00457949(94)00574-5 Computers d Strurrures Vol. 56. No. 6. pp. 993-1007. 1995 Copyright 0 1995 Elwicr Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 $9.50 + 0.00 FINITE ELEMENT FREE VIBRATION ANALYSIS OF ECCENTRICALLY STIFFENED PLATES T. P. Holopainen VTT Manufacturing Technology, Technical Research Centre of Finland, P.O. Box 1705, 02044 VTT, Finland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ (Received 3 February 1994) Abstract-A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach. 1. INTRODUCTION Stiffened plates are structural components consisting of plates reinforced by a system of ribs to enhance their load-carrying capacities. These structural com- ponents have been widely applied to aircraft, ships, bridges, buildings, as well as in many other branches of structural engineering. Thus, there are many circumstances in which a stiffened plate structure is exposed to dynamical loads. Different approaches have been suggested for the vibration analysis of stiffened plate problems [ 1,2]. Many of these approaches can be employed together with the finite element method (FEM). Five main types of finite element models can be identified [l-3]. The early investigators attempted to model the stiffened plate as a plate with orthotropic properties. In this model the stiffeners are assumed to be smeared into the plating, and the structure is replaced by an equivalent plate having different properties in orthogonal directions. The equivalent properties are determined from those of the plate and stiffeners. The second approach is to consider the stiffened plate as a grillage system. In this model the stiffened plate is replaced by a plane structure of intersecting beams. The equivalent properties of beams are deter- mined from those of the stiffeners and by considering the effective breadth of the plate. The third approach is to apply the lumping of the stiffeners. In this model the stiffeners located within a plate element are shifted to the nodal lines of the plate elements. In other words, the layout of the finite element mesh determines the location of the stiffeners. In this approach, as in the previous two, the actual structure is replaced by a modified structure whose topology is different. Subsequently, the num- erical solution is sought for this modified structure. This leads to an idealization error which may be difficult to estimate. Especially when the stiffeners are sparse or irregular, these first three approaches can fail seriously. A more realistic model is achieved through the separate consideration of the plate and the stiffeners, and maintaining the compatibility between the two. In the fourth approach the plating is modelled by plate elements, and the stiffeners by beam elements. In this model the layout of the stiffeners dictates the layout of the finite element mesh of the plating. This may be inefficient and lead to unnecessary fine meshes if only the overall response of the complete structure is required. It may also lead to topological difficulties in mesh generation, where the pattern of the stiffeners is irregular or complex. The fifth approach has been the employment of the stiffened plate element [3-61. The main advantage of this formulation is that the stiffeners can be placed anywhere within the plate element, and they need not necessarily follow the nodal lines. This liberates the user from the rigors of providing a nodal line along every stiffener and, the mesh division can be chosen according to the resolution sought. This is a distinct improvement and considerably simplifies the modelling of stiffened plate structures. The stiffened plate elements presented in Refs [3-61 follow the fifth approach and are based on a quadri- lateral plate-bending element [7]. This isoparametric quadratic element includes eight nodes and is based on the Reissner-Mindlin thick plate model. In order 993