COMPARISON OF COMPUTER-AIDED INTEGRATION TECHNIQUES WHEN OBTAINING INITIAL DISPLACEMENTS MATRIX FOR GEOMETRICALLY NON-LINEAR LAMINATED FINITE ELEMENT /KOMPIUTERINIŲ INTEGRAVIMO TECHNOLOGIJŲ PALYGINIMAS SUDARANT GEOMETRIŠKAI NETIESINIO SLUOKSNIUOTOJO BAIGTINIO ELEMENTO PRADINIŲ POSLINKIŲ MATRICĄ

Abstract

Geometrical non-linearity of the laminated element has not been realized so far in the widely known commercial finite element method packages such as ABAQUS, ALGOR, ANSYS, COSMOS although researches in that field are actively carried out. On the other hand, there is a lot of problems where large displacements and deformations must be dealt with to obtain a precise decision. A wide range of composite orthotopic materials is used in constructions and other fields of technology. Various numerical methods were implemented to handle laminated plates and shells, however most of them are intended for application only with particular types of the structures. The author's aim is to develop a geometrically nonlinear finite element that could be effectively used for analysis of various laminated slabs regardless of their shape, thickness of laminae, properties of materials, direction of orthotropy axes, way of loading and boundary conditions. Obtaining and handling the element's initial displacement matrix used in the iteration process is a highly complicated issue requiring significant amount of computer resources to be involved. One of the most important aims of the research is to develop an element which could be used not only in an expensive multiprocessor mainframes, but also in an usual personal computer. For the structure, a sophisticated finite element TRIPLT having 50 degrees of freedom is used. The geometrical matrix for this element is obtained involving L-coordinates' array while displacements and rotations in the middle of the element are expressed through the nodal displacements (rotations), their derivatives, and displacements (rotations) in the central point. Linear and non-linear components for the geometrical matrix are shown in Eqs 2 and 5. The behaviour of a geometrical non-linear finite elements structure is described by Eq 8. The tangent stiffness matrix consists of the conventional linear elastic stiffness matrix, initial stress matrix and initial displacements matrix which is obtained by Eq 10, using both analytical and/or numerical integrating. The analytical integrating involves expanding of the appropriate expressions into basic matrices (Eqs 11, 12) and using formula 15. The initial displacement matrix in term of constitutive matrix's elements and the basic matrices is shown in Eqs 13 and 14. Numerical integrating is conducted by two methods: those using Hammer and Gauss-Radau weight coefficients. Numerical approach is applied both to the basic matrices and factorised expressions of submatrices involving intermediate arrays and matrices (Eqs 23, 24). Two ways of obtaining the intermediate arrays and matrices are discussed. Because of high complexity of the procedures involved the computer algebra system Mathematica was used for the integrating and recording FORTRAN codes. Comparison of the effectiveness of all the procedures is presented in a table. The investigation results show that the initial displacement matrix obtained by means of numerical integration involves a small amount of arithmetic operations to be handled with a usual personal computer.