The objective of this study is to investigate the effect of the Lode parameter on different material strengths. Recent work has shown that ductile failure highly depends on the stress state characterized by both the stress triaxiality T and the Lode parameter L, which is related to the third deviatoric stress invariant. Thus, for six different steel grades, two different specimen geometries were manufactured to account for two different Lode parameters (L = −1 and L = 0), whereas T is controlled by introducing different sized notches at the center of the specimens. By performing tensile experiments and running finite element simulations, the ductile failure loci of the six materials showed variations between the two specimen geometries, indicating that the failure highly depends on the stress state characterized by both T and L. This indicates the need to reassess the ductile local failure criterion in the ASME codes that only accounts for T as a stress state measure. A Lode sensitivity parameter LS is defined based on the experimental results and revealed that the steel grades with ultimate strength higher than a certain threshold value (450 MPa) exhibit sensitivity to the Lode parameter, and the results showed that the LS increases with increase in the ultimate strength of the steel grade. The results were incorporated to enhance the original ASME local failure criterion by accounting for T, L, and LS to accurately assess ductile failure in high-strength steels.