Second-order Reliability Research Articles

Stochastic multi-scale finite element based reliability analysis for laminated composite structures

This paper proposes a novel multi-scale approach for the reliability analysis of composite structures that accounts for both microscopic and macroscopic uncertainties, such as constituent material properties and ply angle. The stochastic structural responses, which establish the relationship between structural responses and random variables, are achieved using a stochastic multi-scale finite element method, which integrates computational homogenisation with the stochastic finite element method. This is further combined with the first- and second-order reliability methods to create a unique reliability analysis framework. To assess this approach, the deterministic computational homogenisation method is combined with the Monte Carlo method as an alternative reliability method. Numerical examples are used to demonstrate the capability of the proposed method in measuring the safety of composite structures. The paper shows that it provides estimates very close to those from Monte Carlo method, but is significantly more efficient in terms of computational time. It is advocated that this new method can be a fundamental element in the development of stochastic multi-scale design methods for composite structures.

Moving least squares method for reliability assessment of rock tunnel excavation considering ground-support interaction

A practical approach is proposed in this paper for the reliability assessment of rock tunnel excavations using the moving least squares method (MLSM) and the uniform design. The failure probability is computed by the first-order and the second-order reliability method (FORM/SORM), which is based on the generated MLSM response surface (MLSM-RS) via an iterative algorithm. The proposed approach is first implemented in the analysis of a circular tunnel that consists of three limit state functions to illustrate the efficiency and accuracy of the approach. Then, the method is applied to a non-circular tunnel to demonstrate the feasibility and validity of the method for practical problems, in which numerical procedures are commonly employed to solve the implicit limit state functions.

Improve the Accuracy of Asymptotic Approximation in Reliability Problems Involving Multimodal Distributions

Asymptotic approximations such as the first- and second-order reliability method (FORM, SORM) are sometimes preferred in reliability analysis due to their efficiency. Bayesian updating for practical engineering problems can produce unnormalized and multimodal distributions, which cannot be directly used in FORM. The Laplace method can approximate such distributions using a weighted multivariate normal distribution. The weighting scheme for different modes must be carefully made in order to achieve accurate results. In this paper, a new weighting scheme and an optimal weighting scheme are proposed to improve the accuracy of Laplace approximation for multimodal distributions. The proposed weighting schemes are compared with existing weighting schemes in detail, and it is shown that the proposed weighting schemes can produce more accurate results in some cases. Based on that, an asymptotic approximation method for reliability estimation and system response predictions involving unnormalized multimodal distributions is proposed. The method integrates the Laplace method, FORM, and inverse FORM to formulate a completely asymptotic analysis cycle. The overall method is applied to two engineering cases: A simplified bridge model and a steam turbine rotor Bayesian analysis with real ultrasonic inspection data. Results are compared with existing simulation-based methods to investigate the efficiency and accuracy of the method.

Second-Order Fourth-Moment Method for Structural Reliability

AbstractIn the second-order reliability method (SORM), the failure probability is generally estimated based on parabolic approximation of the performance function. In the present paper, the first f...

A Novel Concept for Reliability Evaluation Using Multiple Deterministic Analyses

A novel reliability analysis concept for large Structural/Mechanical systems represented by finite elements using multiple deterministic analyses is presented in this paper. The intent is to extract reliability information by conducing only tens instead of millions of deterministic analyses as an alternative to the classical Monte Carlo Simulation. It is particularly applicable when a deterministic analysis requires a considerable amount of computational time to satisfy the underlying physics. It is developed by integrating the second-order reliability method and an improved response surface method by removing its deficiencies. The efficiency of the integrated scheme is further improved by using advanced statistical and factorial schemes producing compounding beneficial effect. The concept is elaborated using two different illustrative examples. To validate the procedure, the underlying reliabilities are estimated by using the basic Monte Carlo simulation technique to develop the reference or benchmark value. Then, the accuracy and efficiency of the method are compared and verified.

Incidence of residual stresses and steel properties variability on corroded bulk carrier reliability

Time-variant sagging/hogging hull girder bending capacity of a corroded bulk carrier is determined by Monte Carlo simulation, in terms of mean values, variation coefficients and probability density functions; the incidence of welding residual stresses and steel mechanical properties is purposely investigated. Time-variant hull girder reliability is characterized, for both sagging and hogging conditions, by the annual failure probability and is determined by Second-Order Reliability Method. Sensitivity coefficients of all random variables at discrete points in time are also determined. Four different scenarios are considered and analysed, in order to separately and jointly investigate the incidence of welding residual stresses and material properties variability on hull girder capacity and annual failure probability and compare individual results with the reference case, which is relative to no welding stresses and material properties nominal values. The Capesize single side bulk-carrier, benchmarked in the last ISSC Report, is assumed as test case. Finally, obtained results of time-variant hull girder statistical properties and annual failure probabilities are discussed.

