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**A catenoidal patch test for the inextensional bending of thin shell finite elements.**
*(English)*
Zbl 0742.73020

Summary: The history of the problem of inextensional bending of thin shells is briefly introduced. A comparison between Novoshilov’s and Sander’s shell theories is performed and it is shown that, for a catenoid of revolution, both theories will yield identical results. The problem of inextensional bending is then defined and discussed, with the catenoid of revolution as the focus. The theory of inextensional bending of a catenoid of revolution is then established and a simple patch test for constant inextensional bending is created. The critical size of a quadratic thin shell Sanders-type finite element is then predicted, above which parasitic membrane action will prevail. A very surprising result was obtained, that is, it appears to be practically impossible for quadratic thin shell finite elements to achieve a membrane stress free state along the meridional direction of the catenoid under inextensional bending. Finally, the patch test in question is examined, in both an ‘ undistorted’ and a ’distorted’ mode, with quadratic Mindlin isoparametric shell elements and the stress results obtained are presented and discussed. The superiority in the performance of the Gauss point stresses under reduced integration as compared to that of the nodal stresses is highlighted and integrated with results of previous publications.

### Keywords:

distorted mode; Novoshilov’s; Sander’s shell theories; catenoid of revolution; constant inextensional bending; critical size; quadratic thin shell Sanders-type finite element; undistorted modes; quadratic Mindlin isoparametric shell elements; stress results; Gauss point stresses; reduced integration; nodal stresses
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\textit{A. Kamoulakos}, Comput. Methods Appl. Mech. Eng. 92, No. 1, 1--32 (1991; Zbl 0742.73020)

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### References:

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