Reliability based design procedure for better survivability of intact and damaged ships

Transcription

1 Reliability based design procedure for better survivability of intact and damaged ships Prof. P. K. Das, PhD, C Eng., FRINA, FIStructE, FIMarEST,MBCS Director ASRAnet Ltd. Lecture for Maritime Faculty KOTOR 3rd October 2018

2 Outline of Presentation Reliability Component & System Level Ultimate Strength of Unstiffened and Stiffened Panel Parametric Studies on the Ultimate Strength of Imperfect Plate Ultimate Strength of Ships Reliability Analysis of Ships under Grounding and Collisions 2

3 Reliability Component & System Level 3

4 Reliability Component & System level The reliability analysis is essentially the evaluation of probability of failure (P f ) of a component defined by: (1) X thevectorofrandomdesignvariablesforthe component f(x) jointprobabilitydensityfunctionofx. The g(x)=0 is called limit state dividing performance ofthecomponentintofailurestate(i.e.g(x)<0)and safestate(i.e.g(x)>0). Theg(X) 0definesthefailuredomainoverwhich integration of(1) is performed to determine probability of failure of the component. 4

5 The probability of failure of a system is given by: (2) G(X) the limit state function for the system given by: (3) g i (X) the limit state of i th component of the system made up of series of parallel combination of components, appropriately. 5

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7 The main steps in a structural reliability analysis are: Establish a target reliability, or a decision model. Identify all significant modes of failure of the structure or operation under consideration. Formulate a failure criterion in terms of a limit-state function for each mode of failure. Identify stochastic variables and parameters in the limit-state function sand specify their probability distributions. Calculate the reliability against failure for each mode of failure of the structure or operation under consideration. Assess whether the estimated reliability is sufficient and modify the concept if necessary. Evaluate the results of the reliability analysis with respect to parametric sensitivity considerations. 7

8 Risk Analysis and Structural Reliability Uncertainty Consistent level of safety 8

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10 Target Reliability Redundant structure 10-e3-10-e4 Non-redundant structure 10-e4-10-e5 (significant warning) Non-redundant structure 10-e5-10-e6 (no warning) 10

11 Typical modes of failures to be considered are: Yielding Buckling (e.g. of columns, plates, shells and stiffened panels.) Rupture (e.g. of chains) Deformation Fatigue Flooding, foundering, capsizing Wear (e.g. of axles) Corrosion and erosion (e.g. of pipes) 11

12 Limit States Ultimate limit state Fatigue limit-state Progressive collapse limit-state Serviceability limit-state 12

13 Methods for reliability analysis Level-extent of information Moment-order of statistical moments Order-order of the polynomial to approximate limit-sate surface 13

14 Level of Reliability Level-III is an exact probabilistic analysis for the whole structural systems involving the convolution integral. It is conceptually straight forward but in practice difficult to formulate and solve. Moreover, it cannot be directly used for design, for example, for a specified reliability level. It is only level which can satisfactorily incorporate all modes of failure when estimating the total reliability. Very clearly, these methods are not suitable for normal design purposes but there is much scope for the use of Level-III techniques for checking the validity and accuracy of the simplified Level-II and Level-I methods by analysis of specific structures. 14

15 Level of Reliability Level-II methods use means and second moment properties of load and strength distributions for components and structural assemblies in terms of a reliability safety index βwhich corresponds to a notional probability of failure or level or reliability for each failure mode or limit state during the life of the structure. Appropriate partial safety factors may then be derived for particular design situations. These safety checks are made only at selected points on the failure boundary (as defined by the appropriate limit state) rather than as a continuous process, as at Level III. These methods which need make no attempt to find the region of basic variable or state-space which has the highest probability of failure density. This is central to Level-II methods and provides the basis for calculating PSFs at Level-I. 15

16 Level of Reliability Level-I provides a workable design method in which appropriate safety margins are provided usually on a structural element basis by specifying a number of partial safety factors related to some predefined characteristic values of the basic variables. In the strength model these values will usually correspond with the nominal values specified in design such as minimum yield etc. no explicit reliability calculations are undertaken and the levels of risk in different structures are essentially unknown. Design methods involving a number of PSFs are likely to be much greater practical value than Level-II and II methods. 16

17 Level of Reliability Characteristic values are usually given as functions of mean values, coefficients of variation and distribution types. The PSFs may be deduced from Level-II. Level-I methods can be made identical to Level-II methods if the PSFs are continuous functions of the means and variances of the basic variables and of the safety indices. Existing Level-I methods replace this continuous function by discrete values of the factors. In general, PSFs can be associated with each basic variable but this is inconvenient and in practice they are subsumed in cognate groups such as load and strength related, consequences, modelling, systems (or redundancy) etc. 17

18 Choice of probability distribution Wind: gust speed-normal average speed-weibul Extreme-Gumbel. Waves. Current. Fatigue. Yield strength. Ship Data 18

19 System Reliability Load re-distribution (i.e. redundancy) Multiple failure modes (i.e. complexity) Correlation between safety margins for different failure events Types of failure (e.g. brittle/ductile) 19

