scholarly journals Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

2021 ◽  
Author(s):  
Ivo Steinbrecher ◽  
Alexander Popp ◽  
Christoph Meier
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Lagrange Multiplier ◽  
Degrees Of Freedom ◽  
Element Discretization ◽  
Geometrically Exact Beam ◽  
Coupling Constraints ◽  
Rotational Degrees Of Freedom ◽  
3D Solid ◽  
Cosserat Continua

AbstractThe present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

General Beam Cross-Section Analysis Using a 3D Finite Element Slice

2014 ◽  
Author(s):  
Philippe Couturier ◽  
Steen Krenk
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Degrees Of Freedom ◽  
Finite Thickness ◽  
Single Layer ◽  
Element Discretization ◽  
Cubic Polynomial ◽  
Hermitian Interpolation ◽  
Section Analysis

A formulation for analysis of general cross-section properties has been developed. This formulation is based on the stress-strain states in the classic six equilibrium modes of a beam by considering a finite thickness slice modelled by a single layer of 3D finite elements. The displacement variation in the lengthwise direction is in the form of a cubic polynomial, which is here represented by Hermitian interpolation, whereby the degrees of freedom are concentrated on the front and back faces of the slice. The theory is illustrated by application to a simple cross-section for which an analytical solution is available. The paper also shows an application to wind turbine blade cross-sections and discusses the effect of the finite element discretization on the cross-section properties such as stiffness parameters and the location of the elastic and shear centers.

Development of a Nonlinear, C1-Beam Finite-Element Code for Actuator Design of Slender Flexible Robots

2017 ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Takeyuki Ono
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Cross Section ◽  
Beam Model ◽  
Finite Element Code ◽  
Shape Functions ◽  
Lagrange Equations ◽  
Element Discretization ◽  
Flexible Robots ◽  
Actuator Design

For actuator design and motion simulations of slender flexible robots, planar C1-beam elements are developed for Reissner’s large deformation, shear-deformable, curbed-beam model. Internal actuation is mechanically modeled by a rate-form of beam constitutive relation, where actuation curvature is prescribed at each time. Geometrically, a curbed beam is modeled as a frame bundle, whereby at each point on beam’s curve of centroids a moving orthonormal frame is attached to a cross section. After a finite element discretization, a curve of centroids is modeled as a C1-curve, employing cubic shape functions for both planar coordinates with an arc-parameter. The cubic shape functions have already been utilized in linear Euler-Bernoulli beams for the interpolation of transverse displacement. To define the rotation angle of each cross section or the attitude of the moving frame, quadratic shape functions are used introducing a middle node, resulting in three angular nodal displacements. As a result, each beam element has total eleven nodal coordinates. The implementation of a nonlinear finite element code is facilitated by the principle of virtual work, which yields Reissner’s large deformation curbed beam model as the Euler-Lagrange equations. For time integration, the Newmark method is utilized. Finally, as applications of the code, a few inchworm motions induced by different actuation curvature fields are presented.

Beam Element Structural Dynamics Modification Using Experimental Modal Rotational Data

1995 ◽  
Vol 117 (3A) ◽  
pp. 265-271 ◽  
Author(s):  
John A. Cafeo ◽  
Martin W. Trethewey ◽  
H. Joseph Sommer
Keyword(s):  
Finite Element ◽  
Cantilever Beam ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Structural Dynamic ◽  
Modal Data ◽  
Structural Dynamic Modification ◽  
Dynamic Modification ◽  
Rotational Degrees Of Freedom ◽  
Fixed Beam

Structural dynamic modification (SDM) of a fixed-free (cantilever) beam to convert it into a fixed-fixed beam with experimental modal data is presented. The SDM focuses on incorporating experimental rotational degrees-of-freedom (DOF) measured with a novel laser measurement technique. A cantilever beam is tested to develop the experimental modal database including rotational degrees of freedom. A modal database from a finite-element model also is developed for comparison. A structural dynamic modification, with both databases, is performed using a Bernoulli-Euler beam to ground the free end of the cantilever beam. The hardware is then modified and a second experimental modal analysis of the resulting fixed-fixed beam performed. A finite-element model of the fixed-fixed beam also was created. Comparison of results from these four tests are used to assess the effectiveness of SDM using experimental modal rotational data. The evaluation shows that provided high quality experimental rotational modal data can be acquired, SDM work with beam elements can be effective in yielding accurate results.

