Ultimate Strength and Reliability Analysis of a Vlcc

ULTIMATE STRENGTH AND RELIABILITY ANALYSIS OF A VLCC

I. A. Khan, P. K. Das, Universities of Glasgow & Strathclyde, UK G. Parmentier, Bureau Veritas, Paris, France

ABSTRACT

A Very Large Crude Carrier broke its back during discharge of oil in 1980. The accident has been described and the ultimate longitudinal strength based on Smith’s method has been calculated and the results have been compared with the moment at failure estimated by other methods applied to the same case considering the flexural buckling, post buckling and flexural torsional behaviour (tripping) of the compressed part of the structure. Since the ultimate strength of the hull-girder is largely governed by the behaviour of the elements under compression, special attention has been paid to the influences of corrosion, initial imperfection and welding induced residual stress in the elements. Failure probability of the damaged VLCC has been studied assuming the load effects obtained from the Classification Society. The failure probability has been discussed and finally, the survivability of the very large crude carrier has been addressed.

NOMENCLATURE time hardly been formalized. 20 years later W.G. John laid the foundation for longitudinal strength A : Cross-sectional area of stiffeners calculation which was to become a standard B : Breadth of ship procedure in ship structural design thereafter. But C : Curvature applied to the hull-girder John and many subsequent generations of naval Cb : Block coefficient architects accepted that the classical linear theory E : Young’s modulus was a sufficient basis for the longitudinal strength i : Element number calculation. Later in 1965 Caldwell used a Fs : safety factor simplified procedure to calculate the ultimate INA: Moment of inertia about the neutral axis moment of a mid-ship cross-section in the sagging condition, introducing the concept of a structural ICL: Moment of inertia about centre-line h : Wave height instability strength reduction factor for L : Length of ship compressed panels. Faulkner further developed the concept by suggesting a design method to a : plate length calculate this reduction factor. M : Bending moment x,y: Cartesian coordinates of the elements Smith et.al. (1977, 1986) and Dow (1981) Y : Distance from the neutral axis developed an incremental curvature procedure Z : Elastic sectional modulus of hull cross-section which allows the derivation of a moment – Zp : The plastic sectional modulus and curvature relationship for a complete hull. It was ε : Normalised strain in the element based mainly on finite element formulation, where θ : Angle between the neutral axis and base line the plate element strength was obtained from a set φ :Angle the moment vector makes with base line. of empirical curves. Billingsley (1980) used an σ y: Yield stress of the element engineering approach which considered a very b,t,hw,tw,bf,tf : Breadth and thickness of plate, web simple model for each individual beam-column and flange of a stiffened panel respectively. element. While early attempts were based on collapse strength, Adamchack (1984) developed a simplified method, together with a computer program, which implements the simplified 1. INTRODUCTION formula, where ultimate strength of each panel includes flexural-torsional buckling formulations. First evidence of theoretical attempt to evaluate Moment-Curvature relationship curves were built the longitudinal strength, under extreme condition from a set of discrete points corresponding to of a very large ship can be found in the numerous buckling of each panel. sketches of Brunel’s work during 1852. Those sketches and calculations were very remarkable Rutherford & Caldwell (1990) presented a since the theory of longitudinal ship had at that comparison between the ultimate bending moment

1 experienced by very large crude carrier,the Energy reliability analysis software COMREL & Concentration and results of retrospective strength STRUREL. calculations in which a simplified approach to stiffened plates collapse was used, but without 2. PROGRESSIVE COLLAPSE ANALYSIS considering the post-buckling behaviour. Also the importance of lateral pressure, initial The progressive collapse analysis method follows imperfections and corrosion rates were the general approach presented by Smith. The investigated. The validity of the model and the moment-curvature relation is determined by method was confirmed by a non-linear finite imposing a set of curvatures on the hulls girder. element analysis program. Later Gordo et.al. For each curvature the state of average of strain of (1996), calculated the ultimate strength of Energy each beam column element is determined. Concentration using simplified formula proposed Entering with these values in the load shortening by them considering of the effect corrosion and curves, the load sustained by each element may be initial imperfections in flexural buckling mode calculated. The bending moment sustained by the failure. In this present study tripping failure other cross section is obtained from the summation of than flexural buckling failure of the local element the moments of the forces in the individual has been emphasised taking into account of element. The derived state of values defines the corrosion, welding induced residual stress and desired moment curvature relation. The basic imperfection. assumptions of the method are:

During the last few decades, the emphasis in • The elements into which the cross section is structural design has been moving from the sub divided are considered to act and behave allowable stress design to the limit state design, independently. because the latter approach has many more • Plane sections are assumed to remain in plane advantages. A limit state is formally defined as a when curvature is increasing, this condition is condition for which a particular structural member necessary to estimate the strain levels of the or an entire structure fails to perform the function elements, but its validity is doubt full when that it has been design for. It is important to note shear is present at the plate elements. that in limit state design of structures, various • Overall grillage collapse is avoided by types of limit states may be required to have sufficiently strong transverse frames. different safety levels. The actual safety level to be attained for a particular type of limit state is a On the assumption that plane sections remain function of its perceived consequences and ease of plane in bending, the strain corresponding to an recovery to be incorporated in design. The safety applied curvature C can be calculated for each margin of structures can be evaluated by a element of the cross-section using the simple comparison of ultimate strength with the extreme theory of bending. applied loads. To obtain a safe and economic structure, the ultimate load-carrying capacity as y well as the design load must be assessed (xgi,ygi) accurately. The structural designer can perform a x NA (base) structural safety assessment in the preliminary NA (inclined) design stage if there are simple expressions available for accurately predicting the design Horizontal loads, load combinations, and ultimate strength. A φ θ designer may even desire to do this, not only for the intact structure, but also for structures with premised damage, in order to assess and CL categorize their damage tolerance and survivability. Figure 1: Combined Bending of Hull

