Large Deformation Geometric Nonlinear Beam Element Based on U.L. Format

2012 ◽  
Vol 446-449 ◽  
pp. 3596-3603
Author(s):  
Yong Jun Xia ◽  
Qian Miao
Keyword(s):  
Displacement Field ◽  
Strain Tensor ◽  
Quadratic Function ◽  
Beam Element ◽  
Interpolation Polynomial ◽  
Bernoulli Beam ◽  
Displacement Fields ◽  
First Order ◽  
Geometric Nonlinear ◽  
Euler Bernoulli Beam

Based on the geometric deformation of the Euler-Bernoulli beam element, the geometric nonlinear Euler-Bernoulli beam element based on U.L. formulation is derived. The element’s transverse first-order displacement field is constructed using the cubic Hermite interpolation polynomial, and the first-order Lagrange interpolation polynomial is used for the axial displacement field. Then the additional displacements induced from the rotation of the elemental are included into the transverse and longitudinal displacement fields, so those displacement fields are expressed as the quadratic function of nodal displacement. Afterwards the nonlinear finite element formulas of Euler-Bernoulli beam element under the form of U.L. formulation are derived using Cauchy strain tensor and the principle of virtual displacements. The total equilibrium equation and tangent stiffness for large displacement geometric nonlinear analysis of frame are obtained in the total coordinate system. The correctness of this element is proved by typical example.

A New Higher-Order Euler-Bernoulli Beam Element of Absolute Nodal Coordinate Formulation

2020 ◽  
Author(s):  
Peng Zhang ◽  
Jianmin Ma ◽  
Menglan Duan
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Strain Tensor ◽  
Beam Element ◽  
Higher Order ◽  
Position Vector ◽  
Traditional Theory ◽  
Bernoulli Beam ◽  
Geometrically Nonlinear Analysis ◽  
Absolute Nodal Coordinate ◽  
Euler Bernoulli Beam

Abstract In this study, a new higher-order Euler-Bernoulli beam element of Absolute Nodal Coordinate Formulation (ANCF) is developed for geometrically nonlinear analysis of planar structures. The strain energy of the beam element is derived by applying the definition of the Green–Lagrange strain tensor in continuum mechanics. The first contribution of this research is to realize the accurate calculation of curvature on the beam element node by additionally considering the second derivative of the position vector obtained by quintic Hermite interpolation function. Furthermore, in traditional theory, the independent variable of finite formulation is arc-length coordinate s, while in this work, a correction is come up with and proven that it is actually an equivalent parameter. Some benchmark problems of straight beams are solved by the proposed element and accurate results are obtained by just fewer elements when compared with the other works including the traditional ANCF element and B23 element of ABAQUS. What leads to this accuracy result is that the precise calculation of nodal curvature is obtained from higher order interpolation scheme. The correctness and accuracy of the proposed element are validated in this work and it can be further developed for tackling large deformation and large rotation problems of spatial curved beams.

scholarly journals Numerical Solution of Bending of the Beam with Given Friction

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 898
Author(s):  
Michaela Bobková ◽  
Lukáš Pospíšil
Keyword(s):  
Sliding Friction ◽  
Quadratic Function ◽  
Building Construction ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Sustainable Building ◽  
Construction Design ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Fixed Beam

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

scholarly journals Three-Node Euler-Bernoulli Beam Element Based on Positional FEM

2012 ◽  
Vol 29 ◽  
pp. 3703-3707 ◽  
Author(s):  
Liu Jian ◽  
Zhou Shenjie ◽  
Dong Meiling ◽  
Yan Yuqin
Keyword(s):  
Beam Element ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

On a geometrically exact Euler‐Bernoulli beam element

PAMM ◽  
2018 ◽  
Vol 18 (1) ◽  
Author(s):  
Sascha Maassen ◽  
Cátia da Costa e Silva ◽  
Paulo de Mattos Pimenta ◽  
Jörg Schröder
Keyword(s):  
Beam Element ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