Reliability Analysis With Monte Carlo Simulation and Dependent Kriging Predictions

Reliability analysis is time consuming, and high efficiency could be maintained through the integration of the Kriging method and Monte Carlo simulation (MCS). This Kriging-based MCS reduces the computational cost by building a surrogate model to replace the original limit-state function through MCS. The objective of this research is to further improve the efficiency of reliability analysis with a new strategy for building the surrogate model. The major approach used in this research is to refine (update) the surrogate model by accounting for the full information available from the Kriging method. The existing Kriging-based MCS uses only partial information. Higher efficiency is achieved by the following strategies: (1) a new formulation defined by the expectation of the probability of failure at all the MCS sample points, (2) the use of a new learning function to choose training points (TPs). The learning function accounts for dependencies between Kriging predictions at all the MCS samples, thereby resulting in more effective TPs, and (3) the employment of a new convergence criterion. The new method is suitable for highly nonlinear limit-state functions for which the traditional first- and second-order reliability methods (FORM and SORM) are not accurate. Its performance is compared with that of existing Kriging-based MCS method through five examples.

Reliable optimisation design of vehicle structure crashworthiness under multiple impact cases

ABSTRACTThis paper presents the crashworthiness design optimisation of a typical vehicle structure subjected to offset deformable barrier and full rigid barrier impact cases. Several approaches involving sampling techniques, surrogate model, multi-objective optimisation algorithm and reliability analysis are introduced and applied. Four correlation functions of Kriging are adopted for the design optimisation. It shows that the accuracy of the correlation functions is different and the best one is selected for the objective or constraint function. Both the deterministic optimisation and reliability-based design optimisation (RBDO) are conducted. In RBDO, the reliability levels of constraints are computed through the first-order reliability method (FORM) and second-order reliability method (SORM), and are compared with Monte Carlo simulation. It is found that SORM is more accurate than FORM when the constraint functions are highly nonlinear. Finally, the results demonstrate that RBDO solutions are more reliable than that of the deterministic optimisation for real engineering application. RBDO solutions are further validated by finite element analysis model, which shows the effectiveness of the proposed RRDO approach in obtaining the reliable optimal design of the vehicle structures.

Characteristic soil strength for axial pile capacity and its estimation with confidence for offshore applications

Many offshore design codes require the characteristic soil strength for axial pile capacity to be estimated with caution or conservatism, i.e. – in statistical terms – it has to be estimated with confidence. This study is concerned with a reliability-based calibration of the necessary minimum confidence for such estimation. The study demonstrates how to estimate characteristic soil strength for axial pile capacity with confidence and provides an approach to such estimation that will render an incentive to obtain more soil strength data and thereby give credit to the geotechnical designer who opts to test more. The study capitalizes on existing load and resistance models for an offshore example pile from the literature and refers to a number of relevant offshore foundation design codes. In order to accomplish the purpose, a number of prerequisites for a successful study are dealt with before the confidence calibration itself is presented.A suitable approach to represent model uncertainty associated with axial pile capacity predictions is presented and implemented in the reliability analyses that are used for the calibrations. The approach implies that the standard deviation of the ratio between true and predicted capacity is represented as a function of the pile length rather than as a constant. This approach is supported both by theory and by data.Based on stochastic models for load and resistance in conjunction with a method for prediction of axial pile capacity, a second-order reliability analysis of the example pile is carried out. With an assumption of perfect knowledge of the soil strength, with characteristic soil strength defined as the mean value and with prescribed load factors for permanent and environmental loads, the necessary requirement for the material factor on the characteristic soil strength is calibrated by tuning the reliability analysis to meet a prescribed target failure probability.With the calibrated material factor kept unchanged, the reliability analysis is repeated, now with the stochastic model for soil strength altered to include statistical uncertainty owing to limited soil data. The value of the characteristic soil strength is adjusted downward until the result of the reliability analysis again meets the prescribed target failure probability. The resulting value of the characteristic soil strength is interpreted as “the conservatively assessed value” to be used in design when soil data are limited and this value is subsequently used to find the corresponding minimum confidence for characteristic value estimation by capitalizing on the properties of the Student’s t distribution. Results are presented for target annual failure probabilities in the range 10−5–10−4.

Reliability analysis of underground excavation in elastic-strain-softening rock mass

This paper deals with the reliability analysis of a circular tunnel in elastic-strain-softening rock mass. Dilatancy angle which varies with softening parameter in different stress conditions is accounted for. Deterministic and probabilistic analyses of the circular tunnel in elastic-strain-softening rock mass are performed. Computational procedures for the first-order and second-order reliability methods (FORM/SORM) are used in the reliability analyses of the elastic-strain-softening model. The results are in good agreement with those from Monte Carlo simulations incorporating importance sampling. Reliability-based design of the required support pressure for the circular tunnel is efficiently conducted. The effect of positive correlation between compressive strength and elastic modulus of the rock mass on the reliability of the tunnel is discussed. The influence of in situ field stress and support pressure as random variables on the probability of failure of the tunnel is investigated.