In this present study the probability of failure at The bending stress at a point (xgi,ygi) is defined as: scenarios has been calculated. The reliability analyses of Energy Concentration have been M cosϕ.y gi M sinϕ.xgi carried out by using first and second order σ = σ V + σ H = − (1) reliability method using commercially available I NA I CL

2 Where φ is the angle that the bending moment imposed in the vertical and horizontal planes vector makes with the base line and (xgi,ygi) is the respectively is given by: coordinate of a point with respect to a reference εei =C (xgi sinθ - ygi cosθ) (8) located in any point on the neutral axis. M.cosφ and M.sinφ are the vertical and horizontal bending Where εei =the longitudinal edge strain in the moments respectively, M being the resultant element. bending moment. It may be expressed as a 2 2 ½ function of the total moment by C= (Cx +Cy ) (9)

σ y.cosϕ x.sinϕ Where = − (2) M I NA I CL Cx =C.cosθ Cy =C.sinθ (10) Maximum values of stress occur at corners (decks or bilge strake), where both xgi and ygi are Once the strain state of each element is achieved, maximum. the corresponding average stress may be

M cosϕ.y gi−max M sinϕ.xgi−max calculated and consequently the components of σ max = − (3) bending moment at a curvature C can be given as: I I NA CL Equation (3) can be re-written as Mx = ∑ ygi .σi.Ai

My = ∑ xgi. σi. Ai (11) M cosϕ M sinϕ σ max = − (4) Z NA−deckedge Z CL−deckedge The modulus of the total amount is

Maximum stress will occur at an angle of 2 2 1/2 M= ( M + M ) (12) inclination φ, then x y dσ Z max = 0 ⇒ tanϕ = NA−deckedge This is the bending moment on the cross-section if dϕ Z CL−deckedge the instantaneous CG is placed at correct location. (5) Along the step by step increment of the curvature ⎛ Z NA−deckedge ⎞ the neutral axis shifts towards deck during the ϕ σ = tan −1 ⎜ ⎟ ()max ⎜ ⎟ hogging and towards the bottom during sagging. ⎝ Z CL−deckedge ⎠ o Since the new neutral axis shifts to a position For typical ships φ(σmax) ~ 30 . The bending stress where the net load (NL=Σ(Ai.σi)) is zero, taking is zero at the neutral axis of the mid-ship section, compressive and tensile stress with different sign. so equation (3) can be re-written as So it is necessary to calculate the shift between the two imposed curvatures. For this reason a trial and

ygi.cosϕ xgi.sinϕ ⎡INA ⎤ error process need to be implemented, having a − = 0 ⇒ygi = ⎢ tanϕ⎥xgi (6) I I I terminating criterion. For this study the following NA CL ⎣ CL ⎦ criterion proposed by Gordo and Guedes Soares has been used. This is an equation for a straight line in Cartesian -6 coordinate having a slope of (INA/ICL) tanφ. The NL= Σ (Ai.σi) ≤ 10 . σyi. ΣAi (13) slope is given by

3. LOCAL STRENGTH ASSESSMENT y gi ⎡ I NA ⎤ tanθ = = ⎢ tanϕ⎥ xgi ⎣ I CL ⎦ (7) In the stiffened plate panels, the longitudinal stiffeners have the main function of providing the −1 ⎡ I NA ⎤ θ = tan ⎢ tanϕ⎥ necessary support to the plates ensuring that they ⎣ I CL ⎦ retain the required strength. To fulfil this function, stiffeners must have adequate rigidity and the Where θ is the angle the neutral axis makes with spacing between them must be chosen according base (x-axis). The strain at the centroid of an to the main characteristics of the plate namely, its element i (xgi,ygi), when curvature Cx and Cy are thickness and yield stress. The slenderness of the plate has to be designed in such a way that the

3 ultimate average stress is kept closer to the yield column (stiffener and effective associated plate stress as much as possible. acting together) and failure may be towards the plate or towards the stiffener, depending on the The analysis of stiffened plates has been column’s initial shape and the type of loading performed by several researchers and many considered, i.e., eccentrically applied or not, solutions to the problem were presented over the following the shift of the neutral axis or not. In a years. The prediction of the panel behaviour has continuous panel it is usual that the failure is lead to the development of several techniques such towards the plate in one span and towards the as non-linear finite element methods or more stiffener in the adjacent span. The third mode of simplified formulations applying the beam-column failure is the consequence of a lack of torsional concept. Common to all is the need for the rigidity of the stiffener. Interaction with the plate- application of an incremental end shortening if a buckling mode may also occur including realistic description of the post buckling behaviour premature tripping. is required. Also common to later formulation is the use of load end shortening curves for simply Sometimes the first and the second modes are supported plates carried out on separate studies, incorporated in the same group because the which are able to describe the loss of plate buckled shape of the panel is similar and is stiffness after buckling. normally towards the stiffener. To obtain the average load-end shortening curve of the column it Design methods to determine the ultimate load of is assumed that the stiffener has an elastic- the panels were presented among others by perfectly plastic behaviour given by Faulkner et al based on John-Ostenfeld approach, by Carlson, and Dwight and Little based on Perry- ⎧ ⎪ − 1 when ε < −1 (14) Robertson formulation. ⎪ ⎪ Φ (ε ) ≡ Φ e = ⎨ ε when − 1 < ε < 1 ⎪ ⎪ 1 when ε > 1 ⎪ ⎩⎪ Where ε is the normalized strain ratio, i.e. εe/εy = edge strain/yield strain. The slenderness ratio of the plate is given by b σ β = y (15) 0 t E

