This article provides two unique methodologies that may be coupled to study the dependability of multidimensional nonlinear dynamic systems. First, the structural reliability approach is well suited for multidimensional environmental and structural reactions and is either measured or numerically simulated over sufficient time, yielding lengthy ergodic time series. Second, a unique approach to predicting extreme values has technical and environmental implications. In the event of measurable environmental loads, it is also feasible to calculate the probability of system failure, as shown in this research. In addition, traditional probability approaches for time series cannot cope effectively with the system's high dimensionality and cross-correlation across dimensions. It is common knowledge that wind speeds represent a complex, nonlinear, multidimensional, and cross-correlated dynamic environmental system that is always difficult to analyze. Additionally, global warming is a significant element influencing ocean waves throughout time. This section aims to demonstrate the efficacy of the previously mentioned technique by applying a novel method to the Norwegian offshore data set for the greatest daily wind cast speeds in the vicinity of the Landvik wind station. This study aims to evaluate the state-of-the-art approach for extracting essential information about the extreme reaction from observed time histories. The approach provided in this research enables the simple and efficient prediction of failure probability for the whole nonlinear multidimensional dynamic system.

This paper demonstrates the validity of the Naess–Gadai method for extrapolating extreme value statistics of second-order Volterra series processes through application on a representative model of a deep water small size tension leg platform (TLP), with specific focus on wave sum frequency effects affecting restrained modes: heave, roll and pitch. The wave loading was estimated from a second order diffraction code WAMIT, and the stochastic TLP structural response in a random sea state was calculated exactly using Volterra series representation of the TLP corner vertical displacement, chosen as a response process. Although the wave loading was assumed to be a second order (non-linear) process, the dynamic system was modelled as a linear damped mass-spring system. Next, the mean up-crossing rate based extrapolation method (Naess–Gaidai method) was applied to calculate response levels at low probability levels. Since exact solution was available via Volterra series representation, both predictions were compared in this study, namely the exact Volterra and the approximate one. The latter gave a consistent way to estimate efficiency and accuracy of Naess–Gaidai extrapolation method. Therefore the main goal of this study was to validate Naess–Gaidai extrapolation method by available analytical-based exact solution. Moreover, this paper highlights limitations of mean up-crossing rate based extrapolation methods for the case of narrow band effects, such as clustering, typically included in the springing type of response.