An Euler–Bernoulli-like finite element method for Timoshenko beams (original) (raw)
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An improved two-node timoshenko beam finite element
Computers & Structures, 1993
The stiffness, mass, and consistent force matrices for a simple two-node Timoshenko beam element are developed based upon Hamilton's principle. Cub& and quadratic Lagrangian polynomials are used for the transverse and rotational displacements, respectively, where the polynomials are made interdependent by requiring them to satisfy the two homogeneous differential equations associated with Timoshenko's beam theory. The resulting stiffness matrix, which can be exactly integrated and is free of 'shear-locking', is in agreement with the exact Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element exactly predicts the displacement of a short beam subjected to complex distributed loadings using only one element, and the current element predicts shear and moment resultants and natural frequencies better than existing Timoshenko beam elements.
On Exact Analytical Solutions of the Timoshenko Beam Model Under Uniform and Variable Loads
2021
In this research work, we consider the mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem of a system of ordinary differential equations. Instead of numerical solution using finite difference and finite volume methods, an attempt is made to derive the exact analytical solutions of the model with boundary feedback for a better and explicit description of the rotation and displacement parameters of the TB structure model. The explicit analytical solutions have been successfully found for the uniform and real-time variable load cases. The rotation and displacement profiles obtained through the analytical solutions accurately picture the structure of the beam under uniform and variable loads.
Finite Element Method for Vibration Analysis of Timoshenko Beams
2019 9th International Conference on Recent Advances in Space Technologies (RAST), 2019
In industry and lots of engineering applications, rotating components, turbines, helicopter blades, rotors belong to large usage area. Design, material properties and dynamic properties of these structures or components are so significant with respect to efficiency. Frequencies and mode shapes are used to identify the dynamic properties of structures. In this study, a theoretical investigation in free vibration of a functionally graded beam (FGB) is presented with using Finite Element Model. It is assumed that material properties vary along the beam thickness according to power law distributions. Timoshenko beam theory is studied and the FGB are modeled according to this theorem. Free vibration analysis of flap wise bending is studied at symmetrical functionally graded beam. The governing equations of motion and boundary conditions are derived on the basis of Hamilton principle. Analytical solutions of the natural frequencies are obtained with finite element method which the properties of FGB distribution shape functions are used for exponential FG beams with clamped-free end supports. MATLAB code is developed to analyze the free vibration of the functionally graded rotating Timoshenko beam. In the process, finite element formulation (FE) is used and the calculated results are validated with the ones in open literature.
Journal of Scientific Computing, 2015
In this paper we analyze a low-order finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which leads to the conclusion that the method is locking free. For its implementation, displacements and rotations can be eliminated leading to a well posed generalized matrix eigenvalue problem for which the computer cost of its solution is similar to that of other classical formulations. We report numerical experiments which allow us to assess the performance of the method.
Two non-standard finite difference schemes for the Timoshenko beam
African Journal of Mathematics and …, 2012
In this paper, we consider the following Timoshenko beam model. According to the Timoshenko beam theory, the in-plane bending of a clamped uniform beam of length L, cross section A, moment of inertia I, Young's modulus E and shear modulus G, subject to a ...
A new mixed finite element method for the Timoshenko beam problem
ESAIM: Mathematical Modelling and Numerical Analysis, 1991
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Variational Approach of Timoshenko Beams with Internal Elastic Restraints
2013
An exact approach for free transverse vibrations of a Timoshenko beam with ends elastically restrained against rotation and translation and arbitrarily located internal restraints is presented. The calculus of variations is used to obtain the equations of motion, the boundary conditions and the transitions conditions which correspond to the described mechanical system. The derived differential equations are solved individually for each segment of the beam with the corresponding boundary and transitions conditions. The derived mathematical formulation generates as particular cases, and several mathematical models are used to simulate the presence of cracks. Some cases available in the literature and the presence of some errors are discussed. New results are presented for different end conditions and restraint conditions in the intermediate elastic constraints with their corresponding modal shapes.
Numerical modelling of a Timoshenko FGM beam using the finite element method
Functionally graded material (FGM) beams possess a smooth variation of material properties due to continuous change in micro structural details. The variation of material properties is along the beam thickness and assumed to follow the power-law. An exact element based on the first order shear deformation theory was developed. The finite element method is used here to study extensively the static analysis. A cantilever beam subjected to a concentrated force P at the free end for different length-to-thickness ratio has been chosen here for the analysis. For each example, Poisson's ratio of the P-FGM beam is assumed to be varied continuously throughout the thickness direction according to the power law, and other time it is held constant. Timoshenko beam theory is used to capture the shear deformation. The governing equations and boundary conditions are derived from virtual work principle. In this study, the influences of the volume fraction index, length-to-thickness ratio and the Poisson's ratio on the mid plane deflections, stresses distribution and strain energy along the thickness of FGM beam are examined.
Petrov-Galerkin formulations of the Timoshenko beam problem
Computer Methods in Applied Mechanics and Engineering, 1987
Petrov-Galerkin formulations of the Timoshenko beam problem are presented. They are shown to provide the best approximation property, optimal rate of convergence, and nodally exact solution for arbitrary loading, for all values of the thickness of the beam.
The Timoshenko beam theory, 2023
Although it is generally, in the shear deformation beam theory of Timoshenko, recognized the decomposition of the transverse displacement into a bending displacement and a shear subsidence, the explicit introduction of these two variables in the governing equations has not be seen by us in the literature on the subject. This approach has the great advantages to surface all over the developments the close relationship, and differences, between the Euler-Bernoulli and the Timoshenko beam theories, and, mostly, to understand better the beam deformation. In order to obtain coherent equations for these two new variables that we propose, a shift factor required to be introduced. This shift factor can be calculated according to coherent criteria, and places the elastic line of the Timoshenko bending displacement in the best agreement possible with the Euler-Bernoulli.