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### Guest Editorial

ASME J. Risk Uncertainty Part B. 2017;3(3):030301-030301-1. doi:10.1115/1.4036707.
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The fractional calculus, namely, the calculus of integrals and derivatives of real or complex order, has captured considerable popularity and importance in engineering applications during the last decades. The considerable number of scientific publications concerning fractional calculus and its applications in various and widespread fields of science and engineering demonstrates the importance of this mathematical tool. The growing interest in this field is due to the fact that the generalization of differential and integral operators provides some new tools to model, simulate, represent, and solve different kinds of engineering problems. Some examples are viscoelasticity, diffusive transport, electrical networks, probability and statistics, control theory of dynamical systems, chemical physics, optics and signal processing, and so on. Moreover, recently some important applications have regarded various engineering problems in stochastic mechanics. In fact, it has been proved that the fractional calculus is a valuable mathematical tool for the characterization of stochastic processes and random variables, for the simulation of external loads on structures, for the modeling of the internal damping of the medium, for the solution of the Fokker–Planck equation, etc. Basically, the fractional operators are commonly used as purely mathematical tools to solve stochastic differential equations and/or to describe and model mechanical behavior of real structures. In the first case, some contributions regard the solution of Langevin equations, the use of Mellin transform for the evaluation of complex-order spectral moments, the description of random variable and random processes, etc. In the second case, several applications of Caputo and Riemann–Liouville operators have showed the capabilities of these operators in the description of the mechanical behavior of structures and the modeling of real material constitutive laws. In particular, they are able to model nonlocality, viscoelasticity and fading memory effect, multiscale structures, Brownian motions and anomalous diffusion, etc.

Commentary by Dr. Valentin Fuster

### SPECIAL SECTION PAPERS

ASME J. Risk Uncertainty Part B. 2017;3(3):030901-030901-6. doi:10.1115/1.4036700.

In the last decades, the research community has shown an increasing interest in the engineering applications of fractional calculus, which allows to accurately characterize the static and dynamic behavior of many complex mechanical systems, e.g., the nonlocal or nonviscous constitutive law. In particular, fractional calculus has gained considerable importance in the random vibration analysis of engineering structures provided with viscoelastic damping. In this case, the evaluation of the dynamic response in the frequency domain presents significant advantages, once a probabilistic characterization of the input is provided. On the other hand, closed-form expressions for the response statistics of dynamical fractional systems are not available even for the simplest cases. Taking advantage of the residue theorem, in this paper the exact expressions of the spectral moments of integer and complex orders (i.e., fractional spectral moments of linear fractional oscillators driven by acceleration time histories obtained as samples of stationary Gaussian white noise processes are determined.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030902-030902-5. doi:10.1115/1.4036806.

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030903-030903-5. doi:10.1115/1.4036701.

The paper deals with the stochastic dynamics of a vibroimpact single-degree-of-freedom system under a Gaussian white noise. The system is assumed to have a hard type impact against a one-sided motionless barrier, located at the system's equilibrium. The system is endowed with a fractional derivative element. An analytical expression for the system's mean squared response amplitude is presented and compared with the results of numerical simulations.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030904-030904-7. doi:10.1115/1.4036702.

Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030905-030905-7. doi:10.1115/1.4036703.

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030906-030906-7. doi:10.1115/1.4036704.

It is well known that mechanical parameters of polymeric materials, e.g., epoxy resin, are strongly influenced by the temperature. On the other hand, in many applications, the temperature is not known exactly during the design process and this introduces uncertainties in the prevision of the behavior also when the stresses are deterministic. For this reason, in this paper, the mechanical behavior of an epoxy resin is characterized by means of a fractional viscoelastic model at different temperatures; then, a simple method to characterize the response of the fractional viscoelastic material at different temperatures modeled as a random variable with assigned probability density function (PDF) subjected to deterministic stresses is presented. It is found that the first- and second-order statistical moments of the response can be easily evaluated only by the knowledge of the PDF of the temperature and the behavior of the parameters with the temperature. Comparison with Monte Carlo simulations is also performed in order to assess the accuracy and the reliability of the method.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030907-030907-6. doi:10.1115/1.4036705.

In this paper, interval fractional derivatives are presented. We consider uncertainty in both the order and the argument of the fractional operator. The approach proposed takes advantage of the property of Fourier and Laplace transforms with respect to the translation operator, in order to first define integral transform of interval functions. Subsequently, the main interval fractional integrals and derivatives, such as the Riemann–Liouville, Caputo, and Riesz, are defined based on their properties with respect to integral transforms. Moreover, uncertain-but-bounded linear fractional dynamical systems, relevant in modeling fractional viscoelasticity, excited by zero-mean stationary Gaussian forces are considered. Within the interval analysis framework, either exact or approximate bounds of the variance of the stationary response are proposed, in case of interval stiffness or interval fractional damping, respectively.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):030908-030908-8. doi:10.1115/1.4036706.

