This study provides a plate-theory-based sandwich model for delaminated structures that incorporates the third-order shear deformation theory (TSDT) and the Lagrange multiplier methods. The dynamical boundary and continuity conditions are not granted in the TSDT models, therefore, these auxiliary conditions are introduced by the multiplier method. The current model assumes linear elastic orthotropic material behaviour with a compressible core. Hence, the transversal normal stress is not neglected. For particular boundary conditions, a semi-analytical solution could be carried out. However, in order to use the model for more general structural analysis, a special finite element family was developed based on the introduced considerations. This family comprises intact, delaminated and transition elements. The transition elements provide a continuous connection between the intact and delaminated regions. For finite element analysis, the auxiliary conditions are prescribed node-wisely and taken into account by nodal Lagrange multipliers. Thus a physically correct stress field can be acquired in the intact and delaminated regions as well. The developed model is applied to several test problems and the results are compared to previous experiments and well-known solution methods such as 3D finite element analysis and semi-analytical approaches. The new model shows satisfying agreement with the general 3D finite element results, hence the time-consuming 3D modelling can be replaced by the developed 2D model.