## 7 Fibre Stresses under Quasi-Static Loading
### Inclined Fibre Pullout  
The additional bending tensile stresses cannot be derived from the measured values. However, they can be calculated with the aid of the formula according to [PERSON] [20] respectively [PERSON] [21]. The mandrel diameters were determined after the test run, when the inclined fibres were pulled out of the UHPC (Figure 8a). For this purpose, the fibres were carefully removed from the test specimen and examined, respectively, before being measured using a digital microscope. Then, in order to determine the mandrel diameter, circles were iteratively applied to the curvature of the fibres (Figure 8b).  
In addition to the fibers with an orientation of 60\({}^{\circ}\), fibers with an orientation of 30\({}^{\circ}\) were also investigated. The measured mandrel diameters are shown in Table 3. The values for 45\({}^{\circ}\) and 75\({}^{\circ}\) were linearly interpolated and are therefore shown in italics.  
Using the measured mandrel diameters, the bending tensile stresses \(\sigma_{\mathrm{B}}\) were calculated and added to the measured tensile stress from Table 2. It can be seen that the utilization rate of the fibre in the inclined fibre pullout significantly increases from about 0.3 to about 0.5. The utilisation rate in Table 3 is also related to the measured tensile strength of the fibre material [1]. If it is linked to the manufacturer's data, the utilisation rate is even higher (see Equation (9)).  
\[\eta=\frac{1810.0\;\mathrm{N/mm^{2}}}{2000\;\mathrm{N/mm^{2}}}=0.91\;\;\; \mathrm{respectively}\;\;\;\gamma=\frac{1}{0.91}=1.10\;\approx 1.00 \tag{9}\]  
Considering additional bending stresses of fibres and relating them to the manufacturer's tensile strength of the fibre material, the fibre is more or less completely utilized.  
To account for the additional bending stresses within the determination of the utilization rate for fibres in UHPFRC, Equation (7) is extended by Equation (5). In addition, the mandrel diameter \(D\) is expressed as a function of the orientation angle \(\alpha\) following the  
Figure 8: Determining the mandrel diameter: (**a**) bent fibres of specimen and (**b**) iteratively applied circles for measuring with digital microscopy.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Series** & **Orientation \(\alpha\)** & **Measured Mandel** & **Banding Stresses \(\sigma_{\mathrm{B}}\)** & **Max. (Local) Tensile** & **Utilisation Rate** \\  & & **Diameter \(D\)** & **Bent Stress \(\sigma_{\mathrm{st}}^{\epsilon}\)** & **(\(\sigma_{\mathrm{st}}^{\epsilon}+\sigma_{\mathrm{B}}\))/\({}_{\mathrm{st}}^{\epsilon}\)** \\ \hline (-] & [\({}^{\circ}\)] & [mm] & [\(\mathrm{N/mm^{2}}\)] & [\(\mathrm{N/mm^{2}}\)] & [-] \\ VMS30 & 30 & 42.0 \(\pm\) 4.2 & 764.5 & 1810.0 & 0.51 \\ _VMS_45 & 45 & _55.2_ & _582.1_ & _1580.1_ & _0.44_ \\ VMS60 & 60 & 68.3 \(\pm\) 2.4 & 470.1 & 1654.4 & 0.46 \\ _VMS_75 & 75 & _81.5_ & _394.2_ & _1284.3_ & _0.36_ \\ \hline \hline \end{tabular}
\end{table}
Table 3: Calculated mean utilisation rate of pulled out fibres in UHPC under quasi-static loading.  
same schema like setting up an S/N-curve. A linear relationship according to Equation (10) is assumed between the mandrel diameter \(D\) and the orientation angle \(\alpha\) in Table 3.  
\[D=0.88\cdot\alpha+15.7\ \ \ \text{ftir}\ \ \alpha<90^{\circ} \tag{10}\]  
The extension of Equation (7) with the additional bending stress in Equation (5), taking into account the linear relationship between the mandrel diameter \(D\) and the orientation angle \(\alpha\) in Equation (8), yields the utilization rate formulation in Equation (11).  
\[\eta\cdot f_{\text{st}}^{\text{f}}=2\cdot\tau_{\text{st}}^{\text{f}}\cdot \lambda+E_{\text{f}}\frac{\varnothing^{\text{f}}}{D}=2\cdot\tau_{\text{st}}^{ \text{f}}\cdot\lambda+E_{\text{f}}\frac{\varnothing^{\text{f}}}{0.88\cdot \alpha+15.7} \tag{11}\]