20 Reliability Analysis Analytical methods (FORM,SORM) Simulation methods - Basic Monte Carlo Simulation -Importance Sampling -Adaptive sampling 20

21 Choice of Methods Analytical FORM and SORM-verified by simulation methods Simulation samples >100/pf where pf is the failure probability 21

22 Calibration of partial coefficients A set of partial coefficients for a target reliability Design space Code format Optimised partial coefficients 22

23 Design Format Selection for the Limit States A representative design format which fulfils the above requirements is expressed as: 23

24 Load and Load Effects General Aspects Global Loads Global Slamming effects Local Loads Load Combination rules Calculation Methods 24

25 Wave Modelling Short term responses in regular waves are calculated using the principle of linear superposition and wave statistics Short term responses are combined with long term wave statistics for a specific ocean area in order to determine the long-term distribution of VWBM ISSC version of the Pierson-Moskowitz spectrum T m is the average period H S is the significant wave height 25

26 Wave Modelling RAOs for bending moments are obtained either by model test or calculation (e.g. TRIBON) Having RAOs, bending moment response spectra can be calculated by superposition of all selected spectra Pierson and St Denis used a some what generalised form of directional spreading function G 2 G 0 2 cos Angle between an angular wave component and dominant wave direction 26

27 The amplitudes of the Gaussian zero mean stationary stochastic process with a narrow band assumption, follow a Rayleigh distribution such that the probability of exceeding the ampitude x is given by Longuet-Giggins (1952) x Q( x R) exp s 2. R 2 QL ( x) Qs( x R). fr( r) dr 27

28 The probability distribution of the wave induced load effects that occur during long-term operation of these ships in the sea way is obtained by weighing the conditional Rayleigh distribution by the probability of occurrence of the various states in the ship route. f R ( r) dr QL ( x) Qs( x R). fr( r) dr f( h, t s z,, v, c) dh s dt z ddvdc 1exp The resultant probability distribution is fitted to the Weibul distribution given by F X ( x) x w k 28

29 GLOBAL WAVE STATISTICS OCEAN AREAS 29

30 30

31 The design wave bending moment can thus be represented by Gumbel distribution The resultant probability distribution is fitted to the Weibul distribution given by f R ( r) dr f ( h, t s z,, v, c) dh s dt z ddvdc F we ( x we ) expexp x we x w nw 31

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34 Stochastic Combination of Hull Girder Loads Extreme Still Water Bending Extreme Wave Induced Bending Moment Load Combination Factor Ferry-Borges-Castenheta Method 34

35 LOAD COMBINATION RULES FERRY-BORGES METHOD (FERRY- BORGES, 1971 LOAD COINCIDENT METHOD (WEN, 1977) POINT-CROSSING METHOD (LARRABEE, 1981) THE SQUARE ROOT OF THE SUM OF SQUARES (GOODMAN, 1954) 35

36 Load Combination Factor Introducing a load combination factor, the total load might be defined as, F t F SW F W The load combination factor is defined as, W F 1 t 0.5 F 1 W F 1 SW

37 Failure Function for Reliability Analysis g(.) M M u u se n M we M Ultimate Bending Moment Capacity u u M se n M we Uncertainty on Ultimate Strength Still Water Bending Moment Load Combination Factor Uncertainty in Wave Load prediction Non-linear effects Extreme Vertical Wave Bending Moment U M f, A, AS, yd ys yb D A B 37

38 Ultimate Strength of Unstiffened and Stiffened Panel 38

39 IMPERFECTIONS IN THE PLATE AND THE STIFFENED PLATES A ship structure consists of around 80% of stiffened and unstiffened plates. The individual steel plate elements are prone to suffer various types of damages. These damages or imperfections may be considered due to various causes like fall of heavy cargo on the deck, welding, blasts or fatigue cracking etc. 39

40 Short state of the art review of ultimate strength of unstiffened and stiffened plate Author (s) & Year Caldwell (1965) Kmiecik, (1971). Faulkner (1975) Comment The author introduced Plastic Design by considering the influence of buckling and yielding of structural members composing a ship s hull. Performed nonlinear analysis and showed that the strength of rectangular plates were governed by the amplitude of buckling mode component. The author introduced the concept of reduced effective width for defining the plate element stiffness, as required for use in stiffened plate collapse theories. Smith (1977) Determined the load-shortening curves for the stiffened and un-stiffened plate using finite element method and used those curves for the progressive collapse analysis Ueda & Yao (1985) Guedes Soares, Introduced the ISUM method Theoretically included the initial imperfections to the compressive strength calculations. 40

41 Short state of the art review of ultimate strength of unstiffened and stiffened plate Author (s) & Year Comment Pu, Das, Faulkner (1997) Performed ultimate compressive strength and probabilistic analysis of stiffened plate taking account of imperfections. Paik et al (1999) Analytical method for the ultimate compressive strength and effective plating of stiffened panels were presented Sheikh et. al. (2003) Dunbar et al. (2004) Hughes et al. (2004) Khan & Das (2005) studied the stability of steel stiffened plates of T -shape section under uniaxial compression and combined uniaxial compression and bending using FEM analysis. The authors have addressed the influence of local corrosion on stability of a plate and then on a combination of plates forming a stiffened panel. The authors derived modified expressions for elastic local plate buckling and overall panel buckling expressions form 55 Abaqus eigenvalue buckling analyses. The authors studied the influence of location of geometric imperfection on the unstiffened and stiffened plate using FEA. 41

