Transversally higher-order interpolating polynomials for the two-dimensional shear deformable ANCF beam elements based o
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Transversally higher-order interpolating polynomials for the two-dimensional shear deformable ANCF beam elements based on common coefficients Chun H. Zhao1
· Kang W. Bao1 · Yu L. Tao1
Received: 29 January 2020 / Accepted: 8 November 2020 © Springer Nature B.V. 2020
Abstract The polynomial representation for describing the displacement field of the elements is the main factor that determines the performance of the shear deformable beam elements based on the absolute nodal coordinate formulation (ANCF). In order to resolve the locking problem of the ANCF beam elements, the transversally higher-order polynomial representation has been investigated frequently and applied to the displacement field of the elements by increasing the nodal coordinates of the beam elements. In this paper, transversally higher-order interpolating polynomials are added into the polynomial displacement field of the elements by using common coefficients which mean that the coefficients between the higher-order longitudinal and transversal polynomial components are common. The implementation does not require the increase of the nodal coordinates. Two new kinds of two-dimensional transversally higher-order ANCF beam elements are formulated by common coefficients. The effect of transversally higher-order interpolating polynomials on the performance of the proposed ANCF beam elements is studied by means of certain static and dynamic problems. It is shown that the transversally quadratic order polynomial component y 2 introduced by common coefficients can also relieve the problem of Poisson locking, and the proposed beam elements are effective and accurate in the static and dynamic analysis. Keywords Common coefficient · The Poisson locking · Transversally higher-order beam element · Absolute nodal coordinate formulation
1 Introduction The absolute nodal coordinate formulation (ANCF) is developed to study the nonlinear motion of structural components that undergo large displacements [1]. The ANCF employs the mathematical definition of the slopes to define the element coordinates. Therefore, the ANCF elements can be considered as isoparametric elements, and as a result, exact modeling
B C.H. Zhao
[email protected]
1
School of Mechanical and Automotive Engineering, Shanghai University of Engineering Science, Shanghai 201620, China
C.H. Zhao et al.
of the rigid body dynamics can be obtained. In the ANCF, because the element coordinates are defined in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. Continuum mechanics approach [2, 3] was used to derive the expression of the elastic forces because of its simple formulation of the elastic forces and more appropriateness in the nonlinear case. But the proposed elements based on continuum mechanics approach have locking problems, leading to weak performan
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