# Investigations on Interface Shear Fatigue of Semi-Precast Slabs with Lattice Girders
## 3 Small Size Fatigue Tests to Determine S-\(N\) Curves for Lattice Girders
### Evaluation of S-N curves for Lattice Girders  
The small-size fatigue tests were evaluated based on the interactive method, separately for the lower and upper chords as well as for the overall test data. The first analysis only included test data of the own investigations and excluded test results from literature. The yield strength was determined by performing a static test (\(N=1\)). Figure 9a shows the mean functions of the individual evaluations as well as the overall evaluation in double logarithmic illustration.  
By considering standard deviation and quantile factors, the mean function could be transferred to the quantile function with an arbitrary confidence level. In this context,  
Figure 8: Test results of small size fatigue tests for lower and upper chords with reference tests from literature.  
Figure 9: Fatigue test results of lower and upper chords with mean functions (**a**) and 5% quantile functions (**b**) [43].  
Figure 9b depicts the 5% quantile functions for the individual and the overall evaluations. For a confidence level of 90%, the quantile factors \(k_{\text{n}}\) result in 1.69 for the lower chord, 1.71 for the upper chord, and 1.64 for the overall evaluation.  
The analysis confirms that tests with lower chords (black points and black dotted curve) resisted a higher number of load cycles compared to tests with upper chords (grey points and grey dotted curve), but with higher scatter. This influences the 5% quantile function, especially in the range of fatigue strength. The 5% quantile curve of the overall evaluation (black curve) only exceeds the curves of the individual evaluation in the range of short time fatigue strength for the upper chords and in the range of fatigue strength for the lower chords. However, the stress ranges of all test data are located above the overall evaluation.  
For applying the \(S\)-\(N\) curves in design practice, the overall evaluation was used in order to derive consistent design \(S\)-\(N\) curves (Figure 5a) based on the design format of [28; 29], where the trilinear design curves are defined by Equation (4).  
\[\Delta\sigma_{\text{Rsk}}=\Delta\sigma_{\text{Rsk}}(N^{*})\cdot(N^{*}/N)^{1/k} \leq f_{yk} \tag{4}\]  
With  
\(\Delta\sigma_{\text{Rsk}}\) expected stress range  
\(\Delta\sigma_{\text{Rsk}}(N^{*})\) stress range at \(N^{*}\) load cycles  
\(N^{*}\) number of load cycles at break of slope  
\(N\) expected number of load cycles  
\(k\) stress exponents  
with  
\(k=k_{1}\) stress exponent for \(N<N^{*}\)  
\(k=k_{2}\) stress exponent for \(N>N^{*}\) with \(\lceil k_{2}=2\cdot k_{1}\rceil-1\)  
\(f_{yk}\) characteristic yield strength.  
In this context, different strategies of analysis to define a trilinear \(S\)-\(N\) curve were followed [43]. At least, for the revision of the technical approvals [41; 42], the tests from [40; 52] were also taken into account. In analogy to the trilinear \(S\)-\(N\) curves in [28; 29] (Figure 5a), the second point of the trilinear curve was adopted to \(N^{*}=1.0\) million load cycles with the corresponding stress range of \(\Delta\sigma_{\text{Rsk}}=92\) N/mm\({}^{2}\), which was already used for the simplified verification (see Section 2.3). The shape of the design curve was fitted to test data. It adopts the coefficients \(k_{1}=5\) and \(k_{2}=9\) from the design curve in [29] for stirrups as shear reinforcement (Figure 10).  
Figure 10: Assessed \(S\)–\(N\) curve for lattice girders in the latest technical approvals [41; 42].