scholarly journals BEAM ELEMENT UNDER FINITE ROTATIONS

2021 ◽  
Vol 30 ◽  
pp. 87-92
Author(s):  
Emma La Malfa Ribolla ◽  
Milan Jirásek ◽  
Martin Horák
Keyword(s):  
Cross Sections ◽  
Shooting Method ◽  
Small Strain ◽  
Beam Model ◽  
Equilibrium Equations ◽  
Finite Rotations ◽  
Nonlinear Beam ◽  
Linear Elastic ◽  
Beam Elements ◽  
Large Rotations

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Modeling and Experimentation of a Ribbed Caudal Fin for Aquatic Robots

2017 ◽  
Author(s):  
Oscar Rios ◽  
Ardavan Amini ◽  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Beam Theory ◽  
Fe Model ◽  
Beam Model ◽  
Caudal Fin ◽  
Nonlinear Beam ◽  
Beam Elements ◽  
Side Walls ◽  
Thin Beams ◽  
Shear Deformable

Presented in this study is a mathematical model and preliminary experimental results of a ribbed caudal fin to be used in an aquatic robot. The ribbed caudal fin is comprised of two thin beams separated by ribbed sectionals as it tapers towards the fin. By oscillating the ribbed caudal fin, the aquatic robot can achieve forward propulsion and maneuver around its environment. The fully enclosed system allows for the aquatic robot to have very little effect on marine life and fully blend into its respective environment. Because of these advantages, there are many applications including surveillance, sensing, and detection. Because the caudal fin actuator has very thin side walls, Kirchhoff-Love’s large deformation beam theory is applicable for the large deformation of the fish-fin actuator. In the model, it is critical to accurately model the curvature of beams. To this end, C1 beam elements for thin beams are developed by specializing the shear-deformable beam elements, developed by the authors, based upon Reissner’s shear-deformable nonlinear beam model. Furthermore, preliminary experiments on the ribbed fin are presented to supplement the FE model.

Geometrically Exact Analysis of Spatial Frames

1993 ◽  
Vol 46 (11S) ◽  
pp. S118-S128 ◽  
Author(s):  
Paulo M. Pimenta ◽  
Takashi Yojo
Keyword(s):  
Finite Element ◽  
Weak Form ◽  
Small Strain ◽  
Exact Expression ◽  
Galerkin Projection ◽  
Equilibrium Equations ◽  
Rodrigues Formula ◽  
Linear Elastic ◽  
3D Space ◽  
Shear Distortion

A fully nonlinear, geometrically exact, finite strain rod model is derived from basic kinematical assumptions. The model incorporates shear distortion in bending and can take account of torsion warping. Rotation in 3D space is handled with the aid of the Euler-Rodrigues formula. The accomplished parametrization is simple and does not require update algorithms based on quaternions parameters. Weak and strong forms of the equilibrium equations are derived in terms of cross section strains and stresses, which are objective and suitable for constitutive description. As an example, an invariant linear elastic constitutive equation based on the small strain theory is presented. The attained formulation is very convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a finite element code. A mixed formulation of Hu-Washizu type is also derived, allowing for independent interpolation of the displacement, strain and stress fields within a finite element. An exact expression for the Fre´chet derivative of the weak form of equilibrium is obtained in closed form, which is always symmetric for conservative loading, even far from an equilibrium state and is very helpful for numerical procedures like the Newton method as well as for stability and bifurcation analysis. Several numerical examples illustrate the usefulness of the formulation in the lateral stability analysis of spatial frames. These examples were computed with the code FENOMENA, which is under development at the Computational Mechanics Laboratory of the Escola Polite´cnica.

scholarly journals The sustainability of micropolar concrete plasticity model to the finite element size

Vestnik MGSU ◽  
2019 ◽  
pp. 559-569
Author(s):  
Olga N. Pertseva ◽  
Gleb V. Martynov ◽  
Daria E. Monastyreva ◽  
Ekaterina I. Pereladova ◽  
Zaur S. Daurov ◽  
...  
Keyword(s):  
Finite Element ◽  
Cross Sections ◽  
Main Crack ◽  
Maximum Load ◽  
Beam Model ◽  
Element Mesh ◽  
Linear Elastic ◽  
Second Stage ◽  
Element Size ◽  
Micro Cracks

