A Simple and Effective Element for Analysis of General Shell Structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by su~~rn~os~ its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The appl~ation of the element to the analysis of various linear and no&ear problems is demonstrated. t. BOURN Two approaches have basicalIy been employed in the recent efforts on the development of generai sheelf analysis capabilities f I, 21: (, High-order isoparametric elements based on degenerating fully three-dimensional stress conditions have been proposed. # Low-order simple elements that are basically obtained by superimposing plate bending and membrane stiff nesses have been developed. The higher-order isoparametric elements are very versatile (they can be employed as ~ansi~on elementsp, 33) and are quite effective, but they can be costly in use. The element stiffness rna~ is relatively large in size and a su~ciently high enough integration order must be used to avoid the introduction of spurious zero energy modes, The premise of the simple low-order elements lies in that their related matrices can be formed inexpensively. Thus, even when a large number of elements are required to model a complex structure, the overall analysis effort may still be less than with the use of the higher-order isoparametric shell elements. Also, the direct use of stress resultants (moments, membrane forces) may not only decrease the cost of analysis, but also facilitates the interpretation of the computed results. Various simple low-order elements have been proposed recently[&S]. When evaluating these elements for practical analysis, we believe that the following three criteria should be considered: (1) The element should yield accurate solutions when modeling any shell geome~y and under all boundary and loading conditions. In particular, &he element should exactly contain the required 6 zero rigid modes, so that reliable results can always be expected, The theory of the element formulation must be well-understood and should not contain any “numerical fudge factors”. (2) We should be able to use the element in the modeling of general shell structures with beam stiffeners, cut-outs, intersections, and so on. (3) The element should be ~st~ffective in linear as well as in no~inear, static and dyn~ic analysis. In nonlinear analysis, the element should be applicab~ to large displacement, large rotation, and materially nonlinear condi~ns= considering the above criteria we want to emphasize that the reliability aspect in (I) is the most important. Yet, a considerable number of elements that have been published do not satisfy this criterion. Such element developments represent interesting research, but should not be used in actual engineering analyses, because the generated analysis results cannot be interpreted with confidence. The objective in this paper is to present a shell element that is simple and effective and that has been developed with due regard to the above r~u~ements. The element is shown schematically in Fig. I. We observe that the element is flat and has 3 nodes with 6 degrees of freedom per node. The total element stiffness matrix is formulated by superimposing-in the way some of the earliest shell elements were formulated [9E_a plane stress membrane stiffness KM, a bending stiffness Kg and an in-plane rotational stiffness KY In the next section we discuss the derivation of these st#ness matrices for linear analysis. The updated Iranian fo~uIation used in large displacement analysis is then presented in Section 3 and finally in Section 4 we present the results obtained in various demonstrative sample analyses. Since the complete stiffness matrix of the element is obtained by the direct superposition of Kns, Kg and K, we can discuss the formulation of these matrices independently. 2.1 Membrane stiifness matrix The element membrane stiRness Klu is simply the unsent strain plane stress s~ness rna~ of a 3-node element. The bending stiffness matrix is formu~ted using the Mindiin theory of plates with shear deformations included. Using the variables defined in Fig. 2, the displacement components of a point with coordinates x, y, z are in this theory [Sj g = zS&, yl; e = .rB,Cz, ~1; and w = WCX, yI (If where w is the transversal displa~ment, & and 8, are the rotations of the normal to the undefo~ed middle surface about the y and x axes, respectively, The bending strains vary linearly through the thickness &=ZU (2) where K is the three component vector of curvatures