Proban – probabilistic analysis

Proban is a general purpose probabilistic analysis program that covers calculation needs for example in structural reliability. Proban calculates probability, distribution, first passage probability, crossing rate, and related sensitivity measures. The program is not restricted to a particular problem, but allows users to define functions on input and to attach function libraries to the program in a DLL (Windows). Supporting tools such as flexible modeling of variables and events, deterministic analysis, distribution fit, parameter study, function libraries and distribution library come with the program. A variety of methods like FORM, SORM, adaptive response surface, nested reliability analysis, (adjusted) Monte–Carlo simulation, stratified simulation, directional simulation, design point simulation, axis orthogonal simulation and Latin Hypercube simulation are implemented. Proban has a graphical user interface, maintains models on a database, records commands on a journal file for possible reread and displays, plots and prints model and results.     

Probabilistic modelling of overflow, surcharge and flooding in urban drainage using the first-order reliability method and parameterization of local rain series

Failure of urban drainage systems may occur due to surcharge or flooding at specific manholes in the system, or due to overflows from combined sewer systems to receiving waters. To quantify the probability or return period of failure, standard approaches make use of the simulation of design storms or long historical rainfall series in a hydrodynamic model of the urban drainage system. In this paper, an alternative probabilistic method is investigated: the first-order reliability method (FORM). To apply this method, a long rainfall time series was divided in rainstorms (rain events), and each rainstorm conceptualized to a synthetic rainfall hyetograph by a Gaussian shape with the parameters rainstorm depth, duration and peak intensity. Probability distributions were calibrated for these three parameters and used on the basis of the failure probability estimation, together with a hydrodynamic simulation model to determine the failure conditions for each set of parameters. The method takes into account the uncertainties involved in the rainstorm parameterization. Comparison is made between the failure probability results of the FORM method, the standard method using long-term simulations and alternative methods based on random sampling (Monte Carlo direct sampling and importance sampling). It is concluded that without crucial influence on the modelling accuracy, the FORM is very applicable as an alternative to traditional long-term simulations of urban drainage systems.     

基于Beta分布的改进的一次可靠度法研究

为了提高一次可靠度方法的计算精度和计算效率,文章利用Beta分布拟合了功能函数中的随机变量的概率分布,利用VBA语言给出了将Beta分布和对数正态分布转化为标准正态分布的方法,采用蒙特卡洛模拟法比较了采用不同概率分布的一次可靠度计算精度。算例分析表明:当功能函数中随机变量服从对数正态分布时,失效概率被高估;当随机变量服从正态分布时失效概率被低估。利用本文提出的Beta分布拟合随机变量的概率不仅大大提高了计算精度,而且在保证迭代收敛的前提下提高了计算效率。     

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.     

Adaptive hybrid conditional expectation approaches for reliability estimation

Conditional expectation is a simulation procedure that can be used to estimate the reliability of structures. It involves selecting an important random variable, using Monte Carlo simulation to generate sets of sample outcomes of the remaining “unimportant” random variables, estimating the conditional failure probability for each set of sample outcomes and using the average conditional failure probability as the estimate of the failure probability. This direct conditional expectation approach can be extremely effective, but only if the selected important random variable is much more important than the combined effect of the remaining random variables.In this paper, the concept of using importance sampling to generate the unimportant random variables is introduced. With a good sampling density, this greatly improves the efficiency for cases in which the selected important random variable does not dominate. To select such a density, however, one needs informations about the important regions of the failure domain. Typically, such information is not available a priori, and the direct application of this concept is limited.This concept can, however, be used in an adaptive format and two such adaptive hybrid conditional expectation approaches are developed in this paper. They are based on the idea that, as one generates sample outcomes and calculates the conditional failure probabilities at these outcomes, the knowledge of the failure domain increases. If one keeps modifying the importance sampling density to reflect this increasing state of knowledge, one can develop a good sampling density and simultaneously estimate the failure probability efficiently.Examples are used to demonstrate the efficacy of these approches and study the influence of factors like failure probability level and relative importance of random variables. Also discussed are issues such as the efficiency of the new methods (compared to FORM) and the applications to problems involving unions and intersections.     