Introduction. As it is known, deformation of concrete can be divided into several stages. The first stage is characterized by a linear dependence of deformations and stresses, elastic deformations and small loads that, as they increase, lead to the second stage. At the second stage, the dependence becomes curvilinear, while deformations are irreversible, since micro-cracks are formed. Further consolidation of the micro-cracks into meso- and main cracks refers to the third stage and is accompanied by a redistribution of energy to the area of the main crack mouth. However, reaching the ultimate strength is not accompanied by an instant loss of bearing capacity due to the effect of decompression. This phenomenon should be taken into account in the numerical simulation of concrete and reinforced concrete structures, because it significantly affects their strength characteristics. The introduction of such a refinement in the design models will allow reducing cross-sections of the construction components and accordingly getting rid of material overruns. Materials and methods. A digital sample is created for the study using the ANSYS software. A beam model is simulated as a single-span beam with longitudinal reinforcement in the bending zone. The load is applied as a 70 mm offset to the nodes in the line along the application point. Reinforcement is simulated as bilinear isotropic strengthening elements (LINK180). For uniform load distribution, load plates with linear elastic properties are specified at the points where boundary conditions and load are applied. Results. According to the obtained data, stress-deformation curves are constructed identically to the concrete deformation diagram. The values of loads when the first cracking occurs (end of the linear-elastic state), peak loads when the main crack is formed (maximum load for the unreinforced case and the beginning of the steel softening for the reinforced case) as well as ultimate loads and maximum deflections at the mid-span are compared. Conclusions. The results give insignificant (up to 5 %) discrepancies when changing the finite element size. Therefore, when working with calculation software, developers will be able to create correct models with any spacing of the finite element mesh depending on the available computational capabilities. Micropolar theory for simulating the concrete decompression can be considered sustainable to the size of the finite elements.

A Generalized Reissner Theory for Large Axisymmetric Deflections of Circular Plates

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Raymond H. Plaut
Keyword(s):  
Shear Deformation ◽  
Shooting Method ◽  
Plate Thickness ◽  
Uniform Pressure ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Simply Supported ◽  
Reissner Theory ◽  
Large Rotations ◽  
Axisymmetric Problems

A generalized Reissner theory for axisymmetric problems of circular plates is presented. The plate is assumed to be linearly elastic, and large rotations and strains are allowed. Shear deformation and changes in the plate thickness are neglected. Equilibrium equations are formulated, and a shooting method is applied to obtain numerical results for plates subjected to a uniform pressure. The edge of the plate is assumed to be either simply supported or clamped, and is free to move radially. The resulting deflections are compared to those based on the von Kármán theory.

Solution for Geometrically Non-Linear Elastic Deformation of Simple Frames by a Shooting Method

2011 ◽  
Vol 462-463 ◽  
pp. 668-673
Author(s):  
Shi Rong Li ◽  
Ya Dong Hu
Keyword(s):  
Large Deformation ◽  
Shooting Method ◽  
Frame Structure ◽  
Curved Beams ◽  
Equilibrium Equations ◽  
Nonlinear Bending ◽  
Linear Elastic ◽  
Equilibrium Configurations ◽  
Nonlinear Equilibrium ◽  
Plane Frame

Based on an exact geometric nonlinear theory for plane curved beams, geometrically nonlinear equilibrium equations and boundary conditions governing the nonlinear bending of a simple plane frame structure subjected distributed loads were derived. By using the shooting method to numerically solve the boundary value problem of nonlinear ordinary differential equations, large deformation equilibrium configurations of a simple frame with both straight and the curved beam elements subjected uniformly distributed load were obtained. The theory and methodology presented can be used to analyze large deformation of plane simple frames with a variety of geometries and loadings.

