This thesis presents the application of Element Free Galerkin method for analysis of an Elastic Bar. The Element Free Galerkin method is one of the efficient mesh free methods which needs only nodal data and does not need any nodal connectivity, which provides an advantage over the FEM''s. The need for the meshfree methods arose from the fact that the mesh based methods employ the elements for the imposition of the field values. These elements when distorted or misaligned with the back ground mesh give faulty and inaccurate results, further the simulation of explosion or the crack growth is very difficult in FEM''s. The results presented in this thesis are checked for validity and accuracy of numerical solutions and compared with the exact solution. Convergence study has been performed. Results show that the proposed formulations are able to reproduce the exact solution with high accuracy which indicates that these methods would be suitable in more complex environments and domains-settings.