboundary, and 3D effects resultings from the chamferedgeometry of the tank top. It is considered that both theSDA Tank Assessment program (10603) and SDA Fluidsyield a realistic behaviour of the fluid flow level ofpressures, for fill LNG levels which do not involve impactson the tank top.

Sloshing Loads and Scantling Assessment forTanks Partially Filled with Liquids


Chapter 1SECTION 1

LLOYD’S REGISTER2

Figure 1.1

Sloshing Investigation Flow Chart

SHIP DATA

PERIODS CALCULATION

LEVEL OF ASSESSMENTDETERMINATION

LEVEL 1 LEVEL 2 LEVEL 3

SDA Fluids

DYN_STAT

PROCEDUREGUIDELINES

ACCEPTANCECRITERIA

STRENGTH ACCEPTANCECRITERIA

SE

RV

ICE

AC

CE

PT

AN

CE

CR

ITE

RIA

SDA Tank AssessmentProgram (10603)

SDA Ultimate StrengthProgram (10604)

SDA Tank AssessmentProgram (10603)

SDA Ultimate StrengthProgram (10604)

PRESSURECONVERSION

?

SATISFIED?

YES

YES

NO

NO


Scope of Procedure


Chapter 2SECTION 1

Section 1: Scope of Procedure

■■ Section 1: Scope of Procedure

1.1 The procedure applies to tanks and liquid carryingholds of arbitrary shape filled with liquids with theexception of spherical or cylindrical tanks which need to bespecially considered. In addition, some tanks, by virtue oftheir shape, size or degree of internal stiffening, will not besubjected to sloshing loads. If any such tank is likely to bepartially filled, the reasons for exclusion from theinvestigation should be stated and agreed by Lloyd’sRegister. In general, sloshing calculations need not beperformed for peak tanks or bunkers.

1.2 Any scantlings derived as a result of this procedureare to be regarded as additional to the Rule requirementsfor full tanks and liquid carrying holds.


LLOYD’S REGISTER4


Fluid Sloshing Phenomena


Chapter 3SECTION 1

Section 1: Sloshing WavesSection 2: Sloshing Induced Loads

■■ Section 1: Sloshing Waves

1.1.1 As the tank oscillates, different sloshing waves will becreated depending on the fill depth and frequency ofoscillations. An infinite number of different modes of liquidmotion may occur depending upon the conditions ofexcitation and fill depth. However, it is possible to dividethe sloshing phenomena into the following four categoriesto describe the observed modes shown on Figure 1.1.

Standing Wave

The movements of the liquid particles on the surface areessentially vertical, the surface having one or more nodeswhere practically no vertical surface displacement takesplace. Standing waves generally occur when F/Ls ≥ 0,2 andimpart high impact pressures mainly to the tank top.

whereF = Fill height (m)

Ls = Effective horizontal free surface length in thedirection of angular motion (m)

Travelling Wave

The surface has no nodes, a wave crests travels back andforth between vertical tank boundaries. Travelling wavesgenerally occur when F/Ls < 0,2 and impart high impactpressures to both side walls and tank top.

Hydraulic Jump

This Phenomena, which might be considered as a specialcase of a travelling wave, is characterised by a discontinuity(jump) in the surface, forming a vertical front which travelsperiodically back and forth in the tank.

Combination Wave

A combination of standing waves and travelling waves.

For low filling, a standing wave is formed when the tank isoscillating at a frequency far below the fluid naturalfrequency. As the excitation frequency increases, thistransforms into a train of progressive waves having a veryshort wavelength. A hydraulic jump is formed due to asmall disturbance at a range of frequency around the fluidresonance frequency. With further increase in frequencybeyond resonance, the hydraulic jump transforms into asolitary wave. In general, hydraulic jumps are formed onlywhen the fill level is 20 per cent of the horizontal freesurface length of the tank or less.

For high fill levels, the sloshing phenomenon nearresonance is characterised by the formation of standingwaves of large amplitudes. These waves are non symmetricand may be combined with travelling waves at largeamplitude of excitation.



Chapter 3SECTIONS 1 & 2

LLOYD’S REGISTER6

■■ Section 2: Sloshing Induced Loads

2.1.1 Liquid sloshing involves different types ofhydrodynamic loads upon the tank and its internalstructure. There are two types of dynamic pressure whichcan arise from liquid sloshing, namely non-impulsive andimpulsive pressure. Typical time histories for the followingsloshing induced loads are shown in Figure 2.1.

Non Impulsive Dynamic Pressure

These are slowly varying loads, pulsating with a period ofthe order of the sloshing wave period, i.e. period of theorder of the fluid natural period and/or excitation period.

Impulsive dynamic pressure type I

These are due to a rapid but continuous build up of liquidand liquid pressure on the surface of a member which isgradually being immersed. The impulse duration istypically in the order of 1/10 of the sloshing wave period.

Impulsive dynamic pressure type II

These are due to localised impact pressure arising from thecollision between the fluid and the solid surface. Suchpressures can be extremely high and of extremely short risetime duration in the range 1/100 to 1/1000 of the sloshingwave period.

Total Dynamic Forces and Moments

These loads arise from the slowly varying non impulsivehydrodynamic pressure distribution on the tank boundarieswith a period of the order of the sloshing wave period.

Drag and Inertial Forces

These non impulsive forces act on submerged memberswith time fluctuations related to the sloshing wave period.

Vortex Induced Pressure Field

These pressure fields develop around slender memberslocated in the field of oscillating liquid. Interactionbetween the generated pressure fluctuations and naturalmodes of structural vibrations in the member may becomecritical.

Standing Wave

Travelling Wave

Hydraulic Jump

Combination Wave

Fig 1.1

Typical Sloshing Waves


LLOYD’S REGISTER 7


Chapter 3SECTION 2

Figure 2.1

Sloshing Induced Loads and Typical Time Trace


LLOYD’S REGISTER8


Definitions


Chapter 4SECTIONS 1, 2 & 3

Section 1: Filling Factor CoefficientSection 2: Fluid Natural Period

Section 3: Tank DepthSection 4: Maxiumum ‘Lifetime’ Ship MotionsSection 5: Effect of Wash Bulkhead on Sloshing

PressuresSection 6: Pressure at Tank Corners

Section 7: Pressure in Tapered Tanks

■■ Section 1: Filling FactorCoefficient

1.1.1 The filling factor, Fc is defined as follows :

Fc = F/H + 6,0 θo /cosh (Fr) (4.1)

whereF = fill height (m)H = total tank depth (m)θ = θmax or φmax as appropriate (radian)

θo = the greater of θ1 or θ2

θ1 = θe-(Tn – Sn)2/k

θ2 = 0,105 for roll= 0,052 for pitch

k = 4 for roll= 6 for pitch

Fr = the effective filling ratio= π {F – b-[n/(n+1]}/Ls

b = height of internal primary bottom stiffeners (m)n = number of internal primary bottom stiffenersLs = effective horizontal free surface length in the

direction of angular motion (m)g = gravity constant (9,81 m/s2).

A low fill is defined as a filling level for which the factor Fc

is less or equal than 1,02. However, when the fluid andship natural periods are close, Fc will invariably be greaterthan 1,02; in this case, a low fill is defined for F/Ls less orequal to 0,21. Any other filling is defined as high.

■■ Section 2: Fluid Natural Period

2.1.1 The fluid natural period in pitch or roll, Tnp or Tprrespectively is given by :

Tn = - 4πLs/(gtanh(Fr)) (4.2)

■■ Section 3: Tank Depth

3.1.1 The depth of a tank H is measured from the bottomof the tank to the underside of the deck at side. In the caseof holds, the depth is measured from the inner bottom tothe underside of the deck at hatch side, except in doubleskin ships with hatch coaming in line with the inner skin, inwhich case, the depth is measured from the top of thehatch.



Chapter 4SECTION 4

LLOYD’S REGISTER10

■■ Section 4: Maximum ‘Lifetime’Ship Motions

When possible, direct calculation procedures capable oftaking into account the ship’s actual form and weightdistribution should be performed in order to determine theship motions. Such methods will involve the derivation ofthe response to regular waves using strip theory, thederivation of the short term response to irregular wavesusing the concept of sea spectrum, and the derivation oflong term response predictions using statisticaldistributions of sea states.

Otherwise, the following expressions should be used todetermine the approximate maximum ‘lifetime’ shipmotions. These expressions derived on a statistical basiscorrespond to extreme ship motions and accelerations witha probability of occurrence of once in a ship lifetime of 20years for ships of normal proportions.

4.1 Ship’s Natural Rolling Period

The ship’s natural rolling period Snr is given by :

Snr = 2,35 r/ GM (4.3)

wherer = the radius of gyration of roll and may be taken as

0,34 B (m)GM = transverse metacentric height with free surface

correction (m).

For ships for which either r or GM varies significantlybetween loading conditions (for example, bulk carriers andtankers), Snr should be evaluated for each representativeloading condition considered.

4.2 Ship’s Natural Pitching Period

The ship’s natural Pitching period Snp is given by :

Snp = 3,5 TCb = 3.5 ∇ (4.4)LB

whereT = the mean draught (m)

Cb = the block coefficientL = the lenght between perpendiculars (m)B = the ship breadth (m)∇ = the ship displacement (m3)

Similarly, for ships for which either T or Cb variessignificantly between loading conditions (for example, bulkcarriers and tankers), Snp should be evaluated for eachrepresentative loading condition considered.

4.3 Maximum ‘Lifetime’ Roll Angle

The maximum ‘lifetime’ roll angle in degrees is given by :

φmax = (14,8 + 3,7 L/B)e-0,0023.L (4.5)

whereL = the length between perpendiculars (m)B = the ship breadth (m).

4.4 Maximum ‘Lifetime’ Pitch Angle

The maximum ‘lifetime’ pitch angle in degrees is given by :

θmax = (32,7 - 8,2 Cb)e-0,001L(4,9+Cb/2) (4.6)

4.5 Maximum ‘Lifetime’ HeaveAmplitude

The maximum ‘lifetime’ heave amplitude in metres is givenby :

Zmax = 10e-0,0032L (4.7)

but need not be taken greater than 4 metres.

4.6 Maximum ‘Lifetime’ SwayAmplitude

The maximum ‘lifetime’ sway amplitude, in metres, is givenby :

Ymax = 5e-0,0025L (4.8)

but need not be taken greater than 2,50 metres.




Chapter 4SECTIONS 4, 5 & 6

4.7 Maximum ‘Lifetime’ VerticalAcceleration

The maximum ‘lifetime’ acceleration, in m/s2, at alongitudinal position x from midships is given by :

a = ±gao 1+(5,3 – 45/L)2 (x/L + 0,05)2 (0,6/Cb)3/2

(4.9)

whereao = 0,2 V/ L + (34 – 600/L)/Lx = the longitudinal distance from midships to centre

of the tank being considered, with x positiveforwards (m)

V = the ship service speed (knots)g = the acceleration due to gravity (m/s2).

4.8 Centre of rotation

The vertical centre of rotation is to be taken to be at theVCG for the loading condition under consideration. Whenthis is unavailable, the vertical centre of rotation may betaken as at depth (moulded)/2,0 from the keel.

The longitudinal centre of rotation is to be taken to be atthe LCG for the loading condition under consideration.When this is unavailable, the centre longitudinal of rotationmay be taken as at midship.

■■ Section 5: Effect of WashBulkhead on SloshingPressures

Wash bulkheads which represent more than 85% of thetank cross sectional area are taken as being effective assloshing barriers which limits the free surface length.

The effect of a wash bulkhead may be estimated using atotal energy approach applied to the load distribution ascalculated for the tank. The total pressure on the bulkheadwith the estimated effect of the wash bulkhead may beexpressed as follows :

P = Ps + PD /(1 + κ) (4.10)

whereP = the total pressure on bulkhead with estimated

effect of wash bulkheadPs = the static pressure without wash bulkheadPD = (PT – Ps) is the dynamic pressure without wash

bulkhead

PT = the computed total pressure without washbulkhead

λ = (area of openings in wash bulkhead)/(area ofwash bulkhead)

κ = (1–λ)/1+λ).

In the case where frames or transverse members areinstalled instead of wash bulkhead, the pressure on thewatertight bulkhead is observed to decrease to about 80%of the dynamic pressure without frames or transversemembers when only two or three members are installed,but the dynamic pressure no longer decreases withincreasing number of frames or transverse members.

■■ Section 6: Pressure at TankCorners

The pressure at the tank corners may be derived bycombining the corner pressure Proll and PPitch obtainedfrom a level 3 investigation for both rolling and pitchingmotions. The pressure at tank corners is expressed asfollows:

Pcorner = Max [ (Cpp(χ)P2pitch + Cpr(χ)P2

roll)] (4.11)for 0° ≤ χ = Heading ≤ 180°

whereCpp(χ) = the pitch pressure coefficient at χ given in

Figure 6.1 and Table 6.1.Cpr(χ) = the roll pressure coefficient based on the ratio

L/B at χ given in Figure 6.2 and Table 6.2. Forintermediate values of L/B, the factor is to bedetermined by linear interpolation.

The pressure at the corners is to be applied to a distanceextending 0,10 [Tank Breadth] and 0,10 [Tank Length] onthe transverse boundary and longitudinal boundaryrespectively. The pressure value then decreases linearlyover a distance 0,05 [Tank Breadth] or 0,05 [Tank Length]to the pressure value obtained from 2D solution.

The factors Cpp and Cpr are based on the short termmotion responses in long crested irregular seas for pitchand roll. These expressions incorporates both the effect ofmotion amplitude and phase between the components ofmotion.



Chapter 4SECTION 6


Coefficient

Cpp

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Heading (o)

Pitch Pressure Coefficient Cpp

Figure 6.1

Pitch Pressure Coefficient Cpp

Figure 6.2

Roll Pressure Coefficient Cpr

Coefficient

Cpr

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Heading (Degrees)

Roll Pressure Coefficient Cpr

– L/B = 5,50 -– L/B = 6,25 ..... L/B = 7,00 –-– L/B = 7,75 –--– L/B = 8,50




Chapter 4SECTION 6

Cpr Cpr Cpr Cpr Cpr

L/B 5,50 6,25 7,00 7,75 8,50

χ

0 0,000 0,000 0,000 0,000 0,058

5 0,075 0,072 0,078 0,054 0,117

10 0,142 0,139 0,143 0,121 0,221

15 0,229 0,213 0,234 0,186 0,347

20 0,322 0,311 0,334 0,294 0,479

25 0,457 0,451 0,469 0,438 0,604

30 0,614 0,610 0,619 0,597 0,716

35 0,764 0,760 0,758 0,745 0,809

40 0,882 0,878 0,867 0,870 0,900

45 0,958 0,958 0,978 0,975 0,950

50 1,000 1,000 1,000 1,000 0,990

55 0,950 0,941 0,941 0,983 1,000

60 0,852 0,862 0,861 0,900 0,980

65 0,744 0,761 0,774 0,818 0,960

70 0,623 0,647 0,675 0,727 0,920

75 0,503 0,533 0,575 0,635 0,879

80 0,395 0,429 0,483 0,549 0,834

85 0,306 0,344 0,406 0,475 0,785

90 0,240 0,280 0,345 0,415 0,735

95 0,197 0,237 0,301 0,369 0,684

100 0,173 0,211 0,270 0,335 0,634

105 0,162 0,198 0,247 0,307 0,585

110 0,152 0,185 0,229 0,283 0,538

115 0,142 0,171 0,211 0,258 0,494

120 0,129 0,159 0,196 0,231 0,452

125 0,121 0,146 0,181 0,208 0,413

130 0,110 0,133 0,165 0,183 0,378

135 0,100 0,121 0,150 0,167 0,345

140 0,088 0,108 0,135 0,149 0,315

145 0,079 0,096 0,121 0,133 0,288

150 0,067 0,083 0,105 0,117 0,265

155 0,056 0,068 0,088 0,096 0,244

160 0,046 0,058 0,071 0,083 0,226

165 0,033 0,046 0,058 0,069 0,210

170 0,021 0,029 0,040 0,050 0,193

175 0,013 0,021 0,025 0,029 0,175

180 0,000 0,005 0,013 0,013 0,160

χ Cpp

0 0,600

5 0,600

10 0,600

15 0,600

20 0,600

25 0,600

30 0,600

35 0,594

40 0,575

45 0,547

50 0,510

55 0,466

60 0,416

65 0,363

70 0,310

75 0,262

80 0,222

85 0,194

90 0,181

95 0,186

100 0,210

105 0,253

110 0,313

115 0,389

120 0,474

125 0,565

130 0,654

135 0,734

140 0,800

145 0,860

150 0,915

155 0,955

160 0,980

165 0,995

170 1,000

175 1,000

180 1,000

Table 6.1 Table 6.2



Chapter 4SECTION 7


■■ Section 7: Pressure in TaperedTanks

Where tanks are tapered in plan view such as foremost oraftermost tanks, limited model experiments indicated thatin pitching the dynamic pressure on the bulkhead at thenarrow end can be magnified when compared with a tankof uniform section. Lloyd’s Register Fluids two-dimensionalfluid computational procedure cannot take into accountthis aspect. The pressure at the narrow end of the tank canbe expressed in terms of the pressure obtained for a tank ofuniform breadth by using the following expression.

Ptapered = Kt.Pmax.breadth (4.12)

whereKt = 0,8e(0,2235ARb)

ARb = the ratio of the maximum breath to the taperedbreadth.

Kt is also given in Figure 7.1 and Table 7.1.




Chapter 4SECTION 7

Figure 7.1

Tapered Tank Coefficient Kt

ARb 2,100 2,200 2,300 2,400 2,500 2,600 2,700 2,800 2,900 3,000 3,100

Kt 1,279 1,308 1,338 1,368 1,399 1,430 1,463 1,496 1,530 1,564 1,600

1.600

1.500

1.400

1.300

1.200

1.100

1.000

Coefficient

Kt

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Tank Breadth Aspect Ratio ARb

Table 7.1


Units


Chapter 5SECTION 1

Section 1: Units

■■ Section 1: Units

The units used throughout are consistent with the SIstandard where the basic quantities are the metre,Kilogram and Second (MKS system), except for angularmeasurement which is in degrees.


Data Requirements


Chapter 6SECTION 1

Section 1: Data Requirements

■■ Section 1: Data Requirements

The following plans and informations are required toperform a sloshing and scantling investigation:

- General Arrangement- Midship Section Drawing- Longitudinal Bulkhead Drawing- Transverse Bulkhead Drawing- Trim & Stability and Loading Manuals- Material Properties.


Levels of Assessement



Section 1: Pressure DeterminationSection 2: Sloshing Criteria

Section 3: Critical Fill RangeSection 4: Level 1 AssessmentSection 5: Level 2 AssessmentSection 6: Level 3 Assessment

Section 7: Structural Capability

■■ Section 1: Pressure Determination

Three levels of assessment are defined below, eachrequiring a different approach to the estimation of likelymaximum sloshing pressures.

Significant dynamic magnification is considered unlikelyfor the following cases:

- For internally stiffened tanks with two or moredeck girders (rolling)/transverses (pitching)wherethe girder/transverse location is less or equal to25% of the tank breadth/length from thedeck/tank corner, and/or the girder/transverseheight is less than 10% of the tank depth with filllevels greater than the tank depth minus height ofdeck girders/transverses.

or - For fill levels lower than height of any bottomgirders

or - For fill levels in excess of 97 % full for smoothtanks

or - For fill levels less than 10 % for smooth tanks.

■■ Section 2: Sloshing Criteria

Based on Lloyd's Register’s experience, and numericalstudies of a number of cases, it is considered thatsignificant magnification of the fluid motions can occur ifthe following conditions are found:

- The natural rolling period of the fluid and the shipnatural rolling period are within 5 seconds of eachother.

- The natural pitching period of the fluid is greaterthan a value of 3 seconds below the ship naturalpitching period.



Chapter 7SECTIONS 3, 4 , 5 , 6 , & 7


■■ Section 3: Critical Fill Range

The critical fill range may be determined by using thefollowing formula, or using the SDA Tank Assessmentprogram (10603).

Fcrit = 100{Ls 1n [(1 + η)]+ b n } (%)H 2π (1 – η) (n+1)

whereLs = effective horizontal free surface length in direction

of angular motion (m)H = total tank depth (m)b = see Ch 4, 1.1.1g = gravity constant (m/s2)η = 4πLs/[(Snr-5)2g] for fill level at Snr-5 seconds

to upper bound roll critical fill levelor η = 4πLs/[(Snr+5)2g]for fill level at Snr+5 seconds

to lower bound roll critical fill levelor η = 4πLs/[(Snp-3)2g] for fill level at Snp-3 seconds

to upper bound pitch critical fill level.

If η ≥ 1,0 Fcrit is the maximum/minimum value of theupper/lower fill level bound [100%/0%].

If Fcrit ≥ 100,0 Then Fcrit = 100%.

■■ Section 4: Level 1 Assessment

This level of assessment is appropriate where the ship’snatural period in roll differs from the fluid natural periodfor transverse oscillatory flow by more than 5 seconds; andwhere the ship’s natural pitching period exceeds that forthe fluid oscillating longitudinally by more than 3 seconds.


Where the separation of periods defined above is not met,but filling levels are such that impacts on the top of thetank are unlikely, then a level 2 investigation may be usedto assess the sloshing pressures on the tank bulkheads. Thislevel of assessment may also be used for low fill caseswhere the tank has internal stiffening, but the resultingpressures would be considered somewhat conservative. Alow fill is defined when the filling factor coefficient Fc

defined in Ch 4,1.1.1 is equal or less than 1,02. Where Sn(ship natural period) and Tn ( tank natural period) areclose, Fc will invariably give a ‘high’ fill. In the case thatsuch a filling height is equal to or less than 0,21 Ls, a lowfill may be assumed and the case treated as a level 2assessment, otherwise, a level 3 assessment should be used.


Where significant dynamic magnification of fluid pressuresinvolving impacts on the top of the tank is likely, or wherethe effect of internal stiffening is to be taken into account,then a level 3 assessment is required.

■■ Section 7: Structural Capability

The structural capability of the tank boundaries towithstand the dynamic sloshing pressures is to bedetermined using SDA Ultimate Strength program (10604).This program considers the lateral pressure on a stiffenedpanel comprising a single stiffener and attached plating.The ultimate strength of the plating is calculated on thebasis of a defined allowable permanent set taking intoaccount the membrane stress induced in the panel as itdeforms. For the stiffeners, a classical plastic collapsemethod is used taking into consideration both shear andbending strains. Allowance is made for the smallproportion of the pressure load transmitted directly fromthe plating to the supporting primary structure.


Loading Conditions and ShipMotions for Sloshing Analysis



Section 1: General ConsiderationsSection 2: Loading Conditions

Section 3: Level 2 Sloshing Assessment ParametersSection 4: Level 3 Assessment Parameters

■■ Section 1: General Considerations

Where partial fillings are contemplated in all tanks of aship, the following tanks are to be considered in theanalysis together with associated sea conditions given inTable 1.1 provided the following conditions are satisfiedtogether with the relevant level of assessment conditions.

The natural periods of the ship for a given motion typeshould be determined for the service loading conditionsagreed between the builder and the society. When a ship isto be approved for arbitrary tank filling, all approved safeloading conditions should be investigated and the estimationof significant dynamic pressure magnification consideredaccording to the guidelines provided in Chapter 7.

■■ Section 2: Loading Conditions

The following loading conditions are provided as aguideline to the most critical conditions:

- Storm Ballast condition- Segregated ballast condition- All tanks partially filled.

Experience indicates that the shorter the ship naturalperiod, the greater the impact pressure. The procedure for the selection of the critical loadingconditions would therefore suggest that the loadingconditions with the shortest ship natural period shouldtherefore be considered as a production case.

Sea Condition Head Quartering Beam Stern Quartering

Table 1.1

Foremost

Aftermost

Closest to Amidship

Largest



Chapter 8SECTION 2


2.1 Unrestricted Filling Levels - Unspecified Sea-Going Loading Conditions

When a ship is to be approved for Unrestricted FillingLevels - Unspecified Loading Conditions, many arbitraryship loading conditions are possible. In order to cover thecomplete range of loading conditions, the fully loaded andballast condition are to be considered. These twoconditions gives an upper and lower limit for the possiblerange of ship natural period as shown in Figure 2.1. Boththe roll and pitch motion modes are to be examined.

Because of the unrestricted filling level requirement, thecritical sloshing ranges extend from [SnrBallast-5] to[SnrLoaded+5] seconds in roll and from [SnpBallast-3] toinfinity in pitch. Also, because of unrestricted filling levelsthe ship natural period range extends from SnBallast toSnLoaded for both pitch and roll.

For sloshing in the Roll motion mode shown in Figure2.1.a, the critical fill range extends from F1 to F4. All filllevels between F1 to F4 are to be investigated.

- For fill levels between F1 and F2, SnrBallast is to beused.

- For fill level between F3 and F4, SnrLoaded is to beused.

- For fill levels between F2 and F3, Snr is to be equal toTn.

Similarly, for sloshing in the Pitch motion mode shown inFigure 2.1.b, the critical fill range extends from F1 to F4

where F4 = 0,1%. All fill levels between F1 and F4 are to beinvestigated.

- For fill levels between F1 and F2, SnpBallast is to beused.

- For fill levels between F2 and F3, Snp is to be equal toTn.

- For fill levels between F3 and F4 Snploaded is to beused.

2.2 Restricted Filling Levels - Unspecified Sea-Going Loading Conditions

When a ship is to be approved for Restricted Filling Levels -Unspecified Loading conditions, many arbitrary shiploading conditions are possible within the restrictionsimposed. In order to cover the complete range of loadingconditions, the fully loaded and ballast conditions are to beconsidered. These two conditions gives an upper andlower limit for the possible range of ship natural period. Itis recognised that there might be ship natural period bands which will not be applicable as a result of the limitations ofthe fill levels. However, it is recommended to apply theUnrestricted Filling Levels - Unspecified Sea-Going LoadingConditions procedure outlined in Chapter 8, Section 2.1.

2.3 Unrestricted Filling Levels -Specified Sea-Going Loading Conditions

When a ship is to be approved for Unrestricted FillingLevels - Specified Loading Conditions, each specifiedloading conditions is to be examined for the complete fillranges to determine the critical sloshing fill range for eachtank in both roll and pitch motion modes.

2.4 Restricted Filling Levels -Specified Sea-Going LoadingConditions

When a ship is to be approved for Restricted Filling Levels -Specified Loading Conditions, each specified loadingconditions is to be examined for the restricted fill ranges todetermine the critical sloshing fill range for each tank inboth roll and pitch motion modes.




Chapter 8SECTION 2

Fill (%)

Period(s)

100

F1

F2

F3

0

Snr Ballast

Snr Loaded

5 seconds Range of OperatingShip Natural Periods

F4

5 seconds

Figure 2.1.a

Natural Periods Diagram – Roll Motion

Figure 2.1.b

Natural Periods Diagram – Pitch Motion

Fill (%)

Period(s)

100

F1

F2

F3

0

Snp Ballast

Snp Loaded

3 seconds Range of OperatingShip Natural Periods

F4 = 0,1%





■■ Section 3: Level 2 SloshingAssessmentParameters

3.1 Level 2 Ship Motions

The Combination of accelerations and motions to beconsidered for the sea conditions given in Table 1.1 are asfollows and are to be used with the critical loadingconditions:

- Head Seas Vertical acceleration and pitchangle

- Quartering Seas Vertical acceleration and 50% rollangle

- Beam Seas Vertical acceleration and roll angle- Stern Quartering 75% vertical acceleration and roll

angle.

3.2 Level 2 Fill Range

Where a tank is to be approved for arbitrary fillings, fillheights to be investigated are in 5% increments from 15%to 30% and then 10% increments until the low fill heightcriterion Fc defined in Chapter 4, Section 1 is exceeded. Ifa tank is to be approved for particular fillings, these,together with fillings 5% above and below the particularfillings are to be investigated.

■■ Section 4: Level 3 AssessmentParameters

4.1 Level 3 Ships Motions

The combination of acceleration and motion to beconsidered for the sea conditions given in Table 1.1 are asfollows and are to be used with the critical loadingconditions:

– Head Seas Heave and 70% pitch angle– Beam Seas Heave, sway and 70% roll angle.

The investigation of sloshing in head seas requires thatboth aftermost and foremost tank be examined ifhorizontal internal structure are present, as well as thetank closest to amidship.

4.2 Level 3 Investigation Fill Range

Where a tank is to be approved for arbitrary fillings, theupper and lower bound of critical fill heights are to bedetermined according to level 1 procedure. The fill heightsto be investigated are to be taken in 10% increments fromthe lower bound fill height. The fill height at which thefluid natural period matches the ship natural period shouldalso be investigated together with fill level 5% on eachsides.

If a tank is to be approved for particular fillings, togetherwith fillings 5% above and below the particular fillings areto be investigated.

Where horizontal internal structure members are present,fill height coinciding with the location of the girder andwithin a range of 5% above and below the horizontalgirder should be investigated.


Level 1 Sloshing PressureDetermination


Chapter 9SECTION 1

Section 1: Level 1 Sloshing Pressure Determination

■■ Section 1: Level 1 SloshingPressure Determination

Where a level 1 assessment is indicated in accordance withChapter 7, Section 1, the following points need to beobserved:- For oil carrying cargo tanks with dimensions not

departing from standard practice, no furtherevaluation is needed.

- For LNG/LPG ships, sloshing pressures on tankboundaries should be determined according to theRules for LNG/LPG ships.

Otherwise, an ‘equivalent’ static head is to be obtained byassuming the tank to be rolled or pitched to the ‘lifetime’angles ϑ = φmax or ϑ = θmax respectively defined in Section 4.4, and the equivalent pressure is given by:

P = 11,75 (h + (Ls/2)tan ϑ) KN/m2 (9.1)

where h = the static head in upright position (m).

It is not considered necessary to take translational motionsinto account.



■■ Section 1: Smooth RectangularTanks

Where a level 2 assessment is indicated in accordance withChapter 8, Section 1.2, pressures on the tank boundary areto be derived using SDA Tank Assessment program(10603) (Ref. 1) in association with the ‘lifetime’ angles ofroll and pitch and the vertical acceleration defined inChapter 4, Section 4.

The transverse and longitudinal boundaries are to bestudied separately:

- Transverse bulkheads in association with pitch plusvertical acceleration.

- Longitudinal bulkheads in association with roll plusvertical acceleration.

The centre of rotation is defined in Chapter 4, Section 4.8.

■■ Section 2: Smooth Hopper Tanks

In the case of tanks having upper and/or lower hoppertanks, the output pressures from the SDA Tank Assessmentprogram (10603) have to be ‘corrected’ by applying thecorrelation factors derived from experiments (Ref. 2)shown in Figure 2.1.

No correlation factors are given for the knuckle or corner ofthe tank ceiling, as this would be equivalent to a high fillwhich is excluded from a level 2 assessment.

The pressure at the junction of the upper hopper tank andthe vertical tank side, position B shown in Figure 2.1 isgiven by:

PB = K2.P (10.1)

whereK2 = a correction factor depending on filling height F,

the minimum height of the upper hopper tank h,and the angle of the upper hopper tank with thehorizontal.

K2 = (1 + 2,5 F/h)(1 + 2cosβ)/3 for 0,0 < F < 0,8h P = the output pressure from SDA Tank Assessment

program (10603).

The pressure at the junction of the lower hopper tank andthe vertical tank side, position C shown in Figure 2.1, isgiven by:

PC = K3.P (10.2)

whereK3 = a correction factor depending on filling height F,

and the width of the lower hopper tank w, ifw>0,25Ls. If w ≤0,25Ls then no correction isnecessary.

K3 = 1 + 4F/Ls for 0,0 < F Ls/4K3 = 1 + (H-F)/(H – Ls /4) for Ls/4 < F < H

P = the output pressure from SDA Tank Assessmentprogram (10603)

w = the width of the lower hopper tank (w>Ls /4).

10Level 2 Sloshing PressureDetermination

Section 1: Smooth Rectangular TanksSection 2: Smooth Hopper Tanks




Chapter 10SECTION 2


The higher corner pressures are considered to extend overone stiffener spacing from the corner. When this is notapplicable, the extent of influence may be taken as 0,04Heither sides of the corner. Corrected pressure increases atthe corners from equation (10.1) and (10.2) may bereduced linearly to the limit of corner effect defined above.




Chapter 10SECTION 2

Figure 2.1Maximum Pressure Correction Factor Ki

Position B

Correction Factor K2

0.8 h h

1.0

1.0

3.0

H

β

Position C

Ls

w

0.25 Ls

Correction Factor K3

1.0

1.0

2.0



■■ Section 1: General

Where level 3 assessment is indicated in accordance withChapter 7, Section 6, sloshing pressures are to be obtainedusing a finite difference or similar numerical solution to theone implemented in the SDA Fluids program described inReference 1. Alternatively, agreed model experimentswould be accepted as a means of obtaining the maximumdesign pressures.