42 LOCAL STRENGTH Main type of failure observed in the ship structures is given by : a) Plate induced failure b) Stiffener induced failure c) Tripping failure 42

43 Stiffened Panel 43

44 Compressive stress in stiffened plate 44

45 Stiffened Panel Simplified design for optimum stiffened panel design 45

46 Stress Distribution (Faulkner) nn 46

47 POST-BUCKLED STRESS DISTRIBUTION 47

48 Relationship between plate slenderness and strength in compression 48

49 49

50 Stiffened Panel (a) von Karman s effective width vs compressive loads for a square plate with varying the magnitude of initial deflection (without welding induced residual stresses). (b) von Karman s effective width vs compressive loads for a square plate with varying the magnitude of welding induced residual stresses (without initial deflection). 50

51 51 BUCKLING OF PANELS σ ε ) ( when when when e E t b y 0 E t b e for for b for for b 0 To obtain the average load shortening curve of the column it is assumed that the stiffener has an elastic perfectly plastic behaviour, given by

52 Unstiffened Plate Effective width of welded plate elements (Faulkner) 52

53 Plate strength comparison 53

54 BUCKLING OF PANELS (Das, Pu et. al) The interaction co-efficient for the initial residual stress is given by R η 1 Δφ b 1.08 φ b η The interaction co-efficient for the initial imperfection is given by σ σ r y 2η b/t 2η δ0 Rδ β t If the residual stress coexists with the initial imperfection, the combining interaction coefficient is R ηδ δ η0.36 t β Back 54

55 RESIDUAL STRESS (Faulkner) r y 2 b/ t2 nn 55

56 Stiffened Panel Effect of Residual Compressive Stress(σ r ) on Theoretical Critical Stress: σ cr = Unwelded Critical Stress. σ cr = Welded Critical Stress. 56

57 BUCKLING OF PANELS ( Das, Pu et. al.) The effective width is given by b e b 1.08φb Rη Rδ Rηδ β R R R 0β 1 η δ ηδ The stress at a given strain is given by: φ σu σy Φ ε σ σ e y A s b e t Asbt where σ σ e y εσ εσ σ σ E y E y for for σ E σ 0.5ε. E y 0.5ε. y 57 Back

58 TRIPPING FAILURE For tripping failure IACS Rule has been followed The stress,σ T at a strain of ε when the element undergoes tripping failure is given by σ T σ y Φ ε A s σ C A bt.φ s bt b.σ y σ C σ y 1 σ ET ε Φ ε 4σ σ ET y.ε if if σ σ ET ET 0.5ε. 0.5ε. y y 58 Back

59 Mean Value and COV in Das & Pu method for stiffened plate 59

60 Correlation of experimental and predicted Values (Das & Pu) 60

61 Correlation of experimental and numerical date and predicted values (Das & Pu) 61

62 Scantlings of stiffened plates 62

63 Reliability Analysis of stiffened plate (Das & Pu) 63

64 Comparison of results for Stiffened Plate Correlation of ultimate strength prediction with FEA (Hughes et. al., 2004) 64

65 Comparison of results for Stiffened Plate Ultimate strength using FEA and orthotropic surface stress for crossover panels (Hughes et. 65 al. 2004)

66 Faulkner (1975) 66

67 Parametric studies on the ultimate strength of imperfect plate 67

68 IMPOSED DISPALCEMENTS AND BOUNDARY CONDITIONS FOR PLATE MODEL (BASIC CASES) Imposed Displacement (Max = 5 mm) Fixed Z RX y Fixed Y Z RY y z x 0 z Scantlings of the plate b(mm) a/b b/t β Imposed Displacement (Max = 5 mm) Fixed Fixed Y Z RY σo(mpa) E(GPa) x Fixed X Z RX 68

69 MODE OF IMPERFECTIONS FOR DIFFERENT CASES Case 1 Case 2 w w w 0 mx ny sin sinw a b w x/a 0 (2) Mode of imperfection: m=3, n=1 sin mx a sin ny b 0< x/a <1 & 0 <y /b < 1 w Case 2 Case x/a Mode of imperfection: m=1, n=1 (2) 0< x/a <1 0 <y /b < 1 69

70 MODE OF IMPERFECTIONS FOR DIFFERENT CASES (Contd.) Case 3 Case 3 w w w max x/a w w m x n y sin 2 sin a b max sin (3) Mode of imperfection: m=5, n=1 2 m x a n ( y0.2b) 2 sin 0.4 < x/a b< <y /b < 1 4 Cases 4 w Mode of imperfection: m=5, n=5 x/a (4) 0.4 < x/a < <y / b<0.8 70

71 CONCLUSION Summary of the imperfections characteristics 71

72 72

73 Fig.1. Boundary condition for the imperfect stiffened plate Effect of geometric imperfection on the compressive stress of stiffened panel 4m 840mm 73

