Today, an interdisciplinary approach to solving the problems of implantology is key to the effective use of intraosseous dental implantations. The functional properties of restoration structures for the dentition depend significantly on the mechanical stresses that occur in the structural elements and bone tissues in response to mastication loads. An orthopedic design with a bar fixation system connected to implants may be considered to restore an edentulous mandible using an overdenture. In this study, the problem of the mechanics of a complete overdenture based on a bar and four implants was formulated. A mathematical model of the interaction between the orthopedic structure and jawbone was developed, and a methodology was established for the analytical study of the stress state of the implants and adjacent bone tissue under the action of a chewing load. The novelty of the proposed model is that it operates with the minimum possible set of input data and provides adequate estimates of the most significant output parameters that are necessary for practical application. The obtained analytical results are illustrated by two examples of calculating the equivalent stresses in implants and the peri-implant tissue for real overdenture designs. To carry out the final assessment of the strength of the implants and bone, the prosthesis was loaded with mastication loads of different localization. In particular, the possibilities of loading the prosthesis in the area of the sixth and seventh teeth were investigated. Recommendations on the configuration of the distal cantilever of the overdenture and the acceptable level and distribution of the mastication load are presented. It was determined that, from a mechanical point of view, the considered orthopedic systems are capable of providing long-term success if they are used in accordance with established restrictions and recommendations.