Unified and integrated approach in a new Timoshenko beam element
DOI:
https://doi.org/10.1080/17797179.2017.1328643Keywords:
Unified and integrated approach, finite element, Timoshenko beam, shear influence factorAbstract
This paper presents a new unified and integrated approach to construct locking-free finite elements for bending of shear deformable beam element. The new UI (Unified and Integrated) element, with two nodes and three degrees of freedom (d.o.f.) per node, is formulated based on a pure displacement formulation and utilises vertical displacement, rotational and curvature as three d.o.f. at the nodes. A continuity of C2 Hermite shape functions for bending deflection vb is used in which rotation function of θ and curvature of χ are dependently expressed as the first and second derivatives of bending deflection. The formulation of element takes account of the effect of shear transversal forces in order to behave appropriately in the analysis of thin and thick beams. A shear influence factor of ϕ is expressed explicitly, which is a function of length thickness ratio (L/h), as a control for shear deformation. The resulting UI element is absolutely free from locking and preserves the high accuracy of the standard locking-free finite elements and classical Bernoulli Euler element. Finally, several numerical tests are presented to confirm the performance of the proposed formulations.
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References
Bathe, K. J., & Dvorkin, E. N. (1985). A four-node plate bending element based on Mindlin-
Reissner plate theory and a mixed interpolation. International Journal for Numerical Methods
in Engineering, 21, 367–383.
Bletzinger, K.-U., Bischoff, M., & Ramm, E. (2000). A unified approach for shear-locking-free
triangular and rectangular shell finite elements. Computers and Structures, 75, 321–334.
Falsone, G., & Settineri, D. (2011). An Bernoulli-Euler-like finite element method for
Timoshenko beams. Mechanics Research Communications, 38, 12–16.
Hughes, T. J. R., Taylor, R. L., & Kanoknukulchai, W. (1977). A simple and efficient for plate
bending. International Journal for Numerical Methods in Engineering, 11, 1529–1543.
Hughes, T. J. R., & Tezduyar, T. E. (1981). Finite element based upon Mindlin plate theory
with particular reference to the four-node bilinear isoparametric element. Journal of Applied
Mechanics, 48, 587–596.
Kapur, K. (1966). Vibrations of a Timoshenko beam using finite-element approach. The Journal
of the Acoustical Society of America, 40, 1058–1063.
Katili, I. (1993a). A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate
theory and assumed shear strain fields- part I: An extended DKT element for thick-plate
bending analysis. International Journal for Numerical Methods in Engineering, 36, 1859–1883.
Katili, I. (1993b). A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate
theory and assumed shear strain fields- part II: An extended DKQ element for thick plate
bending analysis. International Journal for Numerical Methods in Engineering, 36, 1885–1908.
Katili I. (2006). Metode Elemen Hingga untuk Skeletal. Jakarta: Rajawali Press.
Katili, I., Batoz, J. L., Maknun, I. J., Hamdouni, A., & Millet, O. (2015). The development of
DKMQ plate bending element for thick to thin shell analysis based on Naghdi/Reissner/
Mindlin shell theory. Finite Elements in Analysis and Design, 100, 12–27.
Katili, I., Maknun, I. J., Hamdouni, A., & Millet, O. (2015). Application of DKMQ element for
composite plate bending structures. Composite Structures Journal, 132, 166–174.
Kiendl, J., Auricchio, F., Hughes, T. J. R., & Reali, A. (2015). Single-variable formulations and
isogeometric discretizations for shear deformable beams. Computer Methods in Applied
Mechanic and Engineering, 284, 988–1004.
Li, X.-F. (2008). A unified approach for analyzing static and dynamic behavior of functionally
graded Timoshenko and Bernoulli-Euler beam. Journal of Sound Vibration, 318, 1210–1229.
Maknun, I. J., Katili, I., Millet, O., & Hamdouni, A. (2016). Application of DKMQ element
for twist of thin-walled beams: Comparison with Vlasov theory. International Journal for
Computational Methods in Engineering Science and Mechanics, 17, 391–400.
Timoshenko, S. (1921). On the correction for shear of differential equation for transverse
vibrations of prismatic bars. Philosophical Magazine, 41, 744–746.
Timoshenko, S. (1922). On the transverse vibrations of bars of uniform cross section.
Philosophical Magazine, 43, 125–131.
