In this thesis, the capabilities of two meshless methods have been investigated based on explicit time integration procedure for 2D solid mechanics problems. In the first method, the point collocation method is used and the momentum equation with any damping term can be used. In this investigation, strain rate based damping is used to compensate the errors of low degree polynomial basis functions and enforcement of essential or natural boundary conditions. This method does not require any special treatment for either essential or natural boundary conditions. In the second method, using the weak form of momentum equation, a new method is presented to evaluate the integrals of the Galerkin weak form in meshless methods without any background mesh. Simple Kriging interpolation method has been utilized to obtain the weights at the integration points. Then, the integration of the Galerkin weak form is evaluated using the introduced weights of the integration points. Simple Kriging interpolation has also been used to obtain a diagonal mass matrix. Different types of arrangements for integration points are examined for 2D elasticity problems and the accuracy of the results is compared by numerical examples. The accuracy and efficiency of this integration procedure has been shown by several examples in 2D elasticity problems, 2D free vibration problems and analyzing a plastic large deformation problem. As an example compression of an axisymmetric ring is analysed by the present method. The results show that this method is accurate and can be comparable with FEM. Keywords: Meshless methods, explicit time integration, strong and weak formulations, damping term, weak form integration.