This paper presents a nonlinear rubber spring model for the primary suspension of the railway vehicle, which can effectively describe the amplitude dependency and the frequency dependency of the rubber spring, by taking the elastic force, the fractional derivative viscous force, and nonlinear friction force into account. An improved two-dimensional vehicle–track coupled system is developed based on the nonlinear rubber spring model of the primary suspension. Nonlinear Hertz theory is used to couple the vehicle and track subsystems. The railway vehicle subsystem is regarded as a multibody system with ten degrees-of-freedom, and the track subsystem is treated as finite Euler–Bernoulli beams supported on a discrete–elastic foundation. Mechanical characteristic of the rubber spring due to harmonic excitations is analyzed to clarify the stiffness and damping dependencies on the excitation frequency and the displacement amplitude. Dynamic responses of the vehicle–track coupled dynamics system induced by the welded joint irregularity and random track irregularity have been performed to illustrate the difference between the Kelvin–Voigt model and the proposed model in the time and frequency domain.

Commentary by Dr. Valentin Fuster

### Research Papers

ASME J. Risk Uncertainty Part B. 2017;3(3):031001-031001-12. doi:10.1115/1.4035399.

Corrosion in ship structures is influenced by a variety of factors that are varying in time and space. Existing corrosion models used in practice only partially address the spatial variability of the corrosion process. Typical estimations of corrosion model parameters are based on averaging measurements for one ship type over structural elements from different ships and operational conditions. Most models do not explicitly predict the variability and correlation of the corrosion process among multiple locations in the structure. This correlation is of relevance when determining the necessary inspection coverage, and it can influence the reliability of the ship structure. In this paper, we develop a probabilistic spatiotemporal corrosion model based on a hierarchical approach, which represents the spatial variability and correlation of the corrosion process. The model includes as hierarchical levels vessel–compartment–frame–structural element–plate element. At all levels, variables representing common influencing factors (e.g., coating life) are introduced. Moreover, at the lowest level, which is the one of the plate element, the corrosion process can be modeled as a spatial random field. For illustrative purposes, the model is trained through Bayesian analysis with measurement data from a group of tankers. In this application, the spatial dependence among corrosion processes in different parts of the ships is identified and quantified using the proposed hierarchical model. Finally, how this spatial dependence can be exploited when making inference on the future condition of the ships is demonstrated.

Topics: Corrosion , Vessels , Ships
Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):031002-031002-9. doi:10.1115/1.4035439.

Optimization for crashworthiness is of vast importance in automobile industry. Recent advancement in computational prowess has enabled researchers and design engineers to address vehicle crashworthiness, resulting in reduction of cost and time for new product development. However, a deterministic optimum design often resides at the boundary of failure domain, leaving little or no room for modeling imperfections, parameter uncertainties, and/or human error. In this study, an operational model-based robust design optimization (RDO) scheme has been developed for designing crashworthiness of vehicle against side impact. Within this framework, differential evolution algorithm (DEA) has been coupled with polynomial correlated function expansion (PCFE). An adaptive framework for determining the optimum basis order in PCFE has also been presented. It is argued that the coupled DEA–PCFE is more efficient and accurate, as compared to conventional techniques. For RDO of vehicle against side impact, minimization of the weight and lower rib deflection of the vehicle are considered to be the primary design objectives. Case studies by providing various emphases on the two objectives have also been performed. For all the cases, DEA–PCFE is found to yield highly accurate results.

Commentary by Dr. Valentin Fuster
ASME J. Risk Uncertainty Part B. 2017;3(3):031003-031003-8. doi:10.1115/1.4035704.

Industry has been implementing condition monitoring (CM) for turbines to minimize losses and to improve productivity. Deficient conditions can be identified before losses occur by monitoring the equipment parameters. For any loss scenario, the effectiveness of monitoring depends on the stage of the loss scenario when the deficient condition is detected. A scenario-based semi-empirical methodology was developed to assess various types of condition monitoring techniques, by considering their effect on the risk associated with mechanical breakdown of steam turbines in the forest products (FPs) industry. A list of typical turbine loss scenarios was first generated by reviewing loss data and leveraging expert domain knowledge. Subsequently, condition monitoring techniques that can mitigate the risk associated with each loss scenario were identified. For each loss scenario, an event tree analysis (ETA) was used to quantitatively assess the variations in the outcomes due to condition monitoring, and resultant changes in the risk associated with turbine mechanical breakdown. An application was developed following the methodology to evaluate the effect of condition monitoring on turbine risk mitigation.

Commentary by Dr. Valentin Fuster
Select Articles from Part A: Civil Engineering

### Technical Papers

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2016;3(3):. doi:10.1061/AJRUA6.0000899.
Abstract

Abstract  Traffic congestion is a serious challenge that urban transportation systems are facing. Variable speed limit (VSL) systems are one of the countermeasures to reduce traffic congestion and smooth traffic flow on roadways. The negative impacts of congestion, including road rage, air pollution, safety issues, and traffic delays, are well recognized. The impact of unexpected delays on road users is quantified through travel time reliability (TTR) measures. In this study, a bilevel optimization problem was introduced to determine location, speed limit reduction, start time, and duration of limited number of VSL signs while maximizing travel time reliability on selected critical paths on a network. The upper-level problem focuses on TTR optimization whereas the lower-level problem assigns traffic to the network using a dynamic traffic assignment simulation tool. A heuristic approach, simulated annealing, was used to solve the problem. The application of the methodology to a real roadway network is shown and results are discussed. The proposed methodology could assist traffic agencies in making proper decisions on how to allocate their limited resources to the network to maximize the benefits.

Topics:
Reliability , Simulation , Optimization
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2017;3(3):. doi:10.1061/AJRUA6.0000909.
Abstract

Topics:
Wind velocity , Climate change
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2017;3(3):. doi:10.1061/AJRUA6.0000904.
Abstract

Abstract  In some regions, sea level rise due to climate change is expected to increase saltwater intrusion in coastal aquifers, leading to increased salt levels in drinking water wells relying on these supplies. Seawater contains elevated concentrations of bromide, which has been shown to increase the formation and alter the speciation of disinfection by-products (DBPs) during the treatment process. DBPs have been associated with increased risk of cancer and negative reproductive outcomes, and they are regulated under drinking water standards to protect human health. This paper incorporates statistical simulation of changes in source water bromide concentrations as a result of potential increased saltwater intrusion to assess the associated impact on trihalomethane (THM) formation and speciation. Additionally, the health risk associated with these changes is determined using cancer slope factors and odds ratios. The analysis indicates that coastal utilities treating affected groundwater sources will likely meet regulatory levels for THMs, but even small changes in saltwater intrusion can have significant effects on finished water concentrations and may exceed desired health risk threshold levels due to the extent of bromination in the THM. As a result of climate change, drinking water utilities using coastal groundwater or estuaries should consider the implications of treating high bromide source waters. Additionally, extra consideration should be taken for surface water utilities considering mixing with groundwater sources, as elevated source water bromide could pose additional challenges for health risk, despite meeting regulatory requirements for THM.

Topics:
Public utilities , Groundwater , Shorelines , Climate change , Health risk assessment

### Corrections

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2017;3(3):. doi:10.1061/AJRUA6.0000907.

### Technical Papers

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering. 2017;3(3):. doi:10.1061/AJRUA6.0000902.
Abstract

Abstract  The conventional simulation model used in the prediction of long-term infrastructure development systems such as public–private partnership (PPP)–build-operate-transfer (BOT) projects assumes single probabilistic values for all of the input variables. Traditionally, all the input risks and uncertainties in Monte Carlo simulation (MCS) are modeled based on probability theory. Its result is shown by a probability distribution function (PDF) and a cumulative distribution function (CDF), which are utilized for analyzing and decision making. In reality, however, some of the variables are estimated based on expert judgment and others are derived from historical data. Further, the parameters’ data of the probability distribution for the simulation model input are subject to change and difficult to predict. Therefore, a simulation model that is capable of handling both types of fuzzy and probabilistic input variables is needed and vital. Recently fuzzy randomness, which is an extension of classical probability theory, provides additional features and improvements for combining fuzzy and probabilistic data to overcome aforementioned shortcomings. Fuzzy randomness–Monte Carlo simulation (FR-MCS) technique is a hybrid simulation method used for risk and uncertainty evaluation. The proposed approach permits any type of risk and uncertainty in the input values to be explicitly defined prior to the analysis and decision making. It extends the practical use of the conventional MCS by providing the capability of choosing between fuzzy sets and probability distributions. This is done to quantify the input risks and uncertainties in a simulation. A new algorithm for generating fuzzy random variables is developed as part of the proposed FR-MCS technique based on the $α$-cut. FR-MCS output results are represented by fuzzy probability and the decision variables are modeled by fuzzy CDF. The FR-MCS technique is demonstrated in a PPP-BOT case study. The FR-MCS results are compared with those obtained from conventional MCS. It is shown that the FR-MCS technique facilitates decision making for both the public and private sectors’ decision makers involved in PPP-BOT projects. This is done by determining a negotiation bound for negotiable concession items (NCIs) instead of precise values as are used in conventional MCS results. This approach prevents prolonged and costly negotiations in the development phase of PPP-BOT projects by providing more flexibility for decision makers. Both parties could take advantage of this technique at the negotiation table.

Topics:
Simulation , Chaos