Peridynamic analysis of curved elastic beams

Abstract

Peridynamics is an extended continuum theory which operates with non-local deformation measures as well as long range internal force/moment interactions. The aim of this paper is to present and to analyze peridynamic governing equations for elastic curved beams. To this end the strain energy density is formulated as a function of two non-local deformation measures including the axial deformation and curvature. Applying the Lagrangian formalism, peridynamic equations of motion are derived. To analyze curved beams of finite length peridynamic boundary conditions are required. Examples of boundary conditions including simple support and clamped edge are presented by introducing a fictitious domain. Solutions to PD equations of motion are presented for various problems including static analysis of simply supported and clamped arc beams as well as modal analysis of a slender ring and a simply supported arc beam. To validate peridynamic formulations, solutions for displacements, natural frequencies and vibration modes are compared with those by the classical beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the derived equations of motion and proposed boundary conditions.