scholarly journals Stress-Based FEM in the Problem of Bending of Euler–Bernoulli and Timoshenko Beams Resting on Elastic Foundation

Materials ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 460
Author(s):  
Zdzisław Więckowski ◽  
Paulina Świątkiewicz
Keyword(s):  
Finite Element ◽  
Elastic Foundation ◽  
Degrees Of Freedom ◽  
Elastic Beam ◽  
Linear Functions ◽  
Timoshenko Beams ◽  
Timoshenko Model ◽  
Six Degrees Of Freedom ◽  
Displacement Based ◽  
Posteriori Estimation

The stress-based finite element method is proposed to solve the static bending problem for the Euler–Bernoulli and Timoshenko models of an elastic beam. Two types of elements—with five and six degrees of freedom—are proposed. The elaborated elements reproduce the exact solution in the case of the piece-wise constant distributed loading. The proposed elements do not exhibit the shear locking phenomenon for the Timoshenko model. The influence of an elastic foundation of the Winkler type is also taken into consideration. The foundation response is approximated by the piece-wise constant and piece-wise linear functions in the cases of the five-degrees-of-freedom and six-degrees-of-freedom elements, respectively. An a posteriori estimation of the approximate solution error is found using the hypercircle method with the addition of the standard displacement-based finite element solution.

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.

Curvature-Based Finite Element Method for Euler-Bernoulli Beams

2007 ◽  
Author(s):  
Y. L. Kuo ◽  
W. L. Cleghorn
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Transverse Displacement ◽  
Bernoulli Beam ◽  
Numerical Examples ◽  
Displacement Based ◽  
Curvature Distribution ◽  
Euler Bernoulli Beam ◽  
Element Method

This paper presents a new method called the curvature-based finite element method to solve Euler-Bernoulli beam problems. An approximated curvature distribution is selected first, and then the approximated transverse displacement is determined by double integrations. Four numerical examples demonstrate the validity of the method, and the results show that the errors are smaller than those generated by a conventional method, the displacement-based finite element method, for comparison based on the same number of degrees of freedom.

scholarly journals Solving Galbrun's Equation with a Discontinuous galerkin Finite Element Method

2019 ◽  
Vol 105 (6) ◽  
pp. 1149-1163 ◽  
Author(s):  
Marcus Maeder ◽  
Andrew Peplow ◽  
Maximilian Meindl ◽  
Steffen Marburg
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Early Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linearized Euler Equations ◽  
Displacement Based ◽  
Spurious Modes

Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving Galbrun's equation in a mixed and pure displacement based formulation. The results achieved with the proposed approach are compared with results from previous attempts to solve Galbrun's equation. The numerical determination of acoustic modes and the identification of vortical modes is discussed. Furthermore, case studies for a lined-duct and an annulus supporting a rotating shear-flow are investigated.

STRESS-BASED FINITE ELEMENT METHOD FOR EULER-BERNOULLI BEAMS

2006 ◽  
Vol 30 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Y.L. Kuo ◽  
W.L. Cleghorn ◽  
K. Behdinan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Transverse Displacement ◽  
Bending Stress ◽  
Numerical Examples ◽  
A New Technique ◽  
Displacement Based ◽  
Element Method

This paper presents a new technique, which can apply the stress-based finite element method to Euler-Bernoulli beams. An approximated bending stress distribution is selected, and then the approximated transverse displacement is determined by twice integration. Due to the satisfaction of compatibility, the integration constants are determined by the boundary conditions related to transverse displacement and rotation. To compare with the displacement-based finite element method, this technique provides the continuities of not only transverse displacement and rotation but also stress at nodes. Besides, the boundary conditions related to stress are satisfied. Two numerical examples demonstrate the validity of this technique. The results show that the errors are smaller than those generated by the displacement-based finite element method for the same number of degrees of freedom.