The excitation is to cover conditions that would produce amaximum design pressure envelope on the tankboundaries, taking into account the significantcombinations of ship motions, amplitudes and periods andliquid natural period which could occur simultaneously inthe ship’s lifetime.

■■ Section 2: Limitations andAssumptions of SDAFluids Program

SDA Fluids is based on the ‘Marker And Cell’ (MAC)method and uses a two dimensional finite differencecalculation scheme. Details of the theory can be found inReferences 3, 4 and 5.

The following limitations and assumptions apply to theSDA Fluids program:

a) Any model is idealised as a uniform mesh ofrectangular cells and any attempted modelling isinfluenced by this limitation. Associated with thismesh are the three sets of independent variablesnamely, the pressure at the centre of each mesh cell,

the fluid velocities normal to the horizontal andvertical cell edges (Figure 2.1)Various methods are contained within the logic ofSDA Fluids to reduce the effect of limiting the scopeof modelling to a rectangular mesh and carefulimplementation by the user will render theselimitations insignificant in most cases.

b) Pressures are calculated at the centre of each meshcell only and this must be borne in mind whenhydrostatic loads are of considerable importance suchas when the mesh spacings are large.

c) SDA Fluids does not use a two phase fluid model atthe free surface; the ullage volume is treated as avacuum. Also, the free surface is only representableas a single valued function and consequently cannotexhibit features such as breaking waves. This mayaffect the simulation of low fill, large amplitudeexcitation cases.

d) In cases where the depth is such that the tank bottomis exposed due to the motion of the ship, the sloshingprogram output should be considered with care sincenumerical instability may arise in the solutionprocess. The behaviour of the free surface motionshould therefore be examined to detect anyincongruities.

11Level 3 Sloshing PressureDetermination

Section 1: GeneralSection 2: Limitations and Assumptions of

SDA FLUIDS programSection 3: Data Preparation




Chapter 11SECTION 2


Figure 2.1

JMAX

IMAX

NX

NY

1

I

y

x

dydx

J

1

Ui–1,j Ui–,jPi,j

Vi,j

Vi,j–1

Finite Difference Mesh Arrangement with Fictitious Boundary Cells

Finite Difference Field Variables and their Location




Chapter 11SECTION 3

■■ Section 3: Data Preparation

Further reference may be made to the software usermanual (Ref 1).

3.1 Mesh Spacing

Data Card : PMESH

There is one constant mesh spacing associated with each ofthe two principal axes of the mesh.

For a rectangular tank, it is an easy matter to fit a suitablerectangular grid exactly over the dimensions of the tank.However, if the tank is prismatic in shape, then care mustbe exercised in selecting mesh spacings that enable theslope of any chamfered wall of the tank to be adequatelymodelled. Due to the nature of the discretisation process,chamfers can only be modelled as stepwise boundaries andthe best configuration of mesh to chamfer is such that theboundary of the idealised tank coincides with the modelledtank’s wall at the centre of each appropriate horizontal celledge, or the idealised structure bisects horizontal celledges.

If some internal structure such as stiffeners or deck girdersare to be included, further complications are added sincethese too may only have dimensions corresponding to anintegral number of mesh spacings.

Ideally, each mesh spacing should be a factor of all theimportant tank dimensions in its associated principal axistogether with a cell aspect ratio close to unity.Unfortunately, the mesh spacings cannot usually adopttheir ideal values since this will almost mean that thenumber of mesh cells used by the simulation will beprohibitively expensive in computer time. A compromisehas to be made and it is in this area that the skill of themodeller can be most gainfully employed.

For most applications, using 20-30 cells in the horizontal(i) direction and 15-20 cells in the vertical (j) direction is agood compromise. The minimum number of cells is 20 x15. The maximum number of cells allowed by the programis 60 x 40. It must be borne in mind that modelling with ahigher number of cells than the range recommended willtend to give conservative pressure estimates.

3.2 Fitting the mesh to the tank

Data Card : MESH

The relationship of the tank to the mesh may be bestappreciated by drawing the tank on a coarsely ruled sheetof paper, the ruled spacings reflecting the relativedimensions of each mesh cell as illustrated on Figure 3.1.

The first step is to identify the mesh cells that form the areaof the mesh in which fluid will be present if the tank isconsidered to be completely full. These are the active cellsand all active cells together form the idealised tank.

The bottom left hand cell of the calculation grid containedwithin the idealised tank will always have i,j co-ordinatesof (1,1). Cell numbering is carried out in a similar fashionto the grid-squares on a map.

3.3 Boundary Conditions

Data Card : MESH

The next stage is to identify the active cells through whichor on the edge of which the tank boundary passes. Theseactive cells are also boundary cells. The idealised tank isdefined by specifying the boundary cells on MESH cards.Each mesh card defining one section of the tank boundary.

There are four separate regions of the mesh boundary: theleft and right, which include only the vertical regions at theextreme edge of the mesh, and the top and bottom, whichinclude all other parts (See Figure 3.2). A boundarycondition type may be specified for each region. Care mustbe taken when fitting the mesh to chamfered sections oftank wall to avoid overlapping tank boundary sections.Boundary cells must be defined in a consistent direction,that is anticlockwise round the perimeter of the tank, withthe interior of the tank to the left.

The fluid flow conditions at the tank boundaries imposezero velocity normal to a tank wall, either free flow or zerovelocity normal to a tank wall, and either free flow or zerovelocity along a wall. The former, referred to as ‘free slip’,is the default boundary condition for all mesh boundaryregions and should be used unless the boundary layer isgreater than 2 or 3 mesh divisions thick. Otherwise, thelatter ‘no slip’ condition may be used to force fluid to becompletely stationary on a mesh boundary region.



Chapter 11SECTION 3


3.4 Including Internal Tank Structure

Data Card : BAFFLE

Many tanks have internal stiffeners, transverses or deckgirders. The main effect of such structure is to slow downthe fluid motions but sometimes its effects are moreimportant and less obvious.

Conceptually, the effect of this form of construction is toprevent the passage of fluid through an imaginary linedrawn in the fluid imparting zero fluid velocity across thisline. The modelling of the internal structure is based onmesh cell edges and vertical and horizontal baffles can bemodelled using the appropriate switch in the BAFFLE card.

The cell edge on which the structure lies is to the right orat the top of the specified cells with the tank viewedupright.

If a corner of the tank has a high angle of chamfer, anartificially large effective wave slope may be inducedleading to problems of solution stability. This is particularlylikely if the base of the tank becomes exposed and thenumber of mesh spacings for the chamfer in the verticaldirections exceeds the number in the horizontal direction.This may be overcome by modelling the tank as if it wererectangular at the bottom and using BAFFLE cards tospecify chamfers.

3.5 Properties of the Fluid

Data Card : PFLUID

The properties of the fluid and the amount of fluid in thetank affect the pressure loads obtained from sloshing. Theinitial static fill depth and all the physical properties of theidealised fluid used in the calculation are specified on theFLUID card.

The fill depth is specified as a percentage of the maximumdepth of the tank as defined in Chapter 4, Section 3.

The physical properties of the idealised fluid can bespecified using density, speed of sound in the fluid andkinematic viscosity.

It should also be noted that SDA Fluids does not use a twophase fluid model at the free surface; the ullage volume istreated as a vacuum. Also, the free surface is onlyrepresentable as a single valued function and consequentlycannot exhibit features such as breaking waves. This mayaffect the simulation of low fill, large amplitude excitationcases.

3.6 Pressure Output Sampling Points

Data Card : MESH

The most usual form of analysis required for any tank is anassessment of the maximum pressure loads exerted on itswalls. Pressure data may be calculated and output for everycell referenced on MESH cards. However, the number ofsampling points may be reduced by specifying a samplingrate other than unity.

Pressure data for internal structure defined on BAFFLEcards may be requested by specifying MESH cards for therequired cells and switching off the boundary option.

Similarly, velocity data may be calculated and output forevery cell referenced on the MESH cards.

If the tank structure, the applied excitation and inertialforces are all symmetrical, the calculations may be reducedby requesting output for only half of the tank.

3.7 Ullage Pressure

Data Card : PARAM

A constant pressure may be added to each mesh cellpressure to reflect a difference between the ullage pressureand the pressure of the surroundings.




Chapter 11SECTION 3

Figure 3.1

Mesh Co-ordinate System

18

17

16

15

14

13

1211109876

5

4

321

18

17

16

15

14

13

1211109876

5

4

321

1 2 3 4 5 6 7 8 9 10 11 12 13

J

I

J

I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

BOUNDARY CELLS



Chapter 11SECTION 3


BAFFLE, 18, 9, 20, 9, HBAFFLE, 1, 9, 3, 9, H

BAFFLE, 6, 1, 6, 4, V BAFFLE, 14, 1, 14, 4, V

Cells Defining Horizontal Bending

Cells Defining Vertical Baffles

Figure 3.2

Boundary Conditions Examples

MESH, 4, 18, 1, 15

MESH, 1, 14, 1,4

MESH, 1, 3, 3, 1MESH, 4, 1, 10, 1

MESH, 11, 1, 13, 3

MESH, 13, 4, 13, 14

MESH, 13, 15, 10, 18MESH, 9, 18, 5, 18

(cells marked: )

MESH, M-2, N-1, M-3, N

MESH, M-4, N, .., N

*MESH, M-2, -(N-2), M-2, N-2

*MESH, M-1, -(N-4), M-1, N-4

MESH, M, .., M, N-5

MESH, M-1, -(N-3), -(M-1), N-3*

MESH, M, N-5, -(M), N-5*

* - Single Cell, use negative cell number to indicate direction, right hand side forms boundary

J=N

I=M

(cells marked: )




Chapter 11SECTION 3

3.8 Sloshing Excitation Spectrum

Data Cards: PMESHANGLVERTHORISPEC

These data cards generate sinusoidal components ofmotion according to the following representative equation:

A = Aosin (wt + Φ) (11.1)

The motion is imparted to the idealised tank by the use ofthe relevant motion data cards. The degree of freedomapplied to the tank can be generated using the followingdata cards:

- ANGL Constant amplitude, angular motionabout an axis perpendicular to the meshthrough the centre of rotation as definedin Chapter 4, Section 4.8 and specified inthe PMESH data card.

- VERT Constant amplitude vertical (heave)motion

- HORI Constant amplitude horizontal (sway)motion

- SPEC Special form for varying amplitudemotion with the period of excitationaccording to the following equation:

Ao = Amax.e-(Tp-Tpn)2/2Q but not less than specified Amin

(11.2)

WhereAmin and Amax are specified amplitudes according toChapter 4, Section 4.Tpn and Q are the specified natural period and decayconstant.Tp is the current period.

The amplitude, period and relative phase of the forcedmotions are also specified on the motion card, togetherwith a period increment and an incrementation interval ifthe period is to be varied with time.

For ALL card type :

The initial period may be specified as follows :

Sn’ = Sn + 1 for Tn – Sn < –1Sn’ = Sn + 2 for –1 ≤ Tn – Sn ≤ 1 (11.3)Sn’ = Sn + 3 for Tn – Sn >1

Period increment = - 0,001

Increment interval = Tr ( reference time step).

For SPEC (SPECial) card type :

The maximum amplitude is as defined in Chapter 4,Section 4 and Chapter 8, Section 4.1.

The minimum amplitude should be taken as follows :

Amin = 6° for roll= 3° for pitch.

The decay constant should be taken as follows :

Q = 2 for roll= 3 for pitch.

The natural period is taken as Snr for roll , or Snp for pitch.

Initial phase angle = 0,0°

For VERT (VERTical heave) card type :

Required for assessment of longitudinal and transverseboundaries.Amplitude as defined in Chapter 4, Section 4 and Chapter8, Section 4.

Initial phase angle = both 90,0º and –90º for pitchGravity vector = -9,81 m/s2

For HORI (HORIzontal sway) card type :

Required for assessment of longitudinal boundaries.Amplitude as defined in Chapter 4, Section 4 and Chapter 8,Section 4.Initial phase angle = 180,0°



Chapter 11SECTION 3


3.9 Time Control

Data Card : TIMING

Since SDA Fluids is a transient method of analysis, resultsare obtained in the time domain. One of the mostimportant features of a simulation performed by SDAFluids is the selection of the reference timestep which isdefined on the TIMING card.

The reference time step chosen should not be too small asto make computational time excessive or too large so thatimportant features may be missed out in the datacollection.

The default reference time step is given by:

Tr = (Sn’ – 2 )/200 (11.4)

The length of the simulation is such as to give a finalperiod of excitation 4 seconds less than the initial period,thus the total simulation time is given by:

Tsimul = 4000 Tr (11.5)

The following formula may be used to select the outputwindow. The data output may be stored for a simulationtime range of ±(kSn) the simulation time at which theperiod of excitation is equal to the tank natural period, theoutput time range is given by:

t ± (kSn) = Tsimul(Sn’ – Tn)/4 ± (KSn) (11.6)

± (kSn) represents the output window where k is thenumber of oscillations on each side of the time instantduring the simulation where the instantaneous excitationperiod is equal to the tank natural period. A typical k valueof 2 is usually adequate.

The following formula may be used to determine thespectrum period Tp at a given time of the simulation ts:

Tp = Sn’ + (ts.δinc)/δt (11.7)

whereδinc = the period increment [-0,001]

δt = the time step [Tr].

The following formula may be used to determine thespectrum amplitude Apn at a given time of the simulationts:

Apn = Amaxexp[-(Sn – Sn’ – (ts δinc)/δt)2/2Q] (11.8)

If Ao is less than specified Amin, then Ao = Amin

Equation (11.8) may be rewritten as follows depending onthe starting period.

IfSn’ = Sn + 1 Ao = Amax exp[-(1 + (ts δinc)/δt)2/2Q]Sn’ = Sn + 2 Ao = Amax exp[-(2 + (ts δinc)/δt)2/2Q]Sn’ = Sn + 3 Ao = Amax exp[-(3 + (ts δinc)/δt)2/2Q]

(11.9)



■■ Section 1: Sloshing SimulationQuality AssuranceProcedure

1.1 General

It should be borne in mind that the solution to the sloshingphenomenon is obtained using a complex mathematicalformulation of the fluid flow solved in an iterative method,and that the tank geometry, intervals and fluid aremodelled as cells of finite size over which thevelocities/pressures are evaluated. As a consequence, it ispossible that certain tank geometries and internalstructural arrangements and/or sloshing parametersoutside the extensive range covered during the testing andvalidation of this procedure may present some irregularitiesin terms of fluid flow motion, velocities or boundarypressure.

For these reasons which are solely due to the combinationof modelling assumptions and the type of numericalsolution, the following guidelines have been developed inorder to assist both the novice and experienced user of SDAFluids to detect inconsistencies.

1.2 Minimum Quality Assurance PostProcessing Requirements

The following procedure represents the minimum level ofpost-processing for quality assurance of the simulation. It isrecommended to adhere to these guidelines in order todetect any inconsistencies or unexpected behaviour whichmay occur during the simulation as a result of wrong inputdata or limitations due to assumptions or numericalinstabilities. The following output items are to beexamined:

a) Angular – Horizontal – Vertical Amplitude and PeriodTime History

The excitation spectrum should be examined to confirmthat the applied amplitudes and periods are in accordancewith the intended values specified in the proceduresmanual.

b) Free surface motion simulation

The examination of the free surface motion should beperformed both with and without the ZOOM facility.ZOOM OFF(0)/ON(1-10) allows to observe the behaviourof the free surface with the tank fixed/moving. The freesurface motions should be consistent with applied tankmotion. The types of waves formed during the simulationcan be identified be referring to Chapter 3, Section 1. Thefree surface behaviour is to be examined carefully when thetank bottom is exposed and internal members areimmersed.

12Post-Processing of SDA FluidsData

Section 1: Sloshing Simulation Quality Assurance Procedure

Section 2: Pressure Pulse Time Averaging SchemeSection 3: Dynamic and Static Pressures

Section 4: Structure Natural Frequency CalculationSection 5: Force and Couple

Section 6: Pressure Applied to Internal Structural Members

Chapter 12SECTION 1



Chapter 12SECTION 1


Since the free surface position is obtained at the cell centrealongside the cell vertical j axis, and the free surface isrepresented by line segments joining adjacent cells,excessive free surface slope can be identified. In general,the free surface slope should not exceed the cell aspectratio δy/δx.

c) Free surface envelope

The free surface envelope allows the visualisation of theboundary which is in contact with the fluid over the wholeduration of the simulation. The amplitude of fluid motionon the vertical boundaries, the extent of top boundarypotentially subjected to impulsive pressure, and eventuallythe extent of bottom boundary exposed can be observed.

