CONCEPT AND PRELIMINARY STRUCTURAL DESIGN METHODS FOR THE MODERN MULTI-DECK SHIPS

Vedran Zanic University of Zagreb, Faculty of Mech. Eng. And Naval Architecture, Zagreb, Croatia

SUMMARY

The paper covers the multi-criteria design methods for the practical structural design of the multi-deck ship structures such as Ro-Pax, Ro-Ro, Car carriers, Livestock carriers, Tank-car carriers, Con-Ro ships, Cruise and Passenger ships. Approach combines the fast generation of design variants using the 2.5D transverse strip FEM models and the generic 3D-FEM models (based on macro-elements) into two step decision support procedure based on the topology and scantling optimization. In the concept design phase objectives considered, besides cost and weight, may be the ultimate longitudinal strength safety measure and the system reliability measure in racking mode. Practical imple-mentation of the Decision Support Methodology is presented using (1) the ISSC 2006 benchmark structure of the cruise ship for topology/scantling optimization and (2) the RoPax ship for the reliability based concept design. 1 INTRODUCTION

Objective of the paper is to present the decision support problem (DSP) rationale for the concept design phase where the most far-reaching decisions are made regarding ship's safety and cost. The design environ-ment, capable of imbedding such multiple quality crite-ria (including cost, weight, reliability and nonlinear ultimate strength calculations), is described together with case studies of its application. The problem is particularly demanding for the modern multi-deck ships (cruise ships, passenger fer-ries, RoPax ships, livestock and car carriers, etc.) char-acterized with the extensive superstructure. The influ-ence of the upper superstructure decks to the primary strength considerations for those ships has to be taken into account starting from the concept design phase. Concept design is defined as the phase in structural design when geometry and topology are open to modifi-cation and structural variants are analyzed in accordance with the needs of the head designer. Applied loads are usually taken as deterministic. Selection of appropriate scantlings is important for approximate assessment of structural weight, achievable clearances (regarding height of deck beams and girders etc.) with a goal to define acceptable structural layout. Benefits of optimi-zation procedure in this phase are the biggest (Krueger, 2001). The basic objective is to give designer the ra-tional base for choosing the optimal structural variant for the next phase of design process (where the detailed structural model is developed). Only the full ship 3D FEM analysis (Fig. 1a-d) is considered sufficient (ISSC, 1997) for the correct as-sessment of the global structural response of those ships i.e.: (a) global deformations, (b) effectiveness of upper decks and distribution of long. stresses, (c) transfer of forces between hull and superstructure, (d) shear stress in way of the intermediate recess, (e) shear lag in the decks levels, etc. The main disadvantage of the full-ship 3D FEM model is the large amount of work needed for preparation and evaluation of the model. Therefore it cannot be used efficiently at the concept design stage, but mostly for the verification purposes of the final configuration in preliminary and detail design phases.

The challenge is therefore to establish simpli-fied way of analyzing these complex effects using sim-plified 2.5D transverse strip models and/or generic coarse mesh 3D FE models that can ensure rapid gen-eration and comparison of different structural topologi-cal concepts (Fig. 1e-f). Basic generic structure is a multi-deck model with transverse / longitudinal bulk-heads and provisions for different end restraints, interac-tion of decks and inclusion or exclusion of structural macro-elements (panels, windows, etc), (Andric, 2007). The model cross-section and general layout correspond to the midship section and ship general arrangement. Side panels with big openings are modeled with equiva-lent stiffness using analytical or numerical approaches. For the multi-deck ships with hull-superstructures interaction the design synthesis is particularly complex. It may include two main steps: (1) exploration of topology / geometry concepts (2) scantling / material optimization of the preferred variants from step (1). Each step includes a number of analysis and syn-thesis modules requiring development of the decision support environment for practical usage. Regarding step (1), the main global topological parameters (e.g. size of large side openings, stiffness of longitudinal bulkhead, number of transverse bulkhead, position of recess, etc.) have dominant influence on the shape of primary stress distributions over the ship height and thus on the overall concept efficiency. Taguchi techniques (FFE, orthogonal arrays, ANOVA, etc.) could be used to study multiple topological parameters simultaneously. It enables rational identification of the dominant parameters on topological level and provide designer with the near optimal level of each parameter. Different topological concepts lead to structural design with varying design quality measures but give good starting points for the further multi-attribute scant-lings optimization in the design step (2). Generated Pareto frontiers in steps (1) and (2), containing non-dominated feasible designs, enable the designer to make rational decisions on the selection of the preferred variants / configurations in the concept design phase (Zanic et al. 2006).

