A Discrete-Element Method for Transverse Vibrations of Beam Columns Resting on Linearly Elastic or Inelastic Supports
Authors: Chan Jack H.C. , Matlock Hudson H.C. 
Published: October 2008

ABSTRACT
A discrete-element method of dynamic beam columns, which are rest.ing on linear or nonlinear-elastic, or nonlinear-inelastic supported and subjected to either static fixed loads or dynamic loads, is presented. Two parallel series multielement models, which consist of a collection of elastic and slip elements, are used to simulate the inelastic characteristics of supports. Fracture or softening of the supports is not. considered, but linear viscosity can be included.
Potential applications include study of the dynamic response of offshore piles to wave-induced forces and earthquake-induced forces and prediction of the hysteretic effect of inelastic supports under pavement slabs.

INTRODUCTION
In recent years, a great deal of interest has been focused on the utilization of numerical methods to describe computer models or problems in struct.ural dynamics. This work has been devoted to the development of a beamcolumn computer program which uses multielement models to predict the loading paths of resistance-deflection curves of the nonlinearinelastic supports.
The multielement models for the nonlinearinelastic supports described in this work are not time dependent; therefore, relaxation (creep) of resistance of the support is not included in the model. The retardation, or delayed elasticity of' the support can be considered by installing an external viscous dashpot in parallel with the support m6del. Fracture or softening of the nonlinear-inelastic support are also not included but strain hardening may be considered.
Crank and Nicolson2 have introduced an implicit formula to solve second-order heat flow problems. Salani7 is credited with applying this implicit formula in determining the transverse time-dependent linear deflections of a beam or plate. Essentially, the beam is replaced by an arbitrary number of rigid bars and deformabie' joints, and time is divided into discrete, equal intervals. The representation readily permits the flexural stiffness the elastic restraints, the mass densities, and the applied external loadings to be discontinuous and lumped at the deformable joints that connect the rigid bars. The governing partial differential equation at each joint is approximated by a difference equation that includes several unknown deflections of the joint, whichoccur at specified time intervals. All difference equations are based on the assumption of linear elasticity and elementary beam theory.
The nonlinear characteristics of the supports are considered by using either a spring-load iteration process (adjusting both the stiffness and the load from one iteration to the next, which is known as tangent modulus method) or a load iteration technique (adjusting only the load). The iteration process compares successively computed deflections until a specified tolerance is satisfied. Only three nonlinear characteristics of support curves are considered. The first is exhibiting the same resistance to either upward or downward deflections, hereafter referred to as symmetric resistance-deflection curve.