When the centre of rotation is located on the tankcentreline, e.g. roll and the tank structure is symmetricalabout the tank centreline, the free surface envelope shouldbe approximately symmetrical about the centreline.

Note:Differences in the extent of the free surface envelope onopposite tank side in roll arise from the nature of themotion spectrum which is not linear and symmetric withrespect to time. When harmonic excitation is used, thisbehaviour is still occurring, because of the starting motionand computational round off errors inherent to all finitedifference schemes. Perfect symmetry of the free surface isvery rarely attained, but the difference is often negligible.

d) The velocity animation

The velocity vector animation during the simulation timewindow at which maximum magnification of sloshingeffects occurs, allows the visualisation of both the fluidparticle flow, and the velocity magnitude variations. Forcells adjacent to the boundary, the magnitude of thevelocity vector and its direction reflects the relativemagnitude of the boundary pressure. The behaviour of thefluid particle velocity during fluid/boundary interactioncan also be examined. The velocity vector are given at thecentre of the cell. The flow around internal structure andcorners also needs to be observed for consistency in termsof behaviour and input data.

e) Pressure envelope

The pressure envelope shows the distribution of maximumaverage pressure (see Chapter 12, Section 2) for the rangeof cells selected in the input data (MESH Cards). Ingeneral, it is more explicit to display only the cellsbelonging to one boundary, i.e. vertical RHS/LHS, Top,Bottom or Baffle LHS/RHS.

In general, the following pressure envelope behaviour areshown:

– The pressure envelope on the bottom boundaryshould be fairly uniform except when deep girdersprevent fluid motion. This pressure is mainly due tothe hydrostatic term (ρgF), the heave accelerationand a small angular motion component.

– On vertical boundaries, the pressure envelope belowthe still free surface is mainly hydrostatic with adynamic pressure component due to angular motionand heave and sway (if applicable) motion. Abovethe still free surface, the pressure envelope may risesharply if impacts on the ceiling occur during thesimulation. If deep girders are present, the pressureon the vertical boundary may drop close to thebottom as a result of fluid flow damping due to thegirders.

– On the top boundaries, the pressure envelope mayshown high localised pressure, this may be due to thebehaviour of the free surface, or the interaction of thefluid with an internal structural element.

It is recommended to display simultaneously the pressureenvelope of opposite symmetrical boundaries insymmetrical motion.

e) Pressure time histories at cells exhibiting highpressure values

Cells exhibiting high pressures both on the tank top andvertical boundaries should be examined. Pressure types canbe identified by zooming on the pressure pulses in the timehistory. The occurence of high pressure values should beconsistent with the fluid natural period and the spectrumperiod, so that the maximum pressures occur within theoutput window with a range of ~kSn with respect to thesimulations timescale. In certain cases where the tanknatural period is away from the ship natural period, thepressure time history may exhibit two peak regionscorresponding to excitation at ship natural period andexcitation at tank natural period.

These recommendations are not a self-limiting and the useris encouraged to develop its own post-processing schemebased on these guidelines.





1.3 Inconsistencies and AppliedResults

As mentioned in Chapter 12, Section 1.1, due to thecomplexity and assumptions of the mathematical solutionand idealisation, some unexpected results may occurdespite the extensive range of parameters covered duringboth the validation of this procedure and the softwarealgorithm. Certain problems are known to occur andsymptoms/remedy are shown in Table 12.1.

■■ Section 2: Pressure Pulse TimeAveraging Scheme

During a sloshing simulation, high fluid velocities arise dueto the motion of the tank which require a smaller time stepthan the output time step ∆t in order to satisfy the solutionalgorithm conditions. The program uses an auto time-stepping facility whereby the number of time cycles duringtime interval ∆t (simulation cycle) is normally more thanone and typically about 10. The pressure value of one ofthese simulation cycles therefore would be taken torepresent the pressure values over the time interval ∆t. Thisshould not matter if the pressure distribution is consideredas a whole, but quite often the user considers only the peakvalues without taking the characteristics of the pressureimpulses into account. Consequently this would lead tomisinterpretation of the results, that is, very large pressuresoccurring over very small time intervals being taken torepresent the ‘average’ pressure over the required timeinterval ∆t.

To overcome this problem of excessively large sampledpressure values, a pressure pulse averaging scheme has beendevised to give more ‘realistic’ impact pressures (Ref 1).

The pressure pulse time averaging technique is illustratedin Figure 2.1. The instantaneous pressures which are thedirect solution from the algorithm are replaced by theaveraged pressure of the instantaneous pressure over onetime step. This averaged pressure is referred to as thecomputed pressure in this procedure, and is the pressurewhich is used as the sloshing load for capability assessmentof the structure.

Under certain circumstances, low fluid velocity will makethe solution converge for the reference time step withoutrequirement for smaller time step. However for consistencyof the pressure averaging scheme, SDA Fluids version 3.3and above implement a scheme by which a minimum offour values has to be sampled over each time step toperform the averaging scheme.

Table 12.1 Symptoms and Remedy

Symptoms

Exponential increase of pressure towards end of simulation

Unreasonable behaviour of free surface around horizontal baffle

With a fill level near the baffle level.

Unexpected high pressure on bottom, sides and top boundary

Impulse pressure for all boundary occurring at approx. the same

time instants

Other unexpected phenomenon

Remedy

Set minimum amplitude motion of SPEC card to zero

Model horizontral baffle as a step in the deck ignoring volume

above horizontal baffle

“Water Hammer Effect” – Refer to Lloyd’s Register – London

Modify Idealisation of tank – Refer to Lloyd’s Register – London



Chapter 12SECTION 3


■■ Section 3: Dynamic and StaticPressures

Since the pressures determined by SDA Fluids are dynamicin nature and the assessment of the response of thestructure is based on a static analysis, it is required incertain cases to convert the dynamic pressures toequivalent static pressures. This conversion to staticloading is necessary when the load time history is such thatthe impact period is close to the natural period of theloaded structural component. Then, the equivalent staticpressures can be up to twice the magnitude of the dynamicpressure. Figure 3.1 illustrates the dynamic load factordependence on the natural frequency of the structuralcomponent subjected to dynamic loading of triangularshape and duration t1.

The following guidelines are provided to determine if theconversion of dynamic to static pressure is required :

a) If the factor of safety given by the plastic collapseanalysis (SDA Ultimate Strength 10604) is superioror equal to 2, No pressure transformation is required.

b) If the impact pressure pulse is approximated to atriangular pressure pulse of duration t1 as shown inFigure 3.1, the impact pressure can be considered tobe quasi-static if t1/T>2, where T is the naturalperiod of the structural component. In general, thiscase applies for conventional structures subjected tosloshing pressures.

c) If none of the conditions above are satisfied,conversion of the dynamic to static pressure may berequired.

Figure 2.1

Pressure Time History and Pressure Pulse Averaging Scheme

Figure 3.1

Dynamic Load Factor for Triangular Pulse Load




Chapter 12SECTION 3

3.1 Conversion of Dynamic Pressureto Static Pressure

The response of structures to dynamic loads can be quitecomplicated. It must be borne in mind that couplingbetween structural components can and does occur incomplex structures. For these cases, the analyst mustdecide whether or not the various components of the tankwall can be analysed individually or should a multi degreeof freedom system be used.

If each component of the tank structure is to be analysedindividually then it is usually conservative to assume rigidsupport for each element (neglecting flexibility in thesupporting structure) and that the sloshing forces aretransferred undistorted from one member to the next.Often, the response is attenuated by coupling, but it can beincreased as well.

However, using an equivalent single degree of freedomsystem computed from energy principles (Ref 8 & 9), asimple response calculation method for elasto-plasticbehaviour of tank boundaries has been developed, andimplemented in a computer program DYN_STAT (Ref 10).

The reduction of the dynamic pressure data to a staticpressure data for assessment of the structure is based onthe following points :

a) Solution for plastic behaviour can be based on aDynamic Load Factor (DLF) which is a function of theload time history and the natural period of thestructure.

b) The dynamic load factor when multiplied by the peakpressure gives an equivalent static pressure to beused for design purposes.

c) For plastic behaviour of structures which aresubjected to loads of long duration relative to thestructure fundamental period, the equivalent staticpressure gives good estimates of maximum shearreactions in the structures.

DYN_STAT is a development program restricted at presentto Lloyd’s Register – TPDD/ASRD, and SDA Fluids usersmay contact TPDD/ASRD if required. Alternatively, DLFcharts may be used to convert dynamic pressure to staticpressure.

For the remaining part of this procedure, DYN_STAT refersto the process of converting dynamic pressure to staticpressure using either the software or the charts available inAppendix B5. It should be noted that these charts are basedon an elastic response model whilst DYN_STAT software isbased on an elasto-plastic response model.

3.2 Response Calculation

The design pressure to be used for the assessment of thestructural capability is given as follows :

Pstatic = Pdynamic x DLF (12.1)

3.3 Pressure Conversion Procedure

Generally, the worst case sloshing load for a structuralcomponent is one which has the following properties:

- Highest pressure- Shortest rise time

To establish whether or not a dynamic load factor shouldbe used for a given structural component, the followingprocedure may be used:

1) Calculate collapse strength of all structuralcomponents according to the guidelines provided inChapter 13.

2) Compute the sloshing pressure envelope using SDAFluids.

3) Apply a factor of 2 to the sloshing pressure values forthe structural components which satisfy condition c)of Chapter 12, Section 3 (this is equivalent toapplying the maximum possible dynamic load factor).

4) Identify the areas which require further investigation(i.e. 2 x sloshing pressure is greater than the collapsestrength, in association with the factors of safetygiven in Chapter 13, Section 3).

5) Examine the sloshing pressure time history, andidentify critical impacts according to the guidelinesprovided above.

6) Use the DYN_STAT program or charts to convert thedynamic sloshing pressure into equivalent staticsloshing pressure where required.



Chapter 12SECTIONS 3, 4 , 5 & 6


7) Verify that the equivalent static sloshing pressure isless than the collapse pressure of the structuralcomponents where required with the associatedfactors of safety (Chapter 13, Section 3).

■■ Section 4: Structure NaturalFrequency Calculation

Fundamental frequency of structural component may becalculated using LR.PASS desktop computer program LR20301 (Ref 7), or the formulae available in Appendix B andimplemented in the pressure conversion programDYN_STAT (Ref 10). Charts to determine the naturalfrequency of clamped plates in air and with one sideimmersed are also available in Appendix B.

■■ Section 5: Force and Couple

For some type of analysis, it may be required to obtain theforces acting on the structure. This is particularly importantfor independent cargo tanks for which the forces on thetank supports may be found.

The force and couple are found by integrating the pressureon all cell boundary edges. The centre of integration forcouple calculation may be different from the centre ofrotation for angular motions, and may be used, forinstance, to find the shear force and tripping moment onselected internal structure.

■■ Section 6: Pressure Applied toInternal StructuralMembers

When internal structural members represented by bafflesare present, the sloshing pressure applied to the internalmember is the maximum differential pressure at discretetime instant over the simulation period. The differentialpressure is calculated as the difference between thepressures acting at time instant t at opposite cells about theaxis of the structural element.



■■ Section 1: Pressure andStresses

The SDA Fluids program is used to determine themaximum ‘lifetime’ sloshing pressures. When it isappropriate, to determine the design pressures, theresponse of the structural members also has to be takeninto account. Generally, from the standpoint of structuraldesign, the pressure is important not only for its value butas ’pressure multiplied by area’; that is, even high pressurecauses no damage to the structural member and henceposes no problem in terms of structural strength as long asthe area acted upon is very small. The panel areasurrounded by stiffeners is usually taken as being thesmallest unit area.

Structural members must be strong enough to withstandthese effective loads. The plastic collapse load is used inmany instances to indicate the strength, commonly of bothpanel and stiffener, taking into account their collapsemechanism. For example, the strength of the panelfastened in way of primary and secondary members isobtained as the load required to form a roof shaped hinge.As for the primary members, it is necessary to pay attentionalso to the buckling of panel which composes their girders.

Also an allowance should be made for global bendingstresses which might occurs as a lifetime value and beadded to the sloshing load component.

■■ Section 2: Collapse AnalysisProcedures forClamped StiffenedPanels

2.1 Description

The desk-top program SDA Ultimate Strength (10604) (Ref 1) requires information about the following for theevaluation:

- Bulkhead type- Direction of stiffening- Material properties- Applied pressure envelope- Thickness of plate panels- Spacing, spans and scantlings of stiffeners

The program considers a single stiffener and a breadth ofpanel between that and the next stiffener, or a corner of atank. The panel length is taken as the distance betweenframes, see Figure 2.1.

Stiffeners with the following cross sections may beexamined: angle, bulb plate, flat bar or T cross section, seeFigure 2.2.

Stiffeners are continuous and effectively supported at everyfloor, or girder.

Where brackets are used to reduce the effective length ofthe stiffener, it is assumed that these are arrangedsymmetrically either side of the primary member web, andadequately stiffened.

13Strength Assessment

Section 1: Pressure and StressesSection 2: Collapse Analysis Procedures for Clamped

Stiffened PanelsSection 3: Minimum Factors of Safety

Section 4: Girder Structural Analysis Procedure




Chapter 13SECTION 2


Figure 2.1

Panel Geometry

Figure 2.2

Stiffener Sections & Dimensions

Note: Bulb sections are to be represented as anglesections with dimensions given in accordancewith chapter 4 of the DCPD

Flat bar section T section Angle sectiontp tp

tp

tw

tf

tf

bf

tw

tw

dwdw dw

bf

tp = uniform panel thickness

s = panel breadth (stiffener spacing)l = panel length (frame spacing)pu = uniform pressure applied to

unstiffened side of panelσx = applied axial stress acting in the

panelσy = applied transverse stress acting in

the panelτ = applied shear stress acting in the

panel

s

y z

x

l

σy

σx

PuPu

Pu

tp

τ




Chapter 13SECTION 2

The plate thickness used in the calculation are to be Rulethickness. When a plate panel consist of two or morestrakes, then the plate thickness is to be taken as follows:

t = 0.75 t12 + 0.25 t22

wheret1 is the thickness of the major area of the panel,

greater than 2/3 of the panel breadth.t2 is the thickness of the minor area.

Different yield stresses may be specified for the plating,stiffener web and stiffener flange. These are to be taken asthe minimum specified yield stress or 0.5 per cent proofstress. For normal and higher tensile steel or aluminium,Poisson’s ratio is to be taken as 0.3. The modulus ofelasticity is to be taken as 20.6 E4 N/mm2 for normal andhigher tensile steel and 6.89 E4 N/mm2 for aluminium. Forcargoes carried at cryogenic, or elevated temperatures. Theminimum material properties at the correspondingoperating temperatures of the structure are to be taken.

The panel is allowed an initial shape imperfection and apermanent set. These are the maximum deviations of thepanel from a plane surface for the undeformed panel, priorto the application of the forces, and the deformed panel,respectively.

2.2 Assumptions and Limitations

The initial shape imperfection and the allowablepermanent set are determined by the program from thespecified stiffener spacing. Initial shape imperfection andpermanent set are measured positive towards the stiffenerand negative away from the stiffener.

Default values of panel characteristics are:

- Initial panel imperfection s/(120 kp)kp = (245/σop)

where σop is the uniaxial yield stress of plate material

- Panel permanent sets/400 for longitudinal bulkheads/300 for transverse bulkhead

- Panel membrane stiffness factor10 for longitudinal bulkhead, longitudinal

stiffening4 for longitudinal bulkhead, transverse

stiffening

8 for transverse bulkhead, transversestiffening

4 for transverse bulkhead, vertical stiffening

Corrosion margins are incorporated in the factor of safetygiven in Chapter 13, Section 3.

Other assumptions are as defined in Reference 8.

Note:Corrugated bulkheads cannot be assessed using the currentSDA Ultimate Strength program (10604).

2.3 Applied Loads

a) Sloshing Pressures

For a vertically stiffened bulkhead, subdivided byhorizontal girders or stringers, the applied pressure shouldbe an average of that over a height of

- For the panel, not greater than three times the panelbreadth.

- For the stiffener, the effective length.