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Basic ship types and related structural models : Four extreme models of real ships were selected according to different superstructure/lower hull arrangement (Fig. 1a-1d). They are used for identification of structural prob-lems and calibration of the developed generic design models and procedures to different ship types: Model 1 basic (e.g. car carrier) closed superstructure with sides without openings or with side ramp; Model 2 (livestock carrier) open superstructure on pillar arrangement, high reduction of shear stiffness of superstructure sides, decks weakly restrained fore/aft; Model 3 (cruise ship)- large openings in superstructure sides, internal empty spaces, non alignment of side walls of the superstructure with the hull sides due to life boat recess; Model 4 (tank-car carrier)- very long deck house on the very flexible upper deck. The definition of the design problem (variables, con-straints, attributes) for those ship types may be identi-fied as given in Table 1.

Table 1: Identification of decision support problem MODELS VARIABLES PARAMETERS ATTRIB-

UTES

CON-

STRAINTS

SCANT-

LINGS

COST C.S. RULES

MATERIAL

TRANSV.

STRUC.

(CLASSIC,

HINGE)

SAFETY ULT. STREN.

1

CAR

CARRIER

STIFFENING

TYPE

PILLAR SYST. DWT CLEARANCES

SCANT-

LINGS

BRIDGE

STRUCT

COST C.S. RULES

MATERIAL STERN STRUCT SAFETY ULT. STRENG

2

LIVESTOCK

CARRIER

STIFFENING

TYPE

PILLAR SYST. MAINTE-

NANCE

RACKING

SCANT-

LINGS

RECES POSI-

TION

COST C.S. RULES

MATERIAL TBHD, LBHD SAFETY ULT. STRENGTH

3

CRUISE

SHIP

STIFFENING

TYPE

SIDE OPENINGSVCG WINDOWS

SCANT-

LINGS

DECKHOUSE

LGT

COST C.S. RULES

MATERIAL DH. MATERIAL SAFETY ULT. STRENG

4

TANK-CAR

CARRIER

STIFFENING

TYPE

DH. LAYOUT ROBUST-

NESS

CLEARANCES

(a) Car carrier

(b) Livestock carrier

(c) Cruise ship

(d) Tank car carrier

(e) Simplified cruise ship - ISSC Benchmark -2006

(f) Concept design model of RoPax Fig. 1: Multi-deck ships 3D FEM models

Structural modelling for design synthesis: Classical finite element modelling, giving good insight into stresses and deformations is not capable of giving efficient and fast answers regarding feasibility criteria, particularly in the structural optimization context. And, it is the feasibility that is of primary interest to the designer, not stresses or deformations. Most of the local failure criteria (e.g. different buckling modes of panels) require specified

force and displacement boundary conditions. They are available only if logical structural parts, such as complete stiffened panels between girders / frames, are modeled. Superelements modeling may help in this respect but it is usually impractical except for some particularly complex parts where only the stress or deformation levels are needed. Specially developed macroelements, combining numerical and analytical approaches to logi-

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cal meta-structures (stiffened panels, bracketed and locally reinforced girders, cell elements) could be a fruitful alternative. This mesh size is also sufficient for the coarse mesh vibration analysis. Their use greatly simplifies and speeds up the de-sign work, particularly if the structural modeling is based on general arrangement plans and follows the process of design development from the early stages (Zanic et al. 2001). This is absolutely required when the yard time constraints are imposed on design work. Also, the possible improvement could be achieved by devel-oping efficient link between CAD and global macro-element FE models (ISSC, 2003). Concept design loads: The characteristic of the first three ship types are rather uniform distributions of light-ship/deadweight and concentration of the buoyancy toward the midship portion that results in a very high still water hogging moment. Despite concentration of deadweight amidships the ships are still in hogg. Two loading conditions are usually taken into account considering the combinations of maximum still water and wave hogging and minimum still water hog-ging + sagging wave. First loading condition results in the maximum longitudinal stresses while second can cause buckling problems in upper decks where the scantlings have to be minimized due to stability re-quirements (ISSC, 1997). However, a tank-car (wagon) carrier (in the loading condition with heavy wagons amidships) can suffer high still water sagging bending moment. Combinations with the wave sagging moment can result in the buckling problems, especially for configurations with the long deckhouses. Design problems regarding primary strength: Due to influence of the primary strength on the overall quality of design, and respectively on the decisions made in the concept design phases, the four basic structural configu-rations/models are contrasted in the sequel:

