An alternative procedure in dynamic relaxation

Abstract

In the usual analytical methods for structural analysis, the structure is assumed to be loaded and the internal forces are present in it from the beginning of the calculations. The unknown forces are then found by assembhng and solving a set of simultaneous equations. The conventional methods in structural analysis have been radically altered by the advent of high speed digital computers. A new approach to the calculation of internal forces, made practically possible by this situation, is to follow their development in the structure from its initially unloaded state. This idea has been developed into a method of structural analysis which depends on following the dynamic behaviour of the structure until it it is damped down to its static con&ration. This method, which is presently known as Dynamic Relaxation, was originally conceived by Day [l] during the course of his investigations of complex prestressed concrete pressure vessels for nuclear reactors. The method of dynamic Relaxation (DR) is essentially a step-by-step integration of critically damped vibration using viscous damping to ensure the attainment of a steady-state solution. Dynamic terms involving inertia and viscous damping are added to the static equations of equilibrium. These dynamic equations of equilibrium, after being written in an explicit finite difference form, are solved using a standard substitution technique with the help of a high speed digital computer. When the damping coefficients are chosen to be critical the oscillations die out in the quickest possible time and the function converges to the static value. In practice, a value slightly lower than the critical value is chosen so as to get an oscillatory convergence which approaches the static value in the final limit and also gives bounds between which the true solution is found. The entire formulation is linear in time and consequently the precedure only requires a direct substitution of the value of the function at a previous iteration to calculate the new value of the function. During this direct substitution, it is relatively simple and straightforward to include any nonlinear term appearing in the formulation. For the sake of clarity the application of DR technique to the simple problem of bending of a beam is discussed in Appendix. Dynamic relaxation technique is presently a well established technique to solve linear and nonlinear structural problems. There is considerable amount of literature on its application to the analysis of plates, shells and three-dimensional solids [2-Q. Dynamic relaxation scheme is an explicit formulation and as such the time-wise integration procedure which