Structural reliability analysis by univariate decomposition and numerical integration

A univariate approximation at most probable point for higher-order reliability analysis

Reducing the surrogate model-based method computation without loss of prediction accuracy remains a significant challenge in structural reliability analysis. The unbalanced probability density, important information in critical region and information redundancy of added sample points are ignored in most of traditional surrogate-based methods, resulting in heavy computational burden. In this work, an active learning combining adaptive Kriging method and weighted penalty (AK-WP) is proposed to analyze the reliability of engineering structures. Firstly, an active learning and weighted penalty function (WPLF) is the result of integrating active learning method, weighted function and penalty function, which is proposed to find the most probable point (MPP). Meanwhile, to avoid redundant information, the best suitable MPP is determined by a proposed distance law established between the found MPP and the existing design of experiment (DoE). Secondly, the Kriging model is refined according to best suitable MPP in each iteration. Thirdly, the failure probability is estimated by the Monte Carlo sample points from the n-ball domain until the convergence condition is satisfied. The accuracy and efficiency of the proposed method are demonstrated by some numerical examples including the highly nonlinear, the small probability problems and implicit function, as well as a real engineering application.

Structural Reliability Analysis (SRA) is a useful tool in structural engineering. Uncertainty in input parameters and model uncertainties in the analysis predictions are explicitly modelled by random variables. With this methodology, the uncertainties involved are handled in a consistent and transparent way. Compared to a deterministic analysis, SRA provides improved insight in how the various uncertainties involved influence the results. The main results from SRA is the calculated probability of structural failure, but other useful results such as uncertainty importance factors and design points being the most likely combination of all variables at failure represent helpful information. The present paper illustrates some the features using SRA for two different types of application. The first application is the use of SRA as a tool for code calibration and the second shows the application of SRA to a problem where common practice is likely to be rather conservative and therefore leading to unacceptable results, but where the degree of conservatism is not known. Two examples are chosen to illustrate code calibration; i.e. hull girder ultimate limit state (ULS) for tankers and ULS for mooring design in the ULS for floating offshore vessels. Code calibration involves both SRA and design analysis following the code. It is shown how the design analysis can be modified in order to better reflect a chosen target reliability level across a selected set of test cases representative for what the code should cover. Fatigue of subsea wellhead systems is selected as an example of a special case when application of existing rules may lead to unsatisfactory results which are likely to be rather conservative. It is shown how results can be presented in terms of the accumulated probability of fatigue failure as a function of time. This may be a more suitable basis for decision making than a calculated fatigue life from a standard analysis. It is also illustrated how importance factors from the SRA can be used as guidance on how to prioritize effort in order to improve prediction of the fatigue damage. The present paper is not intended to be detailed in all input and analysis methodology, but draw the attention towards the possibilities and benefits of applying SRA in structural engineering, where the examples are used to illustrate this potential.

Closure to “A new efficient simulation method to approximate the probability of failure and most probable point” (Struct. Safety 2012:39:22–9)

A new structural reliability analysis framework is developed to extend the performance of first-order reliability method, which is low robustness and poor accuracy when dealing with highly nonlinear functions. Initially, an improved conjugate sensitivity factor method is proposed to find the most probable point. The method enhances its robustness of convergence by introducing a conjugate gradient direction based sensitivity factor technique and improves its computational efficiency by putting forward a hybrid conjugate gradient factor and an adaptive step length strategy. Subsequently, the dimension reduction-based saddlepoint approximation method is developed, which uses the dimension reduction approach to construct the limit state function as the additive univariate quadratic functions and applies saddlepoint approximation to obtain the result with higher precision. A comparison analysis from five mathematical and structural examples illustrates that the proposed method is better than most existing methods in terms of robustness, efficiency and accuracy for estimating the failure probability of structures.

This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high‐dimensional model representation (HDMR) generated from low‐order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the lower‐order terms and facilitating lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function. When first‐order HDMR approximation of the original high‐dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of second‐ and higher‐order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high‐dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least‐squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/solid‐mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.

