Modified Mindlin plate theory and shear locking-free finite element formulation

• Dynamic behaviour of thick plate is decoupled into flexural and in-plane vibrations.
• These vibrations are described with one and two equations of motion, respectively.
• Plate in-plane vibrations are analogous to membrane vibrations.
• A flat shell element comprises membrane, plate flexural and plate in-plane element.
• In new FE formulation shear locking does not appear.

The basic equations of the Mindlin theory are specified as starting point for its modification in which total deflection and rotations are split into pure bending deflection and shear deflection with bending angles of rotation, and in-plane shear angles. The equilibrium equations of the former displacement field are split into one partial differential equation for flexural vibrations. In the latter case two differential equations for in-plane shear vibrations are obtained, which are similar to the well-known membrane equations. Rectangular shear locking-free finite element for flexural vibrations is developed. For in-plane shear vibrations ordinary membrane finite elements can be used. Application of the modified Mindlin theory is illustrated in a case of simply supported square plate. Problems are solved analytically and by FEM and the obtained results are compared with the relevant ones available in the literature.