74 Boundary condition for the imperfect stiffened plate Imperfect Stiffened Panel 74

75 Material Properties & Scantlings of the Imperfect stiffened plate Scantlings of the stiffened plate (in mm) a b tp hw tw bf tf wmax Material properties: Young s Modulos (E) =206GPa Poisson s Ratio (υ) = 0.3 Yield Stress (σy) =315 MPa From the above scantlings the slenderness ratios as given in the equation [2] can be calculated as: & 18mm mm

76 w t max p mm 2 18mm Eq. 1 Imperfections in the stiffened plate where and are the plate slenderness of stiffened plates with plate thickness 20mm and 18 mm respectively and is given by: b t p E y (2) Where βand Eare the yield stress and Young s modulus respectively. The figure-1 shows the configuration of an imperfect stiffened plate with 5 equally spaced stiffeners, showing all the scantlings. The shape of the initial imperfections for the stiffened plate structure is given by Equation 3. (3) where w w max my nx sin sin a b 2 2 wmax 0. 15tP20mm tP18mm m 76

77 Figure 2: Average stress distribution of the elements along the y-axis for the stiffened plate with plate thickness 20mm.

78 Fig 3:Average stress on the elements of the stiffened plate with the plate thickness 18mm 78

79 Figure 4: Comparison of the ultimate strength of the stiffened plate with maximum global imperfection of 8.1 mm with plate thickness 20mm 79

80 Figure 5: The comparison of stress-strain relationship of the stiffened plate with the plate thickness 18mm. 80

81 Ultimate Strength obtained using the FE Method for perfect and imperfect plates. Plate US Perfect US Imperfect % Change Thickness Stiffened plate Stiffened plate in US 20mm MPa MPa % 18mm MPa MPa %

82 Fig 6: Load shortening curve for the stiffened plate with thicknesses 18mm and 20mm. 82

83 Fig 7: Comparison of ultimate strength of the imperfect stiffened plate with thicknesses of 18mm and 20mm. 83

84 Ultimate strength using analytical method for the imperfect plate Plate FEM Das & Pu Faulkner s Thickness ANSYS Method Method 20mm MPa MPa MPa 18mm MPa 233 MPa MPa 84

85 Remarks The aim of this present paper has been to investigate the influence of initial imperfection on the collapse behavior of ship structure, using non-linear elastic plastic finite element analyses. Some noteworthy findings are as follows: The Ultimate Strength changes due to the presence of initial imperfections in the unstiffened and stiffened plate structures. The Ultimate Strength mainly decreases due to the initial imperfections. The amount of reduction in the Ultimate Strength only due to the initial imperfection may very between 5-10 %. The ultimate strength varies due to different thicknesses of the plate even though all other scantlings and mode of imperfection have been kept constant. 85

86 Ultimate Strength of Ships 86

87 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP A ship is designed to satisfy the following limit states Ultimate limit state Fatigue Limit State Serviceability Limit State Accidental Limit State?? Ultimate Limit State Fatigue Limit State Accidental Collapse Limit state Ship Designs -Design by Rules - Design by First Principle 87

88 Brief state of the art review of the ultimate strength of ships Authors & Year Caldwell (1965) Smith (1977) Yao et al., (1994) Mansour et al., (1995) Paik et al., (1996) Comments The author is credited to be the first one to attempt to evaluate the ultimate hull-girder strength of a vessel. The author proposed the progressive collapse analysis method to calculate the ultimate strength of ship. Studied on the ultimate hull girder strength interaction relationship under combined vertical and horizontal bending for double hull tanker. Presented an empirical interaction equations based on the results of calculations of one tanker, one container ship and one cruiser. Presented combined vertical and horizontal bending moment using ALPS/ISUM program for eleven vessels. 88

89 Brief state of the art review of the ultimate strength of ships Authors & Year Comments Gordo & Guedes Soares, (1997) The authors proposed another interaction equation based on the results for five tankers and six container ships. Hu et al., (2001) Ozguc & Das et al., (2006) Khan & Das (2007) Analyzed the ultimate longitudinal strength of a typical bulk carrier by using a simplified method under combined vertical and horizontal bending moments. The authors have presented extensive investigation of the hull girder ultimate strength under coupled bending moment. Presented the combined bending moment interaction equations for intact and damaged ships and performed reliability analysis. 89

90 Structural design consideration based on ultimate limit state 90

91 Section modulus based safety measure Vs Ultimate Limit State Based safety measure. 91

92 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP Ship Structure Integrity under normal operating condition A large number of ship accidents continue to occur Accidents caused the loss of cargos, pollution of environments & loss of human lives According to LR,2000: a total of 1336 ships were lost with 6.6million gross tonnage cargo loss between 1995 and people also were reported being killed or missing as a result of total loss during this period. 92

93 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP Total losses of ships in GT during the years

94 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP In Damaged Ships: Horizontal Bending Moment plays a vital role Although vertical BM in quartering sea is smaller than the head seas, horizontal BM is quite large. The ratio of horizontal to vertical BM could be high, because the breadth of a ship is generally higher than its depth. The combined effect of horizontal and vertical bending could be very serious. So the combined effect should be considered in performing the residual strength and reliability analysis. 94

