# A Simplified Methodology to Evaluate the Design Specifications of Hydraulic Components
## 4 Reliability of Hydraulic Component  
Once the load variability is defined, the structural variability of the hydraulic component is considered. Assuming that a hydraulic component manufacturer knows the S-N curves corresponding to any main possible failure modes and according to its know-how (design and manufacturing technology), determining the fatigue resistance reliability of a hydraulic component is possible. Figure 6 shows the typical S-N curves and corresponding P-N curves for a specific cylinder geometry for the same failure mode (F-01) defined by the value of the pressure \(P_{D}\) (or stress \(S_{D}\)), which correspond to \(N=2\ \times 10^{6}\) cycles and \(nn\ \%\) of reliability (known as the pressure class \(nn\), or stress class \(nn\) of the curve) [10]. Figure 4: Statistical distribution of \(P_{2 eq}\). Figure 5: Laboratory test parameters according pressure specifications. Recalling that fatigue damage due to load conditions is represented by \(P_{\phi eq}\), and damage resistance of the hydraulic cylinder is represented by \(P_{D}\):  
\[D\big{[}P_{\phi}\big{]}=N\big{[}m_{p\phi},\ s_{p\phi}\big{]} \tag{14}\]  
\[D[P_{D}]=N\big{[}m_{pD},\ s_{pD}\big{]} \tag{15}\]  
Both distributions have \(P_{S}\) as a common linkage, which meets the conditions  
\[p\big{[}P_{p\phi}\leq P_{S}\big{]}=R_{\phi}\ \text{and}\ p[P_{D}\leq P_{S}]=R_{D} \tag{16}\]  
Any cylinder with a resistance \(P_{D}\) may be mounted in a machine with any of the loads \(P_{\phi}\). The condition for the cylinder to resist is set by the resistance that is superior to the load (see Figure 7). \[y=p_{D}-p_{\phi}>0 \tag{17}\]  
The variable \(y\) follows a normal distribution:  
\[D[y]=N\big{[}m_{y},s_{y}\big{]} \tag{18}\]  
where  
\[ \begin{array}{l}m_{y}=m_{pD}-m_{p\phi}\\ s_{y}=\sqrt{s_{pD}^{2}+s_{p\phi}^{2}}\end{array}  \tag{19}\]  
Figure 6: Typical (**left**) S–N curves and (**right**) P–N curves for specific hydraulic cylinder geometry for this same failure mode (F-01). Figure 7: Cylinder mounted in a machine with any of the loads: (**a**) Load variability and structural variability functions; (**b**) At the end of the life will not fail more than \(\beta\) % of cylinders. fulfilling  
\[R_{A}=p\left[z\geq\frac{m_{y}}{s_{y}}\right] \tag{20}\]  
The design data is \(\beta=(1-R_{A})\), which means \"at the end of the life (desired) will not fail more than \(\beta\)% of cylinders\". If we consider  
\[k_{s}=\frac{s_{p\varnothing}}{s_{pD}} \tag{21}\]  
then,  
\[\frac{m_{y}}{s_{y}}=\frac{-z_{pD}+z_{p\phi}s_{p\phi}}{\sqrt{s_{pD}^{2}+s_{p \phi}^{2}}}=\frac{-z_{pD}+k_{s}z_{p\phi}}{\sqrt{1+k_{s}^{2}}} \tag{22}\]  
Equation (22) allows us to estimate the life expectancy, defined by reliability \(R_{A}\), (or the design data \(\beta\)) as a function of load spectrum variability (defined by the reliability, \(R_{\phi}\)), resistance variability of the hydraulic component (defined by the reliability, \(R_{D}\)) and standard deviation ratio, \(k_{s}\). In practice, the load standard deviation (\(s_{p\phi}\)) is known according the proposed approach, but the resistance standard deviation of the hydraulic cylinder (\(s_{pD}\)) and consequently, the value of the ratio \(k_{s}\) cannot be known at the design stage because, sometimes, the selected cylinder manufacturer is still not known. In Table 3, \(R_{D}\) values have been tabulated applying Equation (22), see diagram Figure 8, for values within the following ranges: \(50\%\leq R_{\phi}\leq 95\%\), \(3\%\leq\beta\leq 10\%\), and \(0.25\leq k_{s}\leq 2\). Figure 8: Scheme to calculate the required resistance reliability (\(R_{D}\)).