In this study advanced low order finite elements for the linear analysis and ultimately, the global optimization of orthotropic shells structures are presented. Low order quadrilaterals are attractive in optimization, since they result in low connectively of the structural stiffness matrix, and hence, reduced computational effort. However, standard 4-node quadrilaterals are notorious for their low accuracy. Both drilling degrees of freedom and assumed stress interpolations have the potential to improve the modeling capabilities of low order quadrilateral finite elements. Therefore, it seems desirable to formulate low order elements with both an assumed stress interpolation field and drilling degrees of freedom, on condition that the elements are rank sufficient and invariant. Firstly, a variational basis for the formulation of two families of assumed stress membrane finite elements with drilling degrees of freedom, is presented. This formulation depends on the formulation of Hughes and Brezzi, and is derived using the unified formulation presented by Di and Ramm. The recent stress mode classification method presented by Feng et al is used to derive the stress interpolation matrices. The families, denoted 8β(M) and 8β(D), are rank sufficient, invariant, and free of locking. The membrane locking correction suggested by Taylor ensures that the consistent nodal loads of both families are identical to those of a quadrilateral 4-node membrane finite element with two translational degrees of freedom per node. Secondly, the rectangular assumed strain plate element presented by Bathe and Dvorkin is combined with the above mentioned membrane families to form flat shell finite elements. The strain-displacement measures of these elements are modified on the element level to incorporate the effect of element warp. Thirdly the constitutive relationship of the flat shell elements is extended to include symmetric orthotropy. In opposition to the general trend to employ quadratic or even cubic elements for orthotropic analyses, it is shown that the simpler 4-node assumed stress families with drilling degrees of freedom presented herein are highly accurate and effective. Finally the influence of the stability parameter γ, the integration scheme order and the effect of the membrane locking correction are evaluated. The numerical value of the parameter γ is shown to be irrelevant in the patch test. The effect of the previously proposed membrane-bending locking correction when included in in-plane analysis is demonstrated. The elements have been incorporated in the EDSAP/CALSAP finite element infrastructure.
|Subject 2||Mechanical engineering|
|Degree Type||Masters degree|