In this study a theory of rods is developed and analyzed by taking into account long-range force interactions between cross sections. The balance of momentum and principle of virtual work are formulated. Different variants of the displacement and force boundary conditions are introduced by the specifying force and virtual power fluxes through a cross section of the rod. With a special constitutive assumption for the interaction force a peridynamic type theory is derived. Based on the analytical solutions under both static and dynamic loadings and for several types of boundary conditions including fixed-fixed and fixed-free edges, benchmark problems are formulated. The numerical mesh free type method is applied to solve the governing equations and numerical results are compared with analytical solutions.