The effective width of a plate is given by

⎧ 2 1 ⎪ − 2 for β 0 ≥ 1 (16) φ b = ⎨ β 0 β 0 ⎪ ⎩ 1 for β 0 ≤ 1

The plate slenderness depends on plate breadth, Figure 2: Possible collapse modes of stiffened panels under thickness, Young’s modulus and the yield stress of compressive loads. (a) Plate induced collapse, (b) Stiffener the plate. The load-shortening curve of the plate induced collapse, (c) Tripping failure can be expressed as function of normalised strain and the slenderness at every level of normalized Failure of panel is usually classified, as shown in strain can be defined as figure 3, as: plate induced failure, column like failure, tripping of stiffeners and over all grillage b σ β = e (17) failure. This last one is normally avoided by t E ensuring that transverse frames are of adequate size therefore it is not considered generally. The where σe is the edge stress of the plate when the first one occurs when the stiffener is sufficiently given strain is εe. Dividing the equations (17) by stocky and the plate has a critical elastic stress equation (15) and replacing σe/σy= εe/εy= ε, the lower than yield stress. The second failure mode is equation (17) can be re written as mainly due to the excessive slenderness of the

4 β = β 0 ε (18) Compression Therefore the effective width at a given strain can be given by, substituting β0 by β in equation (16) t

⎧ 2 1 b ⎪ − 2 for β ≥ 1 (19) φ b = ⎨ β β ⎪ ⎩ 1 for β ≤ 1 Fig.3: Idealized welding induced residual stress in plate During the fabrication process, imperfections are High tensile stresses which develop in the vicinity induced in the structure in the form of initial of a weld due to shrinkage are balanced by non- imperfections and residual stress. Both are uniform compressive stresses across the plate. function of the welding process and have an These compressive stresses affect the yielding influence on the local strength. When stiffening process and hence reduce the pre-collapse stiffness members are welded to the plate, the welding of the structure. Depending on the slenderness of temperatures take on such extreme values that the plate panel involved, the collapse strength may considerable residual stresses resulting from the be affected. The weld induced residual stress process can seriously degrade the plate strength. residual stresses have been shown to induce a As Fig.2 illustrates, the tension block is offset reduction both in stiffness and strength of plate. from the stiffener-plate centroid and thus bending The pattern of residual stresses due to welding occurs as shown. To preserve equilibrium along stiffeners in a plate shows a zone of tension the direction of the stiffener, the tension must be stresses near the welds and a zone of compressive balanced by residual compression which exists stresses in the central region of the plate. largely in the plate. This equilibrium requirement provides a relationship between the magnitude of The interaction co-efficient for the residual stress the compressive residual stress σr in the plating is given by [7]: and the width ηt of the tension zones each side of ⎛ ∆φ ⎞ the weld: R = ⎜1− b ⎟()1+ 0.0078η (21) η ⎜ 1.08 ×φ ⎟ σ 2η ⎝ b ⎠ r = (20) σ y ()b / t − 2η Table 1: Imperfection Levels Level Initial deflection Residual stresses Values of η=4.5 to 6 are typical for as-welded wmax/t σr / σy 2 ships, but values of 3 to 4.5 are more appropriate Slight 0.025β 0.05 for ship design after allowing for the Average 0.1β2 0.15 shakedown[3]. Severe 0.3β2 0.3

δs NA According to the available sources, although the geometric configuration of initial imperfection is quite complex, a simple approach can be adopted to design the initial imperfections on the stiffened plate. Based on the experimental measurements, Smith classified the initial imperfection as slight, average and severe, which is shown in the Table 1. δp Tension The interaction co-efficient for the initial Block at imperfection is given by [7]: Yield Stress δ R = 1− ()0.626 − 0.121β 0 (22) δ t 2ηt 2ηt If the residual stress coexists with the initial b-2ηt imperfection, the combining interaction equation is given by [7]: σy δ Tension σr R = 0.665 + 0.006η + 0.36 0 + 0.14β (23) ηδ t

in which ∆φb is obtained by Equation:

5 (Scantling of stiffed plate: Perfect: 1000x25+797x15+200x33 mm, σr E t σ =315MPa, Corroded: 1000x24+797x14+200x31 mm) ∆φb = y σy E Figure 4: Load-Shortening curve of bottom element in ⎧ 3.62β 2 different scenarios. Et ⎪ for 0 ≤ β ≤ 2.7 = ⎨13.1+ 0.25β 2 (24) E Where ⎩⎪ 1 for β > 2.7 ⎧ 1 εσ y ⎪ 1 − for σ E ≥ 0.5εσ y The radius of gyration is given by σ ⎪ 4 σ e = E (30) ' ⎨ σ 2 I e σ y ⎪ E r = ; (25) for σ E ≤ 0.5εσ y ce ⎪ εσ As + be × t ⎩ y

3 2 3 2 Where σE is the Euler stress and is given by b .t ⎛ t ⎞ t .h ⎛ t h ⎞ I ' = te + b .t.⎜ z − ⎟ + w w + h .t .⎜ z − − w ⎟ e 12 te p 2 12 w w p 2 2 π 2 × E × r 2 ⎝ ⎠ ⎝ ⎠ ce (31) σ E = 2 3 2 a b f .t f ⎛ t t f ⎞ + + b .t .⎜ z − − h − ⎟ f f ⎜ p w ⎟ 12 ⎝ 2 2 ⎠ 3.1 TRIPPING OF STIFFENERS 0.5b .t2 +h .t .(t +0.5h )+b .t .(t +h +0.5.t ) te w w w f f w f (26) zp = Tripping failure is one of the most dangerous b .t +h .t +b .t ()te w w f f failures, since it is always associated with very EIe' is the buckling flexural rigidity of the quick shed of load carrying capacity of the stiffener. The tangent effective width of the plate column. Lateral torsional instability may occur

