# Bending Fatigue Behavior of 17-4 PH Gears Produced by Additive Manufacturing
## 3 Toot Root Stress According to ISO Standard  
To compare the bending fatigue behavior with the ones of the common gear materials it is necessary to define the tooth root stress \(\mathrm{\sigma_{F0}}\). To do that, method B of ISO 6336-3 [22] was used. According to this, and considering that the applied load is not the tangential one, \(\mathrm{F_{t}}\) defined in the standard for two mating gears, but the one coming from the interaction from the gear-sample and the axis, \(\mathrm{\sigma_{F0}}\) can be written as:  
\[\mathrm{\sigma_{F0}}=\frac{\mathrm{F_{t}}}{\mathrm{b}\mathrm{m_{n}}}\mathrm{ Y_{F}}\mathrm{Y_{s}}\mathrm{Y_{s}}\mathrm{Y_{\beta}}\mathrm{Y_{DT}}=\frac{ \mathrm{F}}{\mathrm{b}\mathrm{m_{n}}}\mathrm{Y_{F}}\mathrm{Y_{s}}\mathrm{Y_{ \beta}}\mathrm{Y_{B}}\mathrm{Y_{DT}}\cos\alpha_{\mathrm{n}} \tag{1}\]  
Figure 4: \(\mathrm{\sigma_{F0}}-N\) curve.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test ID** & **Gear Tested** & **Teeth** & \(\mathrm{F_{min}}\) [N] & \(\mathrm{F_{max}}\) [N] & **Cycles** & **Broken Tooth** \\ \hline
1 & A & 1–3 & \(-15\),000 & \(-1500\) & 202,550 & 1 \\
2 & A & 5–7 & \(-14\),000 & \(-1400\) & 229,575 & 7 \\
3 & A & 9–11 & \multicolumn{4}{c}{Interrupted test} \\
4 & A & 13–15 & \(-12\),500 & \(-1250\) & 481,897 & 13 \\
5 & A & 17–19 & \(-12\),250 & \(-1225\) & 581,951 & 17 \\
6 & A & 21–23 & \(-11\),000 & \(-1100\) & 631,460 & 21 \\
7 & B & 1–3 & \multicolumn{4}{c}{Interrupted test} \\
8 & B & 5–7 & \(-9000\) & \(-900\) & 277,690 & 7 \\
9 & B & 9–11 & \(-9000\) & \(-900\) & Runout & - \\
10 & B & 13–15 & \(-9500\) & \(-950\) & 481,897 & 15 \\
11 & B & 17–19 & \(-9000\) & \(-900\) & 581,951 & 19 \\
12 & B & 21–23 & \(-8500\) & \(-850\) & Runout & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experimental points.  
As the tested gears are full-body spur gears, \(\rm Y_{\beta}\) and \(\rm Y_{\rm DT}\) have unitary value. The teeth profile was simply finished with the hobbing machine; therefore, it is impossible to reach an accuracy grade of lower than 5 ([38; 39]) (see [40] for the definition of gear accuracy). As the gear accuracy is surely >4, \(\rm Y_{B}\) is equal to 1. \(\rm Y_{F}\) and \(\rm Y_{S}\) do not have a unitary value and they are calculated as in (2) and (3).  
\[\rm Y_{F}=\frac{\frac{6h_{Fe}}{m_{a}}\cos(\alpha_{Fen})}{\left(\frac{g_{Fe}}{m_ {a}}\right)\cos(\alpha_{n})} \tag{2}\]  
\[\rm Y_{S}=(1.2+0.1L)q_{S}^{\frac{1}{2+2.3L}} \tag{3}\]  
While \(\rm s_{Fn}\) is related only to the gear data, \(\rm h_{Fe}\) and \(\rm\alpha_{Fen}\) are related to the point of load application. As the standards always consider two meshing gears, the outer point of the single contact is considered as the point of the load application. In this case, dealing with the STBF tests, the load application point is defined by the contacts between the gear and anvils (which are shown in Figure 2).  
Starting from the geometrical characteristics of the gear (Figure 4 and Table 1), it is possible to determine all the Equation (1) coefficients; the results are shown in Table 1. Therefore, \(\rm h_{Fe}\) and \(\rm\alpha_{Fen}\) values and, subsequently, \(\rm Y_{F}\) and \(\rm Y_{S}\) have been calculated according to that diameter. \(\rm h_{Fe}\) results to be equal to 0.53 mm,  
\(\rm\alpha_{Fen}\) equal to 13.79\({}^{\circ}\) and, subsequently  
\(\rm Y_{F}=1.5805\) and \(\rm Y_{S}=1.5805\).  
Thus, it is possible to obtain the stress-load relationship by using Equation (4):  
\[\rm\sigma_{F0}=5.17\cdot 10^{-2}F \tag{4}\]  
where F is in N and \(\rm\sigma_{F0}\) is in MPa.