Where the ratio of the unloaded to the effective span a/lshown in Figure 2.3 is less than 0.4, the computed pressureis to be modified by converting the sloshing pressuredistribution to an equivalent uniformerly distributedpressure according to the procedure described in Appendix C.

For fluid sloshing onto the stiffened side of the plating, thefollowing factors may be applied to the sloshing pressures:

- Peffective = Pcomputed x 0.90 for the panel- Peffective = Pcomputed x 0.70 for the stiffener

The panel is subjected to specified lateral pressures appliedto the outside of the plating, i.e. on the unstiffened surface,and to the stiffened surface. These are the maximumpressure differentials applied to each side of the platewhich will occur at different times. All pressures aredefined as positive.

The maximum net pressure which occur at any time is tobe input for the unstiffened surface (Note that wheresloshing can occur in the adjacent tank, this pressureshould be input if it is effectively greater).





b) Applied Stresses

The panel is also subjected to the following applied in-plane stresses: a stress acting in the direction of thestiffener span or axial stress, a stress acting in the directionperpendicular to the stiffener span or transverse stress, anda shear stress, see Figure 2.1. These stresses, with theexception of those given below, are determined by theprogram.

The applied in-plane stresses which may be input by theuser are:

- Axial stress for sloshing loads on transversallystiffened transverse bulkhead or on a longitudinallystiffened longitudinal bulkhead.

- Transverse stress for sloshing loads on a verticallystiffened transverse bulkhead or on a verticallystiffened longitudinal bulkhead.

- Shear stress for sloshing loads on a longitudinalbulkhead.

It should be noted that consideration should be given tothe torsional/lateral bending strength of internal stiffeners.

c) Hull Longitudinal Stresses – longitudinal bulkheads

Longitudinal still water (sw) and wave (w) direct stressesof magnitude (σsw + 0.25 σw) combined with shearstresses (τsw + 0.25 τw) appropriate to the position of eachpanel being considered should be applied. For transverselystiffened bulkhead, tensile longitudinal stresses need not beconsidered.

2.4 Output

The program prints out all input data and calculates theplastic collapse pressures for the plate panels and stiffenersand also safety factors against failure. Details of inputformat and examples can be found in the users manual(Ref 1).

■■ Section 3: Minimum Factors ofSafety

Safety factors are not to be less than the values given in therelevant ship procedural document, or the values givenbelow:

Longitudinal Bulkhead

- maximum allowable pressure on panel 1.1- maximum allowable pressure on stiffener 1.5- maximum allowable pressure range on stiffener 1.5

Transverse Bulkhead

- maximum allowable pressure on panel 1.3- maximum allowable pressure on stiffener 1.5- maximum allowable pressure range 1.1

on stiffener

Deck

- maximum allowable pressure on panel 1.0- maximum allowable pressure on stiffener 1.0

Figure 2.3

Partial Pressure Distribution

awa

wl

RB MBRAMA

l




Chapter 13SECTION 4

■■ Section 4: Girder StructuralAnalysis Procedure

Horizontal/vertical girders are in some cases subjected tohigh impact pressures. The behaviour of girders underlateral pressure loading differ from tank boundaries sincethe girder has a relatively greater deflection in the directionperpendicular to the plane of the girder due to the freeedge at the face plate. The effect of the free edge is tochange the load distribution on the girder flat bar stiffenerand tripping brackets, so that maximum bending momentsand shear forces will be concentrated at the fixed endwhere they are connected to the primary members. Inaddition, submerged girders are subjected to continuouspulsating loads as a result of the fluid oscillatory motion.This phenomenon may lead to fatigue damage of the girderelements, particularly the tripping brackets, stiffeners, andgirder bracket toes.

Examination of damage records indicates that, to-date nodamage to girders attributed to sloshing loads has beenrecorded. The following girder structural analysisprocedure is therefore given as a guide to be used when itis considered necessary to verify the strength of girders. Ifthe application of this procedure indicates that significantreinforcements are required, the requirements should bespecially considered taking into account service experienceof a similar structure on similar size tank onboard a similarship operating under similar partial filling conditions.

4.1 Finite Element Analysis

To assess the strength of the girder, a Finite Element modelof the girder is to be performed. The girder should beidealised using plate and rod elements as shown in Figure4.1. The boundary conditions to be applied to the modelare shown in Table 4.1. The modelling is to be performedaccording to the guidelines in DCPD Section 3, PlaneFrame.

The level of strength of the stiffeners and tripping bracketsis to be checked using both elastic and plastic analysis. Thelevel of strength of the plating stiffeners is also to beexamined. The strength of the plating is to be checked bothon an elastic basis using the results from the F.E. model,and on a plastic basis by using the axial loads determinedfrom the F.E. model to perform the plastic collapse strengthassessment outlined in Chapter 13, Section 2.

4.2 Analytical Structural Analysis

The plating capability is to be assessed using the SDAUltimate Strength program (10604) in association with thesafety factors defined in Chapter 13, Section 3. However,since the SDA Ultimate Strength program (10604) is notapplicable to a cantilever stiffener, an alternative procedureequivalent to the SDA Ultimate Strength program (10604)has been formulated to assess the structural capability ofstiffeners and tripping brackets under sloshing levels. Theprocedure is based on the estimation of the plastic colllapsepressure for the girder stiffeners and tripping brackets. Thisprocedure illustrated in Figure 4.2, and is to be applied inassociation with a Factor of Safety of 2.0.

Structure Types of Constraints Position Constraints

Intersection of Horizontal Boundary All Grid Points 1, 2, 3, 4, 5, 6

Girder and Longitudinal Plating

Intersection of Horizontal Boundary All Grid Points 1, 2, 3

Girder Plating and

Transverse

Intersection of Tripping Boundary All Grid Points 1, 2, 3, 4, 5, 6

Brackets/Face Flat Imposed Moment from

Stiffeners and Trasnverse Frame Analysis

Table 4.1 Boundary Conditions



Chapter 13SECTION 4


Figure 4.1

Typical Girder F.E. Idealisation




Chapter 13SECTION 4

4.3 Applied Loads

a) Sloshing Pressures

When the girder is subjected to specified lateral pressure tothe unstiffened and stiffened surface. The applied lateralpressure is the maximum pressure differentials applies toeach side of the plate at discrete time instant (See Chapter12, Section 6). The applied sloshing pressure is to be taken asthe average of the cell pressure values over the span of thegirder.

For fluid sloshing on the stiffened side of the plating, thefollowing factor may be applied to the sloshing pressures:

- Peffective = Pcomputed x 0.90 for the panel- Peffective = Pcomputed x 0.70 for the stiffener

b) Applied Loads

For bulkhead stringers, the application of the sloshingloads on the bulkhead provide a bending moment at theconnection of the bulkhead stringers. The moment is to bedetermined either using the simple model shown in Figure4.3, or an equivalent procedure.

c) Hull Longitudinal Stresses

For girder in the longitudinal direction such as deckgirders. longitudinal still water (sw) and wave (w) directstresses of magnitude (σsw + 0.25 σw) combined withshear stresses (τsw + 0.25 τw) appropriate to the positionof each panel being considered should be applied.

For girder in the transverse direction such as bulkheadstringers, tensile longitudinal stresses need not beconsidered.

Figure 4.3

Typical Frame to Determine Applied Moment



Chapter 13SECTION 4


24

L (24m + 11l)

Mp3

2m

Figure 4.2



Two acceptance criteria are proposed, and it is requiredthat when ships are to have partial fillings of tanks, at leastone of the acceptance criteria is to be satisfied :

■■ Section 1: Strength basedacceptance criteria

The acceptance criteria for scantlings of partially filledholds shall be such that the structural members forming thetank boundary have a strength capability which exceedsthe loads requirements.

Any scantlings derived as a result of this procedure are tobe regarded as additional to the Rule requirements for fulltanks in cases where partial fillings is requested.

■■ Section 2: Service basedacceptance criteria

Alternatively, sloshing loads may be controlled by theloading requirement which will be specified by the society,this is done by controlling the natural periods of motion ofthe ship such that synchronisation may not occur.

Ship natural periods should therefore be determined by theuse of appropriate method either computational orexperimental as agreed by the society and a polar curve ofship natural period should be produced together with strictguidelines concerning the loading of the ship.

However special considerations will be given to thisacceptance criteria according to ship type, service andother considerations.

14Acceptance Criteria

Section 1: Strength based acceptance criteriaSection 2: Service based acceptance criteria




■■ Applications

Examples of sloshing calculations for all three levels ofanalysis are shown in Appendix A.

15Applications

Chapter 15APPLICATIONS



■■ References

1. ShipRight SDA Sloshing, Software user manual.Lloyd’s Register.

2. Results of Model Sloshing Experiments for Two BulkCarrier Shaped Tanks due to Rolling, DevelopmentUnit Report No 50, Lloyd’s Register.

3. The MAC Method, a Computing Technique forSolving Viscous, Incompressible, Transient Fluid FlowProblems Involving Free Surfaces, Welch J.E, HarlowF.H, Shannon J.P, Daly B.J, Los Alamos ScientificLaboratory, Report LA-5852, 1975.

4. SOLA - A Numerical Solution Algorithm for TransientFluid Flows, Hirt C.W, Nichols B.D, Pomeroy N.C, LosAlamos Scientific Laboratory, Report LA-5852, 1975.

5. Sloshing in Partially Filled Liquid Tanks and Its Effecton Ship Motion : Numerical Simulations andExperimental Verification, Mikelis N.E, Miller J.K,Taylor K.V, RINA Spring Meeting, 1984, Paper No 7.

6. LR.FLUIDS, Theoretical Manual, TPDD Report 90/01,Lloyd’s Register.

7. DTC Program 20301, Users’ Manual for LR.PASSDesktop Computer Programs, Lloyd’s Register.

8. Evaluation of Liquid Dynamic Loads in Slack LNGCargo Tanks, Ship Structure Committee report, SSC297, 1980.

9. The Component Element Method in Dynamics, S.Levy, J. Wilkinson, Mac Graw-Hill, 1976.

10. Conversion of Dynamic to Equivalent Static SloshingLoads, Theoretical & Users Manual, TPDD Report90/09, Lloyd’s Register.

11. Liquid Sloshing in Cargo Tanks, Ship StructureCommittee report, SSC 336, 1990.

References

REFERENCES



■■ Section 1: Level 1 Investigation

The purpose of this example is to perform a level 1 sloshinginvestigation in order to determine whether or not a higherlevel sloshing investigation is required. The exampleperformed on a comparative basis for two vessels of similardimensions is presented for a given filling level based onvessel draught.

Ship Data

Ship A and ship B having the following general particularsare used for this example :

Ship A Ship B

Ship type Tanker TankerDeadweight (tonnes) 64140 64000Length BP (m) 219,00 220,43Breadth mld (m) 32,20 32,21Depth mld (m) 19,03 18,22Draught max (m) 12,821 13,32GM (m) 4,70 5,46Cb 0,8282 0,8135

Tank Dimensions

The values of the free surface breadths and lengths used tocalculate the cargo natural roll and pitch period are givenas follows :

Ship A Ship BMaximum Free Surface Breath (m) 24,08 24,60Maximum Free Surface Length (m) 22,50 21,00

The maximum cargo tank breadths, as obtained from theship plans, were used as the free surface dimensions in thecalculation of cargo natural periods. The use of these cargotank dimensions produces the greatest values of cargonatural roll period and, hence, the minimum values ofseparation between ship and cargo natural roll periods.

In calculating, the cargo natural pitch periods, the choice offree surface lengths requires consideration of the cargotank lengths and where fitted transverse wash bulkheads.Wash bulkheads which represent more than 85% of thetank cross-sectional area are taken as being effective assloshing barriers which limit the free surface length. In asimilar manner to the roll period, the maximum values offree surface lengths produce the maximum values of cargopitch period and the minimum separation periods.

Loading Conditions

Both ships are considered to be loaded to their summermarks with cargo of S.G = 0.878, and the cargo tanks aretaken to be filled to the level equivalent to 1.1675 xsummer draught above the bottom of the tanks. The cargotank filling details are given below :

Ship A Ship BFill Height (m) 14,97 15,55Ullage (m) 4,06 2,67% Fill 78,70 85,30

AExamples

Section 1: Level 1 InvestigationSection 2: Level 2 InvestigationSection 3: Level 3 Investigation

Appendix A SECTION 1



Appendix ASECTIONS 1 & 2


Ship & Tank Natural Period

For both vessels at the specified loading condition, thevalues of natural periods of the ship and tank which arecalculated for both rolling and pitching motion are shownin the following table together with the minimum values ofroll and pitch separation.

Ship A Ship BShip Roll Period (s) 11,87 11,01Max. Cargo Roll Period (s) 5,86 5,93Min. Roll Period Separation (s) 6,01 5,08Ship Pitch Period (s) 11,40 11,53Max. Cargo Pitch Period (s) 5,47 5,37Min. Pitch Period Separation (s) 5,93 6,15

Ship Maximum ‘Lifetime’ Motion Angles

The maximum ‘lifetime’ motion angles calculated accordingto section 4.4 are shown below :

Ship A Ship BMax. Lifetime Roll Angle (deg.) 24,05 24,17Max. Lifetime Pitch Angle (deg.) 8,091 8,081

Level of Assessment

Both vessels meets the roll period criterion of a least 5seconds and the pitch period separation criterion of at least3 seconds, therefore a level 1 assessment is acceptable.

Acceptance Criteria

Both vessels meets the roll period criterion of a least 5seconds. However, ship B, a segregated ballast tanker withcargo in centre tanks only, has a roll period separation of5.08 seconds. Ship B is used as an example for a level 3sloshing investigation shown in Appendix A.3.

Both vessels meets the pitch period separation criterion ofat least 3 seconds.


This example describes a level 2 sloshing investigation.

Ship Data

Fluid sloshing pressure and structural response have beendetermined for a 22500 m3 LPG carrier which principalship particulars are as follows :

Ship Type LPG/Ammonia CarrierLBP 153,50 mB mld 25,90 mD mld 15,40 mdraught 8,30 mCb 0,7569

Tank Dimensions

The tank arrangement is such as the tanks are divided portand starboard by a centreline deep tank bulkhead, and inaddition a wash bulkhead is fitted in each tank at aboutmid-tank length. The principal particulars for the tanks areas follows :

Cargo LPG or Ammonia Max. Breadth (All Tanks) 23,60 mMax. Depth (All Tanks) 12,08 mSteel Yield Stress 36,00 kg/mm2

Length No 1 Tank 31,30 mLength No 2 Tank 35,45 mLength No 3 tank 29,10 m

Loading Conditions

Calculations for the minimum ship periods have been basedon GM and mean draught data given in the buildersloading manual. The loading conditions which requiredconsideration are presented below along with their valuesof GM and mean draught.

Loading GM Mean VCGConditions Draught

(m) (m) (m)

Tank No 1 LPG No 33 3,85 6,06 8,22Ammonia No 31 3,30 6,47 8,35

Tank No 2 LPG No 14 3,55 5,89 8,22No 37 4,17 6,15 8,22

Ammonia No 35 3,80 5,89 8,35Tank No 3 LPG No 46 3,90 6,07 8,11

Ammonia No 44 3,94 6,07 8,01




Appendix ASECTION 2

Ship & Tanks Natural Periods

Natural periods of the fluid and the ship, which have beencalculated for both rolling and pitching motions using theformulae given in Chapter 3 are shown on figure A.2.1.

For pitching motion, since the transverse wash bulkheadsare closed by more than 85% of the tank cross sectionalarea, the natural pitching periods were derived assuming acomplete bulkhead at this position.

Ship Maximum ‘Lifetime’ Motions

The maximum ‘lifetime’ ship motion angles and tanksaccelerations according to Chapter 4, Section 4 are givenbelow :

Roll Angle (deg.) 25,80Pitch Angle (deg.) 11,78

Tank No Vertical Acceleration(xg)

1 +/- 0,7432 +/- 0,4783 +/- 0,509

Level of Assessment

It can be seen that for each tank, the ship roll periodexceeds that for the fluid by more than 5 seconds, exceptfor LPG and Ammonia cargoes at filling levels less than23% and 20% respectively. However, at these filling levels,impact with the tank ceiling is very unlikely and thereforefluid pressures may be determined using a level 2assessment procedure.

Considering the pitching motion of each tank, theseparation between ship and fluid natural periods is lessthan 3 seconds for all filling levels of LPG and Ammonia.Sloshing impacts on the tank ceiling are unlikely due to theseparation between ship and fluid natural periods with thepossible exception of around the 30% - 35% fills where thenatural period is close to that for the ship. Experience fromprevious analyses indicate that ceiling impact would notoccur at this relatively low fluid level and therefore a level2 assessment procedure has been performed.

