# Fatigue Testing of Shear Reinforcement in Prestressed Concrete T-Beams of Bridges
## 4 Comparison with Existing Approaches
Subsequently, the equivalent stress range \(\Delta\sigma_{equ,i}\) corresponding to a damage of 1.0 according to the Palmgren-Miner Rule [6] is calculated from the number of cycles to failure by applying an S-N curve (Step 2). Here, the S-N curve, according to [45], was applied due to its better consistence with the test results (Figure 9). Accordingly, the variable stress range is transformed into a constant stress \(\Delta\sigma_{\text{equ}}\) (Step 3), leading to the same damage. Using Equation (4), derived from Equation (1), in Step 4, the corresponding strut angles under cyclic loading \(\theta_{\text{fat,test}}\) were determined utilizing the shear range \(\Delta V_{test}\) from the test and the previously determined equivalent stress range \(\Delta\sigma_{equ}\). Figure 8: (**a**) Truss model for beams with shear reinforcement; (**b**) Characteristic S–N curve for reinforcing steel (both according to [8,9]). Figure 9: Ratio of the experimental and calculated number of cycles until fatigue failure based on strut inclination, according to Equation (3) and the S–N curve according to (**a**) DIN EN 1992 1-1/NA [7] and (**b**) [PERSON] et al. [45]. For the stirrup legs, which did not fail in the test, the number of load cycles at test stop was assumed to be the respective number of cycles to failure. If several failures occurred at one stirrup leg, the respective smaller number of cycles to failure was used to determine \(\theta_{fat,test}\) and the time of the second stirrup failure was neglected. \[\theta_{fat,test}=arctan\bigg{(}\frac{a_{sw}\cdot z\cdot\Delta\sigma_{out}}{ \Delta V_{test}}\bigg{)} \tag{4}\]  
\[\theta_{fat,EC2}=arctan\big{(}\sqrt{tan\theta_{stat,EC2}}\big{)} \tag{5}\]  
Figure 11a shows the resulting strut angles in test T-33-2 for each stirrup. The strut inclination recalculated from the equivalent stress range of each stirrup leg varied between 23.6\({}^{\circ}\) and 29.8\({}^{\circ}\) with a mean value of 24.5\({}^{\circ}\). The strut angle for fatigue \(\theta_{fat}\) according to Equation (6.65) in EC 2 [6], corresponding to Equation (5) in this paper, amounts to 37.2\({}^{\circ}\) and determines a too-steep strut inclination, overestimating the stress range of the stirrups under fatigue loading. As cyclic loading is expected to reduce the concrete contribution of aggregate interlock due to the local crushing of the crack surfaces, the stress of the shear reinforcement may increase [47]. In the fatigue design of shear reinforcement, this is considered by increasing the angle \(\theta\), determined in the ultimate limit state (ULS) design under static loading, according to Equation (3). On the other hand, the cyclic loading takes place on load level below the characteristic load and is not increased by partial safety factors, as in ULS. To balance these crosscurrent effects, [PERSON] and [PERSON] [17; 40; 46] proposed to calculate the stirrups stress range under cyclic loading for prestressed I-beams with shear reinforcement ratios up to 0.35%, based on the assumption:  
\[1.0\leq cot\theta\leq\frac{1.2+1.4\cdot\frac{\alpha_{cp}}{f_{d}}}{1-\frac{V_{ Radc}}{V_{Ld}}}\leq 3.0 \tag{2}\]0  
In [43], this proposal was extended up to shear reinforcement ratios \(\rho_{w}=0.67\%\). This assumption of Equation (6) results in a compression strut angle of 30.3\({}^{\circ}\) for beam T-33-2, which is on the safe side, as \(\theta_{stat,EC2}\) is higher than the recalculated compression strut angles from the test, which varied between \(\theta_{exp,min}=23.6^{\circ}\) and \(\theta_{exp,max}=29.8^{\circ}\), with a mean value of \(\theta_{exp,mean}=24.5^{\circ}\). The same evaluation was performed for all beams tested with constant load levels and is summarized in Figure 11b. In addition to the mean value of the number of cycles reached before failure, the minimum and maximum values of all stirrups of each beam are also given.