This thesis documents the performance of a p-hierarchical interface-enriched generelized finite element method (p-IGFEM) on weakly discontinuous beams and plates, which are modeled according to the first-order shear deformation theory (FSDT). Here the purpose of the higher-order shape functions is to reduce the shear locking that arises due to FSDT. In this report the examples will be limited to linear, static beams and plates with a single material discontinuity. The method will be assessed in both the convergence rate (in energy norm) and the stability. The method has shown to be very susceptible to small relative distances between enriched and standard nodes. Furthermore, the condition number could successfully be decreased by applying a Jacobi-like diagonal preconditioner.