# Performing Fatigue State Characterization in Railway Steel Bridges Using Digital Twin Models
## 2 Fatigue Analysis System (FAS)  
Summary: This section addresses the fatigue analysis methodologies, focusing on the methods outlined in the Eurocodes, part of which was implemented in this work. Additionally, reference is made to the enhancement implemented in an existing FAS through the use of a DT model. The fatigue analysis methodology in steel bridges is fundamentally based on three approaches, namely the Stress-Life Approach, Strain-Life Approach, and the Fracture Mechanic Approach, each with its field of application, with use depending on the interest of the analysis, which is also associated with the global or local approach [26; 27]. The Stress-Life Approach (also known as the S-N method) is a global method on which the Eurocode 3, part 1-9 (EC3-1-9 [7]) and many other standards are based and is the most widely used in design [2; 26]. The method is based on S-N curves (stress number of cycles to failure) and assumes that the elastic stress state controls the fatigue behavior, generally an accurate representation of high cycle fatigue (HCF). Depending on the stress analysis of the structural detail, this method can be subdivided into the Nominal Stress Method, Hot Spot Stress Method, and Effective Notch Stress Method [25; 26]. The first is the most common, and according to EC3-1-9 [7], is called nominal stress, the stress in the parent material adjacent to a potential crack location, calculated by elastic theory without considering any stress concentration effects. When, however, stress concentration is essential, the last two may be used according to the specific situation. Based on the Stress-Life Method, the Eurocodes establish two different methodologies for assessing fatigue resistance in particular structural detail:  
1. The Equivalent Constant Amplitude Stress Range Method (ECASRM) (also known as \(\lambda\)-coefficient method): the method is significantly simplified as it does not consider variation in stress amplitude and is suitable for evaluations under quasi-static effects when dynamic analyses are not necessary, considering the applicable load models defined in EC1-2 [28]. Safety is considered satisfied when the conditions expressed by the inequalities (1) and (2) below are verified: \[\gamma_{Ff}.\Delta\sigma_{\text{E2}}\leq\frac{\Delta\sigma_{\text{c}}}{\gamma_ {Mf}}\] (1) \[\gamma_{Ff}.\Delta\tau_{\text{E2}}\leq\frac{\Delta\tau_{\text{c}}}{ \gamma_{Mf}}\] (2)In the above expressions, \(\gamma_{Mf}\) and \(\gamma_{Ff}\) represent partial safety factors (resistance and loading sides, respectively), \(\Delta\sigma_{c}\) and \(\Delta\tau_{c}\) represent the fatigue resistance at \(2\times 10^{6}\) cycles, and \(\Delta\sigma_{E2}\) and \(\Delta\tau_{E2}\) express the equivalent constant amplitude stress range at \(2\times 10^{6}\) cycles. When direct and shear stresses act simultaneously, the following condition must be satisfied (Equation (3)): \[\left(\frac{\gamma_{Ff}.\Delta\sigma_{E2}}{\Delta\sigma_{c}/\gamma_{Mf}} \right)^{3}+\left(\frac{\gamma_{Ff}.\Delta\tau_{E2}}{\Delta\tau_{c}/\gamma_{Mf }}\right)^{5}\leq 1.0\] (3)
2. Linear Damage Accumulation Method (LDAM) (also known as Palmgren-Miner rule): the most comprehensive method provided by the Eurocodes and has been widely used in several research projects. Both quasi-static and dynamic analyses can be associated with actual or normative traffic for fatigue specified in EC1-2 [28]. To assess safety, the accumulation of partial fatigue damage corresponding to stress amplitudes \(\Delta\sigma_{i}\) generated from a given stress history is computed (Equation (4)). \[D=\sum\frac{n_{i}}{N_{i}}=\frac{n_{1}}{N_{1}}+\frac{n_{2}}{N_{2}}+\ldots+ \frac{n_{k}}{N_{k}}\leq 1\] (4) where \(D\) represents the accumulated fatigue damage considering a certain cyclic loading; \(n_{i}\) represents the number of load cycles at \(\Delta\sigma_{i}\); and \(N_{i}\) represent the fatigue life expressed in cycles of the detail submitted to the same stress range, \(\Delta\sigma_{i}\), with \(i\) ranging from \(1\) to \(k\).