Ultimate Strength and Reliability Analysis of a Vlcc
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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.