# Risk-Based Hybrid Light-Weight Ship Structural Design Accounting for Carbon Footprint
## 4 Reliability Analysis  
The ships are exposed to different hazards during the service life, where the most critical experience includes extreme loading, capsizing, unacceptable seakeeping, structural failure, and others [32]. Different design solutions may initiate different types of structural collapse. The risk-based evaluation may convert this event into a measure where all consequences for the ships are assessed. The reliability analysis focuses on the progressive ship hull structural collapse and related quantitative consequences associated with the structural failure related to the ultimate compressive strength, progressive corrosion degradation, a possible loss of human life and cargo and environmental pollution, where the last one is seen to be very critical nowadays [32,33]. When a structure or a structural component fails to perform a function for which it was designed, this condition defines its limit state related to serviceability, ultimate, fatigue and accidental limit states [14,34,35]. Figure 2: Bending-curvature relationship. Three possible design solutions are used as a risk control measure to allow the ship hull structure to be designed and to verify the impact of a hybrid lightweight ship structural design solution and carbon footprint. The ultimate compressive strength of the ship hull structure is evaluated based on the ship's hull progressive structural collapse approach initially developed in [36] and widely used for ship hull structural analysis as a function of ship hull loading composed of the vertical still water and wave-induced bending moments. The progressive corrosion degradation magnifies structural failure. A non-linear model [37; 38; 31] is employed for the corrosion degradation assessment of the ship hull as a function of the coating and transition life, as given in Equation (1). The overall hull girder failure consequences are the possible losses, environmental pollution cleaning and related measures to recover losses. The failure consequences are also time-dependent due to the time value of money. The collapse failure is defined as stipulated in [39], where reliability is the probability of keeping the service capacity of the ship during the service life. The objective is to evaluate the reliability of the ship's degrading strength against the demanding loading. The limit state function \(g(\mathbf{X})\) is related to the ultimate load-carrying capacity of the ship structure, define as [40; 41]:  
\[g(\mathbf{X})=\widetilde{x}_{u}\cdot\widetilde{M}_{u}-\left(\widetilde{x}_{ sw}\cdot\widetilde{M}_{sw}+\Psi_{w}\widetilde{x}_{w}\cdot\widetilde{x}_{s} \cdot\widetilde{M}_{w}\right) \tag{2}\]  
where \(\widetilde{M}_{u}\) is the ultimate bending moment capacity, \(\widetilde{x}_{u}\), is the model uncertainty factor of the ultimate bending moment described by the Normal probability density function, \(\widetilde{M}_{w}\) is the vertical wave-induced bending moment. The combination factor between the still water and wave-induced bending moments, \(\Psi_{w}\) is ranging from 0.8 to 0.95, as described in [42; 43]. The model uncertainty factor \(\widetilde{x}_{w}\) accounts for the uncertainties in the linear response calculation and \(\widetilde{x}_{s}\) for the non-linear effects, \(\widetilde{M}_{sw}\) is the still water bending moment with a model uncertainty factor \(\widetilde{x}_{sw}\). The approach employed to define the statistical descriptors of the still water bending moment is given in [44; 45], where the still water bending moment is fitted to a Normal distribution. The regression equations define the mean value and standard deviation as a function of the length of the ship and dead-weight ratio concerning the loading cases as:  
\[Mean(M_{sw}) =\frac{Mean(M_{sw,max})M_{sw,CS}}{100} \tag{3}\] \[StDev(M_{sw}) =\frac{StDev(M_{sw,max})M_{sw,CS}}{100} \tag{4}\]  
where \(M_{sw,CS}\) is given as stipulated by the Classification Societies Rules [46]. The statistical descriptors of the extreme vertical wave-induced bending moment \(M_{ww,e}\) are defined as proposed in [40].