The formulation and implementation of three-dimensional element free Galerkin method (3D EFG) are developed. A simple and efficient scheme for a variable domain of influence stipulates that a constant number of nodal points are visible from each integration location is proposed. This method significantly increases the efficiency of the variable domain of influence by limiting the size of the least-square problem that is solved when computing approximate functions. The 3D EFG methods based on moving least square method use only nodal points to built local and global approximation. Discrete model of the 3D EFG for three-dimensional elastic problems is derived by least potential energy principle. Reference to the 2D EFG, in the 3D EFG, it is enforced to meet displacements boundary conditions by use of limiting nodal point number method and penalty method. The stress concentration of a small column-shaped cavity in a cube subjected to uniaxial uniform tension at two opposing faces in far field. Compared the approximation solutions with theory ones, the results indicate that the 3D EFG is validity in solving three-dimensional elastic problems and the limiting nodal point number method is validity.