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A study on the linear elastic stability of Mindlin plates
Author(s) -
Cheung Y. K.,
Chan A. H. C.,
Tham L. G.
Publication year - 1986
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620220109
Subject(s) - spurious relationship , serendipity , stiffness , stability (learning theory) , vibration , finite element method , structural engineering , flexural strength , element (criminal law) , mathematics , computer science , mathematical analysis , engineering , acoustics , physics , law , statistics , astronomy , machine learning , political science
The debate on the performance of Lagrangian and Serendipity elements in Mindlin's plate theory has been going on for quite some time. Limited published results for static and vibration analysis based on exact integration demonstrated a drastic deterioration in accuracy as the thickness of the plate decreases, and reduced/selective integration schemes have been suggested to improve their performance. Appreciable improvement for Lagrangian elements has been recorded, but it is only marginal for the Serendipity elements. On the strength of such observations one would then be tempted to rule out the exact integration schemes and Serendipity elements. In this paper, the above problem is reviewed for stability analysis of plates. Two elements are chosen from each family, one representing the higher order and the other the lower order element. Contrary to published results, all elements can attain very accurate solutions independent of the integration schemes for a sufficiently restrained plate, although in general the Serendipity elements will require a more refined mesh than the Lagrangian ones. However, for loosely restrained plates, the solution failed when integration is performed by reduced/selective schemes. The failure marks the limitation of the reduced/selective schemes which have somehow introduced spurious modes into the system, but on the other hand it is ironical that these spurious modes in fact contribute to the improvement of performance of the restrained cases. Therefore, one can equally improve the Serendipity elements by a selective scheme which can introduce additional zero modes. A scheme based on (5 × 5) integration points for flexural stiffness and (3 × 3) integration points for shear stiffness improved the 17SE (Serendipity elements) remarkably. However, because of the lack of bounds in most cases, the use of reduced/selected schemes is still not recommended. Finally, this paper also proposed an approximate formulation of the geometric stiffness matrices to replace the full formulation. Such an approximate formulation reduces the number of variables considerably in the eigenvalue search and can still give reasonably accurate results for thin plates with a / t > 15.