Impact Response Based on Timoshenko Beam Theory

Ph.D. University of Hawaii at Manoa 2015.Includes bibliographical references.The elastic impact of a translating flexible pole is studied herein. Three scenarios are considered: 1) transverse impact against a rigid stop, 2) longitudinal impact against a flexible column and 3) transverse impact against a flexible column. Based on Timoshenko beam theory, an analytical solution method using mode superposition for the coupled spring-pole or column-pole system is presented. Any physical set of boundary conditions can be accommodated for the pole and the column. For all cases involving axial impacts, the maximum initial impact force is governed by the local shear deformation in the column and the axial deformation in the pole. However, for transverse impacts, the maximum initial impact force is governed by the local shear deformation in the pole and the column. A simple formula for the maximum initial force is derived and shown to be quite accurate. In no case is the total mass of the pole significant to the initial peak force. Indeed, based on Euler-Bernoulli beam theory the initial impact force is unbounded as the spring stiffness increases whereas Timoshenko beam theory has a clear limiting value for the initial impact force. The impact duration depends on the wave propagation in the pole or the column. In addition, the energy transfer between kinetic energies and strain energies reveals both the initial dependence on shear deformation and the transfer of the associated energy to bending energy. The energy exchange also shows the importance of the inertia of the column in absorbing a significant part of the initial kinetic energy of the pole. It is shown that the moment of inertia has a negligible effect on the impact force, which is an interesting conclusion because some structural finite element codes use a lumped mass matrix that includes translational masses but not rotational inertias. For transverse impact, multiple impacts are considered, and the whole collision event is divided into contact phases and separation phases. It is shown that for all cases the maximum contact force occurs during later contact phases and its value can reach up to 1.6 times the peak force in the first contact phase. The impact duration of the first contact phase depends on the shear wave in the pole or the column according to the mass and wave speed ratios. The total impulse on the pole ranges between 1.5-1.8 times the initial momentum of the pole, depending on the stiffness of the column. The energy exchange during the multiple impacts, while it can be complicated, reveals that for relatively stiff columns the sum of the translational kinetic and bending strain energies of the pole constitutes approximately 90% of the total energy. In all cases considered, relatively little net energy has been transmitted to the column at the time of final separation. For axial impact, multiple impacts depend on the relative stiffness of the column and the pole, and also on the inertia of the pole. Hence, the entire collision event for the stiffest column is characterized by a single impact. However, for the most flexible column all cases involve multiple impacts. For the case of single impact, most of the kinetic energy of the pole is transferred into axial strain energy in the pole. However, for the multiple impacts, most of the kinetic energy of the pole is transferred into bending energy in the column. The maximum impact force reaches up to 1.9 times the initial peak force and the total impulse reaches up to 1.9 times the initial momentum of the pole. For all cases, the duration of the entire collision event depend mainly on the wave propagation in the pole. The impact force and duration depend on the type of impact as well as the end boundary conditions of the column. For all cases, the axial impact yields larger impact force than that for the transverse impact according to the stiffness of the column with similar boundary conditions of the column. The stiffer the column, the larger is the impact force and the smaller the impact duration. In addition, free end column yields the smallest impact force and duration. However, pin-end column gives the largest impact duration and fixed-end column gives the largest impact force. The dynamic amplification factors for shear force and bending moment depend mainly on the stiffness of the column and the inertia of the pole. Cases involving stiffer column and larger pole inertia yield higher dynamic amplification factors. In addition, the dynamic amplification factors significantly increase for cases involving multiple impacts and always reach their maximum values at later impacts