This paper is devoted to the free vibration and buckling analyses of two-phase peridynamic Euler-Bernoulli beams under general boundary conditions. The kernel of the integral peridynamic formulation is chosen in an exponential form, normalized along the finite beam domain. The integro-differential eigenvalue problems (free vibration or buckling analyses in presence of axial load) can be reformulated as a linear sixth-order differential eigenvalue problem associated to six boundary conditions, namely four classical boundary conditions and two additional constitutive peridynamic boundary conditions. These two additional boundary conditions express the local behaviour of the two-phase peridynamic law at the boundaries. Eigen-frequencies and buckling loads are calculated for simply supported, clamped-free, clamped-clamped and clamped-hinge boundary conditions. The transcendental equations for the eigen-frequencies or the buckling loads are explicitly derived for each considered boundary condition. Numerical results obtained by this analytical method are confirmed by two complementary numerical methods, one based on the Rayleigh quotient of the continuous peridynamic problem, and another based on the discretization of the two-phase peridynamic governing equations, through a FDM quadrature approximation. The variational iterative method of Stodola-Vianello is successfully applied to the two-phase peridynamic eigenvalue problems. The paper discusses the order of convergence of each scheme, which depends on the considered adopted numerical method. Finally, the vibration and buckling problems of multi-segmented peridynamic beam and column are analytically and numerically solved, including the connection between a peridynamic segment with a local one.

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