Structural reliability analysis (SRA) has been used to calculate the probability of mooring line failure in an intact mooring system as a function of the magnitude of the safety factor applied in design. A range of different units have been considered, comprising ship shaped units and semisubmersibles at different water depths from 100 m to 2200 m. Environmental conditions representative for the Norwegian continental shelf and the Gulf of Mexico are used in the analyses, and the characteristics of the results in the different environments are compared and discussed. Analyses for Brazilian environment are currently ongoing, but not included here. Time domain analysis is applied to obtain the short-term, extreme value distribution of line tension, conditional on stationary metocean conditions. A large number of different conditions are considered. A response surface is used to interpolate on the distribution parameters in order to describe the tension response in varying conditions. Joint probabilistic models of the metocean environment corresponding to the different geographical locations have been applied, taking account of wind, wave and current and their respective heading angles. A continuous model is used for the metocean conditions at the Norwegian continental shelf, whereas a hurricane model is used in the Gulf of Mexico. The effects of uncertainties in the response calculation are included. The mooring line component strength is based on strength data from break load tests. Conventional catenary chain-wire chain systems as well as polyester moorings are considered. With the probability of failure as a function of the safety factor, it is shown that present regulations result in a significant scatter in reliability level between the cases. Safety factors have been calibrated considering all cases. Alternative design formats are proposed and tested including a format with 2 safety factors. Calibration results are provided as a function of the target reliability level. The final recommendation on target reliability level is given in an accompanying paper at OMAE 2017, comprising both the ULS and the ALS. It is demonstrated that alternative design formats can provide a more consistent safety level across the cases. A different design philosophy is needed for the Gulf of Mexico in order to achieve acceptable risk. Options for design are discussed. The present work provides a unique and comprehensive set of results, where advanced reliability methods are used in combination with detailed response calculations in the time domain. The results provide a basis for calibration of mooring design for ULS and subsequently for regulators to update their rules. The work has been carried out as part of the NorMoor Joint Industry Project, with participants from oil companies, engineering companies, rig-owners, manufacturers and marine authorities. This paper is the first one in a series of three at OMAE 2017, where the second deals with structural reliability analysis of the ALS and the third one provides summary and recommendations for safe mooring design in ULS and ALS.

This paper presents the application of a reliability based design methodology for interference between trawl gear and pipelines, and the associated cost savings in subsea rock installation. As part of Wintershall Norge's Maria development project, a 26 km long 14″ production flowline with DEH (Direct Electrical Heating) and a 46 km (43 km + 3 km infield) long 12″ water injection pipeline has been installed and left exposed on the seabed. The seabed topography in the area is un-even and scarred by iceberg ploughmarks, leading to a significant number of large free spans. Furthermore, fishing activity must be taken into account in the design of these pipelines. According to DNV-RP-F111, if the trawl gear hits in a free span, the pull-over load will be significantly higher than if it hits a section in contact with the seabed. Preliminary analysis indicated that by using the LRFD (Load and Resistance Factor Design) approach as per DNV-RP-F111, a large number of free spans would require rock infill in order to limit the trawl pull-over load. In order to optimize the design and potentially reduce the requirement for free span infill, an optimised methodology based on SRA (Structural Reliability Analysis) was proposed by Wintershall, developed by DNV GL and implemented by Subsea 7 during detail design phase. The optimised methodology involves FE analyses of the sensitivity to various parameters and Monte Carlo simulations, in order to quantitatively assess the probability of failure. Specific FE models analysing the bending moment capacity and response were established. The various input parameters were assessed and included as distributions if deemed required from the sensitivity analysis. Finally, a Monte Carlo simulation was performed to calculate the probability of failure. It was demonstrated that the target safety levels defined by DNV-OS-F101 were reached without free span infill, and hence significant savings in subsea rock installation could be achieved without any deviation from the DNV-OS-F101 code. Cost saving due to reduced subsea rock installation scope is estimated to 7.6 mill Euro. Wintershall facilitated a close dialogue between DNV GL and Subsea 7 throughout the work progress. This resulted in a constructive and open approach, ensuring efficient delivery of the detailed design in line with project schedule. The work carried out has demonstrated two relevant examples where performing a SRA, as opposed to a LRFD, has resulted in a significant reduction of seabed intervention requirements and an associated cost saving, while still being in compliance with DNV-OS-F101 target safety levels. To our knowledge this is the first time the SRA method has been applied for trawl interference design on a live project, and on a CRA (Corrosion Resistant Alloy) lined pipeline with DEH system attached. The success was made possible by close collaboration between the parties involved; Wintershall, DNV GL and Subsea 7.

