Reliability Analysis Procedure Research Articles

Hybrid reliability analysis and optimization for spacecraft structural system with random and fuzzy parameters

In aerospace engineering, the incomplete environment conditions have been realized to be a significant factor affecting the system safety assessment. Considering the hybrid random and fuzzy uncertainties in system inputs, the paper proposes a novel numerical procedure for reliability analysis and reliability-based optimization. Random parameters are adopted to denote the aleatory uncertainties with sufficient sample information; whereas fuzzy parameters are used to quantify the epistemic uncertainties associated with expert opinions. Using the level-cut operation, fuzzy parameters are converted into interval variables, and a satisfaction degree-based interval ranking strategy is utilized to precisely quantify the interval safety possibility. Then the system safety possibility is calculated by the multiple integral, where cut levels of different fuzzy parameters are treated as independent variables. Subsequently based on the given safety index, a hybrid reliability optimization model is established. To avoid the huge computational burden caused by nested-loop optimization, a modified interval Monte Carlo method (MIMC) is proposed for limit state function evaluation. Eventually, a numerical example about the refractory ceramics tile of spacecraft verifies the feasibility of proposed method for hybrid reliability analysis and optimization design in practical aerospace engineering.

An efficient-robust structural reliability method by adaptive finite-step length based on Armijo line search

The robustness of iterative formula as well as its computational efficiency is the essential characteristic of interest for effective reliability analysis of structures by first order reliability method (FORM). A robust and efficient iterative algorithm termed as finite-based Armijo search direction (FAL) method is proposed in the present study for FORM-based structural reliability analysis. A finite-step size is proposed using the Armijo rule and sufficient descent condition to achieve the stabilization of the FORM algorithm. The FAL is adaptively adjusted based on the information obtained from the iterative algorithm at each iteration and Armijo rule. The robustness and efficiency of the proposed FAL method is elucidated using several problems. The results obtained by the proposed method are compared with various existing reliability methods based on steepest descent search direction. The results of the numerical study indicate that the FAL approach is more robust and efficient than the other existing FORM schemes and improves the robustness of FORM formula. Thus, the FAL can be successfully implemented as a robust FORM-based iterative reliability analysis procedure.

Probabilistic safety evaluation of a river bridge substructure against floods

Many factors affect the safety of a reinforced concrete river bridge, and some of these factors might have significant uncertainty that should be evaluated. This study focuses on a system reliability analysis of a reinforced concrete bridge, which is the most common type of river bridge in Taiwan. Specifically, non-linear behaviours of core/cover concrete and steel reinforcement in the substructure of a bridge are considered. Uncertainties of stream surface elevation, water velocity, local scour depth, soil property and record-to-record variations are included in the probabilistic model. The aforementioned parameters of six river geometries are incorporated into a Hydrologic Engineering Center-river analysis system (HEC-RAS) model to assess their impacts on the bridge safety. The effect of river geometry has not been studied thoroughly. Results shown here indicate that it plays an important role. Adopting Bayesian theory to consider the record-to-record uncertainty by combining historic and HEC simulation data is another innovative aspect of this study. Monte Carlo simulation is used to conduct the reliability analysis. The results obtained indicate that the proposed procedure for the bridge reliability analysis is consistent with expert engineering judgement and that the variation in scouring depth and river geometry are pertinent to bridge safety.

Improving the first-order structural reliability estimation by Monte Carlo simulation

The first-order reliability method (FORM) is a well-known procedure for reliability analysis in the engineering community. It provides an efficient way to calculate the failure probability of a structure. However, it introduces an unknown error in the estimations for non-linear limit states. This paper seeks a correction for the FORM by employing Monte Carlo simulation. The proposed method combines the efficiency of FORM with the robustness of the Monte Carlo method. The merits of the proposed method are illustrated through benchmark examples. The results show the high accuracy of the proposed method and a substantial reduction in the number of samples compared with crude Monte Carlo.