()bte is given by: alone by twisting of the stiffener about its line of attachment to the plating; developing a partial or 1 × R × R × R β ≥ 1 full hinge at the intersection, or induced by bte ⎧ η δ ηδ e βe flexural buckling especially if the deflected shape = ⎨ (27) b ⎩ Rη × Rδ × Rηδ 0 ≤ βe ≤ 1 of the column is towards the plate. In that case, the stiffener will be subjected to a higher stress The effective width of the plate is related to the than the average column stress and the critical slenderness as follows: tripping stress could be easily reached, followed by a deep load shedding. Several authors have 1.08 × φb × Rη ×Rδ × Rηδ βe ≥ 1 be ⎧ proposed analytical formula for torsional buckling = ⎨ (tripping), but here those used by Bureau Veritas b 1.08 × R × R × R 0 ≤ β ≤ 1 ⎩ η δ ηδ e (28) [16] has been implemented.

Following Gordo & Soares [6] the ultimate The tripping stress at a given normalised strain is strength of a stiffened plate considering post given by buckling behaviour, modelled as stiffener with an associated width of plate can be given by: ⎡ Asσ C+bt.φb .σ y ⎤ σ T = σ y Φ()ε ⎢ ⎥ (32) A + bt σu σ e ⎡ As + be × t ⎤ ⎣ s ⎦ φ = = Φ()ε ⎢ ⎥ (29) σy σ y ⎣ As + b × t ⎦ Where Φ(ε) and φb are obtained from equations (14) and (19) respectively and 0.9

0.8

0.7 ⎧ σ ET ⎪ if σ ET ≤ 0.5εσ y 0.6 ⎪ ε (33) σ C = ⎨ ⎛ Φ()ε σ y .ε ⎞ 0.5 ⎪ σ ⎜1− ⎟ if σ > 0.5εσ y ⎜ ⎟ ET y 0.4 ⎩⎪ ⎝ 4σ ET ⎠ Stress/Yield Stress 0.3 According to BV rules the Euler torsional 0.2 buckling stress (σ ) can be given as follows: Uncorroded ET 0.1 2 Corroded π .E.I ⎛ K ⎞ I Corroded with Residual Stress & Imperfection w 2 C t 0 (34) σ ET = 2 ⎜m + 2 ⎟ + 0.385.E. 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 I .a m I Strain/Yield strain p ⎝ ⎠ p

6 The torsional co-efficient, KC, can be given as the principal components of the cross-section, [16]: including deep centreline girders at deck and C .a 4 bottom, additional deck and bottom girders as well O (35) K C = 4 as the sides, longitudinal bulkheads, deck and π E.I w bottom, are stiffened by closely-spaced The number of half-waves (m) should be taken longitudinal running between deep transverse equal to be an integer number, such that [16] web frames 5.1 m apart. High tensile steel was used in the upper and lower parts of the hull cross- 2 2 2 2 m (m-1) ≤ KC < m (m+1) (36a) section. and the relationship between the half-waves and the torsional coefficient is as follows [16]

⎧ 1 if 0 ≤ K C ≤ 4 ⎪ m = ⎨ 2 if 4 ≤ K C ≤ 36 (36b) ⎪ ⎩ 3 if 36 ≤ K C ≤ 144 The spring stiffness (C ) of the attached plating is O given by E.t 3 Figure 5: Profile and plan view of Energy Concentration. CO = (37) 2.73b

The net sectorial moment of inertia (I ) of the w Table 2: The Particulars of Energy Concentration stiffener about its connection to the attached plating for T-sections can be given by Length, Overall 326.75m 3 2 t f .b f hw Length Between Perpendiculars 313.0m I = (38) w 12 Molded Breadth 48.19m The net polar moment of Inertia (Ip) of the Molded Depth 25.20m stiffener about its connection to the attached Draft, Summer 19.60m plating for stiffener with face plate is given by Dead Weight 216269 tons. Gross tonnage 98 894 tons t .h3 2 w w Machinery Steam Turbine I p = hwb f t f + (39) 3 The St. Venant’s net moment of inertia (It) of the stiffener without attached plating for stiffeners This VLCC broke her back in July 1980 during a with face plate is given by discharge of oil at Rotterdam. During the period

1 ⎡ ⎛ t f ⎞⎤ December 1979 to March 1980, the ten year old I = h .t 3 + b t 3 ⎜1 − 0.63 ⎟ (40) t ⎢ w w f f ⎜ ⎟⎥ ship, Energy Concentration was surveyed by 3 ⎢ b f ⎥ ⎣ ⎝ ⎠⎦ Bureau Veritas Staffs at Singapore. It was found in the survey report that all cargo and ballast tanks 4. HULL GIRDER FAILURE OF ENERGY were in good condition, with exception of ballast CONCENTRATION tanks No.3, where a greater degree of corrosion was found. The location of these tanks is Energy Concentration, a Very Large Crude Carrier coincident with location of the failure. The last (VLCC), was built by Kawasaki Heavy Industries voyage of Energy Concentration was from the Ltd., Japan. Her construction was completed in Persian Gulf to Rotterdam Europoort. The March 1970 and built to Det Norske Veritas rules. sequence of loading and discharging of the ship A change of rule class to Bureau Veritas took during this voyage was complicated by some place in 1977; and later transferred to Liberian changes of plan by the chatterers, and by the need registration in April 1978. The overall architecture to carry five different grades of cargo, loaded at of the ship was conventional for VLCCs of that three ports for discharge at three other ports. period. Details of these transactions, including the ships’s departure conditions from the Persian Gulf and The structure of the ship can be seen from the cargo discharges at Immingham (in England) and mid-ship section to be of conventional design. All Antifer (in Belgium) have been described in

