In order to evaluate the true triaxial strength properties of rocks, according to the variation of rock strength measured using conventional triaxial tests, a nonlinear strength criterion was developed based on the maximum value of deviatoric stress. The proposed strength criterion was compared with the measured strengths of 12 kinds of rocks and four typical rock strength criterions. The predicted results of the new strength criterion were close to the measured strengths of 12 kinds of rocks, and all the regression coefficients R² were larger than 0.98 and mean absolute percentage errors (MAPE,E_MAP) less than 4% (except 6.83% for No.7 rock). The average E_MAP of the 12 rocks were 2.32%, 2.43%, 5.28%, 7.39%, and 13.74%, calculated by the new strength criterion, exponential criterion, Hock-Brown (H-B) criterion, modified Mohr-Coulomb (MM-C) criterion, and Drucker-Prager (D-P) criterion, respectively. It shows that the strength criterion proposed in this paper could well predict the strength of different types of rocks, and its prediction accuracy was slightly better than that of exponential criterion, an outstanding criterion in rock mechanics, and was far better than the other three typical strength criterions. Based on the conventional triaxial strength criterion, considering the effect of intermediate principal stress, a true triaxial strength criterion was developed by introducing the intermediate principal stress parameter and Rhode stress parameter. Compared with the true triaxial test results of eight rocks, the proposed true triaxial strength criterion well reflected the variation that the maximum principal stress increased firstly and then decreased with the increase of the intermediate principal stress. The R² values of the eight types of rocks were all above 0.9, among which the R² values of five rocks were greater than 0.96. Except for rocks No.13 and No.14 with E_MAP being 7.79% and 4.84%, the E_MAP of the other six rocks were all smaller than 4%. It shows that the proposed strength criterion could well predict the true triaxial test strength of rocks, and reflect the effect of intermediate principal stress well with good universal applicability. The stress spatial characteristics of meridian plane and deviatoric plane also explained that the proposed strength criterion well reflected the effects of hydrostatic pressure and intermediate principal stress on the maximum principal stress.