Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model

This paper presents a new approach, called a nonparametric approach, for constructing a model of random uncertainties in dynamic substructuring in order to predict the matrix-valued frequency response functions of complex structures. Such an approach allows nonhomogeneous uncertainties to be modeled with the nonparametric approach. The Craig–Bampton dynamic substructuring method is used. For each substructure, a nonparametric model of random uncertainties is introduced. This nonparametric model does not require identifying uncertain parameters in the reduced matrix model of each substructure as is usually done for the parametric approach. This nonparametric model of random uncertainties is based on the use of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle. The theory and a numerical example are presented in the context of the finite-element method. The numerical results obtained show the efficiency of the model proposed.     

Wishart Random Matrices in Probabilistic Structural Mechanics

Uncertainties need to be taken into account for credible predictions of the dynamic response of complex structural systems in the high and medium frequency ranges of vibration. Such uncertainties should include uncertainties in the system parameters and those arising due to the modeling of a complex system. For most practical systems, the detailed and complete information regarding these two types of uncertainties is not available. In this paper, the Wishart random matrix model is proposed to quantify the total uncertainty in the mass, stiffness, and damping matrices when such detailed information regarding uncertainty is unavailable. Using two approaches, namely, (a) the maximum entropy approach; and (b) a matrix factorization approach, it is shown that the Wishart random matrix model is the simplest possible random matrix model for uncertainty quantification in discrete linear dynamical systems. Four possible approaches for identifying the parameters of the Wishart distribution are proposed and compared. It is shown that out of the four parameter choices, the best approach is when the mean of the inverse of the random matrices is same as the inverse of the mean of the corresponding matrix. A simple simulation algorithm is developed to implement the Wishart random matrix model in conjunction with the conventional finite-element method. The method is applied vibration of a cantilever plate with two different types of uncertainties across the frequency range. Statistics of dynamic responses obtained using the suggested Wishart random matrix model agree well with the results obtained from the direct Monte Carlo simulation.     

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.     

Probability of Exceedance Estimates for Scour Depth around Bridge Piers

Scour at a bridge pier is the formation of a hole around the pier due to the erosion of soil by flowing water; this hole in the soil reduces the carrying capacity of the foundation and the pier. Excessive scour can cause a bridge pier to fail without warning. Current predictions of the depth of the scour hole around a bridge pier are based on deterministic models. This paper considers two alternative deterministic models to predict scour depth. For each deterministic model, a corresponding probabilistic model is constructed using a Bayesian statistical approach and available field and experimental data. The developed probabilistic models account for the estimated bias in the deterministic models and for the model uncertainty. Parameters from both prediction models are compared to determine their accuracy. The developed probabilistic models are used to estimate the probability of exceedance of scour depth around bridge piers. The method is demonstrated on an example bridge pier. The paper addresses model uncertainties for given hydrologic variables. Hydrologic uncertainties have been presented in a separate paper.     

Updating Properties of Nonlinear Dynamical Systems with Uncertain Input

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to quantify the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is first presented for single-degree-of-freedom systems and then for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator, an elastoplastic system and a four-story inelastic structure are presented to illustrate the proposed approach.     

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.     

Analysis of Life-Cycle Maintenance Strategies for Concrete Bridge Decks

This paper presents the development of a project-level decision support tool for ranking maintenance scenarios for concrete bridge decks deteriorated as a result of chloride-induced corrosion. The approach is based on a mechanistic deterioration model and a probabilistic life-cycle cost analysis. The analysis includes agency and user costs of alternative maintenance scenarios and considers uncertainties in the agency cost and the corrosion rate in the deterioration model. The tool presented in this paper can be used to find the optimal condition index of a given bridge deck that minimizes life-cycle cost. Based on the results obtained on three existing bridge decks, it is shown that the total life-cycle cost (user cost plus agency cost) is a nonlinear function of the maximum tolerable condition of the deck, Sm, and that for a practical range of Sm, the relationship between total life-cycle cost and Sm is convex.     

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.     

Reliability-Based Design Approach for Differential Settlement of Footings on Cohesionless Soils

A probabilistic method is presented to estimate the differential settlements of footings on cohesionless soils, considering the uncertainties in both the load and capacity sides of the design equation. A random field approach is employed to characterize the inherent soil variability. This method is first compared to typical limit values from the literature to denote critical combinations of design parameters that can lead to exceedance of tolerable differential settlements. Then, reliability-based design equations are developed for the serviceability limit state (SLS) design of footings on cohesionless soils. The key parameters controlling the SLS are the allowable angular distortion, site variability, and footing spacing. The results are given in a straightforward design format and indicate that currently suggested deformation factors (resistance factors for SLS) equal to 1.0 are likely to be unconservative for most design situations.     

