The Nonlinear Static Formulation of Variable Cross-Section Beam Element Based on Positional Description

2012 ◽  
Vol 557-559 ◽  
pp. 2367-2370
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Yu Qin Yan
Keyword(s):  
Cross Section ◽  
Large Deflection ◽  
Variable Cross Section ◽  
Beam Element ◽  
Solution Procedure ◽  
Variable Cross ◽  
Hyper Elastic ◽  
Nonlinear Static ◽  
Newton Raphson ◽  
Raphson Iteration

Based on positional FEM (finite element method), the nonlinear static formulation to treat large deflection of variable cross-section beam element is created by using the lowest potential energy theory. Adopting linear constitutive relation for hyper-elastic materials, the formulation and the solution procedure by Newton-Raphson iteration method are very simple.

Study on Kinetic Energy for Saving Materials with Analysis of Lumped Mass Matrix of Variable Cross-Section Beam Element

2013 ◽  
Vol 675 ◽  
pp. 158-161
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Yu Qin Yan ◽  
Xun Lin Diao
Keyword(s):  
Kinetic Energy ◽  
Cross Section ◽  
Mass Matrix ◽  
Nonlinear Problems ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Lumped Mass ◽  
Geometrical Nonlinear ◽  
Properties Of Materials

Based on positional finite element method (FEM), a new, simple and accurate lumped mass matrix to solve dynamic geometrical nonlinear problems of materials applied to variable cross-section beam element has been proposed. According to Hamilton theory and the concept of Kinetic energy, concentrate the beam element mass to the two nodes in certain proportion, the lumped mass matrix is deduced. The lumped mass matrix is diagonal matrix and its calculated quantity is less than using consistent mass matrix about properties of materials under the same calculation precision.

scholarly journals ОПТИМАЛЬНОЕ ПРОЕКТИРОВАНИЕ СТАТИЧЕСКИ ОПРЕДЕЛИМОЙ БАЛКИ ПРИ ОГРАНИЧЕНИИ НА МАКСИМАЛЬНЫЙ ПРОГИБ

2020 ◽  
pp. 28-34
Author(s):  
Сергей Сергеевич Куреннов
Keyword(s):  
Programming Problem ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Variable Cross Section ◽  
Solution Procedure ◽  
Linear Functions ◽  
Problem Solution ◽  
Variable Cross ◽  
Nodal Points

Here is solved the optimization problem for the longitudinal depth distribution in the beam with a limitation on the maximum value of deflection. A review of the references is done, and it is shown that the known solutions are either erroneous, because they are based on false hypotheses, or have a narrow field of application, limited only to symmetrical constructions for which the point of the maximum deflection is known a priori. The paper considers a beam of the rectangular cross-section of constant width. The beam is assumed to be statically determinate, and the load is arbitrary and asymmetric and multidirectional as well. The points (or point) of the beam maximum deflections are unknown in advance and would be determined in the problem-solution procedure. A linear problem is considered. The optimization criterion is the mass of the beam. To find the deflections of the beam, i.e. to solve the differential equation of a variable cross-section beam bending the finite difference method is used. The design problem is reduced to the required beam depths obtaining in the system of nodal points. In this case, the desired solution must satisfy the restriction system for the nodal points shift and the sign of variables as well. Since the restrictions of the shift of each node are considered separately and independently, so the proposed method allows flexible control of the beam shift restrictions. Using the change of variables proposed in the paper, the problem to be solved is reduced to a nonlinear programming problem where the criterion function is separable and restrictions are linear functions. Using linearization, this problem can be reduced to the linear programming problem relatively to new variables. The model problem is solved, and it is shown that the proposed algorithm efficiently allows us to solve the problems of the beam optimal design with the restrictions of the maximally allowed deflection. The proposed approach can be spread for the strength limitations, for beams of variable width, I-beam cross-section, etc.

