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Cross-section deformation, geometric stiffening, and locking in the nonlinear vibration analysis of beams

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Abstract

Stiff behavior of more general finite element (FE) beam formulations in some problems can be misinterpreted as locking based on comparison with simplified analytical and/or less general FE beam formulations. This paper demonstrates that, in more general beam formulations, higher stiffness can be attributed to geometric nonlinearities as result of cross-section deformations, not properly captured by analytical or less general FE beam formulations. Limitations of the ad hoc approaches used in conventional FE beam formulations to account for beam cross-section deformations are identified, their inconsistency with theory of continuum mechanics is explained, and their appropriateness is evaluated in view of a more general approach. Effect of using different constitutive models on the stiff behavior of beams is investigated, and it is demonstrated that the stiff behavior resulting from the geometric stiffening due to the coupling between the cross-section deformations and beam vibrations in more general beam formulations cannot always be interpreted as locking. Relationship between geometric stiffening, cross-section deformation, locking, and constitutive model in more general FE beam formulation is explained. Several numerical examples are used to perform static, dynamic, and thermal analyses; and the results obtained are compared with FE commercial software. These results demonstrate limitations of beam formulations used in commercial FE software, shed light on problems of using simplified analytical solutions for verification, highlight concerns of using conventional FE approaches for soft robots and materials, and caution against misinterpretation of the stiff behavior as locking when using more general beam formulations.

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Data Availability Statement

The data of the models developed in the paper are presented in the numerical example section.

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Acknowledgements

This research was supported by the National Science Foundation (Projects # 1852510).

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL, 60607, USA

    Ahmed E. Eldeeb & Ahmed A. Shabana

  2. Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza, Egypt

    Ahmed E. Eldeeb

  3. School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Dayu Zhang

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Eldeeb, A.E., Zhang, D. & Shabana, A.A. Cross-section deformation, geometric stiffening, and locking in the nonlinear vibration analysis of beams. Nonlinear Dyn 108, 1425–1445 (2022). https://doi.org/10.1007/s11071-021-07102-x

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