The paper investigates the possibility of delamination near the free edges of composites is investigated by the example of a two-dimensional formulation of the Pipes-Pagano problem. A bilinear material damage law based on the cohesive zone model (CZM) is considered. An example of selecting the stiffness of a cohesive layer is given, and a calculation of a laminated composite by the finite element method in the ANSYS program is performed. The method used makes it possible to determine the concentration of interlaminar stresses due to the edge effect, and, using a special set of interface elements simulating mating surfaces and following process of delaminaton, to investigate the composite delamination. When interface elements are added, the problem becomes non-linear and the solution in a two-dimensional formulation significantly reduces labor intensity and simplifies modeling, and also allows you to specify a large number of elements over the thickness of each layer. The problems with the convergence of the solution that have arisen can be eliminated with the help of special numerical modeling techniques and by varying the parameters embedded in the program. The revealed damage to the cohesive layer due to the edge effect can lead to delamination and premature failure of composites. The presented results can be used in the design of products from composite materials in rocket and space technology.