A Study on Computational Efficiency Improvement of Novel SORM Using the Convolution Integration

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.

Methodologies to calculate failure probability and to estimate the reliability of fatigue loaded structures are developed. The applicability of the methodologies is evaluated with the help of the fatigue crack growth models suggested by Paris and Walker. The probability theories such as the FORM (first order reliability method), the SORM (second order reliability method) and the MCS (Monte Carlo simulation) are utilized. It is found that the failure probability decreases with the increase of the design fatigue life and the applied minimum stress, the decrease of the initial edge crack size, the applied maximum stress and the slope of Paris equation. Furthermore, according to the sensitivity analysis of random variables, the slope of Pairs equation affects the failure probability dominantly among other random variables in the Paris and the Walker models.

Maintenance, rehabilitation and replacement of existing bridges by using cost effective management are increasingly important topics in efforts to deal with aging and deteriorating infrastructure. Recently, engineers have shifted the emphasis from design and construction of new bridges to maintenance and rehabilitation of existing bridges. This paper uses the structural reliability method with a predefined performance function to assess the component performance of the main load carrying members of an inner span in a multi-span bridge. The performance of the components is a function of various random variables with certain probabilistic distributions. The analysis is extended using the same performance function to evaluate the reliability of the bay as subsystem and bridge as a system. System reliability in this case is defined as the probability that a system (bridge) performs its intended function under operating conditions for a specified period of time using second order reliability method (SORM) and Monte Carlo simulations. In addition, deterioration in the main carrying elements (components) due to environmental impacts is introduced and the sensitivity parameters for certain random variables are re-evaluated to determine the inspection and maintenance period of the bridge. The deterioration is imposed as a percentage reduction in certain random variables. The paper involves use of multi reliability analysis at component, sub-system and system levels. Results show that probability of failure (Pf) increases with decrease in distance of the bridge from source of deterioration. Reliability index (β) decreased with increase in the age of the bridge because of corrosion of reinforcing steel. Results also show that the rate of probability of failure for bridges exposed to deteriorating effects increased dramatically after 45-50 years.

Reliability analysis of smart laminated composite plates under static loads using artificial neural networks

SummaryIn this study, a post optimization technique for a correction of inaccurate optimum obtained using first‐order reliability method (FORM) is proposed for accurate reliability‐based design optimization (RBDO). In the proposed method, RBDO using FORM is first performed, and then the proposed second‐order reliability method (SORM) is performed at the optimum obtained using FORM for more accurate reliability assessment and its sensitivity analysis. In the proposed SORM, the Hessian of a performance function is approximated by reusing derivatives information accumulated during previous RBDO iterations using FORM, indicating that additional functional evaluations are not required in the proposed SORM. The proposed SORM calculates a probability of failure and its first‐order and second‐order stochastic sensitivity by applying the importance sampling to a complete second‐order Taylor series of the performance function. The proposed post optimization constructs a second‐order Taylor expansion of the probability of failure using results of the proposed SORM. Because the constructed Taylor expansion is based on the reliability method more accurate than FORM, the corrected optimum using this Taylor expansion can satisfy the target reliability more accurately. In this way, the proposed method simultaneously achieves both efficiency of FORM and accuracy of SORM. Copyright © 2015 John Wiley & Sons, Ltd.

