The performance of partition-of-unity based methods such as the generalized finite element method or the extended finite element method is studied for the simulation of cohesive cracking. The focus of investigation is on the performance of bilinear quadrilateral finite elements using these methods. In particular, the approximation of the displacement jump field, representing cohesive cracks, by extended finite element method/generalized finite element method and its effect on the overall behavior at element and structural level is investigated. A single element test is performed with two different integration schemes, namely the Newton-Cotes/Lobatto and the Gauss integration schemes, for the cracked interface contribution. It was found that cohesive crack segments subjected to a nonuniform opening in unstructured meshes (or an inclined crack in a structured finite element mesh) result in an unrealistic crack opening. The reasons for such behavior and its effect on the response at element level are discussed. Furthermore, a mesh refinement study is performed to analyze the overall response of a cohesively cracked body in a finite element analysis. Copyright © 2013 John Wiley & Sons, Ltd.