A curved quadrilateral element for static analysis of shells with geometric and material nonlinearities

1985

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SUMMARYThe developments of an existing 48 degrees-of-freedom (d.o.f.), curved, quadrilateral, thin shell element, for materially and geometrically nonlinear static analysis of shell structures, are extended for the study of dynamic responses of nonlinear shells. The variable-order polynomial representations of the shell surface and the non-axisymmetric definition of the shell boundaries allow the study of the dynamic behaviour of a class of shell structures more general than those treated by using flat plate elements and elements with assumptions of axisymmetry. The equations of motion are based on a Lagrangian frame of reference. A combination of stepby-step and iterative procedures is used for the solution of nonlinear equations. The incremental equations of motion are linearized for computation purposes, and an algorithm for numerical integration based on Newmark's generalized operator for dynamic analysis, using optional iteration, is adopted. The flow theory of plasticity is used in the inelastic range, and perfectly plastic or isotropic strain hardening materials are considered. The spread of plastic zones in the thickness direction is treated by using a layered model. Numerical examples presented include the dynamic analyses of a square plate, a circular annulus, a cylindrical panel and a spherical cap. Comparisons with existing solutions demonstrate the validity and accuracy of the present developments.

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic analysis with a 48 d.o.f. curved thin shell element

Saigal

1985

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Elastic–viscoplastic, large deformation formulations of a curved quadrilateral thin shell element

Saigal

1985

SUMMARYAn elastic-viscoplastic constitutive law is implemented in the formulations of a curved quadrilateral element based on thin shell theory. The element surface is defined using variable-order polynomials allowing representation of a wide range of shell geometries. The large displacement effects are included by using an incremental stiffness procedure together with the Lagrangian mode of description of motion. The time rate effects may be treated by using explicit, semi-explicit or implicit time marching schemes. A scheme for automatic calculation of time step is used. By allowing steady-state conditions to be reached, plasticity solutions are obtained. Both perfect plasticity and strain hardening are considered. The Von Mises yield rule and the associated flow rule are used to determine plastic deformation. The spread of plasticity in the thickness direction is allowed by using a layered model. The procedures developed are applied to solve elastic-plastic, small and large displacement problems of thin plates and shells and the results obtained are compared with existing solutions to illustrate the validity and accuracy of the present developments.

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

1989

SUMMARYA 48-degree-of-freedom (d.0.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical nun-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.