## 3 Environmental Contour Method  
The main idea behind design contours is to identify the most relevant environmental conditions corresponding to a specific return period (or equivalently specified in terms of a probability of exceedance). Subsequently, response analyses are performed for a subset of these conditions. This saves computation time as compared to a more systematic and more accurate long-term response analysis for which the whole range of possible environmental conditions must be considered. In order to perform such a full long-term analysis, the following two-step procedure is relevant:  
1. The ULS or ALS criterion is first expressed on the following form:  
\[G(y_{c},\mathbf{S})=y_{c}-Y(\mathbf{S}) \tag{3}\]  
Here, \(G(\cdot)\) designates the mechanical failure function; the \(n\)-dimensional vector \(\mathbf{S}=(S_{1},S_{2},\ldots,S_{n})^{\mathrm{T}}\) denotes the basic design variables for which the joint probability density function (PDF) is assumed to be known, which is denoted by \(f_{\mathbf{S}}(\mathbf{s})\). \(Y(\mathbf{S})\) is the internal load effect in the structure. The quantity \(y_{c}\) is the corresponding design capacity (which can be a function of several deterministic parameters and/or additional random variables related to the structural properties). Typically, the load effect and structural capacity is formulated in terms of stresses, forces, and bending moments. As an example in relation to the ULS, \(y_{c}\) may, e.g., correspond to the design value of the yield stress of the steel for the ship hull and \(Y(\mathbf{S})\) will then represent the maximum stress load effect in the hull (as a function of the values of the basic design variables). However, since this corresponds to a very strict design criterion, it is more relevant instead to take \(y_{c}\) to designate a maximum permissible plastic strain limit (which for a particular hull member can be translated into a maximum plastic deformation limit). The quantity \(Y(\mathbf{S})\) then designates the maximum strain load effect in the hull. Similarly for the ALS, \(y_{c}\) may refer to a critical plastic strain limit which generally is higher than for the ULS. Depending on the accidental scenario, it could even correspond to the fracture strain of the hull material or a certain maximum allowable fracture damage. However, in the following, a formulation is also provided where both the load effect and capacity are defined in terms of energy levels, i.e., by applying the absorbed impact energy as the load effect and the critical impact energy (corresponding to a given failure mode) as the capacity. This is in accordance with the impact analysis approach outlined in the previous section. 2. The failure probability \(p_{f}\) of the structure corresponding to a given time in operation can next be calculated as:  
\[p_{f}(y_{c})=\int_{G(y_{c},\mathbf{S})\leq 0}f_{\mathbf{S}}(\mathbf{s})ds \tag{4}\]  
Basically, this integral expresses the probability content that corresponds to the \"volume\" of the failure domain in the space which is spanned by the relevant random variables. The boundary of the failure domain is defined as the surface that is obtained by setting the mechanical limit state function in Equation (3) equal to zero. The integral in Equation (4) can be only be expressed in closed form for cases where the joint PDF and the limit state function \(G(y_{c},\mathbf{S})\) are given. However, this is quite rare in practice since the load effect \(Y(\mathbf{S})\) for a given value of the environmental parameter vector \(\mathbf{S}\) usually needs to be obtained by means of numerical simulation and/or by experiments. Evaluated of the integral can also be made by application of the Monte Carlo simulation technique (MCS) or by other reliability methods such as the first order or second order reliability methods (FORM/SORM) [19].