In drilling engineering, the failure of the threaded connection at the drill string joint accounts for 86% of the total failures of the drill string, making it the main risk area for drill string failure. To deeply analyze the stress distribution and stress concentration at the joint, based on the finite element software, the stress analysis of the API standard single-shoulder and double-shoulder drill string joints under axial and torsional loads was conducted. The stress concentration coefficients of axial and torsional for specific joints were calculated, which is convenient for engineering calculation of the maximum stress of the joint and the fatigue of the drill string. The research found that the maximum stress concentration area of the API single-shoulder thread joint is at the root of the first tooth of the thread meshing part. The maximum axial stress and maximum shear stress of the double-shoulder thread joint are significantly lower than those of the single-shoulder thread joint, especially showing superior performance under large torque. For specific joints, the axial stress concentration coefficient of the joint was determined to be 4.21, and the torsional stress concentration coefficient was 3.73. This study provides an important reference for the performance optimization design of the joint, stress calculation of the drilling system, and fatigue life prediction.