95 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP Ultimate Strength Analysis Methodology Analytical Method Finite Element Analysis Progressive Collapse Analysis 95

96 STRUCTURAL SAFETY ASSESSMENT OF DAMAGED SHIP 96

97 PROGRESSIVE COLLAPSE ANALYSIS 97

98 FEATURES OF THE METHOD A progressive collapse analysis based on Smith method (1977) is carried out in this study 1. Overall grillage collapse is avoided by sufficiently strong transverse frames. 2. The ultimate strength is calculated at the hull transverse sections between two adjacent transverse webs. 3. The hull girder transverse section is divided into a set of elements, which are considered to act independently. 4. The hull girder transverse section remains plane during each curvature increment. 5. The hull material has an elastic-plastic behavior. 98

99 TYPE OF ELEMENTS USED IN SHIPS 99

100 ISUM vs. DOW FRGATE EXPERIMENT RESULTS 100

101 PROGRESSIVE COLLAPSE ANALYSIS Combined bending of the hull The neutral axis is given by: y n x n 0 y. A i A i i 101

102 DETAILED STEPS The ship is subjected to curvature in the x and y directions respectively denoted as Cx, Cy. The overall curvature C is related to these two components by: C 2 2 C x C y C x C.cos or and C y C.sin Adopting the right-hand rule, where is the angle between the neutral axis and the x axis and is related to the components of the curvature by: tan C C y x The strain at the centroid of an element i is which depends on its position and on the hull curvature, as given by: i y. C x. C gi x gi y 102

103 PROGRESSIVE COLLAPSE ANALYSIS Once the state of strain in each element is determined, the corresponding average stress may be calculated according to the analytical formula, and consequently the components of the bending moment for a curvature C are given by: M x ygi.. iai y M x.. A gi i i i Where represents the stress of element i at (xgi, ygi). Ai represents the cross sectional area of element i. This is the bending moment on the cross section after calculating properly the instantaneous position of the intersection of the neutral axis associated with each curvature and the centreline (called the centre of force). The condition to determine the correct position of neutral axis is: A i i 0 Atrialanderrorprocesshastobeusedtoestimateitspositioncorrectly. 103

104 Comparison of moment-curvature relationship by different authors 104

105 Longitudinal stress distribution in a hull section at the ultimate limit state, as suggested by Paik and Mansour (1995). (Left) sagging. Right, Hogging (Paik & Thayamballi 2003) 105

106 DOUBLE & SINGLE SKIN TANKERS Mid-ship sections of the double and single skin tankers 106

107 PRINCIPAL PARTICULARS OF THE TANKERS Ship Name L OA (m) L BP (m) B(m) D(m) T (m) C b TD TS TD= Double skin tanker, TS= Single skin tanker Principal particulars of the ships 107

108 COMPARISON OF RESULTS FOR DOUBLE SKIN TANKER Double Skin Tanker MARS Present Method Present Method MARS Hog x 10 9 Nm x 10 9 Nm Sag 8.54 x 10 9 Nm 8.16 x 10 9 Nm Comparision of results for double skin tanker 1.3E+10 The results obtained using the present formula resembles very closely with those from the Bureau Veritas MARS. 1.0E+10 Present Method BV-MARS 7.5E E+09 Bending Moment (Nm) 2.5E E E E+09 Curvature(1/m) -7.5E E E

109 DIFFERENT LOCAL FAILURES This figure shows the kind of local failure each element has undergone during the hogging condition. The tripping failure (red dot) reduces the ultimate strength of the ship in a larger extent. 109

110 COMPARISON OF RESULTS FOR SINGLE SKIN TANKER Single Skin Tanker MARS Present Method Present Method MARS Hog 9.12 x 10 9 Nm 8.82 x 10 9 Nm Sag 8.05 x 10 9 Nm 8.16 x 10 9 Nm Comparision of Results for single skin tanker 1.E+10 Present Method BV-MARS 8.E+09 6.E+09 For the single skin tanker the BV MARS and present method results show less than 4% of difference in intact scenario. 4.E+09 2.E+09 0.E E+09-4.E+09-6.E+09-8.E+09-1.E+10 Vertical Bending Moment (Nm) Curvature (1/m) 110

111 REAL LIFE DAMAGE SCENARIOS 111

112 DAMAGE SCENARIOS (Grounding & Collision) In collision 35% of depth from main deck and in grounding 20% of bottom is considered to be damaged. The damage scenarios are based on the LR Classification Rules 112

113 Typical Moment Curvature Relationship of a ship 113

114 Vertical Bending moment of the DSS Tanker -1.2E E E E E E E E E Sagging Bending Moment Intact Scenario Collision Scenario Grounding Scenario Bending Moment ( Nm ). Curvature (1/m) Hogging Bending Moment The residual strength in grounding and collision scenario has decreased as compared to the ultimate strength 114

115 Horizontal bending moment of double skin tanker in different scenarios 2.0E E+10 Bending Moment (Nm) 1.2E E E+09 Intact Scenario Collision Scenario : Starboard in Tension Collision Scenario: Starboard in Compression Grounding Scenario 0.0E E E E E E E E-0 Curvature (1/m) The pure horizontal bending moment in damaged scenario has also decreased significantly. Due to asymmetry in the collision damage two set of horizontal bending moment has been derived, depending on the direction of bending. 115