Modelling and Experimental Validation of a Novel Piezohydraulic Servovalve

2011 ◽  
Author(s):  
Dhinesh K. Sangiah ◽  
Andrew R. Plummer ◽  
Christopher R. Bowen ◽  
Paul Guerrier
Keyword(s):  
Electrical Power ◽  
Test Results ◽  
Bernoulli Beam ◽  
Element Analysis ◽  
First Order ◽  
Beam Analysis ◽  
Design Tradeoffs ◽  
Fast Flow ◽  
Euler Bernoulli Beam ◽  
Good Agreement

Servovalves are compact, accurate, fast flow modulating valves. However, cost reduction pressures exist, not least due to the electomagnetically actuated pilot stage. This paper describes a servovalve with a jet deflector pilot stage actuated by a multilayer piezoelectric bimorph. The electrical power and voltage requirements are relatively low (+/−30V), and mechanical spool feedback is used as opposed to the more complex electrical feedback alternative. A mathematical model of the valve is presented, which is used to simulate its performance. Finite element analysis is used to model the bimorph actuator and the feedback wire assembly to verify an Euler-Bernoulli beam analysis. A Moog 26 Series servovalve is used as a basis for the prototype. Experimental test results are in good agreement with the simulation results. The high order nonlinear model is also approximated by a first order transfer function to identify the parameters that dictate the main design tradeoffs.

On the Fundamental Transverse Vibration Frequency of a Free-Free Thin Beam With Identical End Masses

2007 ◽  
Vol 129 (5) ◽  
pp. 656-662 ◽  
Author(s):  
A. Erturk ◽  
D. J. Inman
Keyword(s):  
Natural Frequency ◽  
Extreme Case ◽  
Frequency Equation ◽  
Mode Frequency ◽  
Bernoulli Beam ◽  
Thin Beam ◽  
Microelectromechanical System ◽  
First Order ◽  
Order Polynomial ◽  
Euler Bernoulli Beam

Current research in vibration-based energy harvesting and in microelectromechanical system technology has focused renewed attention on the vibration of beams with end masses. This paper shows that the commonly accepted and frequently quoted fundamental natural frequency formula for a beam with identical end masses is incorrect. It is also shown that the higher mode frequency expressions suggested in the referred work (Haener, J., 1958, “Formulas for the Frequencies Including Higher Frequencies of Uniform Cantilever and Free-Free Beams With Additional Masses at the Ends,” ASME J. Appl. Mech. 25, pp. 412) are also incorrect. The correct characteristic (frequency) equation is derived and nondimensional comparisons are made between the results of the previously published formula and the corrected formulation using Euler–Bernoulli beam assumptions. The previous formula is shown to be accurate only for the extreme case of very large end mass to beam mass ratios. Curve fitting is used to report alternative first order and second order polynomial ratio expressions for the first natural frequency, as well as for the frequencies of some higher modes.

Exact formulations of non-linear planar and spatial Euler–Bernoulli beams with finite strains

2011 ◽  
Vol 226 (5) ◽  
pp. 1225-1236 ◽  
Author(s):  
M H Abedinnasab ◽  
H Zohoor ◽  
Y-J Yoon
Keyword(s):  
Strain Field ◽  
Dynamic Models ◽  
Equations Of Motion ◽  
Strain Tensor ◽  
Bernoulli Beam ◽  
Cross Sectional ◽  
Elastic Deformations ◽  
Non Linear ◽  
Lagrange Strain ◽  
Euler Bernoulli Beam

Using Hamilton’s principle, exact equations of motion for non-linear planar and spatial Euler–Bernoulli beams are derived. In the existing non-linear Euler–Bernoulli beam formulations, some elastic terms are dropped by differentiation from the incomplete Green–Lagrange strain tensor followed by negligible elastic deformations of cross-sectional frame. On the other hand, in this article, the exact strain field concerning considerable elastic deformations of cross-sectional frame is used as a source in differentiations. As a result, the achieved closed-form equations are exact and more accurate than formerly reported equations in the literature. Moreover, the applicable dynamic model of inextensional beams which is fully accurate, yet simple has been shown. The planar and inextensional dynamic models have been compared with the existing dynamic models in the literature, and the proposed dynamic models demonstrate significant improvements in the numerical results. Finally, experiments on the carbon fibre rods verify the model presented for inextensional beams.

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