# Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures
## 1 Introduction  
This paper presents statistical analyses performed on fatigue data obtained from [1], where laboratory fatigue tests were performed on reinforcement bars (rebars).  
General methods and techniques utilized for risk and reliability assessment of civil engineering structures are presented [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18].  
Statistical analyses of the data are an essential step for the stochastic modeling of the material fatigue uncertainties, which can next be used as a basis for a probabilistic modeling and reliability analysis [19] of structures with reinforced concrete components, such as wind turbines and bridges [20; 21]. Usually, foundations for onshore wind turbines are constructed by the use of reinforced concrete, which is also used in many bridges. Therefore, the development of stochastic models for the fatigue limit state and estimation of the resulting reliability can be considered as a contribution to reliability assessment of these types of structures, with respect to fatigue failure and also as the basis for the development of optimal strategies for the maintenance of wind turbines and bridges. [22].  
Several methodologies can be used to estimate the statistical parameters. For instance: Maximum Likelihood Method (MLM), moment method, least square method, and Bayesian statistics. In the literature, there are some recommendations indicating which of these methods could be more suitable.  
At the same time, there is no unique answer to this question, especially for a fatigue case study on rebars. On the reliability assessment, choosing a specific method has a direct influence. In the reliability assessment, there is a need to have stochastic modeling for the material-resistance as well as for the loads. In this paper, the material-resistance model is presented in detail, and at the end, using a generic stochastic model for the fatigue load reliability results of a composite bridge are presented.  
The MLM is chosen in this study as it gives an estimate of the statistical uncertainties [23], MLM is considered for fitting the statistical parameters [2] in a regression model for fatigue strength. Typically, the statistical analyses are based on a limited number of data, for which MLM can provide estimates of the uncertainties associated with each of these parameters and the correlation between the parameters [24]. This paper also presents the use of the Bootstrap method, which generates synthetic data based on the available measurements from the experiment.  
Further, Bayesian statistics is considered taking subjective/prior information into account. This is done with application of Bayesian inference with a Markov Chain Monte Carlo implementation [25; 26; 27]. Bayesian updating is an appropriate tool to update the structural performance function for fatigue by applying the information from the structural health monitoring and the prior information about different fatigue parameters. The aim is to compare the results of different methodologies and to provide information in order to select an appropriate method.  
To study the effect of uncertainty of fatigue resistance model on the fatigue reliability of a structure, a case study of Swiss viaduct Cret De l'Anneau is presented. For this structure, long term strain monitoring data on critical reinforcement is available.