Fluid Pressures

Fluid pressures due to sloshing for the partial filling rangeswere calculated using the SDA Tank Assessment program(10603), and the maximum computed pressures areplotted in Figure A.2.2-A.2.4 for tank No 1-3 respectively.

Structural Capability

From Figure A.2.2-A.2.4, it can be seen that the sloshingpressures are substantially less than the Rule designpressures for tank walls for LPG carriers. Therefore, thetank boundaries are covered by Rule requirements. Withrespect to the centreline bulkhead, it may be seen thatsloshing pressures exceed Rule design pressures. Thus, thecentreline and wash bulkheads need to be examined forstructural capability against sloshing loads.

The structural capabilities of individual stiffeners and panelelements in the centreline and wash bulkheads weredetermined using SDA Ultimate Strength program (10604).These are compared in Table A.2.1 to A.2.4 with thedynamic sloshing pressures derived above.

Centreline Bulkhead

The centreline bulkhead scantlings are the same for eachtank. The highest values of sloshing pressures occurred inthe aft region of tank No 1, thus the assessment ofstructural capability can be limited to the aft region of tankNo1 in order to check whether the sloshing loads couldpresent a problem elsewhere. The results for the aft regionof tank No 1 are presented in Table A.2.1 and A.2.2. Fromthe results, it is clear that the ultimate strength of theplating and stiffeners is well in excess of the appliedsloshing pressures. In addition, the factor 3) derived fromthe total pressure range, which in turn is based upon atotal stress range of twice yield considering the maximumpressures applied consecutively to both sides of thebulkhead is satisfactory.

Wash Bulkheads

The structural capability of the wash bulkheads of tank No3 can be taken to be similar to that of tank No 1. The pitchsloshing pressures in the aft and forward regions of thetanks have been used to load the wash bulkheads. FromTable A.2.3 and A.2.4, it is clear that the wash bulkheadsscantlings of each tank as recommended herein aresatisfactory with respect to fluid sloshing.



Appendix ASECTION 2


100

80

60

40

20

0

0 2 4 6 8 10 12 14 16

Tank 1 (fwd)

Tank 1 (aft), 2 and 3

Ammonia

LPG

Roll

Period (secs)

% F

illi

ng

hei

ght

Figure A.2.1

Ship and Fluid Natural Periods

100

80

60

40

20

0

0 2 4 6 8 10 12 14 16

Tank 3 (aft)

Tank 2 (fwd)

LPG and Ammonia

Pitch

Period (secs)

% F

illi

ng

hei

ght

Tank 3 (fwd)

Tank 1 (fwd)




Appendix ASECTION 2

Figure A.2.2

Tank No 1 Fluid Pressure (Partial Fillings)

12 10 8 6 4 2 020

4060

8010

012

014

016

018

020

0

Height above tank bottom (m)

Pre

ssu

re (k

N/m

2 )

Tan

k le

ngt

h =

31,

30m

Des

ign

Pre

ssu

re, t

ank

wal

ls

Fwd

, pit

ch,

Am

mon

iaA

ft, p

itch

, am

mon

ia

Aft

, rol

l, am

mon

ia

Tra

nsv

erse

fra

me

hei

ght

Des

ign

pre

ssu

re,

cen

tre-

lin

e b

ulk

hea

d



Appendix ASECTION 2


Figure A.2.3

Tank No 2 Fluid Pressures (Partial Fillings)

12 10 8 6 4 2 020

4060

8010

012

014

016

018

020

0


Pre

ssu

re (k

N/m

2 )

Tan

k 2

len

gth

= 3

5,45

m

Des

ign

Pre

sure

, tan

k w

alls

Pit

ch, a

mm

onia

Rol

l, am

mon

ia

Tra

nsv

erse

fra

me

hei

ght

Des

ign

pre

ssu

re,

cen

tre-

lin

e b

ulk

hea

d




Appendix ASECTION 2

12 10 8 6 4 2 020

4060

8010

012

014

016

018

020

0


Pre

ssu

re (k

N/m

2 )

Tan

k 3

len

gth

= 2

9.10

m

Des

ign

Pre

sure

, tan

k w

alls

Rol

l, am

mon

ia

Fwd

, pit

ch,

amm

onia

Tra

nsv

erse

fra

me

hei

ght

Des

ign

pre

ssu

re,

cen

tre-

lin

e b

ulk

hea

d

Figure A.2.4

Tank No 3 Fluid Pressures (Partial Fillings)



Appendix ASECTION 2


LocationSpacing x Thickness Capability Applied Pressure Capability/Applied Pressure

(mm x mm) (KN/m2) (KN/m2) (KN/m2)

360 above bottom 725 x 9,0 224,3 156,0 1,44

1807.5 above bottom 725 x 9,0 224,3 140,0 1,60

3257.5 above bottom 725 x 8,5 204,6 122,5 1,67

Location Stiffener Size Span Capability Applied Pressure Capability/Applied Pressure

(mm) (mm) (KN/m2) (KNm/2) (KN/m2)

1) 435,9 152,0 1,48

720 above bottom 350 x 90 x 10/15 IA 3200 2) 411,6 152,0 2,70

3) 536,0 304,0 1,76

1) 301,8 135,5 2,23


3) 370,4 271,0 1,37

1) 300,3 118,0 2,55


3) 368,6 236,0 1,56

1) 268,1 118,0 2,25

3620 above bottom 150 x 150 x 11 FB 3200 2) 261,0 118,0 2,21

3) 332,2 236,0 1,41

LocationSpacing x Thickness Capability Applied Pressure Capability/Applied Pressure

(mm x mm) (KN/m2) (KN/m2) (KN/m2)

Tank 1 or 3

1082,5 above bottom 725 x 10,0 286,1 145,5 1,97

3257,5 above bottom 725 x 8,5 221,7 120,0 1,85

Tank 2

1082,5 above bottom 725 x 10,0 286,1 157,5 1,92

3257,5 above bottom 725 x 8,5 221,7 136,0 1,63

Table A.2.1 Assessment of Centreline Bulkhead Plating Tank No 1 Aft

Table A.2.2 Assessment of Centreline Bulkhead Stiffeners Tank No 1 Aft

Table A.2.3 Assessment of Wash Bulkhead Plating

Notes: 1) Capability for pressure on plane side of plating2) Capability for pressure on stiffener side of plating3) Capability for total pressure range




Appendix ASECTION 2

Acceptance Criteria

From the results, it is clear that the ultimate strength of theplating and stiffeners for the centreline bulkhead is well inexcess of the applied sloshing pressures, the washbulkheads scantlings of each tank as recommended hereinare satisfactory with respect to fluid sloshing.

It should be borne in mind that the computed pressures aresomewhat conservative since as already stated, the methoddoes not include the alleviating influence of the internalstructure particularly the transverse frames and girders.

Further, it is not possible in these calculations to accountfor the pressure reductions caused by the openings in thewash bulkheads which were again assumed to be solidboundaries. Also, with respect to the wash and centrelinebulkheads, it has been assumed that there is no fluid on theother side of the bulkhead to that being considered.

LocationStiffener Size Span Capability Applied Pressure Capability/Applied Pressure

(mm) (mm) (KN/m2) (KNm/2) (KN/m2)

Tank 1 or 3

1) 339,6 141,5 1,48

1445 above bottom 200 x 18 FB 2950 2) 302,2 124,5 2,70

3) 305,7 226,0 1,76

1) 339,6 133,0 2,23

2170 above bottom 200 x 18 FB 2950 2) 302,2 116,0 2,11

3) 305,7 249,0 1,37

1) 339,6 124,0 2,55

2895 above bottom 180 x 18 FB 2950 2) 302,2 107,0 2,41

3) 305,7 231,0 1,56

1) 275,7 116,0 2,25

3620 above bottom 180 x 18 FB 2950 2) 250,2 99,0 2,21

3) 245,7 215,0 1,41

1) 316,3 107,5 2,94

4345 above bottom 120 x 10 FB 1450 2) 290,2 90,0 3,22

3) 274,4 197,5 1,39

1) 142,5 98,0 1,45

5070 above bottom 220 x 12 FB 2950 2) 207,6 81,0 2,56

3) 247,2 179,0 1,38

Table A.2.4 Assessment of Wash Bulkhead Stiffeners

Notes: 1) Capability for pressure on plane side of plating2) Capability for pressure on stiffener side of plating3) Capability for total pressure range



Appendix ASECTION 3



A level 3 investigation is performed for ship B described inAppendix A.1 for a typical tank in rolling motion.

Ship Data

The vessel has the following general particulars :

Ship Type TankerDeadweight (tonnes) 64000Length BP (m) 220,43Breadth mld (m) 32,21Depth mld (m) 18,22Draught max(m) 13,32GM (m) 5,46Cb 0,8135

Figure A.3.1 shows the general arrangement of the cargotanks, and Figure A.3.2 shows a typical transverse sectionamidships.

Tank Dimensions

The free surface breadths used to calculate the cargonatural roll period are given as follows :

Maximum Free Surface Breath (m) 24,60

Loading Conditions

The ship is considered to be loaded to its summer markswith cargo of S.G = 0,878, and the cargo tanks are takento be filled to the level equivalent to 1,1675 x summerdraught above the bottom of the tanks. The cargo tankfilling details are given below :

Fill Height (m) 15,55Ullage (m) 2,67% Fill 85,30

Ship & Tank Natural Period

For the given vessel at the specified loading condition, thevalues of natural periods of the ship and tank which arecalculated for rolling motion are shown in the followingtable together with the minimum values of roll separation.

Ship Roll Period (s) 11,01Max. Cargo Roll Period (s) 5,93Min. Roll Period Separation (s) 5,08

Ship Maximum ‘Lifetime’ Motion Angles

The maximum ‘lifetime’ motion angles calculated accordingto Chapter 4, Section 4 are shown below :

Max. Lifetime Roll Angle (deg.) 24,17Max. Heave Amplitude (m) 4,00Max. Sway Amplitude (m) 2,500.7φmax (deg.) 16,92

The above values of period and motions were used toprovide the standard sloshing excitation spectrumparticular to the subject vessel.

The VCG was taken at 8,59 metres above the baseline andthe kinematic viscosity of fluid cargo was taken to be 5.0Centistokes (5x10-6 m2/s) reflecting the minimum valuefor the range of cargoes carried by this vessel and whichwill produce the greatest values of liquid cargo pressures.

LR.FLUIDS Model

To fit the mesh to the tank and obtain a suitable meshspacing for the tank geometry, the cell size analysis hasbeen performed according to the recommendations inChapter 13, Sections 2.1 and 2.2 as shown in the followingTables A.3.1.

From this analysis, it can be seen that there are twosuitable cell size dimensions :

22 x 16 = 352 cells30 x 22 = 660 cells

Although the 352 cells model would reduce computationaltime significantly compared to the 660 cells model byapproximately half, the 660 cells model has a mesh spacingin the i direction which is equal to the longitudinal deckspacing, this may present an advantage for reducing thepressure data.

The model datafile is shown in Appendix D.

The LR.FLUIDS model of the cargo tank is shown on FigureA.3.3. The LR.FLUIDS model consisted of :

- 30 cell divisions across the tank breadth- 22 cell divisions up the tank height or - 660 cells in total

The girders of the upper deck and bottom were representedby internal baffles as shown on Figure A.3.3.




Appendix ASECTION 3

Post-Processing of LR.FLUIDS Data

The procedure followed is given in Chapters 12 & 13.

a) Collapse Strength

The calculated values of roll sloshing pressures providedlocal pressures to be compared with the collapse strengthof the longitudinal bulkheads and upper deck plating andattached stiffeners forming the cargo tank boundaries. Thecollapse strength of these items of structure was calculatedusing SDA Ultimate Strength program (10604). The resultsfor the deck plating and longitudinal bulkhead plating areshown in Table A.3.3 and A.3.4.

Hull girder bending and shear stresses corresponding to theloading condition investigated, in combination with waveinduced stress, were distributed through the hull sectionand applied to the stiffened panels examined, asappropriate.

b) Adjusted maximum Dynamic Pressure

The pressure envelope obtained by LR.FLUIDS is shown inFigure A.3.4. These results are summarised in Table A.3.3and A.3.4. for the deck and the longitudinal bulkhead.Because of possible dynamic pressure magnification, thepressure are multiplied by a factor of 2 to identify thecritical areas. Also, where applicable, the pressure arecorrected according to Chapter 13, Section 2.3.

Note: Since the tank is symmetrical, the maximum pressure ofthe symmetric cells should be considered as the appliedpressure. Differences in pressure magnitude for oppositecells arise from the nature of the motion spectrum which isnot linear and symmetric with respect to time. Whenharmonic excitation is used, this recommendation is stillvalid. Because of computational round off errors inherentto all finite difference schemes, perfect symmetry of thepressure values is very rarely attained, but the difference isoften negligible.

c) Identification of critical Areas

The critical areas have been identified in Table A.3.3 andA.3.4. The results of these table are shown in Figures A.3.5and A.3.6.

d) Analysis of Critical Pressures

The critical impact pressures identified in the previoussection have been analysed as follows:

- For the relevant cells, a pressure time history hasbeen obtained (Figure A.3.7 to A.3.8)

- All impact pressures with magnitude greater than thecollapse strength have been magnified in order tostudy the time history.

- Impact pressure with duration significantly largerthan the structural component natural period arediscarded.

- The remaining impact pressures have been analysedusing the DYN_STAT program.

The results are summarised in Table A.3.3 and A.3.4.

Results

Longitudinal Bulkhead

Figure A.3.7 shows a typical sloshing pressure history.Because of the fill level, most cells on the longitudinalbulkhead exhibit a non impulsive dynamic pressure timetrace. These pressures time traces have a period of theorder of the tank natural period/excitation period. Fromthe magnified plot, it can be seen that the impact durationis significantly larger than the natural period of the paneland stiffener and associated plating ( 0.0124 s and 0.0142s respectively). Therefore, the pressure on the longitudinalbulkhead can be assumed to be quasi static. The pressurevalues obtained from LR.FLUIDS and adjusted according toChapter 13, Section 2.3 can be assumed to be themaximum lifetime values, and the dynamic load factor isunity. From the strength analysis summarised in TablesA.3.3, and using a Factor of Safety for the panel of 1.1 asgiven in Chapter 13, Section 3. It can be seen that panelnumber 9 to 20 are deficient and require an increase inscantlings.



Appendix ASECTION 3


Deck

Figure A.3.8 shows a typical sloshing pressure history.However, for completeness in this example, the programDYN_STAT has been used. The dynamic Load Factors (DLF)have been calculated with and without heave acceleration.The DLF values remain close to unity. In order to check thesensitivity of the dynamic load factor to the pressuremagnitude and impact duration, the DLF have beencomputed for a range of impact magnitude and duration.Results showed that the level of confidence that the DLFvalue would remain close to unity was large. Therefore, thepressure on the deck can be assumed to be quasi static. Thepressure values obtained from LR.FLUIDS and adjustedaccording to Chapter 13, Section 2.3 can be assumed to bethe maximum lifetime values, and the dynamic load factoris unity.

Note that this example is for illustration purposes since onlythe panel plating has been assessed. In a normalassessment, the stiffener and associated plating should alsobe investigated.

A typical DYN_STAT input data file and output for the deckcell of interest is shown in Appendix D.

The display facility associated with LR.FLUIDS programenables the fluid motion to be shown using velocity vectordisplay at discrete time increments. Figure A.3.9 showsfour typical ‘snap shot’ plots for representative positions inan oscillating cycle.