Engineering criticality analysis on an offshore structure using the first- and second-order reliability method

Due to the uncertainties related to the flaw assessment parameters, such as flaw size, fracture toughness, loading spectrum and so on, the probability concept is preferred over deterministic one in flaw assessment. In this study, efforts have been made to develop the reliability based flaw assessment procedure which combines the flaw assessment procedure of BS7910 and first- and second-order reliability methods (FORM/SORM). Both crack length and depth of semi-elliptical surface crack at weld toe were handled as random variable whose probability distribution was defined as Gaussian with certain means and standard deviations. Then the limit state functions from static rupture and fatigue perspective were estimated using FORM and SORM in joint probability space of crack depth and length. The validity of predicted limit state functions were checked by comparing it with those obtained by Monte Carlo simulation. It was confirmed that the developed methodology worked perfectly in predicting the limit state functions without time-consuming Monte Carlo simulation.

Implementing statistical fitting and reliability analysis for geotechnical engineering problems in R

ABSTRACTReliability analysis and multivariate statistical fitting are valuable techniques that enhance the scientific basis of regulatory decisions in geotechnical problems. This study introduces the use of several R packages specifically developed to assist risk assessors in their geotechnical projects. Firstly, the fitting of parameterised models either to the distribution of observed samples or to characterise the dependence structures among variables, or both is presented. Secondly, the most popular reliability analysis methods, such as the first- and second-order reliability methods and the random sampling simulation method, are implemented in R. The efficiency of implementing these classical approximation methods is demonstrated through two example problems.

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

Time-variant reliability analysis of a corroded bulk carrier in intact and damage conditions is performed by First-Order (FORM), Second-Order (SORM) Reliability Methods and Importance Sampling simulation. Annual failure probabilities are determined up to 25-year ship lifetime, accounting for time-variant corrosion wastage of structural members contributing to hull girder strength. Statistical properties of hull girder capacity are determined by Monte Carlo simulation, applying three correlation models among corrosion wastages of structural members contributing to hull girder strength, namely no correlation, full correlation and full correlation among wastages of structural members belonging to the same category of compartments. A modified incremental-iterative method is applied, to account for instantaneous neutral axis rotation, in case of asymmetrical damage conditions, as for collision and grounding events. Incidence of intact/damage condition, as well as correlation among corrosion wastages, on annual sagging/hogging time-variant failure probability is investigated and discussed. Time-variant sensitivity analyses for intact and damage conditions are also performed, to investigate the incidence of random variables' uncertainties on the attained failure probability. Finally, the bulk carrier section scheme, benchmarked in the last ISSC Report, is applied as test case.

A novel reliability evaluation method for large engineering systems

Abstract A novel reliability evaluation method for large nonlinear engineering systems excited by dynamic loading applied in time domain is presented. For this class of problems, the performance functions are expected to be function of time and implicit in nature. Available first- or second-order reliability method (FORM/SORM) will be challenging to estimate reliability of such systems. Because of its inefficiency, the classical Monte Carlo simulation (MCS) method also cannot be used for large nonlinear dynamic systems. In the proposed approach, only tens instead of hundreds or thousands of deterministic evaluations at intelligently selected points are used to extract the reliability information. A hybrid approach, consisting of the stochastic finite element method (SFEM) developed by the author and his research team using FORM, response surface method (RSM), an interpolation scheme, and advanced factorial schemes, is proposed. The method is clarified with the help of several numerical examples.

A Probabilistic Physics of Failure Approach for Structure Corrosion Reliability Analysis

Corrosion is recognized as one of the most important degradation mechanisms that affect the long-term reliability and integrity of metallic structures. Studying the structural reliability with pitting corrosion damage is useful for risk control and safety operation for the corroded structure. This paper proposed a structure corrosion reliability analysis approach based on the physics-based failure model of pitting corrosion, where the states of pitting growth, pit-to-crack, and cracking propagation are included in failure model. Then different probabilistic analysis methods such as Monte-Carlo Simulation (MCS), First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM), and response surface method are employed to calculate the reliability. At last, an example is presented to demonstrate the capability of the proposed structural reliability model and calculating methods for structural corrosion failure analysis.

