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A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

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Abstract

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives.

The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS.

The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.

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Acknowledgements

The authors would like to thank the Finnish IT Center for Science for the supercomputer time used for calculations, the Academy of Finland (Application No. 259543) for supporting Marko Matikainen, and the National Graduate School of Engineering Mechanics, Finland for supporting Antti I. Valkeapää.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, 53850, Lappeenranta, Finland

    Marko K. Matikainen, Antti I. Valkeapää & Aki M. Mikkola

  2. Laboratory for Engineering Mechanics, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

    A. L. Schwab

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  1. Marko K. Matikainen
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  2. Antti I. Valkeapää
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  3. Aki M. Mikkola
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  4. A. L. Schwab
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Correspondence to Marko K. Matikainen.

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Matikainen, M.K., Valkeapää, A.I., Mikkola, A.M. et al. A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation. Multibody Syst Dyn 31, 309–338 (2014). https://doi.org/10.1007/s11044-013-9383-6

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