Authored by Osama Mohammed Elmardi Suleiman Khayal*
Abstract
The method of dynamic relaxation (DRM) in its early stages of development was perceived as a numerical finite difference technique. It was first used to analyse structures, then skeletal and cable structures, and plates. The method relies on a discretized continuum in which the mass of the structure is assumed to be concentrated at given points (i.e. nodes) on the surface. The system of concentrated masses oscillates about the equilibrium position under the influence of out of balance forces. With time, it comes to rest under the influence of damping. The iterative scheme reflects a process, in which static equilibrium of the system is achieved by simulating a pseudo dynamic process in time. In its original form, the method makes use of inertia term, damping term and time increment.
Keywords: DRM, bending, composite laminates, plate equations, finite differences
Introduction
In the present work, finite differences coupled with dynamic relaxation (DRM) Method, which is a numerical technique, is used. The DRM method was first proposed and developed in 1960; see Rushton KR [1], Cassell AC & Hobbs RE [2], Day AS [3]. In this method, the equations of equilibrium are converted to dynamic equations by adding damping and inertia terms. These are then expressed in finite difference form and the solution is obtained by an iterative procedure as explained below.
The DRM Iterative Procedure
In the DRM technique, explained in the previous sections of this chapter, the static equations of the plate have been converted to dynamic equations i.e. equations (12). Then the inertia and damping terms are added to all of these equations. The iterations of the DRM technique can then be carried out in following procedures:
1. Set all initial values of variables to zero.
2. Compute the velocities from equations (16)-(20).
3. Compute the displacements from equations (22).
4. Apply suitable boundary conditions for the displacements.
5. Compute the stress resultants and stress couples from equations (25) to (32).
6. Apply the appropriate boundary conditions for the stress resultants and stress couples.
7. Check if the convergence criterion is satisfied, if it is not repeat the steps from 2 to 6.
It is obvious that this method requires five fictitious densities and a similar number of damping coefficients so as the solution will be converged correctly.
The Fictitious Densities
The computation of the fictitious densities based on the Gershgorin upper bound of the stiffness matrix of a plate is discussed in Cassel & Hobbs [2]. The fictitious densities vary from point to point over the plate as well as for each iteration, so as to improve the convergence of the numerical computations. The corresponding expressions for the computations of the fictitious densities are given below:
To read more about this article.......Open access Journal of Current Trends in Civil & Structural Engineering
Please follow the URL to access more information about this article
To know more about our Journals...Iris Publishers
No comments:
Post a Comment