Parameter Sensitivity and Importance Measures in Nonlinear Finite Element Reliability Analysis

Finite element reliability methods allow the analyst to define material, load, and geometry parameters as random variables to represent uncertainties in these model parameters. Approximate probabilistic analysis methods produce estimates of the response variance/covariances, probabilities of exceeding specified structural performance thresholds, and parameter importance measures. A necessary ingredient for such analysis is consistent, efficient, and accurate algorithms for computing finite element response sensitivities. In this paper, unified response sensitivity equations with respect to material, load, and geometry parameters are developed for the time- and space-discretized finite element model. The sensitivities with respect to nodal coordinates and global shape parameters in the presence of material and geometric nonlinearities represent an extension of previous work. Practical computer implementation issues are emphasized. The equations are implemented in the comprehensive, open-source, object-oriented finite element software OpenSees. Importance measures from reliability analysis, employing the sensitivity results, are presented to enable the investigation of the relative importance of uncertainty in the parameters of a finite element model. Two example applications demonstrate that the variability in nodal coordinates of a structure can be a significant source of uncertainty along with that in key material and load parameters.     

Beyond lithium in the treatment of bipolar illness

Primary non-Hodgkin lymphoma of the stomach is a rare disorder for which clinical management has not yet been settled completely. Faced with the many uncertainties associated with the selection of a treatment for a patient with this disorder, it is difficult to determine the treatment that is optimal for the patient, as well as the prognosis to be expected. The development of a decision-theoretic model of non-Hodgkin lymphoma of the stomach is described. The model aims to assist the clinician in exploring various clinical questions, among others questions concerning prognosis and optimal treatment. Central to the model is a probabilistic network that offers an explicit representation of the uncertainties underlying the decision-making process. The model has been incorporated in a decision-support system. Preliminary evaluation results indicate that the performance of the model in its present form matches the performance of experienced clinicians.     

Reliability-Based Optimization Considering Manufacturing and Operational Uncertainties

This paper presents approaches for integrating multidisciplinary optimization and probabilistic methods to perform reliability-based multidisciplinary optimization. The approaches are built into a framework that allows solution of optimization problems, wherein system parameters including dimensional tolerances, material properties, boundary conditions, loads, and model predictions are uncertain or variable. This approach directly supports quality engineering because it allows engineers to specify manufacturing tolerances required to achieve the desired product reliability, and it results in robust designs that are optimal over the range of variable conditions because it considers uncertainties during the optimization process. The basic reliability-based multidisciplinary optimization methodology has been demonstrated to design engine components, aircraft lap joints, and transport aircraft wings. Herein this methodology is reviewed and then the focus is on demonstrating a new framework that makes it possible to use these methods with commercial CAD∕CAE tools and support commercial shape parameterization to enable shape optimization and consideration of manufacturing uncertainties.     

Probabilistic Capacity Models and Fragility Estimates for Reinforced Concrete Columns based on Experimental Observations

A methodology to construct probabilistic capacity models of structural components is developed. Bayesian updating is used to assess the unknown model parameters based on observational data. The approach properly accounts for both aleatory and epistemic uncertainties. The methodology is used to construct univariate and bivariate probabilistic models for deformation and shear capacities of circular reinforced concrete columns subjected to cyclic loads based on a large body of existing experimental observations. The probabilistic capacity models are used to estimate the fragility of structural components. Point and interval estimates of the fragility are formulated that implicitly or explicitly reflect the influence of epistemic uncertainties. As an example, the fragilities of a typical bridge column in terms of maximum deformation and shear demands are estimated.     

Fully Probabilistic Framework for Evaluating Excavation-Induced Damage Potential of Adjacent Buildings

This paper presents a framework for a fully probabilistic analysis of the potential for damage to buildings adjacent to an excavation. Herein, the damage potential index (DPI), which is a function of angular distortion and lateral strain, is used to assess building damage potential. A serviceability limit state is established in which the resistance is expressed in terms of the “limiting” DPI, and the load is represented by the “applied” DPI. In this context, damage to the building adjacent to an excavation is said to occur deterministically if the applied DPI is greater than the limiting DPI. For the fully probabilistic analysis, both parameter and model uncertainties of the limiting and applied DPIs are first characterized. The analysis framework is then presented and demonstrated with a case history. Finally, sensitivity analysis is performed to identify the factors to which the probability of damage is most sensitive and to analyze the effect of various assumptions of the input parameters on the computed probability of building damage.     