Development of a nanosatellite de-orbiting system by reliability based design optimization

This paper presents design approaches to develop a reliable and efficient de-orbiting system for the 3USAT nanosatellite to provide a beneficial orbital decay process at the end of a mission. A de-orbiting system is initially designed by employing the aerodynamic drag augmentation principle where the structural constraints of the overall satellite system and the aerodynamic forces are taken into account. Next, an alternative de-orbiting system is designed with new considerations and further optimized using deterministic and reliability based design techniques. For the multi-objective design, the objectives are chosen to maximize the aerodynamic drag force through the maximization of the Kapton surface area while minimizing the de-orbiting system mass. The constraints are related in a deterministic manner to the required deployment force, the height of the solar panel hole and the deployment angle. The length and the number of layers of the deployable Kapton structure are used as optimization variables. In the second stage of this study, uncertainties related to both manufacturing and operating conditions of the deployable structure in space environment are considered. These uncertainties are then incorporated into the design process by using different probabilistic approaches such as Monte Carlo Simulation, the First-Order Reliability Method and the Second-Order Reliability Method. The reliability based design optimization seeks optimal solutions using the former design objectives and constraints with the inclusion of a reliability index. Finally, the de-orbiting system design alternatives generated by different approaches are investigated and the reliability based optimum design is found to yield the best solution since it significantly improves both system reliability and performance requirements.

On Some Methods in Safety Evaluation in Geotechnics

Abstract The paper demonstrates how the reliability methods can be utilised in order to evaluate safety in geotechnics. Special attention is paid to the so-called reliability based design that can play a useful and complementary role to Eurocode 7. In the first part, a brief review of first- and second-order reliability methods is given. Next, two examples of reliability-based design are demonstrated. The first one is focussed on bearing capacity calculation and is dedicated to comparison with EC7 requirements. The second one analyses a rigid pile subjected to lateral load and is oriented towards working stress design method. In the second part, applications of random field to safety evaluations in geotechnics are addressed. After a short review of the theory a Random Finite Element algorithm to reliability based design of shallow strip foundation is given. Finally, two illustrative examples for cohesive and cohesionless soils are demonstrated.

Reliability analysis of a corrugated I-beam girder with ribs

The main aim of this paper is to provide a reliability analysis of a corrugated-web I girder, including the vertical ribs distributed periodically throughout its length. The generalised stochastic perturbation method and its computer implementation – the stochastic finite-element method – are both used to accomplish this goal. They are numerically implemented using the finite-element method system Abaqus with its rectangular, four-noded, structural, thin-shell finite elements and computer algebra system Maple, in which all the probabilistic procedures are programmed. The perturbation-based results are compared with those from traditional Monte Carlo simulation and, separately, an analytical solution is obtained by way of symbolic integration, carried out also in Maple. The plate girder thickness is introduced as the basic input Gaussian random variable in this study, as the most influential geometrical parameter in this structure. The reliability indices are determined according to both first- and second-order reliability methods, and correspond to the serviceability limit state of the structure. The first four basic probabilistic characteristics of the maximum deflection are also computed – expectations, coefficients of variation, skewness and kurtosis. The results obtained here and the entire methodology may be extended further towards computational modelling of the corrosion process, which is frequently mathematically modelled in steel structures using certain time series with random coefficients.

Reliability-based design and its complementary role to Eurocode 7 design approach

Reliability-based design (RBD) can play a useful complementary role to overcome some limitations in the Eurocode 7 (EC7) design approach, for example in situations with parameters not covered in EC7, different parametric sensitivities across different problems, cross-correlated or spatially correlated parameters, design aiming at a target reliability or failure probability, or when uncertainty in unit weight of soil is modeled. The complementary role played by RBD under these circumstances is illustrated and discussed for a shallow foundation, a reinforced rock slope, a Norwegian clay slope with spatial variability, a laterally loaded pile requiring implicit numerical analysis, and an anchored sheet pile wall. A pragmatic RBD approach involving first-order reliability method (FORM) only and a more rigorous RBD approach involving both first-order and second-order reliability method (SORM) are offered. Both approaches are implementable using either spreadsheet-based FORM and SORM procedures, or using various commercially available FORM/SORM packages.

Kriging response surface reliability analysis of a ship-stiffened plate with initial imperfections

The use of structural reliability methods with implicit limit state functions (LSFs) shows the increasing demand for efficient stochastic analysis tools, because the structural behaviour predictions are often obtained by finite element analysis. All stochastic mechanics problems can be solved by Monte Carlo simulation method, nevertheless, in most cases, at a prohibitively high computational cost. Several approximations can be achieved using first-order reliability method (FORM) and second-order reliability method and response surface methods. In this paper, a method that combines the FORM and Kriging interpolation models, as response surface, is proposed. The prediction accuracy of the Kriging response surface obtained from different sampling techniques is assessed, and the failure probability estimates calculated by the FORM using the classical second-order polynomial regression models and the Kriging interpolation models as surrogates of nonlinear LSFs are compared. The usefulness and efficiency of the reliability analysis using the Kriging response surface are demonstrated on the basis of existing results available in the literature and with an application problem of a stiffened plate structure with initial imperfections.