MODEL 1 - CAR CARRIER

Figure 2: Distribution of normal (x) membrane stresses in the ship model and vertically from keel up The structural characteristic of car carrier is the

closed box structure without side openings, except for e.g. large side ramp. This kind of structure is an extreme example where hull and superstructure act as a single body and the primary response is mainly following the extended beam theory (Zanic et al. 2007b) with high effectiveness of the superstructure upper decks. Large number of upper decks ensures relatively large section modulus i.e. low primary stresses. Deck thicknesses are mainly subjected to permissible wheel loading and ex-ternal/internal local pressure. The vertical distribution of normal (x) membrane stresses for the characteristic car carrier structure is presented in the Figure 2 (diagram).

MODEL 2 - LIVESTOCK CARRIER

Livestock carrier (Fig. 1b) is characterized with the superstructure on pillar arrangement with high reduction of stiffness of superstructure sides.

Fr.111

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

24000

-160 -120 -80 -40 0 40 80NORMAL STRESSES (N/mm*2)

poz. 11900 mm from CL

poz. 2800 mm from CL

poz. 9550 mm from CL

Figure 3: Distribution of normal (x) membrane stresses and vertically along hull depth

This structural arrangement causes very small contribution of the superstructure decks to the primary response, so that bottom and lower hull decks (e.g. D6) are highly stressed, see Fig. 3. Rationality of this ap-proach is very questionable. The old question regarding include/exclude superstructure in primary strength is crucial in this case. Rational answer to that problem can be achieved through the use of DSP, as discussed ear-lier. It can provide Pareto frontier of optimal structural configurations and enable rational (objective/subjective) selection of the design that best fits the designers needs (multi-attribute decision making).

MODEL 3 - CRUISE SHIP

Modern cruise ship is characterized with large openings in the superstructure sides, non alignment of side walls of the superstructure with the hull sides due to life boat recess, big atriums, etc.. An overview of some aspects of the structural design of the modern cruise ships is given in (Gudmunsen 1995, Andreau et al. 1988). The superstructure deck effectiveness is

FRAME AMIDSHIPS

0

3000

6000

9000

12000

15000

18000

21000

24000

27000

30000

-100 -80 -60 -40 -20 0 20 40 60 80NORMAL STRESSES (N/mm*2)

CL

7000 mm from CL

SIDE

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around 60-70% with 100% hull effectiveness up to the subdivision deck. The distribution of normal (x) mem-brane stresses is presented in Figure 4.

FRAME AMIDSHIPS

0

3000

6000

9000

12000

15000

18000

21000

24000

27000

30000

-100 -80 -60 -40 -20 0 20 40 60 80NORMAL STRESSES (N/mm*2)

SIDE 7200 mm from CLCL

Figure 4: Distribution of normal (x) membrane stresses in the ship model and vertically from the keel up

MODEL 4 - TANK-CAR (WAGON) CARRIER Tank-car carrier presented in this article is charac-terized with the very long superstructure (deck-house) on the very flexible upper deck without transverse bulk-head (TBHD) bellow. In order to present the superstructure effectiveness in primary strength an overall view of normal (x) stresses are presented in Fig. 5. Due to the stress distri-bution it is obvious that superstructure is very effective amidships. Two basic concepts may be considered: Effective superstructure - results in reduced scantlings

of the hull and increased superstructure scantlings. Non-effective superstructure - exclusion of the super-

structure influence on the primary response. This results in increased scantlings of the hull and a very light superstructure

Distribution of x stresses along hull depth is presented in diagram of Fig. 5, for long superstructure (SS) and alternative concepts (2 / 3 parts with 1 / 2 expansion joints) generated in the sensitivity study. The effective-ness was analyzed w.r.t. max. at main deck using = (NO SS. - WITH SS) / (NO SS - WITH SS-BEAM TH.) (1) from (Caldwell, 1957). Besides that, in general, it is not obvious which structural concept would give higher weight reduction and more efficient material usage, and two step procedure should be used.

