A geometrically non‐linear tensorial formulation of a skewed quadrilateral thin shell finite element

Abstract

AbstractA 48‐degree‐of‐freedom (d.o.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical non‐linearity, is formulated and appropriate numerical procedures are adopted for the development of an efficient approach for the static and dynamic analysis of general thin shell structures. The element surface is described by a variable‐order polynomial in curvilinear co‐ordinates. The displacement functions are described by bicubic Hermitian polynomials in curvilinear co‐ordinates. The directions of the curvilinear co‐ordinates at each nodal point are uniquely defined to coincide with the directions of the boundaries of the element. In the present case of a skewed quadrilateral with non‐orthogonal curvilinear coordinates, the coupling terms of the metric tensor and curvature tensor of the surface no longer vanish, such as in the case of orthogonal co‐ordinates. The 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 is chosen: (i) linear and non‐linear static analysis of square and rhombic plates, cylindrical and spherical shells; (ii) linear vibrations of trapezoidal flat and curved plates; (iii) large amplitude vibrations of a rhombic plate. For the square plate and cylindrical and spherical shell, shewed element meshes with various distortion angles are 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 are compared with alternative available solutions including those obtained using regular rectangular meshes. Pinched thin cylindrical and spherical shells are studied using different skewed meshes and various Gauss integration meshes, and no membrane locking phenomenon is observed.