Biaxial Vibrations of an Elasto-Plastic Beam With a Prescribed Rigid-Body Rotation

In the present work, the development of plastic strains in a flexural beam is studied. The beam is modeled as a Bernoullieuler beam, where large rigid-body rotations and biaxial bending in the small strain regime are studied. The deformation is split into the spatial deformation of a hinged-hinged beam and the movement of the second support. Neglecting axial displacements of the beam, this support moves on a sphere. In the present paper, the latter motion is considered as prescribed. The beam thus is assumed to possess only flexural degrees-offreedom. Such a problem is frequently to be encountered in machine dynamics or robotics. We assume the stiffness of the beam to be considerably lowered due to catastrophic environmental influences, such that the deformations relative to the rigid-body motion, albeit small, reach the plastic regime. The equations of motion are derived by Hamilton’s principle. The potential energy follows from the internal energy due to the elastic part of the deformation and the potential due to gravity. Plastic strains are treated according to the theory of eigenstrains, which act as sources of self-stress upon the linear elastic beam. The biaxial deflections are discretized in space by means of Legendre polynomials. The plastic strains are discretized over length, height and width of the beam by small plastic cells. The plastic strains are computed in every time-step by a suitable iterative procedure. An implicit midpoint rule, which preserves the total energy of the system, is used for integration of the equations of motion. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study.