7 reference [1]. It may be noted here that the sequence of loading, discharging and cargo transfer (exacerbated by the many grades of oil being carried) evidently led to the ship leaving Antifer on 20th July 1980 not only heavily trimmed by stern ,but also with very severe hogging still water bending moment. The ship’s drafts, recorded in the deck logbook indicate a hog of about 43 cm. Subsequent calculations, summarized in reference [1], show that the ship commenced the final leg of the voyage, from Antifer to Rotterdam, with a still water bending moment more than twice the value permitted under the rules by which the ship was classed.

Figure 7: Failure of Energy Concentration (Source Ref.[1])

Figure 6: Hull Failure of Energy Concentration (Source: Ref.[1])

While discharging oil at the Mobil terminal at Rotterdam the ship had broken its back, the event lasted few seconds, with the fore and aft sections trimming forward and aft respectively about a “hinge” in the deck plating near amidships. The water depth along side the terminal was around 20m, and the two ends came to rest on the bottom, thus preventing further jack-knifing of the hull, as seen from the figures 6 and 7. Figure 8: Hull Failure of Energy Concentration (Source: Ref.[1]) It is evident that had this event occurred in the open sea, the hiding process would have-continued There was no fracture in the deck, only a large much further, leading possibly to separation and fold at about frame 76, as seen figure 6. Below the subsequent sinkage of the two halves of the ship. deck level however, extensive crumpling and There was clear evidence that large vertical forces fracture of shell and internal structure, as seen in aced on the fore and aft ends as they settled on the figures 8, show clearly the nature of the structural bottom, diagonal shear buckling of the forward failure. High compressive stresses due to hogging shell side plating, as seen in figure 6 & 7, and initiated buckling in the lower parts of the punching of the rudder stock upwards into the structure causing a reduction in resistance and overhanging stern suggest that the weights of the progressive compressive failure spread upwards two ends of the ship greatly exceeded the through the structure. Gross deformations led to buoyancy available up to the point of grounding the concertina type of failure shown. fore and aft.

8 4.1 ULTIMATE STRENGTH ANALYSIS Detailed information about the corrosion effect was not reported although it was mentioned in the In order to perform the progressive collapse reference [1] that all cargo and ballast tanks were analysis, the mid-ship section of Energy examined and found to be in reasonable condition. Concentration has been divided into stiffened On the basis that the ship was in service for 10 plate and plate elements as shown in figure 12 due years when it failed and assumption that a nominal to symmetry of the mid-ship section, half mid ship-section has been divided into 132 elements. Figures 10, 11 & 12 and table 3 give the relevant information about the dimensions and material properties of the section and elements. Two different type of element has been used in three regions, the deck and bottom are made of high strength steel (HTS, σy=315 MPa) and the main parts of side shell, longitudinals and bulkheads are constructed of mild steel (MS, σy=235 MPa). The frame at the mid-ship section is 5.1m and the spacing between longitudinals varies between 925 mm on the side and bulkheads and 1m on the deck and bottom.

Table 3: Scantling of Elements St. Web Flange Steel St. Web Flange Steel Figure 10: Scantlings of Mid-ship Section of Energy Concentration No. (mm) (mm) Type No. (mm) (mm) Type 1 797x15 200x33 HTS 17 747x12.7 180x25 MS 2 297x115 100x16 HTS 18 797x14 180x25 MS 3 370x16 HTS 19 847x14 180x25 MS 4 425x32 HTS 20 847x14 180x25 MS 5 480x32 HTS 21 847x14 180x25 HTS 6 297x11.5 100x16 HTS 22 847x14 180x25 HTS 7 370x16 HTS 23 897x15 200x25 MS 8 447x11.5 125x22 HTS 24 945x16 200x25 MS 9 549x11.5 125x22 MS 25 897x15 200x25 HTS 10 597x11.5 125x22 MS 26 797x15 180x25 HTS 11 597x11.5 125x22 MS 27 347x11.5 125x22 HTS 12 647x11.5 125x22 MS 28 397x25 HTS 13 350x25.4 MS 29 300x35 MS 14 647x12.7 150x25 MS 30 230x12.7 MS 15 647x12.7 150x25 MS 31 230x12.7 HTS 16 747x12.7 150x25 MS 32 397x11.5 100x25 HTS Figure 11: Scantlings of Mid-ship Section of Energy Concentration

corrosion rate of 0.1 mm per year can be expected, thicknesses of the plating and longitudinal webs were reduced by 1 mm. On the basis that horizontal surfaces are more prone to corrosion NA attack and, unlike the case of shell plating, wastage occurs from both sides, twice the amount of corrosion is assumed to occur in the stiffener flanges. So 2 mm have been deducted from the flange thickness.

Figure 9: Mid-ship Section of Energy Concentration

9 declines smoothly. The calculated bending moment value is very close to the bending moment noted at the time of failure. Again it can be observed that calculated bending moment is smaller than the noted bending moment at the time of failure, which implies that during the failure, failed part of the ship was corroded heavily and the assumed margin of corrosion is nearly equal to the actual amount of corrosion present ain that part of the ship.

