A Novel Second-Order Reliability Method (SORM) Using Non-Central or Generalized Chi-Squared Distributions

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

This paper proposes to apply the convolution integral method to the novel second-order reliability method (SORM) to further improve its computational efficiency. The novel SORM showed better accuracy in estimating the probability of failure than conventional SORMs by utilizing a linear combination of noncentral or general chi-squared random variables. However, the novel SORM requires significant computational time when integrating the linear combination to calculate the probability of failure. In particular, when the dimension of performance functions is higher than three, the computational time for full integration increases exponentially. To reduce this computational burden for the novel SORM, we propose to obtain the distribution of the linear combination using the convolution and to use the distribution for the probability of failure estimation. Since it converts an N-dimensional full integration into one-dimensional integration, the proposed method is computationally very efficient. Numerical study illustrates that the accuracy of the proposed method is almost the same as the full integral method and Monte Carlo simulation (MCS) with much improved efficiency.

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.

In recent years, composite structures have been used in a large number of applications in aerospace, machinery, marine, and civil engineering. However, there are inevitably many uncertainties in the whole life cycle of composite structures, which can easily lead to structural damage and failure. Therefore, it is important to analyze the reliability and sensitivity of composite structures. At present, most of the contributions use the first-order reliability method (FORM) and the second-order reliability method (SORM) to study the reliability of composite structures and compare them with the results of the Monte Carlo simulation. However, both methods have their limitations. FORM cannot guarantee the calculation accuracy for the highly nonlinear limit state equation, and the calculation efficiency of SORM is too low. Therefore, this paper proposes to use importance sampling (IS) and backpropagation neural network-based Monte Carlo (MC-BPNN) to study the reliability, sensitivity, and dispersion of delamination growth of composite laminates. The results show that compared with FORM and SORM, IS and MC-BPNN have higher calculation accuracy and efficiency and can more accurately evaluate the failure degree of composite structures and ensure their safe operation in the field of aerospace equipment. The universality of this method can make it being widely used in the reliability and sensitivity analysis of different composite materials as well as dispersion analysis.

An efficient quasi-Newton approximation-based SORM to estimate the reliability of geotechnical problems

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.

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.

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.

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.

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

An articulated tower is one of the compliant offshore structures connected to the sea bed through a universal joint. In the random sea environment, this joint is subjected to reversal of shear stresses, which makes it susceptible to fatigue damage. In this study, fatigue damage accumulation in articulated joint under random loading is studied. The dynamic analysis of the tower has been carried out for twelve simulated sea states under random wave alone and random wave with wind. Nonlinearities due variable submergence, instantaneous position of the tower and hydrodynamic loading have been taken into account in the derivation of equation of motion. Fatigue life of the joint has been determined by S-N curve and fracture mechanics (F-M) approaches. Advanced First Order Reliability Method (FORM) and Monte Carlo Simulation method have been used for the reliability estimation. The results of the analyses are presented in terms of probability of failure and reliability indices. Sensitivity analysis is carried out to study the effect and participation of various random variables on the joint reliability. Most probable point (MPP) or design points have been located on the failure surface. Important parametric studies have been carried out which yield important information for reliability based design. The results of the study indicate that Miner’s rule, which is generally used in the design against fatigue in steel structures, yields a conservative estimate of probability of failure as compared to the fracture mechanics approach.

A method is presented to efficiently approximate the failure probability of structures subjected to time-variant loads, where the system of loads and structure may have uncertain parameters. The method uses response surface methodology in conjunction with the fast integration technique suggested by Wen and Chen, to provide a limit-state formulation that is computationally simple to solve based on a small number of response time histories. The system reliability may then be quickly computed by first-order reliability method (FORM)/ second-order reliability method (SORM) or Monte Carlo simulation. Sensitivity analysis is performed to determine the effect on the failure probability of changes to the system parameters, which can be important when determining whether uncertainty in a given system parameter is significant. An empirical measure of the accuracy of the response surface approximation is presented.

A probabilistic framework for durability assessment of concrete structures in marine environments was proposed in terms of reliability and sensitivity analysis, which takes into account the uncertainties under the environmental, material, structural and executional conditions. A time-dependent probabilistic model of chloride ingress was established first to consider the variations in various governing parameters, such as the chloride concentration, chloride diffusion coefficient, and age factor. Then the Nataf transformation was adopted to transform the non-normal random variables from the original physical space into the independent standard Normal space. After that the durability limit state function and its gradient vector with respect to the original physical parameters were derived analytically, based on which the first-order reliability method was adopted to analyze the time-dependent reliability and parametric sensitivity of concrete structures in marine environments. The accuracy of the proposed method was verified by comparing with the second-order reliability method and the Monte Carlo simulation. Finally, the influences of environmental conditions, material properties, structural parameters and execution conditions on the time-dependent reliability of concrete structures in marine environments were also investigated. The proposed probabilistic framework can be implemented in the decision-making algorithm for the maintenance and repair of deteriorating concrete structures in marine environments.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis