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The structural performance depends on load and resistance parameters. These parameters are time-varying random variables. Therefore, reliability can be utilized as a rational measure of structural performance. This paper focused on the effect of material related parameters on the reliability of concrete components. Properties of concrete change with time, depending on many factors including proportions of the mix, initial stages such as curing, conditions and maturing, concrete volume and the effect of hydration hear, and exposure to aggressive environments such as pollution and salt.

In this chapter, we present a framework for m-person stochastic games with an infinite state space. Our main purpose is to present a correlated equilibrium theorem proved by Nowak and Raghavan [42] for discounted stochastic games with a measurable state space, where the correlation of the different players’ strategies employs only “public signals” [16]. We will also provide a detailed survey of the literature containing related results, some approximation theorems for general state space stochastic games (the existence of e-equilibria), and the existence of equilibria in some classes of countable state space stochastic games with applications to queueing models.

The finite state space stochastic game model by Shapley [31] covered in [33] was generalized among others by Maitra and Parthasarathy [17], [33] who considered compact metric state spaces. They imposed rather strong regularity conditions on the reward and transition structure in the game and considered the discounted payoff criterion only. Their results have been generalized by many authors. For a good survey of the results which are not reported here the reader is referred to [14], [23], [25], [27], [13].

Corrosion and fatigue are important considerations that affect the structural performance of bridges and this applies to structural steel, reinforced concrete, and prestressed concrete. The optimum management of bridges requires the prediction of the remaining life for partially deteriorated structures. Therefore, this paper deals with the development of reliability-based models for time-varying changes in structural performance due to corrosion and fatigue. Fatigue limit state is one of the important considerations in the design of bridges. Accumulated load cycles can cause cracking or even failure. Reliability analysis is performed for the fatigue limit state function of flexure. Prediction of the remaining fatigue life for steel and concrete beams is illustrated by using examples of existing bridge girders. Corrosion causes a loss of the section and reduction of the load carrying capacity and the rate of corrosion is a random variable. Three levels of corrosion are considered in the paper and the changes in reliability are evaluated as a function of time, which covers the life-time of the bridge.

Evaluation of concrete bridge decks is performed due to natural aging, increasing load spectra, deterioration caused by fatigue, freeze/thaw cycles, shrinkage and temperature gradient. Based on field observations and tests, the analysis is provided for different damage scenarios. The FE model is built to show the behavior of whole superstructure and to check the influence of particular parameters affecting deck degradation. Model is calibrated using strains and deflections measured on a real bridge.

We provide a new characterization of the weighted Banzhaf value derived from some postulates in a recent paper by Radzik, Nowak and Driessen [7]. Our approach owes much to the work by Lehrer [4] on the classical Banzhaf value based on the idea of amalgamation of pairs of players and an induction construction of the value. Compared with the approach in [7] we consider two new postulates: a weighted version of Lehrer's "2-efficiency axiom" [4] and a generalized "null player out" property studied in terms of symmetric games by Derks and Haller [2].

The objective of this paper is to investigate the feasibility of the development of an efficient system to control the highway load effects and bridge deterioration. Such an integrated system may involve the monitoring and control of various parameters including: • vehicle weight, axle loads, axle configuration, speed, multiple presence • traffic volume (average daily truck traffic, ADTT) • vehicle position within the width of the bridge • load effects (moment, shear force, stress, strain and deflection) in bridge components • identification of critically overloaded vehicles • minimum load carrying capacity of the bridge.

This chapter discusses stochastic games. We focus on nonzero-sum games and provide a detailed survey of selected recent results. In Section 1, we consider stochastic Markov games. A correlation of strategies of the players, involving “public signals,” is described, and a correlated equilibrium theorem proved recently by Nowak and Raghavan for discounted stochastic games with general state space is presented. We also resport an extension of this result to a class of undiscounted stochastic games, satisfying some uniform ergodicity condition. Stopping games are related to stochastic Markov games. In Section 2, we describe a version of Dynkin’s game related to observation of a Markov process with random assignment mechanism of states to the players. Some recent constributions of the second author in this area are reported. The chapter also provides a brief overview of the theory of nonzero-sum stochastic games and stopping games.

This chapter presents code development procedures. Design code plays a central role in the building process. It specifies the requirements for the designer so that a minimum acceptable safety level is provided. The current codes specify load models (design loads) and resistance (design load carrying capacity). Safety reserve is implemented through conservative load and resistance factors.

Bridge reliability analysis is based on load and resistance models. In particular, load spectra vary depending on site and component. The objective of this paper is to present the results of measurements of truck loads on girder bridges. The study included several structures located on interstate highways and surface streets.

This paper treats of nonzero-sum average payoff stochastic games with arbitrary state spaces. Such models of games very well fit in some studies in economic theory. A natural uniform geometric ergodicity condition, often used in control theory of Markov chains, is imposed on the transition probabilities of the games. A correlation of strategies of the players, involving “public signals”, is allowed in this paper. The main result is an extension of the correlated equilibrium theorem proved recently by the author and Raghavan for dynamic games with discounting to the average payoff stochastic games. Also some special classes of games that possess Nash equilibria without public signals are discussed. This paper also provides a brief overview of the theory of nonzero-sum stochastic games which is very far from being complete.

Nonzero-sum stochastic games play an increasing role in studying various dynamic economic models. In most mentioned applications it is assumed that the state spaces of the games are uncountable. For a broad discussion on these topics we refer the reader to [2], [3] and their references. The aim of this presentation is to report some correlated and/or Nash equilibrium theorems for a large class of nonzero-sum stochastic games with a continuum of states contained in [11], [12], and [13].

Bridge load and resistance parameters involve a considerable degree of uncertainty. Therefore, the reliability analysis can be used to quantify the risk. There has been a considerable research activity in the area of structural reliability in the last two decades. There are methods available for calculation of the probability of failure for components of buildings, bridges and other structures. They are summarized in several books (e.g. Thoft-Christensen and Baker 1982; Ang and Tang 1986; Madsen, Krenk and Lind 1986; Melchers 1987). For given statistical models of loads and resistance the available methodology provides accurate results. Structural safety was evaluated for bridge components (e.g. Nowak and Grouni 1990).

Planning of inspections is an important part of a bridge management system. It has been based mostly on judgement, and some countries have only one type of periodical inspection of steel bridges. The intervals between inspections are usually between 1 and 6 years. Other countries have superficial inspections every year or more often and more detailed inspections every 3–5 years. However, with limited resources available for maintenance there is a growing interest in optimization of inspection strategies. Sorensen & Thoft-Christensen [1] considered the problem of concrete bridges subjected to corrosion of reinforcement. An optimum approach with two types of inspection was determined for a T-beam. Sommer & Thoft-Christensen [2] described the work done on optimization of inspection strategies.

Stochastic games were first formulated by Shapley in 1953. In his fundamental paper Shapley [13] established the existence of value and optimal stationary strategies for zero-sum β-discounted stochastic games with finitely many states and actions for the two players.

This paper treats of nonzero-sum discounted stochastic games with general state space where the players are allowed to use correlated strategies. The concept of correlated weak equilibrium of Moulin and Vial (being an extension of the Nash equilibrium point) is adopted to such games. An existence theorem is proved for a class of discounted stochastic games with a Borel state space where stationary Nash equilibria are not known to exist.

Reinforcing steel area may be reduced due to corrosion or mechnical damage. Moment-curvature relationships are obtained for reinforced concrete beams using the developed algorithm. The analysis is based on stress-strain curves for materials (steel and concrete). The developed iterative numerical procedure is demonstrated on a typical concrete bridge T-beam. Moment-curvature relationships are obtained for various degrees of reinforcing steel area loss.

Reliability analysis procedure is formulated for girder bridges. Reliability indices are calculated for composite steel girders. System reliability is also calculated and compared to girder reliability. The effect of correlation between the resistances of individual girders is investigated. Sensitivity functions are developed relating girder and system reliabilities with material properties, dimensions and load components. The effect of a partial reduction of girder reliability on the system reliability is evaluated.

Publisher Summary This chapter presents the treats of zero-sum discrete-time stochastic games with Borel state space in which all the primitive data are nonstationary. The transition law in the framework is a sequence of Borel measurable transition probabilities associating with every n-stage history of a game a probability distribution of the (n + l)th state and the payoff is the limit of a nondecreasing sequence of finite stage payoffs. Such games are called nonstationary and constitute a natural extension of discounted as well as positive Markov games. The zero-sum Markov games with time average payoff are not contained in the framework. The chapter discusses two different information structures for nonstationary stochastic games. The first one is the standard information pattern where at each moment both players know the entire history of the play. The second one concern situations where each player, remembering all previous choices, can learn about the actions of his opponent only through the states of the system occurring during the play.

There is a need for efficient diagnostic procedures for evaluation of the current capacity of a structure, monitoring of load and resistance history, and evaluation of the accumulated damage. The available diagnostic techniques are reviewed. They include the methods based on various interrogating media: electromagnetic wave, stress wave, electricity, magnetism, and charged particles. Also considered are the approaches to signal processing. Failure detection and isolation procedures developed in other branches of engineering are summarized. The current trends in the development of new techniques are discussed. The areas considered include microwave, acoustic and optical techniques, integrated sensor systems and motion measurements. Currently used bridge load monitoring procedures are also summarized.

A model for evaluating the fatigue life of a prestressed concrete girder bridge is presented. Experimental studies indicate that fatigue is a random phenomenon. Therefore, the approach is based on probabilistic methods. The fatigue life of the prestressed concrete girder is estimated from the fatigue properties of its components, namely, the cast-in-place slab, precast beam, prestressing steel, and, if partially prestressed, the tension reinforcement. The model utilizes the available S–N curves for structural materials and Miner's rule to evaluate the fatigue life under variable amplitude loading. A model for determining the distribution function of time to first cracking is also presented. To demonstrate the developed models, a typical prestressed concrete girder bridge is analyzed. On the basis of this investigation, it is concluded that fatigue in prestressed concrete girder bridges designed according to the American Association of State Highway and Transportation Officials specifications is not a limiting design criterion. Key words: bridges, fatigue, prestressed concrete, structural reliability.

Steel girder bridges corrode due to environmental exposure. The result is a reduction in both the carrying capacity of a bridge and level of certainty concerning what the capacity may be. As a consequence, the level of safety diminishes with corrosion. A damage model is developed which evaluates the reliability of a corroded steel girder bridge over time. This model is used to evaluate the effects of bridge design and environment on safety. A sensitivity analysis is carried out to identify the most important parameters in corroded bridge safety.

This paper deals with the evaluation of seismic risk for commercial buildings in Memphis, Tennessee. The seismicity of the area is summarized, and commercial buildings are divided into categories with regard to parameters such as number of stories, year of construction, assessed value, total floor area, and structural type. The distributions of these parameters are presented in the figures. During the study, over 15 buildings were examined on site by a team of experts to evaluate their seismic resistances. The quality of the design, materials, and construction was found to be surprisingly good, particularly in those structures built since 1900. Seismic resistance is analytically evaluated for five buildings: a four-story reinforced concrete frame; a four-story steel structure with vertical trusses; a 13-story stell frame; and two multi-story reinforced concrete frames. The loadings from four sources are considered: EI Centro and Taft earthquakes in California (1940 and 1952, respectively) and the forces specified in the 1979 UBC and 1981 BOCA codes. Ratios of load to capacity are calculated. For each building considered, the expected percentage of damage is evaluated for the two earthquakes. The damage prediction is extended to all commercial buildings in Memphis.

Uncertainties in structures result from natural hazards, man‐made hazards, variations within common practice, and accepted departures from common practice. However, the major source of structural uncertainty and failures is departure from accepted practice, i.e., human error. The purpose of an error control strategy is to lower the frequency of errors which have large consequences. Sensitivity functions which relate structural errors to structural safety are bases for a control strategy. Sensitivity analysis is performed for several practical cases to demonstrate calculation of sensitivity functions and its importance in establishing an error control strategy.

System reliability of bridge structures is investigated. Three terms are used to describe the standard reliability: structure function, reliability network, and reliability function. A bridge structure is represented by a reliability network which makes it possible to formulate the reliability function of the system. The reliability function is used to evaluate the reliability of the system. Reliability network and the reliability function are formulated for various types of bridges. Effects of component failures on system reliability are illustrated with examples. Key words: bridges, mathematical models, redundancy, system reliability.

The paper summarizes the calibration procedure used to calculate load and resistance factors for the Ontario Bridge Design Code 1983 edition. The limit states considered include serviceability and ultimate limit states during service and in construction. The acceptance criterion is closeness to a predetermined target safety level. Safety is measured in terms of a reliability index. The results of calibration are discussed for composite steel–concrete girders, pretensioned concrete girders, post-tensioned concrete decks, and timber decks. The analysis of construction design criteria is demonstrated on segmental bridges. Key words: code calibration, bridges, reliability index, load and resistance factors, limit states.

On considere un jeu a somme nulle et a 2 joueurs et on donne certaines conditions generales nouvelles qui assurent l'existence et la mesurabilite de la fonction valeur du jeu et l'existence de strategies mesurables ξ-optimales pour les deux joueurs

Structural reliability depends on uncertainties in resistance and loads. In many practical cases the resistance dominates and a reduction of uncertainty about resistance is an effective way of increasing safety. It can be accomplished by proof loading. A truncated distribution is considered and reliability indices are calculated for various proof load levels. The structural reliability is sensitive to proof loading for larger coefficients of variation of resistance. A Bayesian approach is applied to develop a posterior distribution for resistance, after proof loading. Reliability indices are calculated for various ratios of the coefficients of variation of load and resistance.

The paper deals with a procedure that can be used by a design standards committee to determine safety indices for structures and calculate optimum performance factors for limit states design. Six load components are considered: dead, sustained live, transient live, snow, wind, and earthquake load effects. The procedure is based on an approximation of the probability distributions by normal distributions in the design point situated on the failure boundary. The objective in selection of performance factors is closeness to a target safety level expressed in terms of the target safety index. A computer program was developed to be used in calibration. The procedure is illustrated by examples.

A procedure has been developed to calibrate safe O' indices for various types of structures. The load model is based on normal approximations of non-normal distributions at the design point. For time varying loads, the basic time intervals are considered, with probabilities of occurrence in each interval and conditional distributions of intensity. Load and resistance factor format and allowable stress design are considered. Computations were carried out for the National Building Code of Canada. Safety indices are presented in charts and tables. The results can be used to determine optimum values of resistance factors and/or safety factors.