116 Combined Bending Moment when θ =30 degree for the double skin tanker 1.5E+10 Vertica Bending Horizontal Bending Combined Bending 1.0E E+09 Bending Moment(Nm) 0.0E E E E+10 Curvature (1/m) When the θ =30 degree the horizontal bending moment is higher than the vertical bending moment. 116

117 Vertical Bending momemnts of Single Skin Tanker Intact Scenario Collision Scenario Grounding Scenario 1.E+10-8.E+09-5.E+09-3.E+09 0.E+00 3.E+09 5.E+09 8.E+09 1.E Bending Moment (Nm) Curvature (1/m) 117

118 1.6E+10 Horizontal Bending Moment of Single Skin Tanker 1.4E E+10 Bending Moment (Nm) 1.0E E E+09 Collision Scenario: Starboard in Tension Intact Scenario Collision Scenario : Starboard in Compression Grounding Scenario 4.0E E E E E E E E E E-04 Curvature (1/m) 118

119 Different Scenarios Intact: Grounding Collision RESIDUAL STRENGTH IN DIFFERENT SCENARIOS Bending Type TD (GNm ) TS (GNm) % of Intact Strengt h TD Hog Sag Horizontal Hog Sag Horizontal Hog Sag Horizontal Horizontal % of Intact Strength TS N.B: Horizontal 1 and Horizontal 2 represent the horizontal bending moment when the damaged 119 starboard is in compression and in tension respectively.

120 Typical ultimate hull girder strength interaction equation (ISSC 2000) 120

121 INTERACTION EQUATION FOR THE TANKERS IN Interaction equation for the Double Skin Tanker in Intact scenario INTACT SCENARIOS Normalised Vertical BM Normalised Horizontal BM Interaction equation in hogging condition: (x/m UH ) (y/ MUV ) =1 Interaction equation in sagging condition: (x/m UH ) (y/m UV ) =1 Normalised Vertical BM The vertical and horizontal bending moments are calculated in different curvature ratios and the normalised values are plotted Interaction equation for single skin tanker in intact scenario Interaction equation in hogging condition (x/m UH ) 1.8 +(y/m UV ) 1.8 = Double Skin Tanker (intact) Single Skin Tanker (intact) Interaction equation in sagging condition (x/m UH ) (y/m UV ) 1.73 =1 Normalised Horizontal BM 121-1

122 INTERACTION EQUATION FOR THE TANKERS IN Collision Scenario: Double skin tanker 1 COLLISION SCENARIOS Equation of curve:(x/m UH ) (y/m UV ) 1.48 =1 Hogging: Compression in port side Equation of curve:(x/m UH ) (y/m UV ) 1.73 =1 Sagging: Compression in port side Normalised Vertical BM Equation of curve:(x/m UH ) 1.5 +(y/m UV ) 1.5 =1 Hogging: Compression in starboard side Normalised Horizontal BM Equation of curve:(x/m UH ) (y/m UV ) =1 Sagging: Compression in starboard side Since during the collision, the damage is asymmetric, 2 set of interaction equation has been derived considering the combined bending. Collision Scenario: Single skin tanker 1 Double Skin Tanker (collision) Equation of curve:(x/m UH ) 1.5 +(y/m UV ) 1.5 =1 Hogging: Compression in port side Normalised Vertical BM Equation of curve:(x/m UH ) (y/m UV ) 1.52 =1 Hogging: Compression in starboard side Normalised Horizontal -0.2 Equation of curve:(x/m UH ) (y/m UV ) 1.57 =1 Sagging: Compression in port side -0.4 Equation of curve:(x/m UH ) 1.6 +(y/m UV ) 1.6 =1 Sagging: Compression in starboard side Single Skin Tanker (collision)

123 Normalised Vertical BM INTERACTION EQUATION FOR THE TANKERS IN GROUNDING SCENARIOS Interaction equation for double skin tanker in grounding scenario Interaction equation in hogging condition: (x/m UH ) (y/m UV ) = Normalised Horizontal BM During the grounding scenario the damage is considered to be symmetric, one set of interaction equation has been derived. The interaction coefficients have changed compared to the intact condition Interaction equation in Grounding Scenario Interaction equation in sagging condition: (x/m UH ) (y/m UV ) =1 Normalised Verrical BM Interaction equation in hogging condition (x/m UH ) (y/m UV ) =1-1 Double Skin Tanker (grounding) Normalised Horizontal BM Interaction equation in sagging condition (x/m UH ) (y/m UV ) =1-0.6 Single Skin Tanker (grounding)

