Probabilistic Failure Analysis by Importance Sampling Markov Chain Simulation

A probabilistic approach for failure analysis is presented in this paper, which investigates the probable scenarios that occur in case of failure of engineering systems with uncertainties. Failure analysis can be carried out by studying the statistics of system behavior corresponding to the random samples of uncertain parameters that are distributed as the conditional distribution given that the failure event has occurred. This necessitates the efficient generation of conditional samples, which is in general a highly nontrivial task. A simulation method based on Markov Chain Monte Carlo simulation is proposed to efficiently generate the conditional samples. It makes use of the samples generated from importance sampling simulation when the performance reliability is computed. The conditional samples can be used for statistical averaging to yield unbiased and consistent estimate of conditional expectations of interest for failure analysis. Examples are given to illustrate the application of the proposed simulation method to probabilistic failure analysis of static and dynamic structural systems.     

Nearest Neighbor Probabilistic Model for Aluminum Polycrystals

A real-valued random field {Zi,j} with piecewise constant samples and defined on a lattice L in 2 is developed to characterize two-dimensional metallic polycrystals. The subsets defined by constant values of {Zi,j} are virtual grains and the values of {Zi,j} give Euler angles at the nodes of L. The field {Zi,j} is completely defined by its marginal distribution and conditional probabilities associated with the nearest neighbor model. The defining probabilities of {Zi,j} need to be estimated from measurements of atomic lattice orientation. Random fields {Zi,j} calibrated to the measurements of crystallographic texture in two AA7075 aluminum plates have been used to generate virtual polycrystals. Virtual and actual polycrystals are similar.     

Probabilistic Model for Cost Contingency

This paper proposes a probabilistic model for the calculation of project cost contingency by considering the expected number of changes and the average cost of change. The model assumes a Poisson arrival pattern for change orders and independent random variables for various change orders. The probability of cost overrun for a given contingency level is calculated. Typical input values to the model are estimated by reviewing several U.S. Army Corps of Engineers project logs, and numerical values of contingency are calculated and presented. The effect of various parameters on the contingency is discussed in detail.     

Probabilistic Models and Simulation of Irregular Masonry Walls

A method is proposed for generating samples of irregular masonry walls that capture the essential statistics of a given population. The method first entails characterizing the geometry of scaled star-like inclusions by means of a non-Gaussian random field model and second packing these inclusions together to form a virtual material specimen. The model used in the first step is a nonlinear memoryless mapping of a sum of harmonic functions with Gaussian coefficients while in the second step the model proposed transforms Poisson fields into a domain of inclusions with a sieving curve that matches the sample specimen. The two random field models are used to develop Monte Carlo algorithms which produce virtual material specimens that include two levels of probabilistic characterization, a first level that is correlated to the inclusion geometry, and a second that is dictated by the global morphology of the sample material specimen.     

Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters

In recent years, Bayesian model updating techniques based on measured data have been applied to system identification of structures and to structural health monitoring. A fully probabilistic Bayesian model updating approach provides a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying structural system and to its exclusive foundation on the probability axioms. The plausibility of each structural model within a set of possible models, given the measured data, is quantified by the joint posterior probability density function of the model parameters. This Bayesian approach requires the evaluation of multidimensional integrals, and this usually cannot be done analytically. Recently, some Markov chain Monte Carlo simulation methods have been developed to solve the Bayesian model updating problem. However, in general, the efficiency of these proposed approaches is adversely affected by the dimension of the model parameter space. In this paper, the Hybrid Monte Carlo method is investigated (also known as Hamiltonian Markov chain method), and we show how it can be used to solve higher-dimensional Bayesian model updating problems. Practical issues for the feasibility of the Hybrid Monte Carlo method to such problems are addressed, and improvements are proposed to make it more effective and efficient for solving such model updating problems. New formulae for Markov chain convergence assessment are derived. The effectiveness of the proposed approach for Bayesian model updating of structural dynamic models with many uncertain parameters is illustrated with a simulated data example involving a ten-story building that has 31 model parameters to be updated.     

