A geometrically nonlinear tensorial formulation of a skewed quadrilateral thin shell finite element

A 48 degree-of-freedom (d.o.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical nonlinearity, was formulated and appropriate numerical procedures were adopted for the development of an efficient approach for the static and dynamic analyses of general thin shell structures. The element surface is described by a variable-order polynomial in curvilinear coordinates. The displacement functions are described by bicubic Hermitian polynomials in curvilinear coordinates. Tensorial mathematics is used throughout the formulations. In the present case of skewed quadrilateral with non-orthogonal curvilinear coordinates, the coupling terms of metric tensor and curvature tensor of the surface no longer vanish, such as in the case of orthogonal coordinates. Tensor form is used in the setup of the shape functions, geometric derivatives, stiffness matrix and computer code. This allows for the treatment of shells with irregular shapes and variable curvatures. To evaluate the efficiency and accuracy of this formulation, a systematic list of examples was chosen: (1) linear and nonlinear static analysis of square and rhombic plates, cylindrical and spherical shells; (2) linear vibrations of trapezoidal flat and curved plates; (3) large amplitude vibrations of a rhombic plate. For the square plate and cylindrical and spherical shells, skewed element meshes with various distortion angles were used to study the effect of the distortion angles on the accuracy of the results and to demonstrate the versatility of the present element. All results were compared with alternative available solutions including those obtained using regular rectangular meshes. Pinched thin cylindrical and spherical shells were studied using different skewed meshes and various Gauss integration meshes and no membrane locking phenomenon is observed. Due to the increased popularity of laminated composite plate and shell panels in aerospace and other technical applications, some efforts have been directed towards inclusion of such materials in the existing formulations