The design of any ship structural system or element must provide for adequate safety and proper functioning of that system or element regardless of what philosophy of design is used. The structural systems or elements must have adequate strength to permit proper functioning during their intended service life. The performance of a hull structural element as presented in this paper is defined by a set of requirements stated in terms of tests and measurements of how well the hull girder serves various or intended functions over its service life. Reliability and risk measures can be considered as performance measures, specified as target reliability levels (or target reliability indices, b0’s). The selected reliability levels of a particular structural element reflect the probability of failure of that element. These levels can be set based on implied levels in the currently used design practice with some calibration, or based on cost-benefit analysis. The following three methods can be used to select a target reliability value: (1) agreeing upon a reasonable value in cases of novel structure without prior history, (2) calibrating reliability levels implied in currently used design codes, and (3) choosing a target reliability level that minimizes total expected costs over the service life of the marine structure when dealing with design for which failures result in only economic losses and consequences.
The reliability-based design approaches for a system start with the definition of a mission and an environment for a ship. Then, the general dimensions and arrangements, structural member sizes, scantlings, and details need to be estimated or assumed. The weight of the structure can then be estimated to ensure its conformance to a specified limit. Using an assumed operational-sea profile, the analysis of the ship produces stochastic stillwater and wave-induced responses. The resulting responses can be adjusted using modeling uncertainty estimates that are based on any available results of full-scale or large-scale testing.
The reliability-based design procedure also requires defining performance functions that correspond to limit states for significant failure modes. In general, the problem can be considered as one of supply and demand. Failure of a structural element occurs when the supply (i.e., strength of the element) is less than the demand (i.e., loading on the element). On the other hand, the reliability of this element is achieved when the supply is greater than the demand.
Direct Reliability-Based Design
The direct reliability-based design method uses all available information about the basic variables (including correlation) and does not simplify the limit state in any manner. It requires performing spectral analysis and extreme analysis of the loads. In addition, linear or nonlinear structural analysis can be used to develop a stress frequency distribution. Then, stochastic load combinations can be performed. Linear or nonlinear structural analysis can then be used to obtain deformation and stress values. Serviceability and strength failure modes need to be considered at different levels of the ship, i.e., hull girder, grillage, panel, plate and detail. The appropriate loads, strength variables, and failure definitions need to be selected for each failure mode. Using reliability assessment methods such as FORM, reliability indices b’s for all modes at all levels need to be computed and compared with target reliability indicess. The relationship between the reliability index b and the probability of failure is given by
Pf = 1 - F(b) (10)
where F(.) = cumulative probability distribution function of the standard normal distribution, and b = reliability (safety) index. It is to be noted that Eq. 10 assumes all the random variables in the limit state equation to have normal probability distribution and the performance function is linear. However, in practice, it is common to deal with nonlinear performance functions with a relatively small level of linearity. If this is the case, then the error in estimating the probability of failure Pf is very small, and thus for all practical purposes, Eq. 10 can be used to evaluate Pf with sufficient accuracy (Ayyub and McCuen 1997).
Load and Resistance Factor Design (LRFD)
The second approach (LRFD) of reliability-based design consists of the requirement that a factored (reduced) strength of a structural component is larger than a linear combination of factored (magnified) load effects as given by the following general format:
where f = strength factor, Rn = nominal (or design) strength, gi = load factor for the ith load component out of n components, and Lni = nominal (or design) value for the ith load component out of m components.
In this approach, load effects are increased, and strength is reduced, by multiplying the corresponding characteristic (nominal) values with factors, which are called strength (resistance) and load factors, respectively, or partial safety factors (PSF’s). The characteristic value of some quantity is the value that is used in current design practice, and it is usually equal to a certain percentile of the probability distribution of that quantity. The load and strength factors are different for each type of load and strength. Generally, the higher the uncertainty associated with a load, the higher the corresponding load factor; and the higher the uncertainty associated with strength, the lower the corresponding strength factor. These factors are determined probabilistically so that they correspond to a prescribed level of reliability or safety. It is also common to consider two classes of performance function that correspond to strength and serviceability requirements.
The difference between the allowable stress design (ASD) and the LRFD formats is that the latter uses different safety factors for each type of load and strength. This allows for taking into consideration uncertainties in load and strength, and to scale their characteristic values accordingly in the design equation (Assakkaf 1998). ASD (sometimes called working stress) formats cannot do that because they use only one safety factor as seen by the following general design format:
where R = strength or resistance, Li = load effect, and FS = factor of safety. In this design format, all loads are assumed to have average variability. The entire variability of the strength and the loads is placed on the strength side of the equation. The factor of safety FS accounts for this entire variability.
In the LRFD design format, ship designers can use the load and resistance factors in limit-state equations to account for uncertainties that might not be considered properly by deterministic methods (i.e., ADS) without explicitly performing probabilistic analysis. The LRFD format as described herein is concerned mainly with the structural design of ship hull girder structural components under combinations of different load effects. The intention herein is to provide naval architects and ship designers with reliability-based methods for their use in both early and final design stages and for checking the adequacy of the scantlings of all structural members contributing to the longitudinal and transverse strength of ships. Eq. 11 gives the general form of the LRFD format used in this paper.
The probabilistic characteristics and nominal values for the strength and load components can be determined based on statistical analysis, recommended values from other specifications, and by professional judgment. The LRFD general design formats for hull structural components are given by one of the following two main cases, limit sate 1, and limit sate 2, respectively:
Limit State 1:
Limit State 2:
where f = strength factor, Rn = nominal (or design) strength such as the ultimate stress, gSW = load factor for stillwater load effect such as bending moment, LSW = nominal (or design) value for stillwater load effect such as bending moment, kWD = combined wave-induced and dynamic bending moment factor, and gWD = load factor for combined wave-induced and dynamic bending moment, LWD = nominal (or design) value for wave-induced and dynamic bending moments effect, kW = load combination factor, gW = load factor for waves bending moment load effect, LW = nominal (or design) value for waves bending moment load effect, kD = load combination factor, gD = load factor for dynamic load effect such as bending moment, and LD = nominal (or design) value for dynamic load effect such as bending moment.
The strength and load factors are collectively called partial safety factors (PSF’s). These factors are determined using structural reliability methods based on the probabilistic characteristics of basic random variables for materials, geometry and loads including statistical and modeling (or prediction) uncertainties. The factors are determined to meet target reliability levels that were selected based on assessing previous designs. This process of developing LRFD guidelines to meet target reliability levels that are implicit in current practices is called code calibration.