Fracture Mechanics Model for Analysis of Plain and Reinforced High-Performance Concrete Beams

In developing a one-dimensional analysis and design procedure for reinforced concrete structures, research is generally based on yield phenomena and the plastic flow of steel in tension and concrete in compression. The ability of concrete to resist tension is considered in the form of tension stiffening or is completely disregarded. This procedure does not account for the influence of structural size in changing the failure mode and the stress distribution across the uncracked or cracked ligament. The key factor affecting this stress distribution is found to be the strain-softening modulus. This paper presents an improved model that is based on the fundamental equilibrium equation for the progressive failure of plain concrete beams. The concrete stress-strain relationship in tension is derived by calculating the peak tensile stress and softening modulus for different depths of beams on the basis of the fracture parameters obtained with the size effect law. Thus, the proposed model uses the peak tensile stress and the softening modulus, which vary depending on the size of the beam. To study the effect of the strength of high-performance concrete (HPC) on the concrete tensile stress-strain relationship, the experimental load-deflection plots of different-sized beams are compared with those obtained by using the proposed analytical model for eight different mixes made with locally available fly ash and slag. The model is also extended for lightly reinforced concrete beams, and the results are compared with those in the literature and are found to be in good agreement.     

Probabilistic Estimation and Allocation of Project Time Contingency

Project managers implement the concept of time contingency to consider uncertainty in duration estimates and prevent project completion delays. Some project managers also build a distribution of the project time contingency into the project activities to create a more manageable schedule. Generally, both the estimation and distribution of the project time contingency are conducted by using subjective approaches. Because the project schedule feasibility mainly depends on the variable behavior of the project activities, the estimate of project time contingency and its allocation at the activity level should be obtained by considering the performance variability of each activity rather than basing on human judgment. In this paper, the stochastic allocation of project allowances method, which is based on Monte?Carlo simulation, is proposed to estimate the project time contingency and allocate it among the project activities. The application of this method to a three-span bridge project results in a fair allocation of the project time contingency and provides practical means to control time contingencies at the activity level.     

Probabilistic Stability Analyses of Embankments Based on Finite-Element Method

Conventional limit equilibrium methods are commonly used to assess the stability of embankments. The finite-element method, as an alternative to limit equilibrium methods, is being increasingly used in the deterministic stability analysis of slopes or embankments. In this paper, a practical procedure for integrating the finite-element method and the limit equilibrium methods into probabilistic stability analysis for embankments is presented. The response surface method is adopted to approximate the performance function for the stability problems and the first-order reliability method is used to calculate the reliability index based on an intuitive expanding ellipsoid perspective. The advantages of the response surface method as a bridge between stand-alone numerical packages and spreadsheet-based reliability analysis via automatic constrained optimization are demonstrated and discussed through a hypothetical two-layer slope and an actual case of the James Bay Dykes. The results show the ease and successful implementation of the proposed procedure for reliability analysis of embankments.     

Probabilistic Approach to the Solution of Inverse Problems in Civil Engineering

A wide range of important problems in civil engineering can be classified as inverse problems. In such problems, the observational data regarding the performance of a system is known, and the characteristics of the system and/or the input are sought. There are two general approaches to the solution of inverse problems: deterministic and probabilistic. Traditionally, inverse problems in civil engineering have been solved using a deterministic approach. In this approach, the objective is to find a specific model of a system that its theoretical response best fits the observed data. Obtaining the best fit solution, however, does not provide any information regarding the effect of data and/or theoretical uncertainties on the obtained solution. In this paper, a general probabilistic approach to the solution of the inverse problems is introduced, which provides uncertainty measures for the obtained solution. Techniques for direct analytical evaluation and numerical approximation of the probabilistic solution using Monte Carlo Markov Chains, with and without neighborhood algorithm approximation, are introduced and explained. The presented concepts and techniques and their application are then illustrated in practical terms using a simple example of a modulus determination experiment.     

Probabilistic Approach to Estimation of Urban Storm-Water TMDLs: Regulated Catchment

The present work is concerned with the development of a set of tools for the incorporation of various control measures—best management practices into an analytical probabilistic modeling approach for urban storm-water total maximum daily load (TMDL) estimation. Control measures are divided into two major groups—upstream and downstream, each requiring application of separate modeling principles elaborated in this paper. Applying Monte Carlo simulation to the developed set of expressions allows modeling the “end-of-pipe” parameters of urban storm-water discharges (runoff volume, discharge rate, and pollutant load) on an event average basis, as well as the stream parameters downstream of a storm-water discharge outlet. Model application is illustrated for a catchment regulated with an extended detention dry pond. Representative model results are presented, and a range of potential model applications is discussed. The capability to model the behavior of an urban storm-water system with the application of various control measures is the key precondition for the design of an optimal configuration of a water-protective strategy.     

