## 2 Methodology for Predicting the Remaining Fatigue Life  
Figure 2 shows the methodology for predicting the remaining fatigue life of RC decks based upon site inspected wetting locations on the surface of decks as follows:  
1. Ground penetrating radar is installed on a vehicle that runs about 80 km/h over the bridge, where signal responses of the hidden information of RC decks below the pavement layers are detected [22]. 2. These signals are processed to achieve sound locations of the water especially at the upper surface layer of RC decks. 3. These wetting locations are induced into the finite element model by utilizing the multi-scale simulation program. 4. Travelling wheel load is applied until the failure of RC decks based on a failure criterion that will be stated in a later section. 5. Finally, remaining fatigue life of RC decks is computed. The life-simulation is based upon the multi-scale thermo-hygral analysis [21], where the constitutive laws are upgraded for high cycle fatigue loading and the concrete-water interaction [20; 21] as shown in Figure 3. The micro-fatigue model with cyclic pore pressure was integrated into the multi-scale simulation, where the average stiffness degrades with the disintegration of the aggregates-cement composites during the fatigue simulation of concrete with condensed water. The disintegration phenomenon between aggregates and cement composites is reproduced by integrating the damage evolved with increasing local pressure between aggregates and cement paste. This local pressure, which is also computed by the constitutive model for multi-directional cracked concrete, degrades the bond between cement matrix and aggregates leading to erosion of the composite system [20]. On the basis of previous research of the freeze-thawing mechanism [23] and rate-type of fatigue modelling [24], Equation (1) [21] expresses the overall stresses of disintegrated reinforced concrete, where it is computed by integrating three constitutive models: (1) non-eroded concrete with and without cracking, (2) steel reinforcement and (3) assembly of aggregates. When there is no erosion caused by internal water impact, stresses of the element are rooted in (1) and (2). If full disintegration develops finally, the total stresses come solely from (3). Here, the erosion factor denoted by \(K\) is defined to interpolate these two extreme states. Then, we have  
\[ \begin{array}{l}\sigma_{\rm ij}=\rm K\cdot\sigma_{\rm c,ij}+\sqrt{K}\cdot \sigma_{\rm s,ij}+(1-K)\cdot\sigma_{\rm agg,ij}\\ \rm K=e^{-Z}\\ \rm dZ=-10^{n}\cdot(1+f_{n})\cdot p_{\rm ampl}^{f_{n}}\cdot dp\end{array}  \tag{1}\]  
where \(\sigma_{\rm ij}\) is total compressive stress tensor, \(\sigma_{\rm c,ij}\) and \(\sigma_{\rm s,ij}\) are stress tensors carried by cracked concrete and steel reinforcement, \(\sigma_{\rm agg,ij}\) is the stress tensor representing the fictitious aggregate particles, \(Z\) is set forth as an accumulated damage of concrete in micro-pore structure, (n, f\({}_{n}\)) are coefficients related to the intersection and the slope of S-N diagrams and equal to (2.0, 0.4), respectively, \(\rm p_{\rm ampl}\) is amplitude of pore water pressure. The concrete and steel stresses (\(\sigma_{\text{c,ij}}\) and \(\sigma_{\text{s,ij}}\)) are computed from the given space-averaged strain of finite element by the non-orthogonal multi-directional cracked concrete constitutive model and the aggregate stresses are computed by the multi-spring soil model [21]. During evolution of the disintegration, the cement binder is eroded and the effective stress of the concrete and steel parts is degraded until reaching zero at the same particular time of full erosion (K = 0). Here, the assembly of the remaining aggregates still can sustain compressive stresses from volumetric contraction similar to the behavior of soil particles in the geotechnical field, where the stiffness of the aggregates assembly without cement binder is estimated to be 1/100 of the normal concrete [25].