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This paper presents the numerical simulation of an industrial multi-step deep drawing process. A large strain finite element formulation including a hyperelastic elastoplastic constitutive model and a contact-friction law is used to this end where the steel sheet material parameters considered in the analysis are previously derived through a characterization procedure of its mechanical response. The numerical predictions of the final shape and thickness distribution of the blank are compared and discussed with available experimental values measured at the end of three successive drawing steps. In addition, a plastic work-based damage index is used to assess failure occurrence during the process. The damage values computed at the end of the drawing process are found to be lower than that corresponding to rupture in the tensile test, considered here as the threshold of failure, confirming, as observed experimentally, that neither fracture nor necking is developed in the blank during the whole drawing process. Finally, the possibility to carry out a reduced two-step drawing process, obtained by merging the second and third steps of the three-step process, is precluded since the damage criterion predicts in this case excessively large values that indicate that failure may occur in specific zones of the sheet.