Two accuracy measures of the Kriging model for structural reliability analysis

The Kriging model for structural reliability analysis applications has attracted much attention within recent years. Several Kriging-based strategies of design of experiments are constructed for structural reliability analysis procedure. However, the quantitative accuracy measure of Kriging is still undesirable. Two accuracy measures are introduced. The first one is further studied through derivation and proposed to quantify accuracy of the Kriging-based estimate of the limit state. This paper treats the target failure probability as a variable with epistemic randomness. The second measure is innovatively defined as the standard deviation of the target failure probability. Combining with Chebyshev's inequality, the second measure is available to construct a lower upper bound of the error of the failure-probability estimate. The joint distribution of performance values at untried points of a given Kriging model is derived and proved, which is indeed essential for computing the innovative accuracy measure because it takes the correlation between performance values of untried points into account. Monte Carlo simulation is employed to compute them with acceptable computational cost. To validate the accuracy measures, four benchmark examples are studied. Results demonstrate the availability of them.

A novel evidence theory model and combination rule for reliability estimation of structures

Due to the discontinuous nature of uncertainty quantification in conventional evidence theory(ET), the computational cost of reliability analysis based on ET model is very high. A novel ET model based on fuzzy distribution and the corresponding combination rule to synthesize the judgments of experts are put forward in this paper. The intersection and union of membership functions are defined as belief and plausible membership function respectively, and the Murfhy's average combination rule is adopted to combine the basic probability assignment for focal elements. Then the combined membership functions are transformed to the equivalent probability density function by a normalizing factor. Finally, a reliability analysis procedure for structures with the mixture of epistemic and aleatory uncertainties is presented, in which the equivalent normalization method is adopted to solve the upper and lower bound of reliability. The effectiveness of the procedure is demonstrated by a numerical example and an engineering example. The results also show that the reliability interval calculated by the suggested method is almost identical to that solved by conventional method. Moreover, the results indicate that the computational cost of the suggested procedure is much less than that of conventional method. The suggested ET model provides a new way to flexibly represent epistemic uncertainty, and provides an efficiency method to estimate the reliability of structures with the mixture of epistemic and aleatory uncertainties.

Structural time-dependent reliability assessment of the vibration active control system with unknown-but-bounded uncertainties

The active control system for structural vibration is extremely sensitive to the parametric uncertainty so that more and more concerns of its reliability estimation have been given recently. In view of the insufficiency of the uncertainty information in practical engineering, a non-probabilistic time-dependent reliability method that combines the active vibration control theory with interval analysis is proposed in this paper to effectively estimate the dynamic safety of the controlled structures, in which circumstances the unknown-but-bounded uncertainties in structural parameters are considered. The uncertain structural responses based on the closed-loop control are firstly analyzed and embodied by the interval process model. By virtue of the first-passage theory, an integral procedure of non-probabilistic time-dependent reliability analysis of the active control system for structural vibration is then conducted. Two engineering examples and one experimental application are eventually presented to demonstrate the validity and applicability of the methodology developed.

Non-probabilistic stability reliability measure for active vibration control system with interval parameters

A systematic non-probabilistic reliability analysis procedure for structural vibration active control system with unknown-but-bounded parameters is proposed. The state-space representation of active vibration control system with uncertain parameters is presented. Compared with the robust control theory, which is always over-conservative, the reliability-based analysis method is more suitable to deal with uncertain problem. Stability is the core of the closed-loop feedback control system design, so stability criterion is adopted to act as the limited state function for reliability analysis. The uncertain parameters without enough samples are modeled as interval variables. Interval perturbation method is employed to estimate the interval bounds of eigenvalues, which can be used to characterize the stability of the closed-loop active control system. Formulation of defining the reliability of active control system based on stability is discussed. A novel non-probabilistic reliability measurement index is discussed and used to determine the probability of the stability based on the area ratio. The feasibility and efficiency of the proposed method are demonstrated by two numerical examples.

Exploiting the benefits of multi-scale analysis in reliability analysis for composite structures

This paper investigates two critical issues, namely propagation of multi-scale uncertainty, and selection of failure criteria, related to reliability analysis of composites by using multi-scale methods. Due to the multi-scale architecture of composites, uncertainties exist in both microscale and macroscale parameters. It is necessary, therefore, to consider random variables at various length scales to ensure accurate estimates of the reliability of composites. Three types of homogenization methods, namely rule of mixtures, Mori–Tanaka and computational homogenization, are adopted to link these two scales, and to propagate uncertainty from micro to macro scales. By integrating these homogenization methods with the stochastic finite element method and structural reliability methods, the reliability of composites can be investigated with a limit state function based on a chosen failure criterion. This multi-scale reliability analysis procedure has been applied to analyse laminated fibre reinforced composites made of AS4/3501 carbon/epoxy. Firstly, a comparative study has been conducted to evaluate the performance of the assumed homogenization methods for the reliability of composites, and to identify advantages compared with a single scale analysis. The results show that multi-scale analysis can provide more accurate reliability estimates. Secondly, several popularly used failure criteria for composites have been compared using multi-scale reliability analysis.

