# Fatigue Testing of Shear Reinforcement in Prestressed Concrete T-Beams of Bridges
## 4 Comparison with Existing Approaches  
To check the accuracy of the current approach of Eurocode 2 (EC 2) [6; 7; 8; 9] for shear fatigue of beams with stirrups the measured number of cycles until failure of the stirrups \(N_{test}\) was compared with the calculated number of cycles until failure \(N_{calc}\). For this, a compression strut inclination according to Equations (2) and (3) was used as a basis for determining the stress range under cyclic loading (Equation (1)). The stress in the shear reinforcement under maximum and minimum load calculated with \(\theta_{fat}\) must not exceed the yield strength, and the respective stress range \(\Delta\sigma\) has to be lower than the fatigue resistance--e.g., \(\Delta\sigma_{Rsk}\). \[\Delta\sigma_{calc}=\frac{\Delta V_{test}}{a_{\text{sv}}\cdot z\cdot col\theta _{fat}} \tag{1}\]  
\[1.0\leq cot\theta\leq\frac{1.2+1.4\cdot\frac{\alpha_{cp}}{f_{d}}}{1-\frac{V_{ Radc}}{V_{Ld}}}\leq 3.0 \tag{2}\]  
\[cot\theta_{fat}=\sqrt{cot\theta} \tag{3}\]  
While, according to Eurocode 2, the angle \(\theta\) of the concrete strut to the beam axis (Figure 8a) can be chosen within a range of 21\({}^{\circ}\) to 45\({}^{\circ}\), in the German National Annex \(\theta\) is determined accounting for a concrete contribution by aggregate interlock. Therefore, the angle \(\theta\), according to [6] in conjunction with [7], depends on the compressive stress of concrete due to prestressing \(\alpha_{cp}\) and the ratio of concrete contribution \(V_{Rd,cc}\) to applied shear force \(V_{Ed}\) (Equation (2)). While for buildings, according to [7], \(\theta\) may vary between 18\({}^{\circ}\) and 45\({}^{\circ}\), the German design rules for concrete bridges [9] define a lower limit for the strut inclination of 30\({}^{\circ}\). \(\Delta\sigma_{calc}\) for each test is given in Figure 7. With this, \(N_{calc}\) was determined, on the one hand, for an S-N curve (Figure 8b), specified in EC 2 NA(D) [7] (\(\Delta\sigma_{Rsk}(N^{*})\) = 175 N/mm\({}^{2}\); \(N^{*}\) = 10\({}^{6}\); \(k_{1}\) = 5; \(k_{2}\) = 9) and, on the other hand, for an S-N curve according to [PERSON] et al. [45] (\(\Delta\sigma_{Rsk}(N^{*})\) = 195 N/mm\({}^{2}\); \(N^{*}\) = 10\({}^{6}\); \(k_{1}\) = 4.1; \(k_{2}\) = 7.2), which was derived for reinforcing steel embedded in concrete. Both S-N curves are bilinear with a change of inclination at \(\Delta\sigma_{Rsk}\) and N* = 10\({}^{6}\) load cycles. The inclination of the S-N curve is defined by the stress gradients \(k_{1}\) for \(N\leq N^{*}\) and \(k_{2}\) for \(N\geq N^{*}\). The calculation based on the S-N curve, according to Eurocode 2, clearly underestimates the resisted cycles of the tested stirrups (Figure 9a), especially for low shear reinforcement ratios. The mean value of experimental load cycles was at least six times higher than the calculated number of cycles--in some tests up to almost seventy times. The ratio of test T-22-1 is estimated conservatively, based on the number of cycles when the test was stopped without the fatigue failure of any stirrup. The application  
Figure 7: Resisted number of cycles until fatigue failure \(N_{\text{test}}\) of the shear reinforcement. of the S-N curve, according to [45], reveals a better agreement with ratios of \(N_{\text{test}}/N_{\text{calc}}\) varying between approximately 3 and 29 (Figure 9b). The large difference between observed and calculated load cycles is mainly caused by the overestimation of the calculated stress ranges applying the increased strut inclination for fatigue \(\theta_{\text{fat}}\), according to Equation (3). Therefore, the tests were additionally evaluated using a method proposed in [40,46] with an improved approach to determine the strut angle under cyclic loading. The method is based on an evaluation of the determined number of cycles to failure for each measured leg of the shear reinforcement. The procedure is shown in Figure 10.  
In Step 1, the number of cycles to failure of each individual stirrup \(N_{test,i}\) is determined (Section 3).