A THREE-DIMENSIONAL BEAM ELEMENT UNDERGOING FINITE ROTATIONS BASED ON SLOPES AND DISTANCE MEASURES

2012 ◽  
Vol 09 (04) ◽  
pp. 1250054
Author(s):  
SALVATORE LOPEZ
Keyword(s):  
Three Dimensional ◽  
Distance Measures ◽  
Lagrangian Description ◽  
Finite Rotations ◽  
Transformation Matrices ◽  
Beam Elements ◽  
Large Rotations ◽  
Rotation Parameters ◽  
Strain Measures ◽  
Well Posed

A geometrically nonlinear formulation for three-dimensional beam elements in the hypotheses of large rotations and small strains is presented. In this formulation, based on the total Lagrangian description, the use of the rotation parameters is bypassed. Complex manipulations required to obtain conservative descriptions and well-posed transformation matrices are avoided. In particular, slopes and distances are used instead of rotation parameters to compute the nonlinear representations of the strain measures in the inertial frame of reference. Numerical tests have been carried out to validate the developed technique in the frame structures context.

scholarly journals Asymptotic Behavior of Stable Structures Made of Beams

2021 ◽  
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Cross Sections ◽  
Linear Elasticity ◽  
Circular Cross Section ◽  
Linear Elasticity Theory ◽  
Linear Elastic ◽  
The Cross ◽  
Small Rotation ◽  
Stable Structures

AbstractIn this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

scholarly journals NUMERICAL MODELLING OF THE SOIL BEHAVIOUR BY USING NEWLY DEVELOPED ADVANCED MATERIAL MODEL

2017 ◽  
Vol 57 (1) ◽  
pp. 58-70 ◽  
Author(s):  
Jan Veselý
Keyword(s):  
Numerical Modelling ◽  
Plastic Material ◽  
Material Model ◽  
Small Strain ◽  
Theoretical Background ◽  
Linear Elastic ◽  
In Situ Data ◽  
Modified Cam Clay ◽  
Soil Behaviour

This paper describes a theoretical background, implementation and validation of the newly developed Jardine plastic hardening-softening model (JPHS model), which can be used for numerical modelling of the soils behaviour. Although the JPHS model is based on the elasto-plastic theory, like the Mohr-Coulomb model that is widely used in geotechnics, it contains some improvements, which removes the main disadvantages of the MC model. The presented model is coupled with an isotopically hardening and softening law, non-linear elastic stress-strain law, non-associated elasto-plastic material description and a cap yield surface. The validation of the model is done by comparing the numerical results with real measured data from the laboratory tests and by testing of the model on the real project of the tunnel excavation. The 3D numerical analysis is performed and the comparison between the JPHS, Mohr-Coulomb, Modified Cam-Clay, Hardening small strain model and monitoring in-situ data is done.

Thick General Shells Under General Loading

1989 ◽  
Vol 56 (2) ◽  
pp. 391-394 ◽  
Author(s):  
Lloyd H. Donnell
Keyword(s):  
Cylindrical Shells ◽  
Small Strain ◽  
Equilibrium Equations ◽  
Flat Plates ◽  
Circular Cylindrical Shells

Three equilibrium equations in terms of three displacements are derived in scalar mathematics form, by linear, small-strain elasticity principles, for the case of general thick-walled shells under general loading. These reduce to well-known forms for the particular cases of flat-plates and thick circular cylindrical shells.

scholarly journals Confinement of Vibrations in Variable-Geometry Nonlinear Flexible Beam

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
W. Gafsi ◽  
F. Najar ◽  
S. Choura ◽  
S. El-Borgi
Keyword(s):  
Passive Control ◽  
Inverse Eigenvalue Problem ◽  
Beam Dynamics ◽  
Mode Shapes ◽  
Physical Parameters ◽  
Geometrical Parameters ◽  
Flexible Beam ◽  
Beam Model ◽  
Nonlinear Frequency ◽  
Nonlinear Beam

In this paper, we propose a novel strategy for controlling a flexible nonlinear beam with the confinement of vibrations. We focus principally on design issues related to the passive control of the beam by proper selection of its geometrical and physical parameters. Due to large deflections within the regions where the vibrations are to be confined, we admit a nonlinear model that describes with precision the beam dynamics. In order to design a set of physical and geometrical parameters of the beam, we first formulate an inverse eigenvalue problem. To this end, we linearize the beam model and determine the linearly assumed modes that guarantee vibration confinement in selected spatial zones and satisfy the boundary conditions of the beam to be controlled. The approximation of the physical and geometrical parameters is based on the orthogonality of the assumed linear mode shapes. To validate the strategy, we input the resulting parameters into the nonlinear integral-partial differential equation that describes the beam dynamics. The nonlinear frequency response curves of the beam are approximated using the differential quadrature method and the finite difference method. We confirm that using the linear model, the strategy of vibration confinement remains valid for the nonlinear beam.

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