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Element Stiffness Matrix (element + stiffness_matrix)
Selected AbstractsAccurate eight-node hexahedral elementINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2007Magnus Fredriksson Abstract Based on the assumed strain method, an eight-node hexahedral element is proposed. Consistent choice of the fundamental element stiffness guarantees convergence and fulfillment of the patch test a priori. In conjunction with a ,-projection operator, the higher order strain field becomes orthogonal to rigid body and linear displacement fields. The higher order strain field in question is carefully selected to preserve correct rank for the element stiffness matrix, also for distorted elements. Volumetric locking is also removed effectively. By considerations of the bending energy, improved accuracy is obtained even for coarse element meshes. The choice of local co-ordinate system aligned with the principal axes of inertia makes it possible to improve the performance even for distorted elements. The strain-driven format obtained is well suited for materials with non-linear stress,strain relations. Several numerical examples are presented where the excellent performance of the proposed eight-node hexahedral is verified. Copyright © 2007 John Wiley & Sons, Ltd. [source] Reduced modified quadratures for quadratic membrane finite elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2004Craig S. Long Abstract Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 × 2 Gauss,Legendre integration rule is frequently used, as opposed to full 3 × 3 Gauss,Legendre integration. This ,softens' these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ,hourglass' mode common to Q8 and Q9 elements, since this spurious mode is non-communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non-communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher-order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under-integrated counterparts. Copyright © 2004 John Wiley & Sons, Ltd. [source] Free vibrations of shear-flexible and compressible arches by FEMINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2001Przemyslaw Litewka Abstract The purpose of this paper is to analyse free vibrations of arches with influence of shear and axial forces taken into account. Arches with various depth of cross-section and various types of supports are considered. In the calculations, the curved finite element elaborated by the authors is adopted. It is the plane two-node, six-degree-of-freedom arch element with constant curvature. Its application to the static analysis yields the exact results, coinciding with the analytical ones. This feature results from the use of the exact shape functions in derivation of the element stiffness matrix. In the free vibration analysis the consistent mass matrix is used. It is obtained on the base of the same functions. Their coefficients contain the influences of shear flexibility and compressibility of the arch. The numerical results are compared with the results obtained for the simple diagonal mass matrix representing the lumped mass model. The natural frequencies are also compared with the ones for the continuous arches for which the analytically determined frequencies are known. The advantage of the paper is a thorough analysis of selected examples, where the influences of shear forces, axial forces as well as the rotary and tangential inertia on the natural frequencies are examined. Copyright © 2001 John Wiley & Sons, Ltd. [source] Semi-analytical integration of the 8-node plane element stiffness matrix using symbolic computation,NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2006I. J. Lozada Abstract The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super-parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi-analytical integration and numerical integration were made. It was demonstrated that semi-analytical integration required less CPU time than conventional numerical integration (using Gaussian-Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source] An exact sinusoidal beam finite elementPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008Zdzislaw Pawlak The purpose of the paper is to derive an efficient sinusoidal thick beam finite element for the static analysis of 2D structures. A two,node, 6,DOF curved, sine,shape element of a constant cross,section is considered. Effects of flexural, axial and shear deformations are taken into account. Contrary to commonly used curvilinear co,ordinates, a rectangular co,ordinates system is used in the present analysis. First, an auxiliary problem is solved: a symmetric clamped,clamped sinusoidal arch subjected to unit nodal displacements of both supports is considered using the flexibility method. The exact stiffness matrix for the shear,flexible and compressible element is derived. Introduction of two parameters "n" and "t" enables the identification of shear and membrane influences in the element stiffness matrix. Basing on the principle of virtual work a full set of 18 shape functions related to unit support displacements is derived (total rotations of cross,sections, tangential and normal displacements along the element). The functions are found analytically in the closed form. They are functions of one linear dimensionless coordinate of x,axis and depend on one geometrical parameter of sinusoidal arch, height/span ratio "c" and on physical and geometrical properties of the element cross,section. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | ||||||||||