J.L. Yang

Dynamic behaviour of a rigid, perfectly plastic free–free beam subjected to step-loading at any cross-section along its span

The initial, small deflection response of a free–free beam subjected to a concentrated step-load at any cross-section along its span is examined. The material of the beam is assumed to be rigid, perfectly-plastic. Solutions are obtained for various combinations of the magnitude and location of the load and the partitioning of the initial energy dissipation rates are discussed for some typical situations. It is concluded that (i) when the load is applied at the cross-section 0.6595L (L=half of span of the beam) away from the mid-span, the beam is most difficult to deform plastically, it exhibits its “hardest” behaviour; (ii) when the load is applied at the free end, the beam is most liable to deform plastically, it displays its “softest” behaviour; (iii) no less than 2/3 of the input energy is transferred into kinetic energy of rigid-body motions. This implies that the plastically dissipated energy is always less than 1/3 of the input energy.

Experimental and theoretical study of free–free beam subjected to impact at any cross-section along its span

Abstract A high-speed camera system is adopted to measure the instantaneous deformation of a free–free beam struck by a projectile at any cross-section along its span. The material in the experiment is made of hard aluminum (LY12R) which is less sensitive to the strain rate. A theoretical model is based on an elastic–plastic linear hardening material and the formulation is derived from the minimum principle in dynamics of elastic–plastic continua at finite deformation in association with a numerical technique. Some experimental results and theoretical predictions are given for the beam impacted at mid-span and 1/4 span, and attention is focused on the dynamic behavior of the beam in the early, transient stage of the response.

Deformable body impact: dynamic plastic behaviour of a moving free-free beam striking the tip of a cantilever beam

Abstract A theoretical model based on the rigid, perfectly plastic material idealization is proposed to simulate the dynamic behaviour of two deformable beams colliding with each other. The mid-point of a moving free–free beam is assumed to impinge on the tip of a cantilever beam with the beam axes perpendicular to each other. Complete solutions are obtained for various deformation mechanisms during the dynamic response process for the two deformed beams, and plastic shear sliding is taken into account. Attention is focused on the partitioning of the input energy between the two deformed beams after impact. A deformation map in a governing parameter plane is constructed to permit the calculation of the energy partitioning for a range of the beams’ parameters. This consists of nine regions corresponding to various deformation mechanisms. Typical numerical results are presented to demonstrate the influence of structural and geometrical parameters such as the ratios of the fully plastic bending moments of the two beams, of their fully plastic shear forces, of their masses per unit length and their length, on the energy partitioning after impact. Finally, the severance limit is given for the case of both beams having rectangular cross-sections. This indicates that shear sliding failure may happen in either of the beams if the initial kinetic energy is sufficiently large.

Interaction between reflected elastic flexural waves and a plastic `hinge' in the dynamic response of pulse loaded beams

By employing an elastic-perfectly plastic ideal sandwich beam model and solving the governing equations expressed in finite-difference form, the interaction between the reflected elastic flexural wave and the plastic hinge in the dynamic response of the beam to a suddenly applied force pulse is examined. The evolution of the plastically deforming region in the beam (the plastic hinge) is shown to be closely related to its encounter with reflected elastic waves. The correlation between the formation of a reversed hinge at the root with the notable oscillation in the position of the travelling hinge is also revealed. When the beam is loaded by a step force, the plastic hinge locates at a position close to that predicted by the rigid-plastic analysis and it is relatively stable during its interaction with the reflected elastic waves. When a rectangular pulse is applied, whether the reflected elastic waves can terminate the plastic flow in the plastic zone (i.e. whether the entire plastic zone undergoes complete elastic unloading) is shown to depend on the total impulse imparted to the beam.

DYNAMIC PLASTIC BEHAVIOR OF A FREE-ROTATING HINGED BEAM STRIKING A CANTILEVER BEAM*

A study was made of the dynamic plastic response of two deformable beams colliding with each other. The driving beam, which was a free-rotating hinged beam with a tip mass at the free end, strikes at the tip mass of a target cantilever beam, which was stationary. Complete solutions based on the rigid, perfectly plastic material idealization were obtained for various deformation mechanisms during the dynamic response process of the two plastic beams. Typical numerical examples are presented to demonstrate [[1]] the dynamic plastic behavior of the system after collision between two rectangular beams, as well as between an Ishaped beam and a rectangular beam; and [[2]] the influence of the tip mass ratio on the plastic deformation of the driving and target beams. *Communicated by F. Ziegler.

Dynamic plastic behaviour of a notched free–free beam subjected to step-loading at one end

Abstract A rigid perfectly plastic model is developed to study the initial, small deflection response of a free–free beam with an initial notch along its span under a concentrated step-loading suddenly applied at one end of the beam. Complete solutions are obtained for various combinations of the magnitude of the load, the location of the notch and its defect-severity. The partitioning of the initial energy dissipation rates is discussed for some typical situations. It is concluded that: (i) the different initial deformation mechanisms and the initial energy dissipation rate of the beam depend not only on the magnitude of the load but also on the defect-severity and location of the notch; (ii) because of the influence of the notch, the structural response of the beam is far more complicated than that of the un-notched free–free beam; and (iii) for some cases the maximum rate of energy dissipation in plastic hinges will be more than 1/3 of the total input energy rate, while for an un-notched free–free beam, it has been demonstrated that the plastic dissipated energy is always less than 1/3 of the input energy [Int. J. Impact Engng 21 (1998) 165].

Dynamic behavior of double cantilever beams subjected to impact

Abstract The dynamic plastic behavior of a double cantilever beam system subjected to impact by a rigid mass at their tips is studied. Three approaches are adopted: a rigid, perfectly plastic (r.p.p) complete solution, a mode solution, and an elastic, perfectly plastic (e.p.p) solution based on non-linear beam theory using the Dyna3D finite element code. In the complete solution based on the r.p.p idealization, the dynamic response of the system comprises three successive phases, which are characterized by different sets of plastic hinges in the system and expressed in closed analytical forms. The mode solution based on the r.p.p idealization is presented in a much simpler form. Particular attention is paid to the partitioning of the dissipation of the total input energy between the two beams. Numerical examples are given to demonstrate the influence of some of the structural parameters on the energy partitioning. In particular, the results obtained from the three approaches are compared, and the validity of the r.p.p models in predicting the energy partitioning between the two beams is examined.