Tuesday, March 4, 2014

Formulating the Applied Element Method: Linear 2D (Part I)

I first came to know about the Applied Element Method (AEM) during the one day seminar on "Geotechnics and Geo hazards" 2012 held in Nepal. Ramesh Guragain had mentioned this method in his presentation on the study of collapse of masonry structures (and something to do with fragility functions I believe). The results he showed using AEM were really compelling. I wanted to learn more about this method.

Here I discuss the basic procedures that I have followed to develop an AEM program on Python that can solve Linear 2D structural problems. Poisson's ratio is not considered here. All of the procedures discussed below are based on Meguro and Tagel-Den (2000). The developed program is then used to solve a classic cantilever beam problem. The displacement values obtained using AEM are compared with corresponding theoretical values and values obtained from Finite Element Method (FEM) for plane stress condition. AEM performs better compared to FEM even when the number of elements is small.


Introduction

AEM is modeled by dividing the structure into rigid elements connected with pairs of normal and shear springs as shown in figure 1. The stiffness values of each pair of springs represent the material property of certain area of both the elements as shown in figure 1 b. The stiffness values are determined as shown below:
where E and G are Young's and shear modulus, d is the distance between springs, T is the thickness of the element and a is the length of representative area.