In-plane nonlinear postbuckling analysis of circular arches using absolute nodal coordinate formulation with arc-length method

2020 ◽  
pp. 146441932097141
Author(s):  
Abdur Rahman Shaukat ◽  
Peng Lan ◽  
Jia Wang ◽  
Tengfei Wang
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Experimental Result ◽  
Equilibrium Path ◽  
Arc Length ◽  
Concentrated Load ◽  
Absolute Nodal Coordinate ◽  
Split Method ◽  
Geometric Configurations ◽  
General Continuum

In this study, Absolute Nodal Coordinate Formulation (ANCF) in conjunction with Crisfield’s arc-length method is utilized in order to predict the nonlinear postbuckling behaviour of circular arches. The whole primary equilibrium path in load-displacement space of circular arches under central concentrated load is obtained. Three ANCF based approaches, i.e., the conventional two-dimensional fully parameterized shear deformable ANCF beam element based on the General Continuum Mechanics (GCM) approach, the same element modified by the Strain Split Method (SSM) approach and the ANCF planar Higher Order Beam Element (HOBE) with GCM approach are used. Circular arches with various geometric configurations and boundary conditions such as clamped-clamped, hinged-hinged, clamped-hinged and three-hinged arches are studied which exhibit nonlinear response in the form of snap-through, snap-back and looping phenomenon. The obtained results are compared with the analytical solutions, experimental result (where available in the literature) and numerical approximations (by using the commercially available FEM package). In this paper, the recently proposed ANCF based approaches are successfully implemented which validate and verify the utility of ANCF in nonlinear postbuckling analysis. The characteristics of the three approaches with regard to the adoptability of arc-length method are compared and discussed.

Performance of Curved Beam Elements Using the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Hirohisa Koyama ◽  
Hiroki Yamashita
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Solution Procedure ◽  
Large Displacement ◽  
Curved Beam ◽  
Absolute Nodal Coordinate ◽  
Curved Beam Element ◽  
Strain Components ◽  
The Absolute ◽  
Longitudinal Stretch

In this investigation, a gradient deficient beam element of the absolute nodal coordinate formulation is generalized to a curved beam for the analysis of multibody systems and the performance of the proposed element is discussed by comparing with the fully parameterized curved beam element and the classical large displacement beam element with incremental solution procedures. Strain components are defined with respect to the initially curved configuration and described by the arc-length coordinate. The Green strain is used for the longitudinal stretch, while the material measure of curvature is used for bending. It is shown that strains of the curved beam can be expressed with respect to those defined in the element coordinate system using the gradient transformation and the effect of strains at the initially curved configuration is eliminated using one-dimensional Almansi strain. This property can be effectively used with non-incremental solution procedure employed for the absolute nodal coordinate formulation. Several numerical examples are presented in order to demonstrate the performance of the gradient deficient curved beam element developed in this investigation. It is shown that the use of the proposed element leads to better element convergence as compared to that of the fully parameterized element and the classical large displacement beam element with incremental solution procedures.

A New Type Component Mode Synthesis Beam Element Based on the Absolute Nodal Coordinate Formulation

2003 ◽  
Author(s):  
Riki Iwai ◽  
Nobuyuki Kobayashi
Keyword(s):  
High Efficiency ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Synthesis Method ◽  
Beam Element ◽  
Component Mode Synthesis ◽  
Flexible Beam ◽  
Absolute Nodal Coordinate ◽  
New Type ◽  
The Absolute

This paper establishes a new type component mode synthesis method for a flexible beam element based on the absolute nodal coordinate formulation. The deformation of the beam element is defined as the sum of the global shape function and the analytical clamped-clamped beam modes. This formulation leads to a constant and symmetric mass matrix as the conventional absolute nodal coordinate formulation, and makes it possible to reduce the system coordinates of the beam structure which undergoes large rotations and large deformations. Numerical examples show that the excellent agreements are examined between the presented formulation and the conventional absolute nodal coordinate formulation. These results demonstrate that the presented formulation has high accuracy in the sense that the presented solutions are similar to the conventional ones with the less system coordinates and high efficiency in computation.

