plate bending, penalty method, hybrid-type virtual work, discontinuous Galerkin method 

In present paper, we have given the investigations of the plate bending problem by numerical treatment using hybrid-type penalty method (HPM). The HPM assume linear and nonlinear displacement field with rigid displacement, rigid rotation, strain and its gradient in each subdomain and introduce subsidiary condition about the continuity of displacement into the framework of the variational expression with Lagrange multipliers. For the purpose of this paper, we accepted the Kirchhoff theory that neglects the transversal shear deformation. In the first step of the work, we are giving the equilibrium equations for a deformable body in 3D case and as boundary conditions we are giving geometrical (for displacement field) and kinetic (for surface force) boundary conditions. Secondary we apply Kirchhoff theory to the displacement field of the 3D case for plate bending problem. For this purpose, we use quadratic form that includes rigid, linear and nonlinear parts of the displacements. The parameters used in this displacement field are independently defined in each subdomain. We introduce penalty function that presents strong spring connecting each subdomain. Then, we take the matrix of the subsidiary condition according to the surface integral of the contact surface of each sub-domain. We apply nonlinearity in penalty function such as spring system, which allow us to calculate hinge line. If hinge line makes mechanism then we can calculate limit load. We used load incremental method called r-min method in a material nonlinear analysis. We can calculate growing hinge line systematically using this algorithm for the nonlinear analysis. Finally, we calculate some simple problems to check accuracy of elastic solution and limit load.