An elastic-plastic-fracture constitutive model for concrete materials has been proposed in a previous study by the authors. In this model, a four-parameter failure criterion is used to define the ultimate strength, the associated flow rule displaying a mixed hardening is adopted to represent the nonlinear stress-strain response, a crushing coefficient is introduced to identify the mode of failure, and an anisotropic elastic relationship is used to describe the post-failure behavior. In this paper, matrix constitutive equations for this model suitable for a direct finite element implementation are first presented, and three numerical examples, each representing a typical state of stress, are then solved to illustrate the applicability of the present model in a finite element analysis for the progressive failure studies of three concrete structures: thin-walled concrete cylinders subjected to hydrostatic pressure, the pull-out (Lok) test, and the splitting tension (Brazilian) test. The numerical results are also compared with the available experimental data and/or the available theoretical solutions. A good agreement is generally obtained.