Fr.102

0

2000

4000

6000

8000

10000

12000

14000

16000

-150 -105 -60 -15 30 75 120 165 210

NORMAL STRESSES (N/mm*2)

NO SUPERSTRUCTURE

SUPERSTRUCTUR-1 PART

SUPERSTRUCTUR-2 PART

SUPERSTRUCTUR-3 PART

Figure 5: Distribution of normal (x) membrane stresses for the very long superstructure- different concepts

2 ANALYTICAL MODULES IN SHIP STRUCTURAL DESIGN

For many engineering problems the mathematical model may be decomposed into six meta-systems of which two basic ones (, ) provide physical and envi-ronmental definitions of the problem/process and other four are behavioral systems (, , , ) for modeling response, adequacy, reliability and quality. They can be invoked into the design problem definition modules () and coupled with different optimization solvers () into multi-attribute multi-level hybrid design procedure. Analytical and synthesis modules have been devel-oped and implemented in the ship structural decision support systems: MAESTRO (2006) and OCTOPUS (2006). Table 2 summarizes the analysis modules for all 6 meta-systems. Problem sequencer is used for adapta-tion/selection of analytical modules to be used in the concept synthesis for the multi-deck ships.

2.1 PRIMARY RESPONSE MODELLING (-1)

The 2D OCTOPUS model contains structure be-tween two web frames (one bay) based on data in . For the concept design structural evaluation of primary response (-1 modules: longitudinal strength, warping torsion strength) provide the dominant response fields (Demand) for the design feasibility assessment. Evalua-tions based on the extended beam theory and FEM cal-culation of secondary displacement fields, representing the cross section warping, are sufficiently accurate for this phase. All calculations are performed in the pro-gram module LTOR described in (Zanic et al. 2007b).

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Table 2: Implemented analysis modules for structural design

META-SYSTEM

OCTOPUS ANALYZER (O) AND MAESTRO (M) MODULES (IMPLEMENTED MAPPINGS)

DESCRIPTION OF THE OCTOPUS (O) AND MAESTRO (M) ANALYSIS MATH. MODELS / MAPPINGS

Physical () M: FEM STRUCTURAL MODELER

MAESTRO MODELER is used to define generic 2.5 D/ 3D FEM model with different cross-sections (web-frame, bulkhead).

Environment (-1, ..LC)

O: OCTLOAD - OCTOPUS load module M: MAESTRO 3D FEM loader

Classification Society loads or designer given loads from seakeeping analysis. 3D load distributions are automatically generated.

Response (-1)

(1) O:LTOR- primary strength fields (warping displac.; normal/shear stresses) (1-3) M: MAESTRO 3D FEM solver (3D)

Extended beam theory (cross section warping fields via FEM in vertical / horizontal bending and warping torsion) Full 3D FEM solver using macroelements.

Response (-2, 3)

(2) O: TOKV; (3) O: TBHD-secondary strength fields: transverse and lateral displacements; stresses

FEM analysis of web-frame and bulkhead (beam element with rigid ends; stiffened shell 8-node macro-element) -Fig. 6.

Adequacy / feasibility

(-1, 2)

(1) O: EPAN / M: EVAL library of stiffened panel and girder ultimate strength & serviceability criteria. (2) O: FATCS Rules fatigue calculation

Calculation of macro element feasibility based on super-position of response fields (O: -1, -2, -3) or directly (M: 3D) + libraries of analytical safety criteria - Fig. 7.

Adequacy (-3, 4)

(3) O: LUSA Ultimate longitudinal strength module (3) (M: ALLPS/HULL) (4) O: MIND generator of minimal dimensions

Longitudinal ultimate strength analysis of cross-section using modified Smith (Fig. 9) and modified Hughes/Adamchak (Fig. 8) methods. (Paik procedure is possible via MAESTRO). Class Rules minimal dimensions.

Reliability (-1, 2)

(1) O: US-3 reliability calculation of element and system failure probability (level 1-3, mechanism) (2) O: SENCOR sensit. to correlation of input var.

FORM approach to panel reliability. -unzipping method used to determine system probability of failure (Fig. 24). Sensitivity calculation based on Nataf model.