Appendix ASECTION 3

Figure A.3.1

General Arrangement



Appendix ASECTION 3


Figure A.3.2

Midship Section




Appendix ASECTION 3

Figure A.3.4

Pressure Envelope

Figure A.3.3

Tank Mesh



Appendix ASECTION 3


Figure A.3.6

Deck Loading & Capability

Figure A.3.5

Longitudinal Bulkhead Loading & Capability




Appendix ASECTION 3

Figure A.3.7

Pressure Time History at Centre of Cell (30,1)



Appendix ASECTION 3


Figure A.3.8

Pressure Time History at Centre of Cell (8,22)




Appendix ASECTION 3

Figure A.3.9

Typical Plots of Velocity Vectors



Appendix ASECTION 3


Figure A.3.10

Sloshing Excitation Spectrum




Appendix ASECTION 3

Figure A.3.11

Free Surface Envelope



Appendix ASECTION 3


Table A.3.1 Vertical & Horizontal Cell Size Study

Vertical Dimensions

Tank Height 18,220 m

Total Number of Cells 15 16 17 18 19 20 21 22

Vertical Mesh Spacing (m) 1,215 1,139 1,072 1,012 0,959 0,911 0,868 0,828

Meshed Tank Height (m) 18,225 18,224 18,224 18,216 18,221 18,220 18,228 18,216

Error (%) 0,027 0,022 0,022 0,022 0,005 0,000 0,044 0,022

Bottom Girder Height 4,820 m

Number of Cells 4 4 4 5 5 5 6 6

Meshed Girder Height (m) 4,860 5,556 4,288 5,060 4,795 4,555 5,208 4,968

Error (%) 0,830 5,477 11,037 4,979 0,519 5,498 8,050 3,071

Deck Girder Height 2,400 m

Number of Cells 2 2 2 2 3 3 3 3

Meshed Girder Height (m) 2,430 2,278 2,144 2,024 2,877 2,733 2,604 2,484

Error (%) 1,250 5,083 10,667 15,667 19,875 13,875 8,500 3,500

Cumulative Error (%) 2,107 10,582 21,726 20,668 20,339 19,373 16,594 6,592

Horizontal Dimensions

Tank Width 24,600 m

Total Number of Cells 20 21 22 23 24 25 26 27 28 29 30 31 32

Horizontal Mesh Spacing (m) 1,230 1,171 1,118 1,070 1,025 0,984 0,946 0,911 0,879 0,848 0,820 0,794 0,769

Meshed Tank Width (m) 24,600 24,591 24,596 24,610 24,600 24,600 24,596 24,597 24,612 24,592 24,600 24,614 24,608

Error (%) 0,000 0,037 0,016 0,041 0,000 0,000 0,016 0,012 0,049 0,033 0,000 0,057 0,033

Bottom Girder

to Long BHD 5,740 m

Number of Cells 5 5 5 5 6 6 6 6 7 7 7 7 7

Meshed Spacing (m) 6,150 5,855 5,590 5,350 6,150 5,904 5,676 5,466 6,153 5,936 5,470 5,558 5,383

Error (%) 7,143 2,003 2,613 6,794 7,143 2,857 1,115 4,774 7,195 3,415 0,000 3,171 6,220

Outer Girder

to C.L. Girder 6.560 m

Number of Cells 5 6 6 6 6 7 7 7 7 8 8 8 9

Meshed Spacing (m) 6,150 7,026 6,708 6,420 6,150 6,888 6,622 6,377 6,153 6,784 6,560 6,352 6,921

Error (%) 6,250 7,104 2,256 2,134 6,250 5,000 0,945 2,790 6,204 3,415 0,000 3,171 5,503

Cumulative Error (%) 13,393 9,144 4,886 8,969 13,393 7,857 2,076 7,575 13,448 6,862 0,000 6,398 11,755




Appendix ASECTION 3

A B Cell Centre Maximum Effective 2x

i i j Pressure Pressure Effective

max (A, B) Pressure

(m) (KN/m2) (KN/m2) (KN/m2)

1 30 1 0,415 360,50 360,50 721,00

1 30 2 1,244 326,60 326,60 653,20

1 30 3 2,073 319,80 319,80 639,60

1 30 4 2,901 313,10 313,10 626,20

1 30 5 3,730 306,60 306,60 613,20

1 30 6 4,559 300,50 300,50 601,00

1 30 7 5,388 295,10 295,10 590,20

1 30 8 6,217 290,50 290,50 581,00

1 30 9 7,046 286,40 286,40 572,80

1 30 10 7,875 282,80 282,80 565,60

1 30 11 8,704 279,80 279,80 559,60

1 30 12 9,534 277,50 277,50 555,00

1 30 13 10,363 275,90 275,90 551,80

1 30 14 11,192 275,30 275,30 550,60

1 30 15 12,021 280,50 280,50 561,00

1 30 16 12,850 287,50 287,50 575,00

1 30 17 13,679 295,70 295,70 591,40

1 30 18 14,508 304,50 304,50 609,00

1 30 19 15,337 313,20 313,20 626,40

1 30 20 16,166 320,50 320,50 641,00

1 30 21 16,995 324,50 324,50 649,00

1 30 22 17,824 347,00 347,00 694,00

A B Cell Centre Maximum Effective 2x

i i j from Deck Pressure Pressure Effective

Edge max (A, B) Pressure

(m) (KN/m2) (KN/m2) (KN/m2)

1 30 22 0,410 347,00 312,30 624,60

2 29 22 1,230 338,10 304,29 608,58

3 28 22 2,050 335,30 301,77 603,54

4 27 22 2,870 354,30 318,87 637,74

5 26 22 3,690 393,10 353/79 707,58

6 25 22 4,510 410,10 369,09 738,18

7 24 22 5,330 418,80 376,92 753,84

8 23 22 6,150 447,00 402,30 804,60

9 22 22 6,970 117,00 105,30 210,60

10 21 22 7,790 107,60 96,84 193,68

11 20 22 8,610 108,90 98,01 196,02

12 19 22 9,430 113,00 101,70 203,40

13 18 22 10,250 121,90 109,71 219,42

14 17 22 11,070 141,00 126,90 253,80

15 16 22 11,890 171,60 154,44 308,88

Panel Panel Collapse Adjusted Critical Min.

No Center Strength Sloshing Pressure Safety

Pressure Factor

(m) (KN/m2) (KN/m2)

1 0,450 541,421 359,05 1,508

2 1,350 541,421 325,73 1,662

3 2,140 516,601 319,25 1,618

4 2,280 517,676 313,76 1,650

5 3,575 343,934 307,82 1,117

6 4,405 344,649 301,64 1,143

7 5,235 345,232 296,10 1,166

8 6,065 345,232 291,35 1,185

9 6,895 304,948 283,15 _***_ 1,062

10 7,725 304,948 283,45 _***_ 1,076

11 8,555 304,948 280,34 _***_ 1,008

12 9,385 285,763 277,91 _***_ 1,028

13 10,215 285,763 276,18 _***_ 1,035

14 11,045 285,146 275,41 _***_ 1,035

15 11,875 285,146 279,59 _***_ 1,020

16 12,705 247,522 286,28 _***_ 0,865

17 13,535 247,522 294,28 _***_ 0,841

18 14,365 228,073 302,99 _***_ 0,753

19 15,195 228,073 311,72 _***_ 0,732

20 16,025 228,073 319,26 _***_ 0,714

21 16,762 456,681 323,38 1,412

22 17,406 456,681 335,67 1,361

23 18,050 456,681 353,15 1,293

Panel Panel Collapse Adjusted Critical Min.

No Center Strength Sloshing Pressure Safety

Pressure Factor

(m) (KN/m2) (KN/m2)

1 0,410 596,649 312,30 1,910

2 1,230 596,649 304,29 1,961

3 2,050 596,649 301,77 1,977

4 2,870 596,649 318,87 1,871

5 3,690 596,649 353,79 1,686

6 4,510 596,649 369,09 1,617

7 5,330 596,649 376,92 1,583

8 6,150 596,649 402,30 1,483

9 6,970 596,649 105,30 5,666

10 7,790 596,649 96,84 6,161

11 8,610 596,649 98,01 6,088

12 9,430 596,649 101,70 5,867

13 10,250 596,649 109,71 5,438

14 11,070 596,649 126,90 4,702

15 11,890 596,649 154,44 3,863

Table A.3.3 Longitudinal Bulkheads Loading & Capability

Longitudinal Bulkhead Sloshing Pressure Analysis Longitudinal Bulkhead Plating Strength Analysis

Minimum Factor of Safety 1,10

Table A.3.4 Deck Loading & Capability

Deck Sloshing Pressure Analysis

Deck Plating Strength Analysis

Minimum Factor of Safety 1,00



■■ Section 1: Natural Frequency ofPlate

The natural frequency of a clamped plate in air is given by

fair = 1/(2 π)-(1/(ρ h)(A4Dx/a4+B4Dy/b4

+2CDxy/(a2b2)) Hertz

whereA = 4,730B = 4,730C = 151,30h = plate thicknessρ = plate material density

Dx = (Exh3)/(12(1-νxνy))Dy = (Exh3)/(12(1-νxνy))

Dxy = Dxνy + Gh3/6ν = Poisson’s ratio

Ex, Ey= Elastic moduli parallel to x and y axes,respectively

G = Shear modulusa = panel lengthb = panel breadth

The natural frequency in air of a clamped plate of aspectratio a/b can be obtained from the graphs shown on FigureB.1.1. These figures have been computed using thefollowing variables.

Ex = 206,00 E9 N/m2

Ey = 206,00 E9 N/m2

ν = 0,30ρ = 7800 kg/m3

The natural frequency of an isotopic charged plate in air fair may be reduced to the followng expression:

fair

= 55375t + + 0,6045 Hertz

where

a panel lenght (metres)b panel breadth (metres)t panel thickness (mm)

fair

may be rewritten as follows:

fair

=

where

K = 5.5375 + + 0,6045

For a selection of plate panel aspect ratio a/b, thecoefficient k is given in Table B1, and illustrated in FigureB.1.1.

Table B1

1.00 1.25 1.50 2.00 3.00 4.00 5.00 6.00 7.00 8.00

K 8,937 9,278 10,058 12,201 17,260 22.607 28,042 33,516 39,009 44,514

BNatural frequencies ofStructural Components

Section 1: Natural Frequency of PlateSection 2: Natural Frequency of Plate Stiffener

Section 3: Effect of SubmergenceSection 4: Dynamic Load Factor Charts

Appendix B SECTION 1

ba

2( ) ab

2( )

ba

2( ) ab

2( )

ba

Ktab



Appendix BSECTION 1


Figure B.1.1




Appendix BSECTIONS 2, 3 & 4

■■ Section 2: Natural Frequency ofPlate Stiffener

The natural frequency of a plate stiffener in air is given by

fi = Ki/(2πL2) (EI/(m(1+ π2El/L2GA))) Hertz

whereEI = Flexural rigidity of plate stiffener combination

GA = Shear rigidity of the plate stiffener combinationL = Beam length

m = Mass per unit length of the stiffener andassociated plating

Ki = Constant where i refers to the mode of vibration

Mode Ki

1 22.40

2 61.70

3 121.0

4 200.0

5 299.0

■■ Section 3: Effect of Submergence

To obtain the frequency fwater of a plate with one sideexposed to air and the other side exposed to a liquid, thefrequency calculated in air fair may be modified by thefollowing formula :

fwater = fair. ψ

whereψ = (p/(p + ρ1/ρp))p = π t (1/a2 + 1/b2)

ρ1 = density of the liquidρp = density of the plate

t = plate panel thickness

The coefficient Ψ is illustrated for various and ratio inFigure B.3.1.

ab

tb

■■ Section 4: Dynamic Load FactorCharts

4.1 Gradually Applied Load

For a gradually applied load as shown in Figure B.4.1., theinstantaneous dynamic load factor DLF can be calculated asfollows:

0 ≤ t ≤ t1 DLF = t/t1 – sin(ω t)/ω t1 = t/t1– T/(2πt1).sin(ω t)

t1 < t DLF = 1 + T/(2πt1)[sin(ω(t–t1)) – sin(ω t)]

wheret1 is the rise of the applied load as shown in Figure

B.4.1.T is the natural frequency of the structural

componentω = 2π/T

Typical instantaneous DLF time histories are shown inFigure B 4.2. The maximum DLF is the value to be used forconversion of dynamic pressure to static pressure. DLFmax is given in Figure B.4.3 and Table B.4.1 Linearinterpolation may be performed to obtain DLF values forintermediated t1/T values.

Figure B.4.1

Typical Gradually Applied Load

P

P1

t1Time



Appendix BSECTION 3


Figure B.3.1




Appendix BSECTION 4

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

DLF

Time t1

t1/T = 0.250 t1/T = 3.333

2.000

1.800

1.600

1.400

1.200

1.000

0.800

0.600

0.400

0.200

0.000

DLF

max

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000

t1/T

Figure B.4.2Gradually Applied Load Typical Instantaneous DLF Time Histories

Figure B.4.3Gradually Applied Load DLF max - t1/T



Appendix BSECTION 4


4.2 Triangular Pulse Load

For a triangular pulse load as shown in Figure B.4.4, theinstantaneous dynamic load factor DLF can be calculated asfollows:

0 ≤ t ≤ 0.5t1 DLF = 2t/t1 – T/πt1).sin(ωt)0.5t1 ≤ t ≤ t1 DLF = 2 – 2t/t1 + T/(πt1)

[2sin(ω(t–t1/2)) – sin (ωt)]t1 ≤ t DLF = T/(πt1)[–sin(ω(t–t1))

+ 2sin(ω (t–t1/2)) – sin (ωt)]

wheret1 is the rise time of the applied load as shown in

Figure B.4.4.T is the natural frequency of the structural

componentω = 2π/T

Typical instantaneous DLF time histories are shown inFigure B.4.5. The maximum DLF is the value to be used forconversion of dynamic pressure to static pressure. DLFmax is given in Figure B.4.6 and Table B.4.2 Linearinterpolation may be performed to obtain DLF values forintermediate t1/T values.

Figure B.4.4

Typical Triangular Pulse Load

P

P1

2t1t1

Time




Appendix BSECTION 4

1.5

1

0.5

0

-0.5

-1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

DLF

Time t1

t1/T = 0.250 t1/T = 1.250

2

Figure B.4.5Triangular Pulse Load Typical Instantaneous DLF Time Histories

2.000

1.800

1.600

1.400

1.200

1.000

0.800

0.600

0.400

0.200

0.000

DLF

max

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000

t1/T

Figure B.4.6Triangular Pulse Load DLF max - t1/T



Appendix BSECTION 4


t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

0,00

00,

000

1,00

01,

508

2,00

01,

000

3,00

01,

158

4,00

01,

000

5,00

01,

092

0,10

00,

312

2,10

01,

481

2,10

00,

960

3,10

01,

168

4,10

00,

978

5,10

01,

083

0,20

00,

608

2,20

01,

441

2,20

00,

953

3,20

01,

167

4,20

00,

976

5,20

01,

069

0,30

00,

875

2,30

01,

397

2,30

00,

980

3,30

01,

155

4,30

00,

989

5,30

01,

050

0,40

01,

098

2,40

01,

342

2,40

01,

016

3,40

01,

132

4,40

01,

007

5,40

01,

056

0,50

01,

273

2,50

01,

282

2,50

01,

055

3,50

01,

103

4,50

01,

029

5,50

01,

058

0,60

01,

391

2,60

01,

228

2,60

01,

089

3,60

01,

084

4,60

01,

050

5,60

01,

054

0,70

01,

465

2,70

01,

165

2,70

01,

115

3,70

01,

070

4,70

01,

069

5,70

01,

045

0,80

01,

501

2,80

01,

104

2,80

01,

126

3,80

01,

060

4,80

01,

083

5,80

01,

032

0,90

01,

514

2,90

01,

052

2,90

01,

135

3,90

01,

025

4,90

01,

091

5,90

01,

017

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

t1/T

DLF

max

0,00

00,

000

1,00

01,

000

2,00

01,

000

3,00

01,

000

4,00

01,

000

5,00

01,

000

0,10

01,

984

2,10

01,

089

2,10

01,

047

3,10

01,

032

4,10

01,

024

5,10

01,

019

0,20

01,

935

2,20

01,

156

2,20

01,

085

3,20

01,

059

4,20

01,

045

5,20

01,

036

0,30

01,

858

2,30

01,

198

2,30

01,

111

3,30

01,

078

4,30

01,

060

5,30

01,

049

0,40

01,

757

2,40

01,

216

2,40

01,

125

3,40

01,

089

4,40

01,

069

5,40

01,

056

0,50

01,

637

2,50

01,

212

2,50

01,

127

3,50

01,

091

4,50

01,

071

5,50

01,

058

0,60

01,

504

2,60

01,

189

2,60

01,

116

3,60

01,

084

4,60

01,

066

5,60

01,

054

0,70

01,

368

2,70

01,

151

2,70

01,

095

3,70

01,

070

4,70

01,

055

5,70

01,

045

0,80

01,

232

2,80

01,

104

2,80

01,

067

3,80

01,

059

4,80

01,

039

5,80

01,

032

0,90

01,

109

2,90

01,

052

2,90

01,

034

3,90

01,

025

4,90

01,

020

5,90

01,

017

Table

B.4

.2 T

riangula

r P

uls

e L

oad D

LF m

ax

- t1

/T

Table

B.4

.1 G

radually

Applie

d L

oad D

LF m

ax

- t1

/t



■■ Section 1: General

The conversion of an arbitrary distributed loading is basedon the assumption that failure occurs at fixed ends in theplastic mode. Arbitrary distributed loading end momentsand shear forces are used to determine the equivalentuniformly distributed loading.