2.0E+10

1.8E+10

1.6E+10

1.4E+10

1.2E+10

1.0E+10

8.0E+09

Figure 12: Division of Energy Concentration half mid-ship 6.0E+09 section into 132 elements. Hogging Moment(Nm) Bending 4.0E+09 Uncorroded Scenario Corrosion Scenario 2.0E+09 Corrosion with Residual Stress & Imperfection

0.0E+00 0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 Although different corrosion rates could be Curvature(1/m) considered more appropriate for application to vertical structure where the stiffener webs are Figure 14: Moment Curvature Relationship of Energy oriented horizontally, the above reductions were Concentration in Hogging Condition for different scenarios. 0.0E+00 assumed to apply on a global basis. Since Energy -0.00035 -0.0003 -0.00025 -0.0002 -0.00015 -0.0001 -0.00005 0 Concentration was in service for ten years it is -2.0E+09 assumed that it has average level of imperfections in the local elements. For Energy Concentration -4.0E+09 Uncorroded Scenario η=3 has been assumed, which correspond to σr= Corrosion Scenario Corrosion with Residual Stress & Imperfection 0.167 in a plate having a width thickness ratio of -6.0E+09

40. -8.0E+09

Moment(Nm) Bending Sagging 4.1.1 SUMMARY OF RESULTS -1.0E+10

2.0E+10 -1.2E+10 Curvature(1/m) 1.6E+10 1.2E+10 Figure 15: Moment Curvature Relationship of Energy

8.0E+09 Concentration in Sagging Condition for different scenarios.

4.0E+09 Table 4: Ultimate Strength of Energy 0.0E+00 -0.0004 -0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003 0.0004 Concentration without corrosion Curvature(1/m) -4.0E+09 Ultimate Strength (MNm) -8.0E+09 Method Initial %

-1.2E+10

Bending Moment(Nm) Hog Sag Hog Sag -1.6E+10 Failure 17940 92.8 Yield Moment 19332 100 Figure 13: Moment Curvature Relationship of Energy Concentration considering corrosion in the local elements. Plastic Moment 22618 117 Rutherford et. al. 18979 15450 98.2 79.9 MARC-FEM 20630 106.7 Referring to the moment-curvature relationship, Gordo et. al 19164 16392 99.1 84.8 figure 13, it is observed that maximum bending moment of 17261 MNm occurs at 1.2 x 10-4 m-1 Present Study 18300 11418 94.7 59.1 and after that point the bending moment curve considering Tripping

10 Inclusion of a tripping formulation in the 5 STRUCTURAL RELIABILITY ANALYSIS behaviour of stiffened panels is seen to be very OF THE ENERGY CONCENTRATION important in this case. The deck stiffeners, made of bar, don’t have much flexural torsional rigidity Traditionally, in the design process, practitioners and calculated tripping stress is lower than the and designers have used fixed deterministic values flexural buckling stress which associated the plate. for loads acting on the girder and for its strength. This fact leads to a very high reduction of the In reality these values are not unique values but ultimate strength in sagging compared to hogging, rather have probability distributions that reflect where the deck is in tension, as observed from the many uncertainties in the load and strength of the figure 13 that ultimate strength in sagging is about girder. Structural reliability theory deals mainly 37 % lower than the ultimate strength in hogging. with the assessment of these uncertainties and the methods of quantifying and rationally including Table 5: Ultimate Strength of Energy them in the design process. The load and strength Concentration with corrosion are thus modelled as random variables. Fig. 16 shows the frequency density functions of load and Ultimate Strength (MNm) the strength of the girder in terms of applied Method Corrosion % bending moment and ultimate moment capacity of the girder, respectively. Both, the load L and Hog Sag Hog Sag strength R are assumed to follow the normal Failure 17940 - 97.2 - probability distribution. Yield Moment 18457 100 Plastic Moment 21533 116.7 Rutherford et. al. 17860 14420 96.8 78.1 Density Function MARC-FEM 18520 100.3

Gordo et. al 17885 15232 96.9 82.5 Present Study 17261 10671 93.5 57.8 Failure Area considering Tripping Strength(S) Load (L) Table 6: Ultimate Strength of Energy

Concentration with corrosion, imperfection and residual stress in the elements Mean L Mean S Ultimate Strength (MNm) Figure 16: Frequency Distribution of Strength(R) & Load(L) Corrosion, Imperfection % Method & Residual Stress Now, a simple function g(S, L) can be constructed, the limit state function describes the safety margin Hog Sag Hog Sag M between the strength of the girder and the load Failure 17940 -- acting on it. Yield Moment 18457 100 Plastic Moment 21533 116.7 M=g(S,L) = S-L (41) Rutherford et. al. ------MARC-FEM ------Both S and L are random variables and may Gordo et. al ------assume several events or conditions describe the Present Study 16364 10016 88.7 54.3 possible states of the girder considering Tripping Case 1. M=g(S, L)<0 Represents a failure In this case the ship had very slender panels and as state since that means that load(L) a consequence they promoted the tripping of exceeds the strength(S) stiffeners, at an early stage of the loading followed Case 2. M=g(S,L)>0 Represents Safe State by the premature collapse of the section at a low Case 3. M=g(S,L)=0 Represents the limit state curvature. Again from the figures 14 and 15 it can surface or border surface between the safe be observed that residual stress, imperfection in and failure state. the local members decreases the ultimate strength of the ship up to a noticeable margin.

