AbstractIn this study, a method for completely eliminating the presence of transverse shear locking in the application of the element‐free Galerkin method (EFGM) to shear‐deformable beams and plates is presented. The matching approximation fields concept of Donning and Liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope (first derivative of displacement) fields and is adopted. Implementation of the matching fields concept requires the computation of the second derivative of the shape functions. Thus, the shape functions for displacement fields, and therefore the moving least‐squares (MLS) weight function, must be at least C1 continuous. Additionally, the MLS weight functions must be chosen such that successive derivatives of the MLS shape function have the ability to exactly reproduce the functions from which they were derived. To satisfy these requirements, the quartic spline weight function possessing C2 continuity is used in this study. To our knowledge, this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element‐free Galerkin method. Several numerical examples confirm that bending analyses of thick and thin beams and plates, based on the matching approximation fields concept, do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields. Copyright © 2001 John Wiley & Sons, Ltd.