The demand for using thin-walled composite structures for instance, in aircraft and ships are rising in the industry. These structures contain material interfaces, and at these interfaces, the structure exhibits non-smooth field behavior which is known as weak discontinuity. It is essential to study the effects of such discontinuities on the performance of the structure. These structures are usually modeled with standard finite element method (FEM) with fitted or geometry conforming meshes to the discontinuities. Shear locking is a key hurdle to overcome in FEM when using first-order shear deformation theory (FSDT) for thin structures. Classical beam (Euler-Bernoulli) or plate (Kirchhoff-Love) theories do not face such issues but ignore shear deformation and set the requirement of shape functions that are C1 continuous. The popular mesh independent method, eXtended/Generalized finite element method (X/GFEM) has also been used to capture the kinematics of discontinuous thin structures that are also devoid of locking. The presented work introduces a high-order discontinuity-enriched finite element formulation for linear and nonlinear (large deformations) discontinuous FSDT beam and plate elements. High-order interpolations are used to mitigate shear locking. The high-order enrichment functions are constructed by using hierarchical shape functions of the p-version of FEM (p-FEM), where the linear enrichment function is obtained from the interface enriched generalized finite element method (IGFEM) is combined with hierarchical shape functions of p-FEM. Convergence study shows that the proposed method accurately captures the discontinuity kinematics leading to an exact solution when compared with the analytical solution. A high condition number of the stiffness matrix is observed in the stability study which leads to loss of stability. A Jacobi like diagonal preconditioner is used to reduce the condition number but does not work well for high-order interpolations.