Nonlinear Stain of Variable Cross-Section Beam Element Based on Positional FEM

2012 ◽  
Vol 557-559 ◽  
pp. 2371-2374
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Xiao Dong Jia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cross Section ◽  
Beam Width ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Nonlinear Fem ◽  
Beam Elements ◽  
Beam Depth

Based on positional finite element method, this paper attempts to find beam elements that can show the characteristics of the variable cross-section beam and can be practically applied. In this paper, the stain on a random point of the variable cross-section beam element is obtained when beam depth changes in a linear or quadratic parabolic way and beam width is fixed. The calculation is different and simpler than the classical nonlinear FEM.

Study on Flexible Spin-Up Maneuver with Variable Cross-Section Based on Positional FEM

2014 ◽  
Vol 670-671 ◽  
pp. 834-837
Author(s):  
Lv Zhou Ma ◽  
Yu Qin Yan ◽  
Xun Lin Diao ◽  
Jian Liu
Keyword(s):  
Cross Section ◽  
Mass Matrix ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Lumped Mass ◽  
Nonlinear Fem ◽  
Calculation Results ◽  
Properties Of Materials ◽  
Consistent Mass Matrix

Based on positional finite element method (FEM) and MATLAB platform, program VBEP (Variable cross-section Beam Element based on Positional FEM) is compiled. Flexible spin-up maneuver is calculated. The calculation results show that positional FEM uses fewer elements and gains higher calculation precision and efficiency when compared with traditional nonlinear FEM, and that calculated quantity using lumped mass matrix is less than using consistent mass matrix about properties of materials under the same calculation precision.

A Higher-Order Variable Cross-Section Viscoelastic Beam Element Via ANCF for Kinematic and Dynamic Analyses of Two-Link Flexible Manipulators

2017 ◽  
Vol 09 (08) ◽  
pp. 1750116 ◽  
Author(s):  
Haidong Yu ◽  
Chunzhang Zhao ◽  
Hui Zheng
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Variable Cross Section ◽  
Beam Element ◽  
Higher Order ◽  
Flexible Manipulator ◽  
Variable Cross ◽  
Viscoelastic Beam ◽  
Shear Locking ◽  
Large Deformation Problems

A new viscoelastic beam element with variable cross-sections is developed based on the absolute nodal coordinate formulation, in which the higher-order slope coordinates are used to describe the variable geometric boundaries and circumvent possible shear-locking problem. The mass and stiffness matrices of the new element are derived by considering the variable geometrical boundary in the integration functions. The modified Kelvin–Voigt viscoelastic constitutive model for large deformation problems is introduced into the stiffness matrix. The dynamic model of a typical two-link manipulator with variable cross-section links is established where the constraint equations of revolute joints are considered with Lagrange multipliers. The kinematic trajectories of the manipulator with various materials and geometrical parameters are numerically studied. It is shown that the new element could circumvent shear-locking problem and yield improved accuracy and convergence compared with the conventional beam elements for solving large deformation problems. Also, the viscosity of the structural material helps to reduce the deformation of the links and improve the kinematic precision of the manipulator, hence the trajectory of the flexible manipulator could be controlled by changing the geometrical shape of the cross-section of links under the constraint of same mass.

Numerical Examples of Variable Cross-Section Beam Elements

2012 ◽  
Vol 557-559 ◽  
pp. 822-825
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Xiao Dong Jia
Keyword(s):  
Exact Solution ◽  
Cantilever Beam ◽  
Cross Section ◽  
Beam Width ◽  
Variable Cross Section ◽  
Beam Element ◽  
Variable Cross ◽  
Numerical Examples ◽  
Beam Elements ◽  
Beam Depth

Based on MATLAB platform, program VCBEP (Variable Cross-section Beam Element based on Positional FEM) is compiled, and the cantilever beam with linear profile and the parabolic simple supported beam are calculated. The variable cross-section beam element is proposed to analyze rectangular beam whose beam depth changes in a linear or quadratic parabolic way and beam width is fixed and the exact solution can be obtained.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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