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A piezolaminated composite degenerated shell finite element for active control of structures with distributed piezosensors and actuators

This paper deals with the formulation of a nine-noded piezolaminated degenerated shell finite element for modeling and analysis of multilayer composite general shell structures with bonded/embedded distributed piezoelectric sensors and actuators. The distributed PZT sensors and actuators used in the composite smart structures are relatively thin and could have arbitrary variation of curvatures and thicknesses. They cannot be modeled with shell elements based on curvilinear shell theories which would need the specification of constant shell curvatures and thicknesses. Modeling them with piezo finite elements available in popular commercial codes like ABAQUS, ANSYS, MARC, etc, would need relatively greater computational effort as they are based on solid element formulation. In view of these, the present proposed degenerated piezoelectric shell element would be a better choice giving good computational accuracy and efficiency. The main advantage of a degenerate shell element is that it is not based on any shell theories and is applicable over a wide range of curvatures and thicknesses. This element is developed by using the degenerate solid approach based on Reissner–Mindlin assumptions which allow the shear deformation and rotary inertia effect to be considered and the 3D field is reduced to a 2D field in terms of mid-surface nodal variables. Uniformly reduced integration is carried out to overcome membrane locking and shear locking and the numerical integration is carried out in all three directions to obtain accurate results. The present element has 45 elastic degrees of freedom and 10 electric degrees of freedom per piezoelectric layer in the element. The potential induced due to bending deformation is more accurately represented by assuming quadratic variation of the electric potential through the thickness of each piezoelectric layer. This is achieved by interpolating using nodal mid-plane electric potentials and one electric degree of freedom representing the potential difference between the top and bottom surfaces of the piezoelectric layer. Few case studies of composite general shells with piezoelectric sensors and actuators have been considered by modeling them with the above elements and the active vibration control performance has been studied using linear quadratic Gaussian (LQG) control.