AbstractIn this work, meshless methods based on the local Petrov–Galerkin approach are proposed for the solution of dynamic problems considering elastic and elastoplastic materials. Formulations adopting the Heaviside step function and the Gaussian weight function as the test functions in the local weak form are considered. The moving least‐square method is used for the approximation of physical quantities in the local integral equations. After spatial discretization is carried out, a non‐linear system of ordinary differential equations of second order is obtained. This system is solved by Newmark/Newton–Raphson techniques. At the end of the paper numerical results are presented, illustrating the potentialities of the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.