Dynamic Response of Closed-Loop System with Uncertain Parameters Using Interval Finite-Element Method

Using the interval finite-element method, the vibration control problem of structures with interval parameters is discussed, which is approximated by a deterministic one. Based on the first-order Taylor expansion, a method to solve the interval dynamic response of the closed-loop system is presented. The expressions of the interval stiffness and interval mass matrix are developed directly with the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of dynamic responses is developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The effect of the different levels of uncertainties of interval parameters on responses is discussed. The comparison of the present method with the classical random perturbation is given, and the numerical results show that the present method is valid when the parameter uncertainties are small compared with the corresponding mean values.     

Probabilistic Seismic Demand Models and Fragility Estimates for Reinforced Concrete Highway Bridges with One Single-Column Bent

In performance-based seismic design, general and practical seismic demand models of structures are essential. This paper proposes a general methodology to construct probabilistic demand models for reinforced concrete (RC) highway bridges with one single-column bent. The developed probabilistic models consider the dependence of the seismic demands on the ground motion characteristics and the prevailing uncertainties, including uncertainties in the structural properties, statistical uncertainties, and model errors. Probabilistic models for seismic deformation, shear, and bivariate deformation-shear demands are developed by adding correction terms to deterministic demand models currently used in practice. The correction terms remove the bias and improve the accuracy of the deterministic models, complement the deterministic models with ground motion intensity measures that are critical for determining the seismic demands, and preserve the simplicity of the deterministic models to facilitate the practical application of the proposed probabilistic models. The demand data used for developing the models are obtained from 60 representative configurations of finite-element models of RC bridges with one single-column bent subjected to a large number of representative seismic ground motions. The ground motions include near-field and ordinary records, and the soil amplification due to different soil characteristics is considered. A Bayesian updating approach and an all possible subset model selection are used to assess the unknown model parameters and select the correction terms. Combined with previously developed capacity models, the proposed seismic demand models can be used to estimate the seismic fragility of RC bridges with one single-column bent. Seismic fragility is defined as the conditional probability that the demand quantity of interest attains or exceeds a specified capacity level for given values of the earthquake intensity measures. As an application, the univariate deformation and shear fragilities and the bivariate deformation-shear fragility are assessed for an example bridge.     

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.     

Efficient Probabilistic Back-Analysis of Slope Stability Model Parameters

Back-analysis of slope failure is often performed to improve one’s knowledge on parameters of a slope stability analysis model. In a failed slope, the slip surface may pass through several layers of soil. Therefore, several sets of model parameters need to be back-analyzed. To back-analyze multiple sets of slope stability parameters simultaneously under uncertainty, the back-analysis can be implemented in a probabilistic way, in which uncertain parameters are modeled as random variables, and their distributions are improved based on the observed slope failure information. In this paper, two methods are presented for probabilistic back-analysis of slope failure. For a general slope stability model, its uncertain parameters can be back-analyzed with an optimization procedure that can be implemented in a spreadsheet. When the slope stability model is approximately linear, its parameters can be back-analyzed with sensitivity analysis instead. A feature of these two methods is that they are easy to apply. Two case studies are used to illustrate the proposed methods. The case studies show that the degrees of improvement achieved by the back-analysis are different for different parameters, and that the parameter contributing most to the uncertainty in factor of safety is updated most.     

Model Uncertainty in Finite-Element Analysis: Bayesian Finite Elements

In this paper, probabilistic models for structural analysis are put forward, with particular emphasis on model uncertainty. Context is provided by the finite-element method and the need for probabilistic prediction of structural performance in contemporary engineering. Sources of model uncertainty are identified and modeled. A Bayesian approach is suggested for the assessment of new model parameters within the element formulations. The expressions are formulated by means of numerical “sensors” that influence the model uncertainty, such as element distortion and degree of nonlinearity. An assessment procedure is proposed to identify the sensors that are most suitable to capture model uncertainty. This paper presents the general methodology and specific implementations for a general-purpose structural element. Two numerical examples are presented to demonstrate the methodology and its implications for probabilistic prediction of structural response.     

Probabilistic Model for Polycrystalline Microstructures with Application to Intergranular Fracture

A two part probabilistic model for polycrystalline microstructures is described. The model utilizes a Poisson–Voronoi tessellation for the grain geometry and a vector random field model for the crystallographic orientation. The grain geometry model is calibrated to experimental data through the intensity of the Poisson point field underlying the Poisson–Voronoi tessellation and the orientation random field is calibrated to experimental data through its marginal distributions and second moment properties. Realizations of the random microstructure are generated by use of translation methods and are used, with simplified mechanical models, to investigate the problem of intergranular fracture. It is found that intergranular cracks exhibit some statistical properties of a scaled Brownian motion process.