•A new analytical model is established to calculate the maximum tooth root stress and critical section location of gear based on mechanics theory.•The new model can determine the critical section location accurately and quickly by solving the extreme value.•Both bending moment stress and compressive axial stress are taken into account in the model with the accurate profile equation.•The result of the new model is consistent with that of finite element method and more reliable than international standards. An accurate calculation of the maximum tooth root stress (TRS) and critical section location (CSL) provides a basis for predicting and improving gear performance. The irregular profile represented by the implicit function may cause the calculation to be more complex. In current research, finite element methods (FEM model) and experimental test methods (ET model) can obtain accurate results but need large computational resources and time. The results from ISO 6336:2006 (ISO model) and AGMA 2101-D04 (AGMA model) are obtained conveniently but sometimes not reliable. Therefore, a new analytical model based on the mechanics theory with an accurate profile equation is established to calculate the maximum TRS and corresponding CSL quickly and accurately by solving the extreme value. Finally, the results of the spur gear in five cases with different parameters are obtained and compared to those of the FEM, ISO and AGMA models. It is shown that the results of the new model are in agreement with those of the FEM model, even under different parametric conditions.