## 7 Fibre Stresses under Quasi-Static Loading
### Straight Fibre Pullout  
The maximum utilization rate according to the test program appeared at test series VES90-2 with \(\eta=0.38\). The value applied to a straight fibre pullout with a measured embedded depth of \(l_{\text{e}}=6.60\) mm and the assumption that no additional bending stress occurred. Because of the low utilization rate of one third of the fibre's tensile strength, the design of the fibre (geometry, tensile strength) seems to be uneconomical at first. In case the utilization rate of the fibre should be increased, a modification of the fibre design is necessary. For this purpose, the bond model of the embedded fibre must be selected. According to [18], the course of the bond stresses along an embedded fibre can be assumed in good approximation as rigid plastic. With the maximum measured tensile stress from Table 2, the following Equation (6) results for the bond stress.  
\[\tau_{\text{st}}^{\text{f}}=\frac{\sigma_{\text{st},90}^{\text{f}}\cdot A_{ \text{f}}}{\pi\cdot\Box^{\text{f}}\cdot l_{\text{e}}}=\frac{1374.1\,\text{N/mm ${}^{2}$}\cdot\pi\cdot(0.19\text{ mm}/2)^{2}}{\pi\cdot 0.19\text{ mm}\cdot 6.60\text{ mm}}\approx 10.0\,\text{N/mm${}^{2}$} \tag{6}\]  
Following, the tensile stress \(\sigma_{\text{st},90}^{\text{f}}\) is replaced by the tensile strength of the fibre material \(f_{\text{st}}^{\text{f}}\), representing the balance point in regard to fibre rupture. Furthermore, the slenderness \(\lambda=l_{\text{f}}/\varnothing^{\text{f}}\) has to be inserted into Equation (6). This is followed by Equation (7).  
\[\tau_{\text{st}}^{\text{f}}=\frac{\sigma_{\text{st},90}^{\text{f}}\cdot A_{ \text{f}}}{\pi\cdot\Box^{\text{f}}\cdot l_{\text{e}}}=\frac{\sigma_{\text{st },90}^{\text{f}}\cdot\varnothing^{\text{f}}}{2\cdot l_{\text{f}}}\rightarrow \eta\cdot f_{\text{st}}^{\text{f}}=2\cdot\tau_{\text{st}}^{\text{f}}\cdot \lambda=20\cdot\lambda \tag{7}\]  
With a tensile strength of \(f_{\text{st}}^{\text{f}}=3575.8\) N/mm\({}^{2}\), a bond strength of \(\tau_{\text{st}}^{\text{f}}=10.0\) N/mm\({}^{2}\) and a utilization rate of \(\eta=1.0\) the slenderness of the fibre is maximized to approximately \(\lambda=179\). As the pullout behaviour is accompanied by scatter [17], it is not practical to utilize the full tensile strength of the fibre without a safety margin. Therefore, a reduced utilization rate should be considered. Figure 7 shows possible slendernesses of fibres related to the chosen utilisation rate.  
In this context, it must be mentioned that the given slendernesses in Figure 7 is only valid for the investigated fibre type and a straight fibre pullout.  
The calculation of the fibre slenderness related to the utilisation rate in Figure 7 depends on a fibre tensile strength of approximately 3576 N/mm\({}^{2}\) that has been measured in [1]. However, the manufacturer states a tensile strength of approximately 2000 N/mm\({}^{2}\)  
Figure 7: Slederness of investigated high-strength micro steel fibre in relation to the utilisation rate.  
If the utilisation rate is now correlated to the manufacturer's data, it results in a significantly higher rate and a smaller safety margin (see Equation (8)).  
\[\eta=\frac{965.52\;\mathrm{N/mm^{2}}}{2000\;\mathrm{N/mm^{2}}}=0.48\;\;\;\mathrm{ respectively}\;\;\;\gamma=\frac{1}{0.48}=2.08\;\approx 2.00 \tag{8}\]