A finite element investigation of the post-buckling strength of thin-walled structural members under compression

1978 ◽  
Vol 5 (4) ◽  
pp. 595-610
Author(s):  
H. P. Lee ◽  
P. J. Harris
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Step Method ◽  
Six Degrees Of Freedom ◽  
Modified Cholesky Decomposition ◽  
Compressive Loads ◽  
Post Buckling ◽  
Lagrangian Coordinate ◽  
Thin Walled

By employing the finite element displacement method using the tangent stiffness approach, the paper presents results of post-buckling analysis of plates and three-dimensional thin-walled members subjected to uniaxial compressive loads. A simple rectangular element with six degrees of freedom at each node, suitable for the analysis of nonplanar prismatic members with slope discontinuities (folded plates), is employed. Both geometric and material nonlinearities have been considered based on a Lagrangian coordinate system and. the flow theory of plasticity. The nonlinear equations are solved using the Newton–Raphson method in the elastic range and the step-by-step method with equilibrium corrections in the plastic range. A modified Cholesky decomposition technique is employed to solve the basic stiffness equations.

Computerized Three-Dimensional Finite Element Reconstruction of the Left Ventricle from Cross-Sectional Echocardiograms

1984 ◽  
Vol 6 (1) ◽  
pp. 48-59 ◽  
Author(s):  
P. E. Nikravesh ◽  
D. J. Skorton ◽  
K. B. Chandran ◽  
Y. M. Attarwala ◽  
N. Pandian ◽  
...  
Keyword(s):  
Finite Element ◽  
Left Ventricle ◽  
Finite Element Model ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Element Model ◽  
Left Ventricular Geometry ◽  
Six Degrees Of Freedom ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

A computerized method for the generation of a three-dimensional finite element mesh of left ventricular geometry is presented. The technique employs two dimensional echocardiographic images of the left ventricle. The echocardiographic transducer is attached to an articulated, computerassisted, position registration arm with six degrees-of-freedom. These six degrees-of-freedom record the location and orientation of the transducer, when images are obtained, referenced to an external point. Eence, the images are digitized and aligned relative to one another, then several interpolation and curve fitting steps are used to reconstruct a threedimensional finite element model of the left ventricle. The finite element model can be used for volume determination, stress analysis, material property identification, and other applications.

scholarly journals Isogeometric Implementation of High-Order Microplane Model for the Simulation of High-Order Elasticity, Softening, and Localization

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
Erol Lale ◽  
Xinwei Zhou ◽  
Gianluca Cusatis
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Control Point ◽  
Three Dimensional ◽  
Local Strain ◽  
High Order ◽  
Particle Model ◽  
Displacement Fields ◽  
Displacement Based ◽  
The Continuum

In this paper, a recently developed higher-order microplane (HOM) model for softening and localization is implemented within a isogeometric finite-element framework. The HOM model was derived directly from a three-dimensional discrete particle model, and it was shown to be associated with a high-order continuum characterized by independent rotation and displacement fields. Furthermore, the HOM model possesses two characteristic lengths: the first associated with the spacing of flaws in the material internal structure and related to the gradient character of the continuum; the second associated with the size of these flaws and related to the micropolar character of the continuum. The displacement-based finite element implementation of this type of continua requires C1 continuity both within the elements and at the element boundaries. This motivated the implementation of the concept of isogeometric analysis which ensures a higher degree of smoothness and continuity. Nonuniform rational B-splines (NURBS) based isogeometric elements are implemented in a 3D setting, with both displacement and rotational degrees-of-freedom at each control point. The performed numerical analyses demonstrate the effectiveness of the proposed HOM model implementation to ensure optimal convergence in both elastic and softening regime. Furthermore, the proposed approach allows the natural formulation of a localization limiter able to prevent strain localization and spurious mesh sensitivity known to be pathological issues for typical local strain-softening constitutive equations.

scholarly journals Free Vibration Analysis of a Rotating Composite Shaft Using the -Version of the Finite Element Method

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
A. Boukhalfa ◽  
A. Hadjoui ◽  
S. M. Hamza Cherif
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Critical Speeds ◽  
Six Degrees Of Freedom ◽  
Composite Shaft ◽  
Composite Layers ◽  
Gyroscopic Effects

This paper is concerned with the dynamic behavior of the rotating composite shaft on rigid bearings. A -version, hierarchical finite element is employed to define the model. A theoretical study allows the establishment of the kinetic energy and the strain energy of the shaft, necessary to the result of the equations of motion. In this model the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. A hierarchical beam finite element with six degrees of freedom per node is developed and used to find the natural frequencies of a rotating composite shaft. A program is elaborate for the calculation of the eigenfrequencies and critical speeds of a rotating composite shaft. To verify the present model, the critical speeds of composite shaft systems are compared with those available in the literature. The efficiency and accuracy of the methods employed are discussed.

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