# Pullout Response of Ultra-High-Performance Concrete with Twisted Steel Fibers
## 4 Numerical Analysis
### Numerical Results  
The FEA results of fiber pullout force were then compared with the average experimental results for the fiber embedment length of 20 mm, as shown in Figure 12. The numerical results match well with the testing results in the Linear debonding stage and the early stage of fiber untwisting; while, the pullout force slightly decreased and stabilized with further pullout displacements from 6 mm to 20 mm, causing a lower simulated pullout force compared to the testing value. The higher testing  
Figure 11: The Elastic Perfectly Plastic model of steel fiber.  
pullout force may be caused by the creep-induced resistance from the 28-day curing concrete, while this creep was not considered in our FEA model. In addition, the tangential frictional resistance after the debonding between the UHPC matrix and the twisted steel fiber is not considered in the FEA model; while minor tangential frictional resistance may still exist, producing the extra pullout force in the testing results.  
Figure 13 shows the evolution of Mises stress response over the pullout course at pullout displacements of 0, 4.0 mm, 8.0 mm, 12.0 mm. 16.0 mm and 20.0 mm. Interestingly, the untwisting phenomenon was found within the effective untwisting fiber length of around 1.7 mm away from the exterior concrete surface. The untwisting was completed when the whole fiber was pulled out at the end of the simulation. The above observation is consistent with our experimental study and others such as [PERSON] et al. [14].  
The above numerical and testing results reveal the strain hardening behavior of UHPFRC with TSFs is caused by the fiber untwisting at the effective untwisting length near the exterior concrete  
Figure 12: Comparison between numerical and experimental results for the case with fiber embedment length of 20 mm.  
Figure 13: The Mises stress evolution over the pullout process.  
surface. This supports the assumption in our theoretical model that only considers the friction on the fiber-fiber interface at the effective untwisting length, neglecting the tangential frictional resistance after the debonding between the UHPC matrix and the twisted steel fiber. Table 1 displays that the tested effective untwisting length is insensitive to the fiber embedment length. To further optimize the pullout behavior, further study on the strategies of extending effective untwisting length can be thus performed.  
The friction at matrix-fiber interface and its adhesive bond are critical to increase the pullout force. Consequently, the matrix-fiber interface with optimum property is needed. For the same mix proportion, an optimum mixing and casting is thus required to produce an intact interface with a minimum number of pores and other defects. Figure 5 shows a number of pores that were caused during the casting process, though we conducted a careful casting procedure, similar to studies such as [23]. The dense mixture with a low water-binder ratio might increase the difficulty of escaping the bubble during the casting. Future study can be carried out to improve the casting quality of UHPFRC and investigate the influence of air bubbles on the matrix-fiber interface and thus the pullout force.  
The significant strain hardening in the plastic untwisting stage causes an optimized property of UHPFRC with TSFs. This strain hardening causes a significant increase in energy dissipation area enclosed by the pullout curves. Thus, the twisting is an ideal way to improve the strain hardening property and the UHPFRC with TSFs may be applied well in the concrete structures subjected to earthquakes or shocking to absorb dynamic energy.