Quality

(-1, ..,9)

(1) O & M: WGT / (2) CST - cost/weight modules (3) O: MUH / MUS-ult. hull girder bending moment (4) O: URL - ultimate racking load (5) O: FLIFE-fatigue life (6) O: UDBP / (7) UDBR - reliability measures (8) O: GML / (9) TSN - robustness measures (in dev.)

Minimal structural weight = max. DWT increase; Min. initial cost Calculations using LUSA (sagging, hogging-Figs.8, 9, 24) and SORM Deterministic calculation using US-3 and TOKV. Fatigue life calculation for longitudinal-web intersection Upper Ditlevsen bound: panel or racking (Fig. 24) failure probabilities Information context measure / Taguchi S/N ratio via FFE.

2.2 TRANSVERSE STRENGTH RESPONSE (-2)

The response of transverse elements of ships structure (particularly transverse strength in racking) is also provided by the use of FEM. This calculation is performed in the separate -2 program module TOKV by use of 2D FEM model. Eight types of finite elements and macro-elements are included in the present module i.e. : two bracketed beams, two bars, two quadrilateral stiffened shell elements (8-node), triangle and spring element (Fig. 6).

Fig. 6: RoPax shear stress field &racking displacements

2.3 STRUCTURAL SAFETY MEASURES (-1, 2)

The procedure for determination of structural fea-sibility is at the heart of design procedure since satisfac-tion of those criteria is the guarantee of structural integ-rity. Failure functions g(x)=0 (Class Rules and/or direct

calculation (Hughes, 1983) define nonlinear failure surfaces dividing feasible from infeasible designs.

Figure 7: Structural adequacy for RoPax load case 2

The form of functions is analytical since they are applied to regions of regular shape between girders or frames (macro-elements). The normalized form of fail-ure criteria is given in the format: ,1)()()(1 += DSFCDSFCxg (2) Demand D (load effect) is obtained from the macro-elements in FEM response modules (1-3) and C is macro-element Capability calculated from the library of failure criteria . SF is the applied safety factor. For safe structures g 0. Applied library is presented in Table 3. Number of feasibility (safety) checks is usually very large. It is defined as product of (no. of macroelements * no. of failure criteria * no. of design load cases).

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Table 3: Library of failure criteria in -1

NAME CRITERIA DESCRIPTION - PLATE

PCMY Panel Collapse Membrane Yield (Von Mises) PYLS Panel Yield Longitudinal Strength PCAPS Panel Collapse Arched Plate Yield PCAPT Panel Collapse Arched Plate Shear PFLB Panel Failure. Local Buckling PCES Panel Collapse Edge Shear S-BCES SLS, Biaxial Compressive Edge Stress U-BCES ULS, Biaxial Compressive Edge Stress S-BCAES SLS, Biaxial Compression & Edge Shear U-BCAES ULS, Biaxial Compression & Edge Shear SYCF Stiffener Yield Compression Flange SYTF Stiffener Yield Tension Flange SYCP Stiffener Yield Compression Plate SYTP Stiffener Yield Tension Plate STBCL Stiffener Elastic Buckling

SLBSCW Stiffener Local Buckling Shear (Compr.Web)

U_CB ULS, Column Buckling U_BCB ULS, Beam Column Buckling U_TFB ULS, Torsional/Flexural Buckling FCPB Frame Collapse Plastic Bending FYCF Frame Yield Compression Flange FYTF Frame Yield Tension Flange FYCP Frame Yield Compression Plate FYTP Frame Yield Tension Plate FYSW Frame Yield Shear Web S_UCLLS SLS, Uniaxial Compressive Load LS S_BCSLLS SLS, Biaxial Compressive & Shear Load LS U_CBCLLS SLS, Combined Biaxial & Lateral Load LS

