COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, zyxwvu VOl. 10, 11-19 (1994) SEMI-LOOF ELEMENT FOR PLATE INSTABILITY J. JBNSSON, zyxwvu S. KRENK zyxwvu Department zyxwvutsr of Building Technology and Structural Engineering, University of Aalborg, Sohngaardsholmsvej zyx 57, Aalborg, Denmark AND L. DAMKILDE Department of Structural Engineering, Technical University of Denmark, Lyngby, Denmark SUMMARY In the formulation of the semi-Loof element the rotation of the tangent plane is derived from the interpolation of the transverse displacement, while the rotation of the normal is interpolated separately by another set of shape functions. The geometric stiffness matrix can be formulated by use of either of the two rotation representations. It is demonstrated that the use of the tangent plane representation in the geometric stiffness matrix is far superior to the common form at present. INTRODUCTION In plate theory, translation as well as rotation of the normal to the plate must satisfy continuity requirements. These requirements complicate the formulation of displacement-based finite elements for plates. In the semi-Loof elements zyxwvu - originally proposed by Irons - the transverse displacement component and the rotation components of the normal to the undeformed plate are interpolated independently in terms of the values at suitably located nodes. The tangent plane of the deformed plate is defined in terms of derivatives of the interpolation of the transverse displacement. As a result of this procedure the slopes of the tangent plane and the rotations of the normals are represented by different sets of shape functions, and a point- matching procedure must be used to relate them. In Kirchhoff plate elements equality is imposed at selected points, while Mindlin elements use these points to evaluate the shear strain. The intention is that after matching at selected points the two sets of rotations are equal to within the order of shear strains. In the formulation of the geometric stiffness matrix for use in stability analysis a rotation field is needed. In the analytical formulation of plate theory it does not matter which set of rotations is used, because the difference between them is of the order of shear strains, i.e. by assumption an order of magnitude smaller than the rotations associated with instability. However, in the semi-Loof element the use of different functions to represent the two sets of rotations may lead to differences in the performance of the resulting elements. Indeed, as demonstrated in the Examples Section of this paper the use of the slopes of the tangent plane in the formulation of the geometric stiffness matrix leads to far better performance of the element. 0748-8025/94/010011-09$09.50 zyxwvu 0 1994 by John Wiley & Sons, Ltd. Received 29 May 1991 Revised zyx 1 July 1993