Assessment of epistemic uncertainties in the shear strength of slender reinforced concrete beams

Selection of an appropriate model for shear capacity of reinforced concrete (RC) slender beams poses a number of challenges. First, different models with different levels of conservatism have been proposed in an attempt to describe shear resistance. Second, according to Reineck et al. (2014), code provisions for shear capacity of RC beams with shear reinforcement have been primarily derived from test data with respect to the required amount of shear reinforcement and the calculation of maximum shear capacity. Third, current models have been developed based on databases presenting two major drawbacks: (i) most data points are crowded in the small size range, and (ii) the means of the subsidiary influencing parameters are very different within different intervals of beam size (or beam depth). In this study, a filtered database is used in such a way to circumvent the drawbacks mentioned above. A random variable “model error”, i.e. ratio experimental to predicted shear strength, is associated to each of the shear models analyzed in this work (NBR 6118, ACI 318, EUROCODE 2, and CSA A.23.3). It was observed that in some cases, most notably for the effective depth, a trend exists for a decrease in the “model error” as the effective depth increases. Considering the limitations of the four analyzed models, a nonlinear regression model was proposed. The database presented by Reineck et al. (2014) was used in the assessment of the effectiveness and accuracy of the proposed regression model. No trend was found associated to the most significant variables in the shear strength prediction, i.e. a uniform level of conservatism is attained throughout the range of these variables. The regression model proposed herein and the attendant statistics of the model error (mean, coefficient of variation and type of distribution) can be easily used in a reliability analysis procedure to assess safety levels implicit in different design procedures.

Statistical calibration of safety factors for flexural stiffness of composite columns

Composite column design is strongly influenced by the computation of the critical buckling load, which is very sensitive to the effective flexural stiffness (EI) of the column. Because of this, the behaviour of a composite column under lateral loading and its response to deflection is largely determined by the EI of the member. Thus, prediction models used for composite member design should accurately mirror this behaviour. However, EI varies due to several design parameters, and the implementation of high-strength materials, which are not considered by the current composite design codes of practice. The reliability of the design methods from six codes of practice (i.e., AS 5100, AS/NZS 2327, Eurocode 4, AISC 2010, ACI 318, and AIJ) for composite columns is studied in this paper. Also, the reliability of these codes of practice against a serviceability limit state criterion are estimated based on the combined use of the test-based statistical procedure proposed by Johnson and Huang (1997) and Monte Carlo simulations. The composite columns database includes 100 tests of circular concrete-filled tubes, rectangular concrete-filled tubes, and concrete-encased steel composite columns. A summary of the reliability analysis procedure and the evaluated reliability indices are provided. The reasons for the reliability analysis results are discussed to provide useful insight and supporting information for a possible revision of available codes of practice.