2.4 ULTIMATE STRENGTH MODULE (-3)

Since the ship's longitudinal ultimate strength might be viewed as it's most acceptable safety measure, prediction of the ultimate longitudinal bending moment becomes essential and unavoidable part of the ship structural concept design (Figs. 8, 9). Methods em-ployed should support multiple failure modes and their interactions, while giving precise prediction of collapse and post-collapse behavior of the structural members involved (particularly those under compression). On the other hand, multiple executions within design loop de-mand utilization of stable, robust and sufficiently fast algorithms. Considering the above stated demands, OCTOPUS was provided with two distinct methods for ultimate strength assessment, namely: modified Smith method, and modified Hughes/Adamchak method. In-corporated Smith method particularities include con-temporary advances which improve the accuracy during multi-deck ship application, as well as the ability to consider vertical shear force influence on the ultimate hull girder strength. Longitudinal ultimate strength is usually analyzed at the hull girder cross-section with maximum bending moment, where the shear force is nonexistent. Accounting for shear might be interesting when there is a cross-section along the hull girder with less then maximum value of bending moment, but with

significant value of the shear force (cases of alternate loading conditions). Hughes/Adamchak method has been modified to incorporate possibility of very exten-sive FEM analysis application which improved struc-tural member collapse predictions and thus improved accuracy of the basic method.

Figure 8: Collapse sequence for RoPaxin hogging (Hughes/Adamchak)

Influence of primary shear and transverse loading on overall longitudinal strength is also incorporated in the analysis, since OCTOPUS Analyzer program sup-plies the required structural response data (-1, 2).

Figure 9: Hull girder ultimate capacity (modified Smith)

2.5 RELIABILITY CALCULATION (-1) Reliability measure, as calculated at component level, is not appropriate for redundant multi-deck struc-tures. Therefore the system reliability measure is used in the optimization procedure. The system reliability measure is calculated using the -unzipping method. For this purpose, -module US3 (Zanic, Stipcevic, Hozmec, 2006) is developed inside the OCTOPUS software (see Fig. 10 and associated Table 4). The purpose of the -unzipping procedure (devel-oped by Thoft-Christensen and Sorensen) is the practi-cally applicable identification of the probabilistically dominant collapse scenarios. They are selected from the enormous set of potential collapse scenarios of redun-dant structures. Probabilistically dominant collapse scenarios are identified at the first, second, third and mechanism level. The system reliability measure at third level was found sufficient for the design purposes. Collapse scenario identification is based on arbi-trarily defined criteria, for each system level [min-lev1, min-lev1+ -lev1] ... [min-mech_lev, min-mech_lev+ -mech_lev]. System reliability measure is modeled as a serial system

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of all identified, probabilistically dominant collapse scenarios. Structural redundancy can be also assessed from the most dominant failure scenarios.

Table 4 Module description for Figure 10

OCTOPUS-US3s module description Module

Reliability measure calculation using the -unzipping procedure

UNZIPP

Statistical input (load and resistance variables, correlation matrix for loads and resistance variables)

STATInp

Automatic generation of potential failure element model. Automatic generation of potential collapse scenarios. Identification of probabilistically dominant collapse scenarios.

PFE-BISrch

Calculation of safety margin for potential failure elements

SafMar-CALC

Equivalent safety margin calculation for each collapse scenario.

EquivPSM

Reliability measure calculation for each identified collapse scenario. Murotsu bounds, Dunnet-Sobel method, Simple bounds)

PFParSys

System reliability measure calculation of the structure - modeled as serial system of identified, probabilistically dominant collapse scenarios.(Ditlevsen bounds, Dunnet-Sobel method, Simple bounds)

PFSerSys

2.6 SENSITIVITY CALCULATION ( -2) Correlation may play an important role in the reli-ability analysis (Zanic, Stipcevic, 2005) Basic sensitiv-ity-to-correlation matrices are used for the selection and simplification of the model definition (or in the failure mode analysis) for FORM / SORM analyses They are: B//R = [B ]- sensitivities of failure mode safety

indices i,km

i to elements of correlation matrix R [ ]km P//R = [P ] - sensitivities of modal failure prob-

abilities (Pi,km

iP )fi to elements of R, G//R = [G ] - sensitivities of bimodal correlation

coefficients ij,km

ij to elements of R, H//R = [H ] - sensitivities of joint failure prob-

abilities P (for modes i & j) ij,km

ij to elements of R, PPB//R= [PU,km] - sensitivities of failure probability

bounds (eg. Ditlevsen upper bound) to R, BG//R= [BG,km] - sensitivities of generalized safety

index ( )B1G P = to elements of R. The expressions for all important sensitivity matri-ces with respect to modified correlation matrix R (Na-taf model) are given in a very simple form and for e.g. the Ditlevsens upper bound sensitivity matrix and the safety index sensitivity matrix read:

=