124 Reliability Analysis of Ships under Grounding & Collision 124

125 g( x) Where k us x sw M sw LIMIT STATE FUNCTION In general the interaction equation can be represented as: 1 So the limit state function can be given as 1. k x M k us, k uw = Combination factor for damage scenarios (using ABS 95 rule) x sw = model uncertainty for predicting the still water bending moment; x wv,x wh = the error due to the analysis over prediction in vertical and horizontal wave induced bending moment; x sv, x sh = uncertainty of a model that takes nonlinearities into account; uv uw uv x x uv and x uh = the random variables representing the model uncertainty in ultimate vertical strength and horizontal strength respectively. These random variables consider uncertainties for the different 125 formulae used and the uncertainties in the yield strength of steel. wv x sv M wv m M M v uv k uw m x x wh uh x M sh M M M uh h uh wh n n

126 The Properties of parameters (Mansour et.al & 1994) Parameter Distribution Mean Value COV x uv Normal x uh Normal x sw Normal x wv Normal x wh Normal x sv Normal x sh Normal M sw Normal Table 0.05 M wv Weibull Table M wh Weibull Table where x sw = model uncertainty for predicting the still water bending moment; x wv,x wh = the error due to the analysis over prediction in vertical and horizontal wave induced bending moment; x sv, x sh = uncertainty of a model that takes nonlinearities into account; x uv and x uh = the random variables representing the model uncertainty in ultimate vertical strength and horizontal strength respectively.these random variables consider uncertainties for the different formulae used and the uncertainties in the yield strength of steel. 126

127 Safety & Reliability of Damaged Bulk Carrier Structures Condition Coefficient Intact Groundi ng Collisio n k us Hogging k uw k us Sagging k uw Combination factors for bending moment (ABS, 1995) 127

128 STILL WATER MINIMUM LOADS Following IACS Rule (Jan 2006), the minimum still water bending moment is given by Msw CwvL B C wv Cb KNm hogging 2 L BC 0.7KNm sagging b C wv 10.75((300L)/100) ((L350)/150) 3/2 3/2 100L300(m) 300L350(m) L350(m) 128

129 WAVE INDUCED VERTICAL & HORIZONTAL LOAD Following IACS Rule (Jan 2006), the minimum wave induced vertical and horizontal bending moment is given by M wv 110.C 190.C wv wv L L 2 2 B BC C b b knm 3 knm for for sagging hogging M wh (0.3 L 2000 )F M.f p.c wv.l 2 T.C b knm 129

130 Ships SAFETY INDEX AT DIFFERENT SCENARIOS FOR DOUBLE SKIN TANKER Interaction coefficients and Probability of failure for the double skin tanker Still water BM &Wave Induced Loads Bending type M SW M WV M WH TD Hog Sag TS Hog Sag All values are in GNm (= 10 9 Nm). Msw-still water BM, Mwv-wave induced vertical load, Mwh-wave induced horizontal load. The interaction coefficients changes with respect to the damages Differen t Scenari os Bendin g Type Intact: Hog Ground ing: Collisio n: Sag Hog Sag Coeff, m & n P f E E E E- 04 Hog E- 03 Sag E Hog E β

131 SAFETY INDEX AT DIFFERENT SCENARIOS FOR SINGLE SKIN TANKER Interaction coefficients and Probability of failure for the single skin tanker Different Scenarios Bending Type Coefficients, β m & n P f Intact: Hog E Sag E Grounding: Hog E Sag E Collision: Hog E Sag E Hog E Sag E

132 SENSITIVITY ANALYSIS GROUNDING DOUBLE SKIN TANKER α i 1 ( g(x)/ x i1 i ) 2 g(x) x i Mwh 0.1 Muh 0.06 Ms 0.23 The sensitivity analysis identify the random variables that have most important effect on the reliability estimates. Mwv 0.5 Muv

133 It is important to consider the effect of torsion when ship is damaged, since effect of torsion becomes prominent due to presence of large holes in the side shell. 133

134 134 LIMIT STATE FUNCTION CONSIDERING TORSION l n m U UH H UV V T T M M M M 1 g(x) The limit state function considering the torsional effect may be given as follows:

135 CONCLUSION 1. It has been observed that the residual strengths in the damage scenarios show significantly lower values compare to intact scenario 2. The reliability index changes due to the damages. Since the residual strength doesn t decrease as fast as the still water and wave induced loads, the reliability index is found to be higher in damage scenarios. 3. the probability of failure in sagging condition for the double skin tanker is higher than that in hogging condition and subsequently in sagging the ships have low reliability index (β) for this ship. 4. It can also be also noted that the probability of failure is dependent on the extent of damage and the environmental load during the damage scenario 5. It has been observed that horizontal load combing with the vertical one is very dangerous from structural safety point of view, so for the safety assessment of ships during damage scenario, the horizontal bending should be considered. 6. Finally when there are large holes created in the hull due to damages, torsion combining with vertical and horizontal bending can be devastating. Briefly the effect of torsion and the limit state equation combining torsion with vertical and horizontal bending moment has been presented. 135