11 The probability of failure implied in case (1) can external pressure. Therefore, the stochastic models be computed from: of still water bending moment, vertical wave- induced bending moment, and their combination P = P[M = g(S, L) ≤ 0] = f dSdL (42) are briefly described in the coming section. f ∫∫ S ,L g (S ,L)≤0 The IACS(‘95) unified formula for estimating the where fS,L(S,L) is the joint probability density function of S and L, and the domain of integration design wave–induced bending moments is expected as is over all values of S and L where the margin M 2 −3 ⎧ 110.CwL B(Cb + 0.7)×10 kNm for sagging (46) is not positive, i.e. not in the safe state. If the M w = ⎨ 190.C L2BC ×10−3 kNm for hogging applied load on the girder is statistically ⎩ w b independent from the girder strength the above Eq. (42) can be simplified and interpreted as: where L. B and CB are ship length (in m), moulded breadth and block coefficient, respectively, and Cw ∞ is the wave coefficient given by:

Pf = FS (L) f L (L)dL (43) ∫ ⎧ 10.75 − ((300 − L) /100)1.5 100 < L ≤ 300 0 ⎪ (47) C = 10.75 300 < L ≤ 350 w ⎨ ⎪ 10.75 − ((L − 350) /150)1.5 L > 350 where FS(.) and fL(.) are the cumulative ⎩ distribution function of S and the probability density function of L, respectively. Eq. (43) is the Applying these expressions to Energy convolution integral to L. As mentioned above, the Concentration the magnitude of wave induced S and L are both statistically independent and bending moment is calculated to be 7754 MNm in normally distributed. Eq. (43) can be thus shown hogging and 8398 MNm in sagging conditions. to reduce to The still water bending moment is due to the action of the ship’s lightweight, deadweight and Pf =Φ(-β) (44) buoyancy. From the information given in the reference [1], its hull geometry, load conditions on here Φ(.) is the standard normal cumulative its final voyage and sequence of cargo tank distribution function and β is called a safety index discharge, the distribution of still water bending defined as: moment can be calculated. The permissible still water bending moment in that condition is calculated to be 5524MNm. µ S − µ L β = (45) 2 2 σ S + σ L 5.2 LIMIT STATE FUNCTION Where µ, σ represent the mean value and standard deviation of the random variable, respectively. It could be noticed that, as the safety index (β) The limit state function for the Energy increases, the probability of failure Pf as given by Concentration can be defined as equation (44) decreases. g(X ) = xu M u − xsw M sw − xw xs M w (48)

5.1 LOAD EFFECTS AND LOAD where: xu= the random variable representing the

COMBINATIONS model uncertainty in ultimate strength; xsw= model uncertainty for predicting the still water bending moment; x = error in wave induced bending To assess the survivability of Energy w Concentration taking into account of different moment due to analysis over prediction; scenarios, it is necessary to compare the values of xs=uncertainty of model that takes nonlinearities the load effects in the various components with into account Mu, Msw and Mw represent the their residual strengths. Determining the load ultimate strength of the Energy Concentration hull effects is one of the major objectives in the girder, still water bending moment and wave- structural reliability assessment. Ship hull girders induced bending moment respectively in 3 are predominantly subjected to combined actions different scenarios, viz. (a) without corrosion, (b) of still water, wave-induced bending moments and with corrosion and (c) with corrosion, residual

12 stress and imperfection. It may be noted here that scenarios in hogging and sagging condition, taking impact of dynamic moment induced, has not been into account the change of loading conditions. taken into account, since the Energy 1.6E-04 Without Corrosion Concentration was still during the moment of 1.4E-04 With Corrosion failure. For the present study model uncertainties 1.2E-04 With Corrosion+Res. St.+Imperfection factors have been assumed as 1. 1.0E-04

Table 7: Properties of parameter 8.0E-05

6.0E-05

Probability of Failure

Mean 4.0E-05 Parameter Distri- Value(MNm) COV bution Hog Sag 2.0E-05 M * Normal 18300 11418 0.10 0.0E+00 u Scenarios(Sagging) Mu** Normal 17261 10671 0.10 Figure 18: Probability of failure at different scenarios in sagging condition Mu*** Normal 16364 10016 0.10 4.3 Without Corrosion Msw Normal 5524 5524 0.04 4.2 With Corrosion

Mw Normal 7754 8398 0.20 4.1 With Corrosion+Res. St.+Imperfection ) β *= without corrosion 4 **=with corrosion 3.9 ***=with corrosion, residual stress and imperfection 3.8 1.0E-01

Without Corrosion 3.7 9.0E-02

With Corrosion 3.6 8.0E-02 Second Order Reliability Reliability Order Index( Second With Corrosion+Res. St.+Imperfection 3.5 7.0E-02

3.4 6.0E-02

3.3 5.0E-02 1 Scenarios(Sagging) 4.0E-02

Probability of Failure Failure of Probability Figure 19: Reliability index at different scenario in sagging 3.0E-02

2.0E-02 It can be noticed that failure probability 1.0E-02 considering the corrosion, residual stress and 0.0E+00 1 imperfection in hogging condition is about 8.65% , Scenarios(Hogging) Figure 16 : Probability of failure at different scenarios in where as for same scenario in sagging condition hogging condition it’s 0.014%. So it answers the question for ship

2.5 Without Corrosion being failed in hogging condition rather than in With Corrosion sagging condition, even though the ultimate

2 With Corrosion+Res. St.+Imperfection

) strength in hogging is much higher than that in β sagging condition. In order to compare with 1.5 failure probability of intact ship, wave induced moment assumed to those determined according to 1 Classification rules. The corresponding reliability index (β) using FORM and SORM has been Second Order Reliability Index( Order Second 0.5 presented in figures 17 and 19 for hogging and sagging conditions respectively. The reliability 0 1 index in case of corrosion with residual stress and Scenarios(Hogging) imperfection found to be 3.63 in sagging Figure 17: Reliability index at different scenario in hogging condition, compared to 1.36 in hogging condition.