Force Vibration Analysis of Thick Beams on Two Parameter Elastic Foundation by Finite Element Method

2002 ◽  
Author(s):  
Ali Bahcivan ◽  
Vedat Karadag
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Cross Section ◽  
Vibration Analysis ◽  
Degrees Of Freedom ◽  
Beam Element ◽  
Thick Beam ◽  
Two Parameter

In the present work, the free vibration analysis of rectangular cross-section uniform beams on two-parameter elastic foundation, considering shear deformation and rotatory inertia is made by the finite element method. In this analysis, two different thick beam elements are used. The first 4 degrees of freedom thick beam element has two nodes with two degrees of freedom at each node such as transverse displacements and cross-section rotations. In the second beam element, the nodal variables are the transverse displacement, the cross-section rotation and shear deformation. The elastic foundation is idealized as a constant two-parameter model characterized by two moduli, i.e., the Winkler foundation modulus k and the shear foundation modulus kG. In the case kG = 0, this model reduces to the Winkler model, i.e., the elastic foundation is idealized as a constant one-parameter model. Axial displacement of the beam is also considered. Three kinds of end conditions, i.e., simply-supported, clamped-clamped and clamped-free ends are considered in this study. The effects of axial force, foundation stiffness parameters and partial elastic foundation on the natural frequencies of the beam are examined. In this analysis, the vibration calculation results are presented in the tables and their importance in design are discussed. The numerical results obtained from this analysis are compared with the exact or available solutions, wherever possible. Numerical results and comparisons show the effectiveness of the proposed method.

A Four-Node Tetrahedral Finite Element Based on Space Fiber Rotation Concept

2019 ◽  
Vol 11 (1) ◽  
pp. 67-78
Author(s):  
Kamel Meftah ◽  
Lakhdar Sedira
Keyword(s):  
Finite Element ◽  
Strain Field ◽  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Field Tensor ◽  
Tetrahedral Element ◽  
Numerical Studies ◽  
Rotational Degrees Of Freedom ◽  
Tensor Approximation

Abstract The paper presents a four-node tetrahedral solid finite element SFR4 with rotational degrees of freedom (DOFs) based on the Space Fiber Rotation (SFR) concept for modeling three-dimensional solid structures. This SFR concept is based on the idea that a 3D virtual fiber, after a spatial rotation, introduces an enhancement of the strain field tensor approximation. Full numerical integration is used to evaluate the element stiffness matrix. To demonstrate the efficiency and accuracy of the developed four-node tetrahedron solid element and to compare its performance with the classical four-node tetrahedral element, extensive numerical studies are presented.

scholarly journals Coupled Bending-Bending-Torsion Vibration of a Pre-Twisted Beam with Aerofoil Cross-Section by the Finite Element Method

2003 ◽  
Vol 10 (4) ◽  
pp. 223-230
Author(s):  
Bulent Yardimoglu ◽  
Daniel J. Inman
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Higher Order ◽  
Coupling Coefficients ◽  
Cross Sectional ◽  
Torsion Vibration ◽  
Proper Order ◽  
Proposed Model

The present study deals with a finite element model for coupled bending-bending-torsion vibration analysis of a pretwisted Timoshenko beam with varying aerofoil cross-section. The element derived in this paper has two nodes, with seven degrees of freedom at each node. The nodal variables are transverse displacements, cross-section rotations and the shear angles in two planes and torsional displacement. The advantage of the present element is the exclusion of unnecessary derivatives of fundamental nodal variables, which were included to obtain invertable square matrix by other researchers, by choosing proper displacement functions and using relationship between cross-sectional rotation and the shear deformation. Element stiffness and mass matrices are developed from strain and kinetic energy expressions by assigning proper order polynomial expressions for cross-section properties and considering higher order coupling coefficients. The correctness of the present model is confirmed by the experimental results available in the literature. Comparison of the proposed model results with those in the literature indicates that a faster convergence is obtained. The results presented also provide some insights in the formulation by clearly indicating that higher order coupling terms have considerable influence on the natural frequencies.

Application of Rotational Measurements in Stiffness Reconstruction of Beams and Frames

2009 ◽  
Vol 413-414 ◽  
pp. 189-194
Author(s):  
Zbigniew Zembaty ◽  
Seweryn Kokot
Keyword(s):  
Inverse Problem ◽  
Finite Element ◽  
Damage Detection ◽  
Minimization Problem ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Frame Structure ◽  
Reconstruction Method ◽  
Harmonic Vibrations ◽  
Rotational Degrees Of Freedom

A stiffness reconstruction method is tested when rotational degrees of freedom are added to the dynamic model of the structure. The inverse problem is formulated as a minimization problem in terms of harmonic vibrations of the structure and its finite element model. An example of frame structure is analyzed by numerical simulations. The results of these numerical analyses show that the damage detection appeared to be much more effective when the angular amplitudes of harmonic vibrations are acquired. This makes very good prospects for the future applications of angular sensors in damage detection of structures.

A Slice Finite Element Model for Bending Analysis of Curved Beams

2011 ◽  
Vol 338 ◽  
pp. 282-285 ◽  
Author(s):  
Wen Guang Jiang ◽  
Li Juan Yan
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Finite Element Modelling ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Curved Beams ◽  
Pure Bending ◽  
Cross Sectional ◽  
Bending Analysis ◽  
Constraint Equations

The pure bending analysis of curved beams may be performed by finite element modelling of only a representative slice sector of the beam cross-section, by establishing exact deformation relationships between degrees of freedom of corresponding nodes on the corresponding artificial cross-sectional boundaries. These deformation relationships can be conveniently realized using constraint equations between nodal degrees of freedom. Numerical example has been given to demonstrate the accuracy and effectiveness of the proposed method.

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