A 4-degree-of-freedom Kirchhoff beam model for the modeling of bending–torsion couplings in active-bending structures

2017 ◽  
Vol 32 (2) ◽  
pp. 69-83 ◽  
Author(s):  
Baptiste Lefevre ◽  
Frédéric Tayeb ◽  
Lionel du Peloux ◽  
Jean-François Caron
Keyword(s):  
Cross Sections ◽  
Beam Theory ◽  
Relaxation Method ◽  
Mathematical Work ◽  
Beam Model ◽  
Dynamic Relaxation ◽  
High Interaction ◽  
Slender Beams ◽  
Internal Forces ◽  
Dynamic Relaxation Method

Gridshells are lightweight structures made of interconnected slender beams. Due to large displacements, high interaction between the beams, and bending–torsion coupling, modeling gridshells requires specific non-linear numerical tools to reach convergence within a reasonable time. In this article, the development of such a tool is presented. It is based on the Kirchhoff beam theory and uses the dynamic relaxation method. First, from Kirchhoff’s equations, the internal forces and moments acting on a beam are obtained. Once this mathematical work is done, the dynamic relaxation method is used in order to get the static equilibrium configuration of the beam. This new approach is tested on several examples and validated for slender beams with arbitrary rest-state configuration and cross sections. In particular, results for ribbons with high bending–torsion coupling are presented. Finally, this process enables the fast and precise modeling of gridshells including bending–torsion coupling.

scholarly journals Estimation Of Young’s Modulus Of Elesticity By The Form Finding Of Grid Shell Structures By The Dynamic Relaxation Method

2015 ◽  
Vol 23 (4) ◽  
pp. 25-30 ◽  
Author(s):  
Ivana Grančičová ◽  
Ján Brodniansky
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Relaxation Method ◽  
Dynamic Relaxation ◽  
Form Finding ◽  
Grid Shell ◽  
Important Input ◽  
Internal Forces ◽  
Considerable Impact ◽  
Dynamic Relaxation Method

Abstract The paper is basically focused on the process of form finding by the dynamic relaxation method (DRM) with the aid of computational tools that enable us to make many calculations with different inputs. There are many important input values with a significant impact on the course of the calculations and the resulting displacement of a structure. One of these values is Young’s modulus of elasticity. This value has a considerable impact on the final displacement of a grid shell structure and the resulting internal forces.

scholarly journals Analysis of Tensegrity Structures with Redundancies, by Implementing a Comprehensive Equilibrium Equations Method with Force Densities

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Miltiades Elliotis ◽  
Petros Christou ◽  
Antonis Michael
Keyword(s):  
Relaxation Method ◽  
Force Density ◽  
System Of Equations ◽  
Dynamic Relaxation ◽  
Equilibrium Equations ◽  
Tensegrity Structures ◽  
Linear System Of Equations ◽  
Internal Forces ◽  
Dynamic Relaxation Method ◽  
2D And 3D

A general approach is presented to analyze tensegrity structures by examining their equilibrium. It belongs to the class of equilibrium equations methods with force densities. The redundancies are treated by employing Castigliano’s second theorem, which gives the additional required equations. The partial derivatives, which appear in the additional equations, are numerically replaced by statically acceptable internal forces which are applied on the structure. For both statically determinate and indeterminate tensegrity structures, the properties of the resulting linear system of equations give an indication about structural stability. This method requires a relatively small number of computations, it is direct (there is no iteration procedure and calculation of auxiliary parameters) and is characterized by its simplicity. It is tested on both 2D and 3D tensegrity structures. Results obtained with the method compare favorably with those obtained by the Dynamic Relaxation Method or the Adaptive Force Density Method.

Dynamic-relaxation solution for the large deflection of plates with specified boundary stresses

1969 ◽  
Vol 4 (2) ◽  
pp. 75-80 ◽  
Author(s):  
K R Rushton
Keyword(s):  
Large Deflection ◽  
Relaxation Method ◽  
Plane Boundary ◽  
Dynamic Relaxation ◽  
Simply Supported ◽  
Von Karman ◽  
Relaxation Solution ◽  
Von Kármán Equations ◽  
Mesh Spacing ◽  
Dynamic Relaxation Method

The von Kármán equations for the large deflection of plates are solved by the dynamic-relaxation method. Detailed results are presented for square plates having simply supported edges with zero in-plane boundary stresses. The results show that high stresses occur towards the corners of the plates. The mesh effect is investigated and recommendations are made for the optimum mesh spacing.

Form-finding of deployable mesh reflectors using dynamic relaxation method

2018 ◽  
Vol 151 ◽  
pp. 380-388 ◽  
Author(s):  
Xinyu Wang ◽  
Jianguo Cai ◽  
Ruiguo Yang ◽  
Jian Feng
Keyword(s):  
Relaxation Method ◽  
Dynamic Relaxation ◽  
Form Finding ◽  
Dynamic Relaxation Method

A Digital Computer Solution of the Laplace Equation Using the Dynamic Relaxation Method

1968 ◽  
Vol 19 (4) ◽  
pp. 375-387 ◽  
Author(s):  
K. R. Rushton ◽  
Lucy M. Laing
Keyword(s):  
Laplace Equation ◽  
Iterative Process ◽  
Basic Equation ◽  
Relaxation Method ◽  
Dynamic Relaxation ◽  
Computer Solution ◽  
Relaxation Solution ◽  
Simple Substitution ◽  
Dynamic Relaxation Method ◽  
Explicit Finite Difference

SummaryThe Dynamic Relaxation solution of the Laplace equation introduces dynamic terms into the basic equation. When this is written as an explicit finite difference formulation it can be solved by an iterative process which only requires a simple substitution routine. The method is easy to programme and requires small storage in the computer. By studying problems involving wind tunnel interference in steady flow, the potentialities of the method are demonstrated.

Cable-Membrane Structures - Numerical Analysis and Practical Application

2016 ◽  
Vol 837 ◽  
pp. 99-102
Author(s):  
Milos Huttner ◽  
Jiří Maca ◽  
Petr Fajman
Keyword(s):  
Numerical Analysis ◽  
Relaxation Method ◽  
Computation Method ◽  
Practical Application ◽  
Dynamic Relaxation ◽  
Membrane Structures ◽  
Form Finding ◽  
Correct Operation ◽  
Commercial Program ◽  
Dynamic Relaxation Method

This paper presents a practical application of form-finding process of cable-membrane structures. The dynamic relaxation method with kinetic damping is used as the computation method for numerical analysis. A brief description of the construction, a description of the models and the way of solving tasks will be introduced. The correct operation of the implemented algorithm will be compared with a commercial program.

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