# Effect of Crack Orientation on Fatigue Life of Reinforced Concrete Bridge Decks
## 2 Methodology and Studied Cases  
### Data Assimilation Technology  
For fatigue-lifetime simulation of RC decks based on site inspected cracks, the data assimilation technology of the multi-scale simulation program and the pseudo cracking method, which are broadly validated [23; 24; 25; 26], were utilized. The constitutive laws of the multi-scale simulation were upgraded in the last few decades to deal with high cycle fatigue, so they can account for fatigue damage of concrete by a decrease in both strength and stiffness, and increase in time dependent deformations on the basis of the direct path integral method [24]. As concrete is a cementitious composite, a single  
Figure 1: Site inspection for bottom surface cracks of reinforced concrete (RC) decks. crack opening (Mode-I) though cement paste solid exhibits tension softening and the fracture energy is much less than that of crack shear transfer of Mode-II (see Figure 2) owing to the interlocking of rough crack surfaces. Then, blunt multiple cracks are consequently dispersed, and the direction of subsequently propagating cracks is assumed to coincide with that of the updated principal stress, which satisfies the equilibrium with non-orthogonal preceding cracks. It means the minimum fracture energy of the whole analysis domain associated with the shear transfer along crack planes and tension softening normal to cracks [24]. The high shear transfer along preceding crack planes of concrete is a fatigue resistant mechanism which differs from the case of smooth preceding crack planes without shear transfer [32]. Generally, the existing crack will propagate if its propagation is associated with a decrease in the total energy of the system [33; 34]. In the case of RC members, the post-shear cracks propagate through the preceding cracks when the shear can transfer from one crack's plane to another. Therefore, the system may easily reach the point of minimum potential energy. On the other hand, when the preceding crack's arrest mechanism occurs, the system is in a stable state, however, more energy is required to create independent shear cracks until forming a failure path. These are so-called cracks interactions, which are explained in detail in later sections. On the other hand, the pseudo cracking method [26] is a numerical technique for fatigue life of RC decks based upon site-inspected cracks on the surfaces of RC decks. Internal unknown cracks are generated in the finite element model in the early cycles of fatigue loading by the corrector-predictor approach based on energy principles in order to satisfy dynamic equilibrium and deformational compatibility [26]. The multi-scale simulation program can successfully capture the response of a multi-cracked element under multi-stress conditions [18; 23], where the global response is an assembly of local crack responses of all the cracks inside the element. The loading conditions of each crack must satisfy the deformational compatibility and equilibrium in its local direction. Through this process, some cracks are activated, while others are idle, resulting in a mechanism called anisotropic crack interaction. The activated cracks dominate the overall behavior, while other cracks are dormant. The activation of pre-cracks reduces the stress concentration in the diagonal direction, where it may affect the response of new cracks. Thus, crack interactions (activation or dormancy) may govern the global structural behavior of RC members. It should be noted that the kinematics of cracks of concrete including opening/closing, slip, and anisotropic interactions are already integrated into the multi-scale simulation program [18; 23]. Figure 3 shows examples of the validation of data assimilation technology. Figure 3a shows an actual RC deck from a site with current damage [26] that was tested experimentally to check its remaining fatigue life. Next, it was analyzed by the data assimilation technology for validation. It is clear that the simulation results are in good agreement with the experimental ones.