Location A B A’ B’Reactions RA RB RA’ = WL/2 RB’ = WL/2Moments MA MB MA’ = WL2/12 MB’ = WL2/12

The equivalent uniformly distributed loading is obtained asfollows:

a) Determine the equivalent uniformely distributedloads due to the end reactions.

WRA = 2.RA/LWRB = 2.RB/L

b) Determine the equivalent uniformly distributed loadsdue to the end moments.

WMA = 12.MA/L2

WMB = 12.MB/L2

c) The equivalent uniformly loading load W is given by

W = Max[WRA; WRB; WMA; WMB]

CDetermination of EquivalentUniformly Distributed Loading

Section 1: GeneralSection 2: Determination of Equivalent Uniformly

Distributed Loading

Appendix C SECTION 1

RB MBRAMA

l

RBRAMA

l

weq



Appendix CSECTION 2


■■ Section 2: Determination ofEquivalent UniformlyDistributed Loading

2.1 Trapezoidal Distributed Loading

To determine the end moments and reactions for a simpletrapezoidal loading shown below, the following formulaemay be used.

RA = wa(l–a)3(l+a)/(2l3) + (w1–wa)(l–a)3(3l+2a)/2013)RB = –wa(l-a)3(1+3a)/(12l3) + (w1–wa)(l–a)3(2l+3a)/(60l2)RB = (wa+w1)(l–a)/2–RA

MB = RAl+MA–wa(l–a)2/2–(w1–wa)(l–a)2/6

For the simple trapezoidal case, a regression analysis hasbeen performed to express the equivalent loading weq interms of w1, wa/w1, and a/l. The equivalent loadingis as follows:

Weq = w1[0.702+0.291α) + (–0.307 + 0.557α)β+ (–0.400–0.850α)β 2]

whereα = wa/w1

β = a/l

2.2 Arbitrary Distributed Loading

To determine the end moments and reactions for anarbitrary distributed loading, it is assumed that the loadingdistribution can be broken down into n segments. Eachsegment is considered as a component load case yieldingend moments and reactions are obtained by summingindividual components. End moments and reactions areobtained using the formulae for the simple trapezoidaldistributed loadcase.

For a distributed loading made of three segments as shownbelow, the total loadcase may be divide in three individualloadcase components as shown below.

w1 = wA – w3

w2 = wD – w2

w3 = wB – a(wB–wC)/(a–b)w4 = l(wB–wC)/(a/b) +wB – a(wB–wC)/(a/b)

For each component loadcase, the reactions and momentsat A and B can be calculated.

Reaction MomentLocation A D A DLoadcase

1 RA1 RD1 MA1 MD1

2 RA2 RD2 MA2 MD2

3 RA3 RD3 MA3 MD3

Total RAt RDt MAt MDt

Hence, the equivalent uniformly distributed loads due tothe end reactions.

WRAt = 2.RAt/lWRDt = 2.RDt/l

awa

wl

RB MBRAMA

l

A

WAWC

RBT MBTRATMAT

LB

WBWD

A

W1

+

B

W2

+

W4

L

W3




Appendix CSECTION 2

The equivalent uniformly distributed loads due to the endmoments.

WMAt = 12.MAt/l2

WMDt = 12.MDt/l2

Thus, the equivalent uniformly loading load W is given by

W = Max[WRAt; WRDt; WMAt; WMDt]

For a distributed loading made of four segments as shownbelow, the total loadcase may be divide in four individualloadcase components as shown below.

w1 = wA –(w3+w5)w2 = wD –(w4+w6)w3 = wB – a(wB–wC)/(a–b)w4 = l(wB – wC)/(a/b) + wB – a(wB–wC)/(a/b)w5 = wC – b(wC–wD)/(b/c)w6 = l(wC – wD)/(b–c) + wC – b(wC–wD)/(b–c)

For each component loadcase, the reactions and moments at A and E can be calculated.

Reaction MomentLocation A E A ELoadcase1 RA1 RE1 MA1 ME1

2 RA2 RE2 MA2 ME2

3 RA3 RE3 MA3 ME3

Total RAt REt MAt MEt

Hence, the equivalent uniformly distributed loads due tothe end reactions.

WRAt = 2.RAt/lWREt = 2.REt/l

The equivalent uniformly distributed loads due to the endmoments.

WMAt = 12.MAt/l2

WMEt = 12.MEt/l2

Thus, the equivalent uniformly loading load W is given by

W = Max[WRAt; WREt; WMAt; WMEt]

The same principles can be extended to more than 4segments, and it would be recommended to implement thiscalculation procedure on a spreadsheet to speed up thecalculation process.

A

WAWC

RBT MBTRATMAT

CB

WBWE

W4

L

W3

L

WD

W6

L

W5

C

W2

+A

W1

+

+



DSDA FLUIDS Data File DYN_STAT Data File & Output

Appendix D DATA FILE AND OUTPUT



Appendix DDATA FILE & OUTPUT


TITLE A.3 LEVEL 3 INVESTIGATION / ROLL 70.05%

$

PMESH 0.82 0.828 0.0 -0.81 1 1 1 1

$

MESH 1 1 30 1 2001 BDRY NOTF NOTC




$

MESH 8 1 8 6 2001 NOTB

MESH 9 1 9 6 2001 NOTB

MESH 15 1 15 6 2001 NOTB

MESH 16 1 16 6 2001 NOTB

MESH 22 1 22 6 2001 NOTB

MESH 23 1 23 6 2001 NOTB

MESH 8 20 8 22 2001 NOTB

MESH 9 20 9 22 2001 NOTB

MESH 15 20 15 22 2001 NOTB

MESH 16 20 16 22 2001 NOTB

MESH 22 20 22 22 2001 NOTB

MESH 23 20 23 22 2001 NOTB

$

BAFFLE 8 1 8 6 VERT

BAFFLE 15 1 15 6 VERT





$

IMAGE 0.0 0.0 24.60 0.0 24.60 18.22 0.0 18.22

IMAGE 030 18.22 0.0 .0.0

IMAGE 6.56 0.0 6.56 4.82

IMAGE 12.30 0.0 12.30 4.82

IMAGE 18.04 0.0 18.04 4.82

IMAGE 6.56 18.22 6.56 15.82

IMAGE 12.30 18.22 12.30 15.82

IMAGE 18.04 18.22 18.04 15.82

$

PFLUID 85.30 878.0 5.OE-6

$

MOTION SPEC 16.92 12.01 -0.001 0.0 6.0 11.01 2.0

MOTION VERT 4.0 12.01 -0.001 90.0 -9.81

MOTION HORI 2.5 12.01 -0.001 180.0 -9.81

$

TIMING 0.0451 180.2

$

OUTPUT 170.00 180.2 1 1 YYYYYYYYY

SDA FLUIDS Model Data File





DECK CELL (8,22)

APPLICATION EXAMPLE

20600.,3,7850.,3500.,830.,18.5,878.,0.

A

284.,11.,90.,16.,3500.

4

0.,0.,.0456,447.,.0913,65.,.13692,0.0

4

0.0,0.,0.4756,497.154,1.4848,677577,2.692,691.46

4

0.0,0.,0.4756,497.154,1.4848,677.577,2.692,691.46

2

11.01,4.0,12.01,8.01,-0.001,0.0451,2.0,180.2,179.84

1

0.001,0.6,0.025

1

200.,600.,25.

DYN_STAT Model Data File





CONVERSION OF DYNAMIC SLOSHING PRESSURE

TO STATIC PRESSURE

FOR STRUCTURAL CAPABILITY ASSESSMENT

Title : DECK CELL (8,22)

Subtitle : APPLICATION EXAMPLE

PANEL DATA

1. YOUNGS MODULUS = 206000.0000 N/MM2

2. POISSON S RATIO = .3000

3. MATERIAL DENSITY = 7850.0000 KG/M3

4. FRAME SPACING = 3500.0000 MM

5. STIFFENER SPACING = 830.0000 MM

6. PLATE THICKNESS = 18.5000 MM

7. CARGO DENSITY = 878.0000 KG/M3

8. COEFFICIENT OF VISCOUS DAMPING = .0000 Ns/M

STIFFENER DATA

9. STIFFENER TYPE = A

10. WEB DEPTH = 284.0000 MM

11. WEB THICKNESS = 11.0000 MM

12. FLANGE WIDTH = 90.0000 MM

13. FLANGE THICKNESS = 16.0000 MM

14. STIFFENER EFFECTIVE SPAN = 3500.0000 MM

PLATE NATURAL FREQUENCY & ADDED MASS

PLATE NATURAL FREQUENCY IN AIR = 148.7652 Hz

PLATE NATURAL FREQUENCY IN WATER = 93.0842 Hz

ONE SIDE IMMERSED

PLATE MASS = 421.8786 KG

PLATE MASS + ADDED MASS = 1077.5530 KG

PLATE ADDED MASS = 655.6744 KG

PLATE & STIFFENER NATURAL FREQUENCY & ADDED MASS

PLATE & STIFFENER NATURAL FREQUENCY IN AIR = 72.4130 Hz

PLATE & STIFFENER NATURAL FREQUENCY IN WATER = 46.0999 Hz

ONE SIDE IMMERSED

PLATE & STIFFENER MASS = 547.2745 KG

PLATE & STIFFENER MASS + ADDED MASS = 1350.3151 KG

PLATE & STIFFENER ADDED MASS = 803.0406 KG

HEAVE ACCELERATION TYPE : 1

NO HEAVE ACCELERATION





IMPACT DATA

Point Time Pressure

(s) (KN/M2)

1 .0000 .0000

2 .0456 447.0000

3 .0913 65 .0000

4 .1369 .0000

PLATE LOAD DEFLECTION DATA

Point Deflection Load

(M) (KN/M2)

1 .0000 .0000

2 .0005 497.1540

3 .0015 677.5770

4 .0027 691.4600

PLATE STIFFENER LOAD DEFLECTION DATA


1 .0000 .0000

2 .4756 497.1540

3 1.4848 677.5770

4 2.6920 691.4600

PLATE RESPONSE

DYNAMIC LOAD FACTOR = 1.0146

MAXIMUM DEFLECTION = .4328 MM @ TIME = .0457310000 Seconds

MAXIMUM INSTANT. DLF = 3.1913 @ TIME = .1354140000 Seconds

DEFLECTION @ MAX DLF = .0066 MM

PLATE STIFFENER RESPONSE









CONVERSION OF DYNAMIC SLOSHING PRESSURE

TO STATIC PRESSURE

FOR STRUCTURAL CAPABILITY ASSESSMENT

Title : DECK CELL (8,22)

Subtitle : APPLICATION EXAMPLE

PANEL DATA

1. YOUNGS MODULUS = 206000.0000 N/MM2

2. POISSON S RATIO = .3000

3. MATERIAL DENSITY = 7850.0000 KG/M3

4. FRAME SPACING = 3500.0000 MM

5. STIFFENER SPACING = 830.0000 MM

6. PLATE THICKNESS = 18.5000 MM

7. CARGO DENSITY = 878.0000 KG/M3

8. COEFFICIENT OF VISCOUS DAMPING = .0000 Ns/M

STIFFENER DATA

9. STIFFENER TYPE = A

10. WEB DEPTH = 284.0000 MM

11. WEB THICKNESS = 11.0000 MM

12. FLANGE WIDTH = 90.0000 MM

13. FLANGE THICKNESS = 16.0000 MM

14. STIFFENER EFFECTIVE SPAN = 3500.0000 MM

PLATE NATURAL FREQUENCY & ADDED MASS

PLATE NATURAL FREQUENCY IN AIR = 148.7652 Hz

PLATE NATURAL FREQUENCY IN WATER = 93.0842 Hz

ONE SIDE IMMERSED

PLATE MASS = 421.8786 KG

PLATE MASS + ADDED MASS = 1077.5530 KG

PLATE ADDED MASS = 655.6744 KG

PLATE & STIFFENER NATURAL FREQUENCY & ADDED MASS

PLATE & STIFFENER NATURAL FREQUENCY IN AIR = 72.4130 Hz

PLATE & STIFFENER NATURAL FREQUENCY IN WATER = 46.0999 Hz

ONE SIDE IMMERSED

PLATE & STIFFENER MASS = 547.2745 KG

PLATE & STIFFENER MASS + ADDED MASS = 1350.3151 KG

PLATE & STIFFENER ADDED MASS = 803.0406 KG

HEAVE ACCELERATION TYPE : 2

LR. FLUIDS HEAVE ACCELERATIONSPECTRUM





15. SHIP NATURAL PERIOD = 11.0100 S

16. MAX. AMPLITUDE OF MOTION = 4.0000 M

17. INITIAL PERIOD = 12.0000 M

18. FINAL PERIOD = 8.0100 S

19. PERIOD INCREMENT DTP = -.0100 S

20. PERIOD INCREMENTATION TIME TX = .0451 S

21. DECAY CONSTANT = 2.0000

22. TOTAL STIMULATION TIME = 180.2000 S

23. TIME @ IMPACT = 179.2000 S

24. TOTAL ANALYSIS TIME = .1369 S

IMPACT DATA

Point Time Pressure

(s) (KN/M2)

1 .0000 .0000

2 0.0456 447.0000

3 .0913 65.0000

4 .1369 .0000

PLATE LOAD DEFLECTION DATA


(M) (KN/M2)

1 .0000 .0000

2 .0005 497.1540

3 .0015 677.5770

4 .0027 691.4600

PLATE STIFFENER LOAD DEFLECTION DATA


(MM) (KN/M2)

1 .0000 .0000

2 .4756 497.1540

3 1.4848 677.5770

4 2.6920 691.4600





PLATE RESPONSE





PLATE STIFFENER RESPONSE





PLATE DLF ENVELOPE FOR IMPACT DURATION VARIATION

IMPACT DURATION DLF

(s)

.001000 .637729

.026000 1.103836

.051000 .996494

.076000 1.034846

.101000 1.002531

.126000 1.019406

.151000 1.006817

.176000 1.011162

.201000 1.008416

.226000 1.006442

.251000 1.009668

.276000 1.003518

.301000 1.007718

.326000 1.000228

.351000 1.006481

.376000 1.000335

.401000 1.003092

.426000 1.003824

.451000 1.012421

.476000 1.008774

.501000 1.014042

.526000 1.005778

.551000 1.006800





PLATE & STIFFENER DLF ENVELOPE FOR IMPACT DURATION VARIATION

IMPACT DURATION DLF

(s)

.001000 .591189

.026000 1.074144

.051000 1.064706

.076000 1.020206

.101000 .995725

.126000 1.019618

.151000 1.002431

.176000 1.002431

.201000 1.003371

.226000 1.010839

.251000 1.012997

.276000 1.009454

.301000 1.009546

.326000 1.013475

.351000 1.016578

.376000 1.004241

.401000 1.002680

.426000 .999372

.451000 1.010281

.476000 1.006869

.501000 1.019607

.526000 1.009253

.551000 1.022801

PLATE & STIFFENER DLF ENVELOPE FOR IMPACT PEAK MAGNITUDE

VARIATION

IMPACT PEAK VALUE DLF

(KN/M2)

200.000000 .994511

225.000000 1.024422

250.000000 1.028570

275.000000 1.014918

300.000000 1.015779

325.000000 1.019875

350.000000 1.014645

375.000000 1.010340

400.000000 1.010458

425.000000 1.013678

450.000000 1.009959

475.000000 1.014371

500.000000 .993092

525.000000 .775871

550.000000 .675292

575.000000 .588068





PLATE DLF ENVELOPE FOR IMPACT PEAK MAGNITUDE VARIATION

IMPACT PEAK VALUE DLF

(KN/M2)

575.000000 .588068

200.000000 1.014381

225.000000 1.020108

250.000000 1.008922

275.000000 1.013400

300.000000 1.009135

325.000000 1.006221

350.000000 1.015807

375.000000 1.011072

400.000000 1.014197

425.000000 1.007106

450.000000 1.008405

475.000000 1.012052

500.000000 1.026342

525.000000 .795785

550.000000 .641544

575.000000 .698342

SDA Sloshing Cover clean - Lloyd's Register · PDF fileLLOYD’S REGISTER Chapter Contents...

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