Influence of micro-scale uncertainties on the reliability of fibre-matrix composites

Efficient sampling methods for global reliability sensitivity analysis

This paper proposes a novel second-order reliability method (SORM) using non-central or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point (MPP) in the standard normal U-space, (2) error due to approximating the quadratic function in U-space by a hyperbolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two errors by describing the quadratic failure surface with the linear combination of non-central chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function (PDF) over the linear failure surface and the second approach uses the cumulative distribution function (CDF) of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method (FORM), conventional SORM, and Monte Carlo simulation (MCS) results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

The paper describes a quantitative model for risk-based integrity maintenance planning for offshore pipelines. The model calculates the risk level as the product of failure consequences and failure probabilities. Failure consequences considered include financial aspects relating to repair, interruption and restoration; safety aspects related to platform and vessel occupants; and environmental aspects associated with spills. Distinction is made between small leaks, large leaks and ruptures, and the major failure causes (namely, corrosion and mechanical damage) are considered. The model quantifies the benefits associated with different maintenance alternatives by calculating their impact on the probability of failure and risk level. A formal decision analysis approach is then used to evaluate the costs and benefits associated with each alternative, and identify the best overall maintenance strategy. This model differs from previous work in risk-based planning in that it:utilizes structural pipe failure models and reliability analysis techniques to calculate failure probabilities;quantifies the impact of maintenance on risk; andutilizes a methodology based on decision theory to identify optimal choices. It is applicable to various aspects of maintenance planning for offshore pipelines such as the selection of inspection tools, inspection intervals and defect repair criteria; and the optimization of damage prevention activities including mechanical protection, cover monitoring and public awareness programs. The paper will include the results of simulations dealing with maintenance optimization for corrosion and mechanical damage. These results demonstrate that risk-based planning is a powerful tool for minimizing risk by maintaining high reliability at the lowest possible cost. The most significant aspect of this work is that it provides a novel approach to managing pipeline integrity in a cost-effective manner. Introduction The goal of an effective pipeline integrity maintenance program is to ensure that the limited resources available are utilized most effectively in reducing the risks posed by pipeline failures to life, the environment and financial assets. The uncertainties associated with making such an assessment has led to a recognition of risk analysis as a sound basis for making decisions on integrity maintenance. Risk, in this context, is defined as the probability of line failure multiplied by a measure of the associated adverse consequences. The risks associated with a pipeline system and its individual segments, and the potential risk reductions resulting from different maintenance activities can be used as a basis for evaluating different alternatives and developing an optimal maintenance plan. This paper describes a comprehensive methodology for risk-based integrity maintenance optimization of offshore natural gas, crude oil and petroleum product pipelines. The methodology has been developed and implemented in a software package called PIRAMID?, (Pipeline Risk Analysis for Maintenance and Inspection Decisions) under a joint industry program sponsored by a number of transmission companies and government agencies (see Acknowledgements). The types of decisions addressed by the methodology include the choice of inspection methods (e.g., cover depth survey, in-line inspection) and inspection intervals, as well as the choice of maintenance actions (e.g., corrosion inhibition, hydro-test, coating reconditioning and replacement).

An extensive study of the flexural performance of monoblock prestressed concrete sleepers in a light rail system was conducted as part of a research program funded by the Federal Transit Administration. Five consecutive sleepers deployed on the track were instrumented with strain gauges at their critical design cross-sections (center and rail seats) to obtain relevant flexural information during an uninterrupted period of 14 months. Results were compared with the projected design capacities obtained from the application of current design standards, resulting in glaring differences. The current design methodologies were deemed insufficient for the development of optimal design solutions for light rail applications. Furthermore, structural reliability analysis is employed to study the flexural capacity of the sleeper design. A capacity model based on the material and geometric properties of the sleeper design was developed. The demand model was derived from the field flexural data of over 27,000 train passes, fitting this information to predefined probability distributions. Four limit-state functions were defined to represent the typical flexural failure modes. The probability of failure was calculated using first-order reliability method, second-order reliability method, and Monte Carlo simulation. Ultimately, the analysis yielded consistent results for the three methods, showing largely low probability of failure at both design cross-sections under the studied demand level. In conclusion, the sleeper's capacity was higher than the existing field demands, indicating an overly conservative design approach.

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

The system probability of failure calculation of the series system entails multi-dimensional integration, which is very difficult and numerically expensive. To resolve the computational burden, the narrow bound method, which accounts for the component failures and joint failures between two failure modes, has been widely used. For the analytic calculation of the component probability of failure, this paper proposes to use the most probable point (MPP)-based dimension reduction method (DRM). For the joint probability of failure calculation, three cases are considered based on the convexity or concavity of the performance functions. Design sensitivity analysis for the system reliability-based design optimization (RBDO), which is the major contribution of this paper, is carried out as well. Based on the results of numerical examples, the system probability of failure and its sensitivity calculation show very good agreement with the results obtained by Monte Carlo simulation (MCS) and the finite difference method (FDM).