Development of a Mobility-Based Service Measure for Freeway Facilities

The Highway Capacity Manual (HCM) currently uses a bipolar approach to defining level of service (LOS) for freeway facilities: either (1) weighted density or (2) assigning LOS F if one or more segments experience LOS F. The major shortcoming of this approach is that density is a poor indicator of travelers’ experiences under congested conditions; speeds or travel times are more relevant to travelers and are consistent with how agencies measure and report congestion using field data. Likewise, saying a facility operates at LOS F if one only segment experiences LOS F is misleading. Further, five of the six LOS ranges exist where speeds are relatively high (above approximately 50 mph) and only one LOS range is used to define the congested regime, which is of the highest interest in large urbanized areas. Especially with regard to operations improvements, many congestion management techniques will improve congestion (e.g., delay) but the facility will still be classified as LOS F under the current definition.This paper deals with this issue by defining a travel time-based service measure for freeway facilities. Two HCM applications are explored: (1) traditional(static) freeway analysis and (2) the new travel time reliability (TTR) analysis procedure. Several performance measures are explored for the service measures by analyzing field data from seven U.S. urban areas. Although the research is not complete as of this writing, the mean travel time index is probably the best measure for the static procedure and either the 80th or 95th percentile travel time index should be used with the reliability procedure. Analysis of field data revealed strong correlation between the mean and upper percentiles of the travel time distribution, so these measures are internally consistent.A shift away from the current density-based LOS structure is recommended. The new structure uses ranges of the selected travel time measures that indicate different levels of the user experience. This approach is similar to what is done in the HCM for urban streets. Reconciling LOS concepts between freeways and urban streets will make the HCM more usable for the emerging field of performance management.It is recommended that travel time reliability be the service measure for freeway facilities which are part of an extensive freeway network in large urbanized areas, those over 500,000 population. An extensive freeway network is defined as having at least three freeway facilities. Analysts should use their judgment in cases that do not meet these Density would continue to be used on all other freeway facilities as well as for other aspects of freeways covered by the HCM (e.g., freeway segments).Recognizing that agencies may want to define their own service standards, a procedure for setting locally-defined standards is developed, based on the steps used in this research.It is expected that the freeway facility service measures will be an aid in setting transportation system and freight performance targets, a a demonstration of how they would be used for target setting is presented.

Lifecycle Costs for Energy Equipment FMECA for Gas Turbine

The procedure of reliability analysis for gas turbine unit (GTU) cogeneration-type SGT-800 is discussed. The example of creating a functional structure and categories of the failures criticality for GTU is provided. The qualitative matrix for GTU structural elements analysis is constructed and the criticality values are calculated.

Hydraulic reliability analysis of an urban loop high-pressure gas network

Scientific analyses and the provision of practical ways of improving the hydraulic reliability of urban high-pressure gas networks are essential for the safe and stable supply of natural gas by a supplier. A reasonable estimation of the actual obtainable flow for each consumer in the event of a failure is key to the quantitative analysis of hydraulic reliability, which is not an easy task for a loop network. Based on an analysis of the hydraulic characteristics of a loop network, this paper discusses the limitations of existing methods and presents a feasible method for determining the actual nodal flow in a high-pressure gas network loop. This method adopts a process whereby the hydraulic regimes are first analysed according to the designed nodal flow. Next, the nodal pressures are checked, and the actual nodal flow is gradually rectified. Finally, the hydraulic regimes are repeatedly analysed according to the rectified flow until each nodal pressure and each nodal flow satisfy the nodal pressure equations. Moreover, a detailed procedure for a hydraulic reliability analysis of a loop network is presented, and its feasibility is confirmed with an example. The example results show that the network system hydraulic reliability is only 0.879 using the proposed method when pipe 2 fails, whereas the reliability would become 1.0 if using the existing method. The factors affecting the reliability are discussed exhaustively from the aspects of the network system and the consumer, and a number of measures are suggested for improving the hydraulic reliability.

Reliability Analysis of Earth Slopes Considering Spatial Variability

The paper presents a computational procedure for reliability analysis of earth slopes considering spatial variability of soils under the framework of the Limit Equilibrium Method. In the reliability analysis of earth slopes, the effect of spatial variability of soil properties is generally included indirectly by assuming that the probabilistic critical slip surface is the same as that determined without considering spatial variability. In contrast to this indirect approach, in the direct approach, the effect of spatial variability is included in the process of determination of the probabilistic critical surface itself. While the indirect approach requires much less computational effort, the direct approach is definitely more rigorous. In this context this paper attempts to investigate, with the help of numerical examples, how far away are the results obtained from the indirect approach from that obtained from the direct approach. In both the approaches, it is required to use a model of discretization of random fields into finite random variables. A few such models are available in the literature for one-dimensional (1D) as well as two-dimensional (2D) spatial variability. The developed computational scheme is based on the First Order Reliability Method (FORM) coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. The study includes bringing out the computational advantages and disadvantages of the three commonly used discretization models. The sensitivity of the reliability index to the magnitudes of the scales of fluctuation has also been studied.

