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Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity

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This study thoroughly examines various higher-order three and four-node beam elements for use in the absolute nodal coordinate formulation (ANCF). The paper carefully investigates which potential benefits and drawbacks the utilization of higher-order ANCF beam elements without in-slope vectors has in the case of the usage of full three-dimensional elasticity. When the elastic forces for shear-deformable ANCF beam elements are calculated using full three-dimensional elasticity—especially in the form of the St. Venant–Kirchhoff material law—Poisson locking severely deteriorates the accuracy of the numeric results. As shown in this paper, an existing approach to preventing this locking phenomenon for three-node beam elements can still produce unsatisfying results in load cases involving bidirectional bending. The results of this study show that enriching the polynomial basis used to approximate the beam kinematics provides a natural solution to this issue. As will be seen, these findings for three-node elements can also be extended to four-node elements. When using a sufficient approximation order in transverse directions, satisfying accuracy can be achieved both in conventional one-dimensional bending and in the above-mentioned bidirectional load case.
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Nonlinear Dyn (2017) 88:1075–1091
DOI 10.1007/s11071-016-3296-x
ORIGINAL PAPER
Higher-order beam elements based on the absolute nodal
coordinate formulation for three-dimensional elasticity
Henrik Ebel ·Marko K. Matikainen ·
Vesa-Ville Hurskainen ·Aki Mikkola
Received: 4 December 2015 / Accepted: 14 December 2016 / Published online: 28 December 2016
© Springer Science+Business Media Dordrecht 2016
Abstract This study thoroughly examines various
higher-order three and four-node beam elements for use
in the absolute nodal coordinate formulation (ANCF).
The paper carefully investigates which potential ben-
efits and drawbacks the utilization of higher-order
ANCF beam elements without in-slope vectors has in
the case of the usage of full three-dimensional elas-
ticity. When the elastic forces for shear-deformable
ANCF beam elements are calculated using full three-
dimensional elasticity—especially in the form of the
St. Venant–Kirchhoff material law—Poisson locking
severely deteriorates the accuracy of the numeric
results. As shown in this paper, an existing approach
to preventing this locking phenomenon for three-node
beam elements can still produce unsatisfying results
in load cases involving bidirectional bending. The
H. Ebel (B)
Institute of Engineering and Computational Mechanics,
University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart,
Germany
e-mail: henrik.ebel@itm.uni-stuttgart.de
M. K. Matikainen ·V.-V. Hurskainen ·A. Mikkola
Mechanical Engineering, Lappeenranta University of
Technology, Skinnarilankatu 34, 53850 Lappeenranta,
Finland
e-mail: marko.matikainen@lut.fi
V.-V. Hurskainen
e-mail: vesa-ville.hurskainen@lut.fi
A. Mikkola
e-mail: aki.mikkola@lut.fi
results of this study show that enriching the polyno-
mial basis used to approximate the beam kinematics
provides a natural solution to this issue. As will be
seen, these findings for three-node elements can also
be extended to four-node elements. When using a suffi-
cient approximation order in transverse directions, sat-
isfying accuracy can be achieved both in conventional
one-dimensional bending and in the above-mentioned
bidirectional load case.
Keywords Continuum beam elements ·
Three-dimensional elasticity ·St. Venant–Kirchhoff
material ·Princeton beam experiment ·Numerical
locking
1 Introduction
The computer analysis of multibody system dynam-
ics has become increasingly important in advanced
machine system design. Increased computational power
and enhanced formulations are making it possible
to solve progressively more sophisticated mathemat-
ical models describing the dynamic behavior of com-
plex systems. Nevertheless, computational efficiency
is an important consideration for the multibody system
dynamics analyst, and the level of detail for a mechan-
ical system model should not be any greater than is
needed to yield a sufficiently accurate numerical result.
In multibody system dynamics, an acceptable simpli-
fication for the analysis of motion and forces in many
123
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... С учетом (19) получим ...
... Рис. 1. Неструктурированные шестигранные сетки для модели пружины и пневматического актуатора (в том числе в разрезе) (а), эталонная шестигранная для задачи деформирования куба (б), перемещения вдоль осей Ozи Oy для теста Принстон (в, г) Пример 2. Перейдем к тесту, осуществленному в Принстонском университете, США (далее тест Принстон) [10,19] о деформировании прямоугольной балки длиной 0.508 м вдоль оси Ox, шириной 3.2024×10 −3 м вдоль Oy и высотой 12.377×10 −3 м вдоль Oz. Параметры линейно-упругого материала E = 2.07 × 10 11 Па, ν = 0.3. ...
... На рисунке 1в представлены результаты перемещения конца балки вдоль оси Oz, где присутствует наибольшая жесткость конструкции на изгиб и, как следствие, наиболее явно проявляются эффекты сдвигового запирания. При расчетной шестигранной сетке, содержащей 700 слоев ячеек вдоль оси Ox (где каждый слой состоит из 8 ячеек, равномерно распределенных в плоскости Oyz), расхождение полученных МКЭ в терминах ANCF значений u 1 z с данными u [19] z , представленными в статье [19] (на основе ячеек типа 3843 и других элементов более высокого порядка), не превышает 1.5%. Для сетки, состоящей из 300 слоев, расхождение значений u 2 z с данными u [19] z не превышает 12%, при этом для 150 слоев максимальное отличие в результатах u 3 z достигает 40%. ...
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... Compared with the eight-node solid element (or the twenty-node hexahedron element), the ANCF beam elements can be computationally more efficient (Obrezkov et al., 2021). In addition, with the cross-sectional higher-order polynomial (Matikainen et al., 2014;Shen et al., 2014;Ebel et al., 2017;Orzechowski and Shabana, 2016) or warping function (Tang et al., 2022) interpolations on the cross sections, the in-plane and out-of-plane warping deformations can be obtained computational effectively while needing fine cross-sectional mesh via dozens of solid elements on the section. ...
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