The article examines the efficiency of using the finite element method in the form of the classical mixed method to solve a subclass of problems in structural mechanics, such as problems with the elastoplastic behavior of materials according to the Prandtl diagram. For the first time, an algorithm based on the advantages of the finite element method in the form of the classical mixed method, essential for problems of the considered class, has been developed. The essence of the proposed algorithm is that the system of resolution equations of the system, when implemented, is supplemented by a vector of unknowns, which allows for switching the state of the system’s sections without changing the computational scheme. A comparison of the algorithm proposed by the authors with existing algorithms based on the finite element method in the form of the classical mixed method is presented. The developed algorithm has been verified on well-known examples of nonlinear structural mechanics, both for the calculation of a bar system and for the problem of forming a plastic hinge in a frame. The advantages and disadvantages of the proposed algorithm are outlined, as well as prospects for its further modernization.