Probabilistic Control of Project Performance Using Control Limit Curves

This study introduces a new probabilistic project control concept to assure an acceptable forecast of final project performance, in terms of not exceeding planned budget and schedule risk levels. This concept consists in the implementation of performance control limit curves for both actual cost and elapsed time, obtained with a probabilistic approach and a graphical representation referred to as Stochastic S curves (SS curves). In order to facilitate the project control process, control limit curves can be used to display and evaluate actual project performance status without the need of actualizing at completion performance forecasts. Three different approaches (quality, benchmarking, and incremental variance) are proposed in this paper for obtaining the project performance control limit curves. In order to find the control limit curve definition with more conservative acceptable performance variations, these approaches are tested in an example project. A further managerial advantage is found in the recommended approach, as it allows monitoring the use of both cost and scheduling contingencies, along the project execution.     

Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging

This paper presents a newly developed simulation-based approach for Bayesian model updating, model class selection, and model averaging called the transitional Markov chain Monte Carlo (TMCMC) approach. The idea behind TMCMC is to avoid the problem of sampling from difficult target probability density functions (PDFs) but sampling from a series of intermediate PDFs that converge to the target PDF and are easier to sample. The TMCMC approach is motivated by the adaptive Metropolis–Hastings method developed by Beck and Au in 2002 and is based on Markov chain Monte Carlo. It is shown that TMCMC is able to draw samples from some difficult PDFs (e.g., multimodal PDFs, very peaked PDFs, and PDFs with flat manifold). The TMCMC approach can also estimate evidence of the chosen probabilistic model class conditioning on the measured data, a key component for Bayesian model class selection and model averaging. Three examples are used to demonstrate the effectiveness of the TMCMC approach in Bayesian model updating, model class selection, and model averaging.     

Model Selection Using Response Measurements: Bayesian Probabilistic Approach

A Bayesian probabilistic approach is presented for selecting the most plausible class of models for a structural or mechanical system within some specified set of model classes, based on system response data. The crux of the approach is to rank the classes of models based on their probabilities conditional on the response data which can be calculated based on Bayes’ theorem and an asymptotic expansion for the evidence for each model class. The approach provides a quantitative expression of a principle of model parsimony or of Ockham’s razor which in this context can be stated as “simpler models are to be preferred over unnecessarily complicated ones.” Examples are presented to illustrate the method using a single-degree-of-freedom bilinear hysteretic system, a linear two-story frame, and a ten-story shear building, all of which are subjected to seismic excitation.     

Probabilistic Approach to Predict Cracking in Lightly Reinforced Microconcrete Panels

The ultimate strength of structures made of brittle materials—such as microconcrete—strongly depends on microstructural defects, the structure size, and the loading pattern. Probabilistic approaches allow one to take account of such dependencies. By using a Weibull model, cracking of ferrocement panels is analyzed. Provided the behavior of the reinforcement remains elastic, it is shown that the Weibull parameters identified on unreinforced microconcrete samples tested in flexure may be used to predict multiple cracking in ferrocement panels tested in tension. A key aspect of the analysis is related to the understanding and modeling of the stress heterogeneity effect on the local failure probability of unreinforced as well as reinforced microconcrete by the use of a so-called Weibull stress.     

Evaluation of Probabilistic Point Estimate Methods in Uncertainty Analysis for Environmental Engineering Applications

This paper begins with a comprehensive review of various point estimate methods, with an emphasis on their differences and similarities in theory and application. The Rosenblueth probabilistic point estimate method is a computationally straightforward technique for the uncertainty analysis of engineering problems. It is capable of estimating a statistical moment of any order of a model output involving several stochastic variables that are correlated or uncorrelated, symmetric, or asymmetric. However, in multivariate problems with more than two stochastic variables involved, the Rosenblueth method is not able to provide a unique solution, rather an approximate solution to indeterminate problems. This is attributed to the fact that the number of unknowns to be solved is larger than the number of governing equations provided. An improved modified Rosenblueth point estimate method is proposed to circumvent the drawback of the nonunique solution of the original Rosenblueth method and to increase the computational efficiency in modeling. One example application on the particle terminal velocity computation is presented for illustration. A quantitative performance index is introduced to assess the performance of various point estimate methods. It is concluded in this study that the modified Rosenblueth method has a comparable performance to the Rosenblueth method and yet resolves the nonuniqueness problem in solutions.     