Sensitivity estimation within first and second order reliability methods

An efficient method to compute the sensitivity of failure probability estimates to changes in distribution parameters is proposed. The method is applicable to failure probability estimates obtained by first and second order reliability methods (FORM and SORM). It is based on the fact that current FORM and SORM approaches require transforming the problem into standard normal space and then numerically fitting first/second order approximations. To compute the change in failure probability due to a change in parameter, one can transform the points that were used to numerically fit the approximation into a new standard normal space corresponding to the changed parameter, fit a new approximation and use the new approximation to compute the changed failure probability.The proposed method is computationally efficient because it does not require any new structural response evaluations. In contrast to conventional methods that are accurate for FORM estimates of component problems, the proposed method is accurate for FORM and SORM estimates of both component and system problems.     

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.     

Comparison of response surface and neural network with other methods for structural reliability analysis

The evaluation of the failure probability and safety levels of structural systems is of extreme importance in structural design, mainly when the variables are eminently random. Some examples of random variables on real structures are material properties, loads and member dimensions. It is necessary to quantify and compare the importance of each one of these variables in the structural safety. Many researchers studied structural reliability problems and nowadays there are several approaches for these problems. Two recent approaches, the Response Surface (RS) and the Artificial Neural Network (ANN) techniques, have emerged attempting to solve complex and more elaborated problems. In this work, these two techniques are presented, and comparison are carried out using the well known First Order Reliability Method (FORM), Direct Monte Carlo Simulation and Monte Carlo Simulation with Adaptive Importance Sampling technique with approximated and exact limit state functions. Problems with simple limit state functions (LSF) and closed form solutions of the failure probability are solved in order to highlight the advantages and shortcomings using these techniques. Some remarks are outlined regarding the fact that RS and ANN techniques have presented equivalent precision levels. It is observed that in problems where the computational cost of structural evaluations (looking for the failure probability and safety levels) is high, these two techniques may turn feasible the evaluation of the structural reliability through simulation techniques.     

Reliability analysis of rock slopes involving correlated nonnormals

A two-dimensional rock slope in Hong Kong and a three-dimensional hypothetical tetrahedral wedge are analyzed probabilistically using an intuitive and transparent constrained optimization approach for the first-order reliability method (FORM). The results are compared with Monte Carlo simulations. Simple procedures for incorporating truncated non-normal distributions in reliability analysis are described. The effects of statistical correlations on the computed reliability index are studied. The difference between probabilities of failure inferred from reliability index and from Monte Carlo simulations are investigated via the response surface method. It is shown that the efficiency of reliability-based approach can be combined with the robustness of Monte Carlo simulation. The meanings of reliability index and probability of failure are also discussed.     

Reliability analysis of ground–support interaction in circular tunnels using the response surface method

The first-order and the second-order reliability method (FORM/SORM) are used to evaluate the failure probability of three performance functions of the ground–support interaction in circular tunnels subjected to hydrostatic stresses. The response surface method (RSM) is used to enable reliability analysis of the implicit convergence-confinement method. The friction angle, cohesion and elastic modulus of the rock mass are considered as basic random variables and are first assumed to obey normal distributions. The quadratic polynomial with cross terms is employed as response surface function to approximate the limit state surface (LSS) at the design point. The strategies for the RSM are presented. The failure probability with respect to different criteria are obtained from FORM/SORM and compared to those generated from Monte Carlo simulations. The results show that the support installation position has great influence on the probability of the three failure modes under consideration. Comparison between analysis using correlated and uncorrelated friction angle and cohesion indicates that the influence of the correlation on the reliability analysis depends on the support installation position and the orientation of the LSS. The reliability analysis involving correlated non-normal distributions and the reliability-based design of the support are also investigated.     