Probabilistic Stability Analysis of the Bois Brule Levee Considering the Effect of Spatial Variability of Soil Properties Based on a New Discretization Model

The paper presents a computational procedure for reliability analysis of earth slopes for which the probabilistic critical slip surfaces have been determined considering spatial variability of soils. The effect of spatial variability has been taken into account based on a newly developed discretization model which is free from the shortcomings of the discretization models available in the literature and all these shortcomings are demonstrated numerically in this paper. The developed algorithm is based on the First Order Reliability Method coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. A case study of the Bois Brule Levee has been reanalyzed using the developed technique and several observations have been discussed. The result obtained in this study clearly shows that the proposed model is considerably more conservative than other available models.

A probabilistic non-linear finite element analysis for slope stability problem

This paper deals with the slope stability problem by a finite element reliability analysis considering the spatial variation of soil strength parameters. In this work, the spatial variation is described using random field theory. To simplify the implementation of the proposed procedure for slope reliability analysis, elastic perfectly plastic constitutive model describing the soil behaviour using the Mohr coulomb yield criterion is adopted. The application of the proposed approach for the slope reliability analysis is performed using a performance function expressed in terms of stress fields mobilised along a circular failure surface. For this purpose, a computation of the stress gradient based on the incremental theory of plasticity is used. The results of the proposed procedure are checked by a probabilistic method based on the combination of the Bishop's model and the Monte Carlo simulation. The numerical examples have elucidated the efficiency and the validity of the present procedure to slope reliabi...

Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability

The objective of this paper is to investigate the effect of copulas for constructing the bivariate distribution of shear strength parameters on system reliability of geotechnical structures. First, the bivariate distribution of shear strength parameters is constructed using copulas. Second, the implementation procedure of system reliability analysis using direct Monte Carlo simulation (MCS) is developed. Finally, the system reliability of a retaining wall and a rock wedge slope is presented to explore the effect of copula selection on geotechnical system reliability. The results indicate that the system reliability of geotechnical structures under incomplete probability information could not be determined uniquely because the bivariate distribution of cohesion and friction angle with given marginal distributions and correlation coefficient could not be determined uniquely. The copulas for modeling dependence structure between cohesion and friction angle have a significant influence on the system reliability of geotechnical structures. Such an influence includes two phases separately. The first phase is that the dependence structure between shear strength parameters characterized by copulas affects the reliability of single failure mode, depending on the marginal distributions, dependence structure between shear strength parameters, and reliability level of each failure mode. The second phase is that the reliability of each failure mode influences on system reliability, only depending on reliability level of each failure mode and correlations among various failure modes. It is important to distinguish between the effect of copula selection on reliability of each failure mode and that on geotechnical system reliability.

Bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability

This paper aims to propose a bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability. The procedure for a traditional slope reliability analysis with fixed distributions of shear strength parameters is presented first. Then, the variations of the mean and standard deviation of shear strength parameters and the Akaike Information Criterion values associated with various distributions are studied to characterize the uncertainties in distribution parameters and types of shear strength parameters. The reliability of an infinite slope is presented to demonstrate the validity of the proposed method. The results indicate that the bootstrap method can effectively model the uncertain probability distributions of shear strength parameters. The uncertain distributions of shear strength parameters have a significant influence on slope reliability. With the bootstrap method, the slope reliability index is represented by a confidence interval instead of a single fixed index. The confidence interval increases with increasing factor of slope safety. Considering both the uncertainties in distribution parameters and distribution types of shear strength parameters leads to a higher variation and a wider confidence interval of reliability index.

Simulation and Optimization of an Actuator

This paper presents a new methodology for the Reliability Based Particle Swarm Optimization with Simulated Annealing. The reliability analysis procedure couple traditional and modified first and second order reliability methods, in rectangular plates modelled by an Assumed Modes approach. Both reliability methods are applicable to the implicit limit state functions through numerical models, like those based on the Assumed Mode Method. In modified approaches, the algorithms are based on heuristic optimization methods such as Particle Swarm Optimization and Simulated Annealing Optimization. Numerical applications in static, dynamic and stability problems are used to illustrate the applicability and effectiveness of proposed methodology. The results of example show that the predicted reliability levels are accurate to evaluate simultaneously various implicit limit state functions with respect to static, dynamic and stability criterions.