An assessment of flat triangular plate bending elements with displacement degrees-of-freedom at the three comer nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees-of-freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three-node plate bending elements are the DKT and HSM elements. 1.A particular shell theory is used and discreti~,ed.'*~ 2. Three-dimensional continuum equations are used and discretized (isoparametric elements) (References 6 and 7, etc.). 3. Plate bending and membrane element stiffnesses are superimposed and assembled in a global co-ordinate system (References 8 and 9, etc.).The three approaches have advantages and disadvantages, and it is still difficult to state which of the three approaches is most effective based on criteria combining accuracy, computational cost and simplicity in use (in the data input phase as well as in the interpretation of results). Approach 3 received a great deal of attention for the linear analysis of shell structures in the mid-l960s," but the activities related to approaches 1 and 2 have dominated the past 10 years. It is only recently that a new impetus has been given to the analysis of shells using approach 3. 0029-598 1 /80/ 12 15-177 1 $01 .OO Considering this approach, triangular flat elements having displacements and rotations at the corner nodes as degrees-of-freedom-the engineering dof-are particularly appealing for many practical reasons; for example, arbitrary shell geometries, general supports and cut-outs, and beam stiffeners can be modelled. These elements have a total of 18 dof (3 translations and 3 rotations at each node) or 15 dof (3 translations and 2 rotations) depending on whether the rotation about the normal is included as a dof. The element formulation is based on a superposition of membrane and bending actions. Among the most recent papers on this subject, References 12 and 13 deal with the linear analysis, whereas References 9,14,8 and 15 deal with the geometrically nonlinear analysis of shells with large displacements and rotations. In References 1 2 , 9 and 14 the hybrid stress formulations are used, whereas in References 13, 8 and 15, displacement-type formulations are employed.A very important consideration in the development of these shell elements is the representation of the bending behaviour. Although several theoretical and numerical studies on plate bending finite elements have appeared in the past 15 years, a detailed recent study and comparison of triangular plate bending elements with only 3 dof at the corner nodes (displacement w and rotations & and By...
Abstract-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. BOURNTwo 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.
SUMMARYWe present a new surface-intrinsic linear form for the treatment of normal and tangential surface tension boundary conditions in C '-geometry variational discretizations of viscous incompressible free-surface flows in three space dimensions. The new approach is illustrated by a finite (spectral) element unsteady Navier-Stokes analysis of the stability of a falling liquid film.
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