To investigate the influence of corrosion, residual stress and initial imperfection on the ship hull 6 CONCLUSION reliability a series of analyses are conducted both The ultimate strength of Energy Concentration has in hogging and sagging conditions using been calculated using the simplified method. The COMREL Pro. The figures 16 and 18 reveal the moment-curvature relationship obtained shows, probability of failure of the VLCC at different smooth decrease in the bending moment after the

13 highest point in the curve. The ultimate strength of [3]. FAULKNER D. 1975: A Review of Effective Plating the Energy Concentration shows significant for the Analysis of Stiffened Plating in Bending and difference due to consideration of tripping Compression. Journal of Ship Research, Vol. 19, No.1, formulation. The deck stiffeners are prone to pp 1-17. [4]. CALDWELL J.B., 1965: Ultimate Longitudinal tripping failure as they are flat bars. This fact leads Strength. RINA Transaction, Vol. 107, pp 411-430. to a very high reduction of ultimate bending [5]. GEUDES SOARES C., SOREIDE T.H.,1983: moment in sagging compared to hogging, where Behaviour and Design of Stiffened Plates Under the deck is in tension. Predominantly Compressive Loads. International Shipbuilding Progress, Vol. 30, Nº 341, pp. 13-27. The presence of corrosion can’t be avoided, since [6]. GORDO J.M., GUEDES SOARES C.1993: the ship was in service for 10 years. The ultimate Approximate Load Shortening Curves for Stiffened Plates Under Uniaxial Compression, Integrity of strength of the ship significantly decreased both in Offshore Structures-5, D. Faulkner, M.J. Cowling, A. hogging and sagging due to corrosion. Incecik, P.K. Das (Eds.) EMAS, pp. 189-211. [7]. PU Y., DAS P.K., FAULKNER D. Nov. 1997: Ultimate The consideration of welding induced initial Compression Strength and Probabilistic Analysis of imperfection and residual stress in local element Stiffened Plates. Journal of Offshore Mechanics and have degrading effect on the ultimate strength the Arctic Engineering, 119, pp 270-275. ship. However, these effects occur at separate [8]. GORDO J.M., GUEDES SOARES C., FAULKNER D. times in ship’s life; the residual stress and 1996: Approximate Assessment of the Ultimate Longitudinal Strength of the Hull Girder. Journal of imperfection are present from the very early stage Ship Research, 1996, Vol. 40, Nº 1, pp. 60-69 of ship’s life, where as the corrosion increases [9]. RAHMAN M.K., CHOWDHURY M.1996.Estimation with passage of time. If an allowance for corrosion of Ultimate Longitudinal Bending Moment of Ships and is included in the design, then reduction allowance Box Girders. Journal of Ship Research, Vol. 40. No. 3. for residual stress and imperfection should be pp 244-257. considered. [10]. KHAN IA, DAS PK, ZHENG Y, 2005: Structural Response of Intact and Damaged Stiffened Plated High compressive stresses due to the hogging Structure for Ship Structures. Proceedings of Maritime Transportation and Exploitation of Ocean and Coastal initiated buckling in the lower parts of the Resources-Guedes Soares, Garbatov & Fonseca structure causing a reduction in resistance, and (eds),Lisbon,Portugal, © 2005 Taylor & Francis Group, progressive compressive failure spread upwards London. Pages- 455-460. through the structure. The probability of failure in [11]. KHAN I.A.2004: Ultimate Strength of Ship: A Case the hogging case has been significantly high Study, Bachelor of Technology Thesis. Dept. of Naval compared to that in sagging. So it can be observed Architecture & Ocean Engineering, Indian Institute of that the VLCC is more prone to failure in hogging Technology Madras, India. [12]. SMITH C.S.1977: Influence of local compression compared to sagging, as actually happened during failure on ultimate longitudinal strength of a ship’s hull. its failure. PRADS ’77, Tokyo. [13]. OZGUC O, DAS P.K., BARLTROP N.2005.A 7 ACKNOWLEDGEMENT comparative study on the structural integrity of single and double side skin bulk carriers under collision This study has been partially funded by the EU damage. Marine Structures. Vol. 18. pp 511-547. [14]. DAS P.K., FANG C., 2006: Residual Strength and Project MARSTRUCT-Network of Excellence on Survivability of Ships after Grounding and Collision, Marine Structures. The first author is grateful to Article in Press, Journal of Ship Research. Dr. Özgür Özgüç and Mr. M. Shahid for their [15]. DAS P.K., FANG C., 2005: Survivability and reliability valuable comments and suggestions. of damaged ships after collision and grounding. Ocean Engineering, Vol. 32, pp 293-307. REFERENCES [16]. BUREAU VERITAS, DECEMBER 2003: Rules for the Classification of Steel Ships, Bureau Veritas, France. [17]. COMREL & STRUREL Manual, RCP GmbH, Federal [1]. BOWEN P.T., 1981: Decision of the commissioner of Republic of Germany. the maritime affairs, R.L. In the matter of the Major [18]. SMITH C.S., DOW R.S. 1986: Ultimate Strength of Hull Fracture of ENERGY CONCENTRATION, Ship’s Hull under Biaxial Bending, ARE TR86204, ARE Republic of Liberia, Ministry of Finance, Monrovia, Dunfermline, Scotland. Liberia.

[2]. RUTHERFORD S.E., CALDWELL J.B.,1990: Ultimate Strength of Ships: A Case Study. SNAME Transaction, Vol.98, pp 441-471.