# Pullout Response of Ultra-High-Performance Concrete with Twisted Steel Fibers
## 2 Analytical Model
### Model Derivation
#### 2.2.1 Fiber with a Rectangular Section  
As shown in Figure 2, for a twisted steel fiber with the \(2a\times 2b\) rectangular section, the contact stress of the matrix on fiber surface is assumed to be linearly distributed on a section. The contact stress at the corner point is \(N_{1}\) (\(y\)) and its distribution is antisymmetric in the other parallel face. Take \(dx\) to be the increment at the long side and it is subjected to compression stress from the cementitious matrix as  
\[p=\frac{x}{b}N_{1}(y) \tag{1}\]  
where \(p\) is the compression stress, \(b\) is the length of the long side. The line load can be obtained by integrating p along the long side as  
\[p_{X}=\int_{0}^{b}\frac{x}{b}N_{1}(y)dx \tag{2}\]  
Considering the angle \(\theta\) of the long side twisted surface (Figure 2), the component of the pressure in the fiber axial \(y\) directions  
Figure 1: Typical pullout curve and its simplified trilinear model for twisted steel fiber-UHPC matrix: (**a**) typical pullout curve; and (**b**) simplified trilinear model for twisted steel fiber-UHPC. \[p_{y}=\int_{0}^{b}\frac{x}{b}N_{1}(y)sin\theta dx \tag{3}\]  
The normal force \(p_{x}\) on the twisted surface is  
\[p_{x}=\int_{0}^{b}\frac{x}{b}N_{1}(y)cos\theta dx \tag{4}\]  
As the fine sands were mixed in the matrix without coarse aggregates, the contact between TSF and matrix is relatively uniform. Thus, we assume that the sliding friction coefficient \(\mu\) between the matrix and the steel fiber is constant, the friction force in the fiber axial direction is  
\[f_{y}=\mu\cdot\int_{0}^{b}\frac{x}{b}N_{1}(y)cos\theta dx \tag{5}\]  
The friction coefficient \(\mu\) is determined by referring to the experimental results by [PERSON] and [PERSON] [19]. The friction coefficient at concrete-steel interface is mainly determined by the processing methods of steel and it was ranged from 0.35 to 0.58. In this model, the friction coefficient of 0.5 at concrete-steel interface for UHPFRC was set in the model in this study. The twisted surface angle at the short side is \(\theta^{\prime}\). Thus, the axial resultant force of the four sides (line load) \(P_{f^{\prime}}\) is  
\[P_{f^{\prime}}=2\int_{0}^{b}\frac{x}{b}N_{1}(y)(sin\theta+\mu\ cos\theta)dx+2 \int_{0}^{a}\frac{x}{b}N_{1}(y)\big{(}sin\theta^{\prime}+\mu\ cos\theta^{ \prime}\big{)}dx \tag{6}\]  
The untwisting pullout force \(P_{f}\) is thus obtained by integrating \(P_{f^{\prime}}\) on the effective untwisting length \(\Delta l\):  
\[P_{f}=\int_{0}^{\Delta l}\bigg{[}2\int_{0}^{b}\frac{x}{b}N_{1}(y)(sin\theta+ \mu\ cos\theta)dx+2\int_{0}^{a}\frac{x}{b}N_{1}(y)\big{(}sin\theta^{\prime}+ \mu\ cos\theta^{\prime}\big{)}dx\bigg{]}dy \tag{7}\]  
Similarly, the twisted torque \(T\) can be expressed as  
\[T=\int_{0}^{\Delta l}\bigg{[}2\int_{0}^{b}\frac{x}{b}N_{1}(y)(\cos\theta-\mu \ sin\theta)\cdot xdx+2\int_{0}^{a}\frac{x}{b}N_{1}(y)\big{(}\cos\theta^{ \prime}-\mu\ sin\theta^{\prime}\big{)}\cdot xdx\bigg{]}dy \tag{8}\]  
It assumed that the compressive stress evenly distributed along the fiber axial direction (\(\Delta l\) direction), that is, \(N_{1}(y)=N_{1}\). Equations (7) and (8) can be simplified to  
\[P_{f}=2\bigg{[}\int_{0}^{b}\frac{x}{b}N_{1}(sin\theta+\mu\ cos\theta)dx+\int _{0}^{a}\frac{x}{b}N_{1}\big{(}sin\theta^{\prime}+\mu\ cos\theta^{\prime}\big{)} dx\bigg{]}\cdot\Delta l \tag{9}\]  
\[p_{X}=\int_{0}^{b}\frac{x}{b}N_{1}(y)dx \tag{2}\]0 Force analysis sketch of twisted steel fiber of rectangular section. \[p_{X}=\int_{0}^{b}\frac{x}{b}N_{1}(y)dx \tag{2}\]1  
Figure 3 shows the geometric relation of a pitch. For any point \(K\) on the steel fiber surface, it travels through a pitch \(l_{0}\) in the embedment length direction (\(y\) direction) by a distance of \(2\pi\sqrt{x^{2}+a^{2}}\).