136 Thank You! 136

137 References Zheng, Y and Das, P.K.: 'Reliability Analysis of Stiffened Plates by Improved Response Method', Journal of 'Engineering Structures', vol. 22, no. 5, May 2000, pp Maerli, A., Das, P.K. and S. Smith: 'A Rationalisation of Failure Surface Equation for the Reliability Analysis of FPSO Structures', Intl Shipbuilding Progress, Vol. 47, no. 450, July 2000, pp Das, P.K. and Zheng, Y.: 'Cumulative Formation of Response Surface and its use in Reliability Analysis', Journal of Probabilistic Engineering Mechanics, Vol. 15, Issue 4, pp , October Das, P.K. and Dow, R.: 'Hull Girder Reliability of a Naval Ship under Extreme Load', Journal of Ship Technology Research, Germany, Vol. 47, no. 4, October Yu, L., Das, P.K. and Zheng, Y.: 'Stepwise Response Surface Method and its Application to Reliability Analysis of Ship Structures'. OMAE Journal Vol. 124, November, 2002 Das, P.K., Thavalingam, A. and Bai, Y.: Buckling and Ultimate Strength Criteria of Stiffened Shells under Combined Loading for Reliability Analysis Thin-walled Structures vol.41, issue 1, page 69-88, January 2003 Fang C L, Das, P K Survivability& Reliability of Damaged Ships after 137 Collision and Grounding Journal of Ocean Engineering Vol. 32 Issue 3-4

138 References Ozguc O, Das PK, Barltrop NDP. A comparative Study on the Structural Integrity of Single and Double Side Skin Bulk Carriers under Collision Damage. Marine Structures 18 (2005) Ozguc O, Das PK, Barltrop NDP. A proposed method to evaluate hull girder ultimate strength. Journal of Ships & Offshore Structures Volume 1 Number Ozguc O, Das PK, Barltrop NDP. The new simple design equations for the ultimate compressive strength of imperfect stiffened plates. Ocean Engineering Vol. 34, Issue 7, May 2007, p Das, P K & Fang, C L Residual Strength and Survivability of ships after grounding and Collision. Journal of Ship Research, June 2007, Vol. 51, No 2 Pu, Y., Das, P.K. and Faulkner, D.: 'Ultimate Compression Strength and Probabilistic Analysis of Stiffened Plates', Journal of Offshore Mechanics and Arctic Engineering (OMAE), November 1997, vol. 119, 6 pp. Pu, Y., Das, P.K. and D. Faulkner: 'A Strategy for Reliability-Based Optimisation', Journal of Engineering Structures, vol. 19, no. 3, pp , Pu, Y., Das, P.K. and D. Faulkner: 'Structural System Reliability Analysis of SWATH Ships', Journal of Engineering Structures, U.S.A., vol. 18, no. 12, pp , Dec. 1996, 5 pp. 138

139 References Das, P.K. and Dow, R.: 'A Rationalisation of Failure Surface Equation for the Reliability Analysis of a Naval Hull Girder', Proceedings of OMAE 2000, New Orleans, February Yu, L., Das, P.K. and Barltrop, N.: 'Importance Sampling Method with Kernel Density Estimate A General Approach', Proceedings of the 8th Intl Conference on Structural Safety and Reliability (ICOSSAR 2001), California, U.S.A., June Yu, L., Das, P.K. and Zheng, Y.: 'Stepwise Response Surface Method and its Application to Reliability Analysis of Ship Structures'. Proceedings of the 20th Intl Conference on Offshore Mechanics and Arctic Engineering (OMAE 2001), Rio de Janeiro, Brazil, June Paul, E., Thavalingam, A., Das P.K.: Ultimate Strength and Structural Reliability Analysis of FPSO Hull Girders. OMAE-2002, June 2002, Oslo. Yu, L., Das, P.K. and Zheng, Y: Reliability Analysis of Marine Structures by Response Surface Methods. ASRANet 2002 Colloquium, July 2002 Das, P.K. and Thavalingham, A.: Strength& Reliability of FPSO s ISC 2002 October St Petersburg, Russia Fang, C. and Das, P.K. Hull Girder Ultimate Strength of Damaged Ships. Proceedings of PRADS 2004, September 2004, Lubeck. 139

140 References Das,P K. and Fang, C Safety & Reliability of Damaged Bulk Carrier Structure. ICOSSAR 2005, June 2005, Rome Zheng,Y. Das,P.K, Yu,L and Leira, B. A Benchmark Study on Response Surface Method. IMAM 05 Sept Lisbon. Khan,I.A., Das,P.K and Zheng,Y. Structural Response of Intact and Damaged Stiffened Plated Structures for Ship Structures. IMAM 05 Sept 2005 Khan, I., Das, P.K., Survivability and residual strength of tankers after collision damage. Proceedings of the International Conference on Towing and Salvage of Disabled Tankers (TSDT 2007) pp 21-30, March 2007, Glasgow UK Ozguc O, Samuelides M, Das PK. A Comparative Study on the Collision Resistance of Single and Double Side Skin Bulk Carriers. International Congress of International Maritime Association of the Mediterranean (IMAM 2005), Lisbon, Portugal, September. Shahid, M, Das, P.K., Structural System Reliability Using Finite Element Methods and FORM/SORM Reliability Processors, International conference on Advancements in Marine Structures, March 2007, Glasgow, United Kingdom Shahid M., Das P.K, Ship Transverse Frame Structural System Reliability Based on FEM International Congress of International Maritime Association of the Mediterranean (IMAM 2007) 2-6 September 2007, Varna 140