Probabilistic Critical Excitation Method for Earthquake Energy Input Rate

Since earthquake ground motions and their input effects on structures are very uncertain even with the present state of knowledge, it is desirable to develop a “robust” structural design method taking into account these uncertainties. Approaches based on critical excitation methods have been proven to be promising for such robust structural design. A new critical excitation method is developed here in which the mean earthquake energy input rate is chosen as a measure of criticality. The earthquake energy input rate is closely correlated with the story deformation and this supports the suitability of the energy input rate as a criticality measure in the case where the deformation is crucial in the design. The ground motion is described as a uniformly modulated nonstationary random process. The power [area of power spectral density (PSD) function] and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. The key for finding the new random critical excitation is the interchange of the order of the double maximization procedures with respect to time and to the PSD function. Examples for a specific envelope function of the ground motion are presented for demonstrating the validity of the proposed method. Extension of the proposed method will be discussed for a more general ground motion model, i.e., nonuniformly modulated nonstationary models, and for a more general problem for variable envelope functions and variable frequency contents.     

Probabilistic Forecasting of Project Performance Using Stochastic S Curves

This study presents a new methodology for evaluating at-completion project performance status. This new procedure uses the concept of stochastic S curves (SS curves) to determine forecasted project estimates as an alternative to using deterministic S curves and traditional forecasting methods. A simulation approach is used for generating the stochastic S curves, and it is based on the defined variability in duration and cost of the individual activities within the process. Stochastic S curves provide probability distributions for the budget and time values required to complete the project at every selected point of intermediate completion. Final project performance is determined by comparing the planned budget and project duration, with the expected forecasted final cost and elapsed time, respectively. The SS-curve methodology permits objective evaluation of project performance without the limitations inherent in a deterministic approach. The probabilistic characteristics of this approach enable users to more accurately determine at-completion cost and duration variations and evaluate the performance improvement of proposed corrective actions.     

Probabilistic Approaches for Pavement Fatigue Cracking Prediction based on Cumulative Damage Using Miner’s Law

In mechanistic-empirical (M-E) pavement design, pavement damage is modeled as a random variable with a pre-specified distribution (normal or lognormal). The extent of fatigue cracking in terms of percentage cracking is computed as the probability of cumulative damage exceeding unity. This paper provides a methodological framework for characterizing damage distribution under mixed traffic loading (multiple strain levels) with an improved forecast of traffic spectrum based on renewal theory. Using the linear Miner’s law for damage accumulation, analytical representation of damage distribution is obtainable owing to the proportional relationship between maximum tensile strain of pavement and traffic load under linear elasticity condition. Numerical computation shows that percent of cracking from derived damage distribution is greater than that from hypothetical normal or lognormal distributions traditionally used in the M-E pavement design. The method developed here and the derived model can be used in pavement design and pavement management systems.     

Probabilistic Assessment of Slope Stability That Considers the Spatial Variability of Soil Properties

In this paper, a numerical procedure for probabilistic slope stability analysis is presented. This procedure extends the traditional limit equilibrium method of slices to a probabilistic approach that accounts for the uncertainties and spatial variation of the soil strength parameters. In this study, two-dimensional random fields were generated based on a Karhunen-Loève expansion in a fashion consistent with a specified marginal distribution function and an autocorrelation function. A Monte Carlo simulation was then used to determine the statistical response based on the generated random fields. This approach makes no assumption about the critical failure surface. Rather, the critical failure surface corresponding to the input random fields of soil properties is searched during the process of analysis. A series of analyses was performed to verify the application potential of the proposed method and to study the effects of uncertainty due to the spatial heterogeneity on the stability of slope. The results show that the proposed method can efficiently consider the various failure mechanisms caused by the spatial variability of soil property in the probabilistic slope stability assessment.     

多晶体晶粒尺度三维组织建模及可视化

采用基于Monte Carlo Potts模型的仿真技术对单相多晶体各向同性晶粒组织进行了三维建模.利用OpeGL图形接口,在所建多晶体组织模型数据结构的基础上进而实现了三维可视化和任意角度观测.结果表明,所建几何图像模型可以很好地仿真预定细观组织结构参数的三维多晶体组织,不但满足模型化研究所要求的统计意义上的准确性,而且非常符合实际材料显微组织的多变性特征,可望用作细观尺度多晶材料加工过程数值模拟和细观力学计算的基本组织模型.