A general procedure for first/second-order reliabilitymethod (FORM/SORM)

First/second-order reliability method (FORM/SORM) is considered to be one of the most reliable computational methods for structural reliability. Its accuracy is generally dependent on three parameters, i.e. the curvature radius at the design point, the number of random variables and the first-order reliability index. In the present paper, the ranges of the three parameters for which FORM/SORM is accurate enough are investigated. The results can help us to judge when FORM is accurate enough, when SORM is required and when an accurate method such as the inverse fast Fourier transformation (IFFT) method is required. A general procedure for FORM/SORM is proposed which includes three steps: i.e. point fitting limit state surface, computation of the sum of the principal curvatures K_s and failure probability computation according to the range of K_s. The procedure is demonstrated by several examples.     

A Refined Methodology for Durability‐Based Service Life Estimation of Reinforced Concrete Structural Elements Considering Fuzzy and Random Uncertainties

Abstract: A reliable method for service life estimation of the structural element is a prerequisite for service life design. A new methodology for durability‐based service life estimation of reinforced concrete flexural elements with respect to chloride‐induced corrosion of reinforcement is proposed. The methodology takes into consideration the fuzzy and random uncertainties associated with the variables involved in service life estimation by using a hybrid method combining the vertex method of fuzzy set theory with Monte Carlo simulation technique. It is also shown how to determine the bounds for characteristic value of failure probability from the resulting fuzzy set for failure probability with minimal computational effort. Using the methodology, the bounds for the characteristic value of failure probability for a reinforced concrete T‐beam bridge girder has been determined. The service life of the structural element is determined by comparing the upper bound of characteristic value of failure probability with the target failure probability. The methodology will be useful for durability‐based service life design and also for making decisions regarding in‐service inspections.     

Probabilistic stability analysis of earth dam slope under transient seepage using multivariate adaptive regression splines

Earth dams are widespread throughout the world and their safety has gained increasing concern from geotechnical engineering societies. Although probabilistic stability analysis approach has been widely applied to the safety assessment of geotechnical structures, few studies have been performed to investigate the effects of water level fluctuations on earth dam slope stability considering uncertainties of soil parameters. This study proposes an efficient probabilistic stability analysis approach by integrating a soft computing algorithm of multivariate adaptive regression splines (MARS). The calibration of a MARS model generally requires a large number of training samples, which are obtained from repeated runs of deterministic seepage and slope stability analyses using the GeoStudio software. Based on the established MARS model, the earth dam slope failure probability can be conveniently evaluated. As an illustration, the proposed approach is applied to the probabilistic stability analysis of Ashigong earth dam under transient seepage. The effects of the uncertainties of soil parameters and water level fluctuation velocity on the earth dam slope failure probability are explored systematically. Results show that the MARS-based probabilistic stability analysis approach evaluates the earth slope failure probability with satisfactory accuracy and efficiency. The earth dam slope failure probability is significantly affected by the water level fluctuation velocity and the coefficient of variation of the effective friction angle.

     

Slope reliability and back analysis of failure with geotechnical parameters estimated using Bayesian inference

A Bayesian approach is proposed for the inference of the geotechnical parameters used in slope design. The methodology involves the construction of posterior probability distributions that combine prior information on the parameter values with typical data from laboratory tests and site investigations used in design. The posterior distributions are often complex, multidimensional functions whose analysis requires the use of Markov chain Monte Carlo (MCMC) methods. These procedures are used to draw representative samples of the parameters investigated, providing information on their best estimate values, variability and correlations. The paper describes the methodology to define the posterior distributions of the input parameters for slope design and the use of these results for evaluation of the reliability of a slope with the first order reliability method (FORM). The reliability analysis corresponds to a forward stability analysis of the slope where the factor of safety (FS) is calculated with a surrogate model from the more likely values of the input parameters. The Bayesian model is also used to update the estimation of the input parameters based on the back analysis of slope failure. In this case, the condition FS = 1 is treated as a data point that is compared with the model prediction of FS. The analysis requires a sufficient number of observations of failure to outbalance the effect of the initial input parameters. The parameters are updated according to their uncertainty, which is determined by the amount of data supporting them. The methodology is illustrated with an example of a rock slope characterised with a Hoek-Brown rock mass strength. The example is used to highlight the advantages of using Bayesian methods for the slope reliability analysis and to show the effects of data support on the results of the updating process from back analysis of failure.     