This study presents a parallelized framework for the grid-and-radius-based Morgenstern–Price method to enable an efficient reliability analysis of unsaturated finite soil slopes under correlated and non-Gaussian conditions. To address the limitations of deterministic techniques and the significant computational demands associated with traditional Monte Carlo simulation, the framework incorporates surrogate modeling approaches, namely polynomial chaos expansion–enhanced Monte Carlo simulation (PCE-MCS) and Kriging-based Monte Carlo simulation (K-MCS). This research conducts a comprehensive evaluation of these surrogate-assisted methods with the gradient-based approaches for estimating the probability of failure (pf) under correlated and non-Gaussian distributions. The surrogate models exhibit exceptional accuracy (R2 > 99.7%, mean absolute error <0.6%) and drastically reduce computational cost compared with traditional methods. Parametric and variability studies show that increasing the mean stability number (0.005–0.065) and friction angle (20°–30°) reduces the pf from 19.19% to 0.005% and 14.9% to 0.01%, respectively, while higher variability increases pf. Similarly, increasing the air-entry value reduces pf from 29.8% to 0.01%, whereas pf rises to 15.1% and 21.8% with increases in soil–water characteristic curve parameters nf (0.1–3.25) and mf (0.25–1.65). A detailed comparison against gradient-based techniques such as the first-order reliability method (FORM) and the second-order reliability method (SORM) underscores the superiority of surrogate-based approaches. For instance, the FORM shows errors up to 92.59%, while SORM errors reach 62.75%, particularly in highly nonlinear or variable scenarios. In contrast, PCE-MCS and K-MCS errors remain generally below 10%. Furthermore, reliability-based design contour charts are developed using K-MCS. These charts align with the USACE reliability classification, enabling rapid and reliable assessment of unsaturated slope stability without the need for large-scale simulations. Therefore, by effectively bridging traditional geotechnical engineering and advanced surrogate techniques, the proposed approach enhances practical decision-making in unsaturated slope design and supports the advancement of reliability-based design optimization in geotechnical practice.

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.

In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n →∞ is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.

In this work, a method for the application of multi-fidelity modeling to the reliability analysis of 2D elastostatic structures using the boundary element method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a center circular hole subjected to uniaxial tension using Monte Carlo simulations (MCS), the first-order reliability method (FORM), and the second-order reliability method (SORM). Two BEM models were investigated, a low-fidelity model (LFM) of 20 elements and a high-fidelity model (HFM) of 100 elements. The response of these models at several design points was used to create multi-fidelity models (MFMs) utilizing second-order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were significantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM.

An asymptotic approximation is developed for evaluating the probability integrals that arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on second-order reliability methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte Carlo simulation. Simple structures subjected to white noise stochastic excitation are used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method.

Time-variant bulk carrier reliability analysis in pure bending intact and damage conditions

The efficiency of the first-order reliability method (FORM) and the accuracy of Monte Carlo simulations (MCS) are coupled in probability-based designs of reinforced rock slopes, including a Hong Kong slope with exfoliation joints. Load–resistance duality is demonstrated and resolved automatically in a foundation on rock with a discontinuity plane. Other examples include the lengthy Hoek and Bray deterministic vectorial procedure for comprehensive pentahedral blocks with external load and bolt force, which is made efficient and more succinct before extending it to probability-based design via MCS-enhanced FORM. The FORM–MCS–FORM design procedure is proposed for cases with multiple failure modes. For cases with a dominant single failure mode, the time-saving importance sampling (IS) and the fast second-order reliability method (SORM) can be used in lieu of MCS. Two cases of 3D reinforced blocks (pentahedral and tetrahedral, respectively) with the possibility of multiple sliding modes are investigated. In the case of the reinforced pentahedral block, direct MCS shows that there is only one dominant failure mode, for which the efficient method of importance sampling at the FORM design point provides fast verification of the revised design. In the case of the reinforced tetrahedral block, there are multiple failure modes contributing to the total failure probability, for which the proposed MCS-enhanced FORM procedure is demonstrated to be essential. Comparisons are made between Excel MCS and MATLAB MCS.

Probabilistic analysis of corroded pipeline under localized corrosion defects based on the intelligent inspection tool

Flexible operation of coal-fired power plants contributes to the intensification of the life consumption processes, which is a serious problem especially in the case of units with a long in-service time. In steam turbine rotors, the crack propagation rate and material wear caused by low-cycle fatigue increase. The aim of the research is an attempt to forecast the development of these processes and to estimate the probability of critical elements damage, such as the high-pressure and intermediate-pressure rotors. In the stress state analyses, the finite element method (FEM) is used, the Monte Carlo method and the second order reliability method (SORM) is apply to calculate the probability of failure. It is proposed to use risk analysis to plan preventive maintenance of the turbine. The optimal intervals for carrying out diagnostic tests and prophylactic repairs is determined for various operating scenarios and various failure scenarios. This enables a reduction of the costs while ensuring the safety of the turbine's operation.

A Critical Study on the Haldar and Mahadevan's Reliability Analysis