A Procedure for the Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

2007 ◽  
Author(s):  
Aki Mikkola ◽  
Oleg Dmitrochenko ◽  
Marko Matikainen
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.

Shear Correction for Thin Plate Finite Elements Based on the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola
Keyword(s):  
Shear Deformation ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Plate Elements ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

This study is an extension of a newly introduced approach to account transverse shear deformation in absolute nodal coordinate formulation. In the formulation, shear deformation is usually defined by employing slope vectors in the element transverse direction. This leads to the description of deformation modes that, in practical problems, may be associated with high frequencies. These high frequencies, in turn, could complicate the time integration procedure, burdening numerical performance of shear deformable elements. In a recent study of this paper’s authors, the description of transverse shear deformation is accounted for in a two-dimensional beam element, based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. In this study, the approach to account for shear deformation without using transverse slopes is implemented for a thin rectangular plate element. In fact, two new plate elements are introduced: one within conventional finite element and another using the absolute nodal coordinates. Numerical results are presented in order to demonstrate the accuracy of the introduced plate element. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable plate elements.

An improved variable-length beam element with a torsion effect based on the absolute nodal coordinate formulation

2017 ◽  
Vol 232 (1) ◽  
pp. 69-83
Author(s):  
Shuai Yang ◽  
Zongquan Deng ◽  
Jing Sun ◽  
Yang Zhao ◽  
Shengyuan Jiang
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Variable Length ◽  
Ball Screw ◽  
Position Vector ◽  
Drilling Process ◽  
Rotational Angle ◽  
Absolute Nodal Coordinate ◽  
One Dimensional ◽  
Torsion Effect

This paper proposes an improved variable-length beam element based on absolute nodal coordinate formulation and arbitrary Lagrangian–Eulerian description to build dynamic model of a one-dimensional medium with mass transportation and a non-ignorable torsion effect. The rotational angle of the presented element is interpolated using the same Hermite polynomials as the position vector such that the change rate of the rotational angles of the two nodes are also introduced into generalized coordinates of the element, which ensures the continuity of the nodal torque. Numerical examples demonstrate that the proposed element can precisely describe the dynamic behaviour of a one-dimensional medium. Furthermore, its ability to describe the torsion effect is significantly enhanced compared to earlier element in the literature. In engineering applications, the proposed element can be used in the dynamic analysis of drill stems in the drilling process, slender workpieces of cylinder shafts in turning processes and leading screws in ball screw mechanisms.

Absolute Nodal Coordinate Formulation Coupled Deformation Modes

2006 ◽  
Author(s):  
Bassam A. Hussein ◽  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Structures ◽  
Mindlin Plate ◽  
Elastic Line ◽  
Absolute Nodal Coordinate ◽  
Geometric Stiffening ◽  
General Continuum ◽  
Deformation Modes

The finite element absolute nodal coordinate formulation (ANCF) leads to beam and plate models that relax the assumption of the classical Euler-bernoulli and Timoshenko beam and Mindlin plate theories. In these more general models, the cross section is allowed to deform and it is no longer treated as a rigid surface. The coupling between the bending and cross section deformations leads to the new ANCF-coupled deformation modes that are examined in this study. While these coupled deformation can be source of numerical and convergence problems when thin and stiff beam models are considered, the inclusion of the effect of these modes in the dynamic model is necessary in the case of very flexible structures. In order to examine the effect of these coupled deformation modes in this investigation, three different large deformation dynamic beam models are discussed. Two of these models, which differ in the way the beam elastic forces are calculated in the absolute nodal coordinate formulation, allow for systematically eliminating the coupled deformation modes, while the third allows for including these modes. The first of these models is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second and third methods that can be used to eliminate the coupled deformation modes are based on the elastic line approach and the Hellinger-Reissner principle. It is shown in this study that the inclusion of the ANCF-coupled deformation modes introduces geometric stiffening effects that can not be captured using other finite element models.