Estimation of failure probabilities by local approximation of the limit state function

For failure probability estimates of large structural systems, the numerical expensive evaluations of the limit state function have to be replaced by suitable approximations. Most of the methods proposed in the literature so far construct global approximations of the failure hypersurface. Rather than concentrating on the construction of the failure hypersurface, an adaptive local approximation scheme for the limit state function that is based on the moving least squares method is proposed in this study. It integrates well with existing importance sampling schemes and yields both efficient and robust estimates of the failure probability.     

EVALUATION OF THE PROBABILITY OF FAILURE WITH UNCERTAIN DISTRIBUTION PARAMETERS

Formulations are presented for estimating probability of failure considering uncertainty of distribution parameters in time invariant and time variant reliability analyses. Based on the formulations the probability of failure can be calculated by recursively using the first order reliability method. Also, a more efficient approximate analysis procedure by using the point estimate method to estimate the probability of failure is given. In this analysis procedure, the point estimate method is used to discretize the uncertain distribution parameters in the time invariant reliability analysis, and to discretize the time-independent random variables and uncertain distribution parameters in the time variant reliability analysis. The probability of failure is then obtained by weighting the probability of failure conditioned at each of discrete points. The conditional probability of failure can be calculated by using first order reliability method, second order reliability method or any other convenient methods. The use of point estimate method to treat uncertain distribution parameters in calculating probability of failure is less computer time consuming than the one of recursively using FORM. Illustrative numerical examples of calculating probability of failure are presented.     

Sensitivity analysis of vertically loaded pile reliability

The purpose of this paper is to examine the influence of geotechnical uncertainties on the reliability of vertically loaded pile foundations and the use of this information in decision-making support, especially when gathering the information necessary for reliability analyses. Two case studies of single pile foundations were selected, and each uncertainty source was investigated to identify which are the most important and influential in the evaluation of vertical pile resistance under axial loading. Reliability sensitivity analyses were conducted using FORM (the first-order reliability method) and MCS (Monte Carlo simulations). The characterisation of uncertainties is not an easy task in geotechnical engineering. The aim of the analyses described in this paper is to optimise resources and investments in the investigation of the variables in pile reliability. The physical uncertainties of actions, the inherent variability of soil and model error were assessed by experimental in situ standard penetration tests (SPT) or from information available in the literature. For the cases studied, the sensitivity analysis results show that, in spite of the high variability of the soils involved, model error also plays a very important role in geotechnical pile reliability and was considerably more important than soil variability in both case studies. From a comparison of the two reliability methods (FORM and MCS), it was concluded that FORM is applicable in simple cases and as a first approach because it is an approximate method and sometimes does not have the capability to incorporate every detail of the problem, namely a specific probability density function or more specific limit conditions.     

基于单变量降维模型和坐标旋转的可靠度混合分析方法

经典的一次可靠度方法对于隐式功能函数和强非线性功能函数的可靠度问题存在适用性问题,尽管二次可靠度方法可以一定程度上处理强非线性功能函数的问题,但理论基础和计算过程均颇为复杂,不利于实用。为克服上述问题,将一次可靠度确定验算点的过程与响应面法的思路相结合是一种行之有效的思路。为此,文中首先引入具有普适性的一次可靠度法,其中考虑了相关非正态随机变量的Nataf变换,并引入单边差分法针对性地解决了隐式功能函数求偏导数的问题|其次,根据梯度值引入坐标旋转向量,并对旋转后的功能函数引入单变量函数降维近似模型|再次,结合验算点的函数值、梯度值以及附加点的函数值,确定各分量函数的二次多项式近似,从而获得近似的整体功能函数|然后,采用重要性抽样法计算近似功能函数的失效概率|最后,分别通过数值算例和工程算例对建立方法的精度和效率进行了验证。结果表明建议方法具有高精度、高效率的特点,且无论对于显式和隐式功能函数均具有广泛适用性。