Coupled Deformation Modes in the Large Deformation Finite Element Analysis: Generalization

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Oleg N. Dmitrochenko ◽  
Bassam A. Hussein ◽  
Ahmed A. Shabana
Keyword(s):  
Continuum Mechanics ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Structures ◽  
Three Dimensional ◽  
Numerical Results ◽  
Elastic Line ◽  
Absolute Nodal Coordinate ◽  
Plate Elements ◽  
General Continuum ◽  
Deformation Modes

The effect of the absolute nodal coordinate formulation (ANCF)–coupled deformation modes on the accuracy and efficiency when higher order three-dimensional beam and plate finite elements are used is investigated in this study. It is shown that while computational efficiency can be achieved in some applications by neglecting the effect of some of the ANCF-coupled deformation modes, such modes introduce geometric stiffening/softening effects that can be significant in the case of very flexible structures. As shown in previous publications, for stiff structures, the effect of the ANCF-coupled deformation modes can be neglected. For such stiff structures, the solution does not strongly depend on some of the ANCF-coupled deformation modes, and formulations that include these modes lead to numerical results that are in good agreement with formulations that exclude them. In the case of a very flexible structure, on the other hand, the inclusion of the ANCF-coupled deformation modes becomes necessary in order to obtain an accurate solution. In this case of very flexible structures, the use of the general continuum mechanics approach leads to an efficient solution algorithm and to more accurate numerical results. In order to examine the effect of the elastic force formulation on the efficiency and the coupling between different modes of deformation, three different models are used again to formulate the elastic forces in the absolute nodal coordinate formulation. These three methods are the general continuum mechanics approach, the elastic line (midsurface) approach, and the elastic line (midsurface) approach with the Hellinger–Reissner principle. Three-dimensional absolute nodal coordinate formulation beam and plate elements are used in this study. In the general continuum mechanics approach, the coupling between the cross section deformation and the beam centerline or plate midsurface displacement is considered, while in the approaches based on the elastic line and the Hellinger–Reissner principle, this coupling is neglected. In addition to the fully parametrized beam element used in this study, three different plate elements, two fully parametrized and one reduced order thin plate elements, are used. The numerical results obtained using different finite elements and elastic force formulations are compared in this study.

A piecewise beam element based on absolute nodal coordinate formulation

2014 ◽  
Vol 77 (1-2) ◽  
pp. 1-15 ◽  
Author(s):  
Zuqing Yu ◽  
Peng Lan ◽  
Nianli Lu
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Absolute Nodal Coordinate

A new higher-order locking-free beam element based on the absolute nodal coordinate formulation

2017 ◽  
Vol 232 (19) ◽  
pp. 3410-3423 ◽  
Author(s):  
Haidong Yu ◽  
Chunzhang Zhao ◽  
Bin Zheng ◽  
Hao Wang
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Higher Order ◽  
Continuity Condition ◽  
Large Rotation ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
The Cross

The beam elements based on the absolute nodal coordinate formulation are widely used in large deformation and large rotation problems. Some of them lead to shear and Poisson locking problems when the continuum mechanics method is employed to deduce the generalized elastic force of the element. To circumvent these locking problems, a new higher-order beam element is proposed that may capture the warping and non-uniform stretching distribution of the cross-section by introducing the trapezoidal cross-section deformation mode and increasing the order of interpolation polynomials in transverse direction. The curvature vectors are chosen as the nodal coordinates of the new element that improve the continuity condition at the element interface. Static and dynamic analyses are conducted to investigate the performance of the new element. Poisson locking phenomena may be eliminated effectively for the new element even when Poisson’s ratio is greater than zero. Meanwhile, the distortion deformation of the cross-section may be described directly. The new element has a better convergence performance compared with the spatial absolute nodal coordinate formulation beam element for that shear locking issue is eliminated. The results also show that the new element fulfills energy conservation and may be applied to the dynamics of both straight and initial curved structures with large deformation.

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