Solution for Geometrically Non-Linear Elastic Deformation of Simple Frames by a Shooting Method

2011 ◽  
Vol 462-463 ◽  
pp. 668-673
Author(s):  
Shi Rong Li ◽  
Ya Dong Hu
Keyword(s):  
Large Deformation ◽  
Shooting Method ◽  
Frame Structure ◽  
Curved Beams ◽  
Equilibrium Equations ◽  
Nonlinear Bending ◽  
Linear Elastic ◽  
Equilibrium Configurations ◽  
Nonlinear Equilibrium ◽  
Plane Frame

Based on an exact geometric nonlinear theory for plane curved beams, geometrically nonlinear equilibrium equations and boundary conditions governing the nonlinear bending of a simple plane frame structure subjected distributed loads were derived. By using the shooting method to numerically solve the boundary value problem of nonlinear ordinary differential equations, large deformation equilibrium configurations of a simple frame with both straight and the curved beam elements subjected uniformly distributed load were obtained. The theory and methodology presented can be used to analyze large deformation of plane simple frames with a variety of geometries and loadings.

Numerical Solution of Bending Problem of a FGM Cantilever Beam with Large Deformation

2010 ◽  
Vol 160-162 ◽  
pp. 503-506
Author(s):  
Qing Lu Li ◽  
Shi Rong Li
Keyword(s):  
Boundary Value Problem ◽  
Large Deformation ◽  
Cantilever Beam ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Equilibrium Equations ◽  
Nonlinear Bending ◽  
Distributed Load ◽  
Fgm Beam

Nonlinear bending problem of FGM cantilever beam under distributed load are discussed in this paper. Based on the large deformation theory and considering the axial extension of the beam, the equilibrium equations with geometric nonlinearity of a FGM beam subjected to distributed load are established. They consist of a boundary value problem of ordinary differential equations with strong non-linearity, in which seven unknown functions are included and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and analytical continuation, the nonlinear boundary-value problem is numerically solved as well as the non-linear bending characteristic curves of the deformed beam versus the load are presented.

scholarly journals BEAM ELEMENT UNDER FINITE ROTATIONS

2021 ◽  
Vol 30 ◽  
pp. 87-92
Author(s):  
Emma La Malfa Ribolla ◽  
Milan Jirásek ◽  
Martin Horák
Keyword(s):  
Cross Sections ◽  
Shooting Method ◽  
Small Strain ◽  
Beam Model ◽  
Equilibrium Equations ◽  
Finite Rotations ◽  
Nonlinear Beam ◽  
Linear Elastic ◽  
Beam Elements ◽  
Large Rotations

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Non-Linear Analysis of a FGM Cantilever Beam Supported on a Winkler Elastic Foundation

2014 ◽  
Vol 602-605 ◽  
pp. 131-134
Author(s):  
Qing Lu Li ◽  
Qian Hong Shao
Keyword(s):  
Elastic Foundation ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Functionally Graded ◽  
Equilibrium Equations ◽  
Nonlinear Bending ◽  
Distributed Load ◽  
Winkler Elastic Foundation ◽  
Non Linear ◽  
Graded Material

In the present study, nonlinear bending problem of functionally graded material (FGM) cantilever beams resting on a Winkler elastic foundation under distributed load are discussed. Based on the large deformation theory and considering the axial extension of the beam, the equilibrium equations with geometric nonlinearity of FGM beams subjected to distributed load are established. In the analysis, it is assumed that the material properties of the beam vary continuously as a power function of the thickness. By using shooting method, the nonlinear boundary-value problem is numerically solved as well as the non-linear bending characteristic curves of the deformed beam versus the load are presented. The effects of material gradient property and foundation stiffness parameter on the bending deformation of the beam are discussed in detail.

Instability of Discrete Pseudo-Conservative Structural Systems

1991 ◽  
Vol 44 (11S) ◽  
pp. S194-S198 ◽  
Author(s):  
Anibal E. Mirasso ◽  
Luis A. Godoy
Keyword(s):  
Classical System ◽  
Potential Energy Function ◽  
Structural Systems ◽  
Equilibrium Path ◽  
Equilibrium Equations ◽  
Total Potential ◽  
Approximate Form ◽  
Nonlinear Equilibrium ◽  
Mechanical Models ◽  
New Formulation

Critical and postcritical states of pseudo-conservative discrete structural systems are studied by means of a new formulation leading to a classification of critical states and to an approximate form of the postcritical equilibrium path. The nonlinear equilibrium equations are derived from the total potential energy function of a classical system, but with the addition of at least one control parameter. The follower force effect is thus included by nonlinear constraints to the equilibrium equation. The nonlinear equations are solved by perturbation techniques. Finally the theory is applied to investigate the instability of some simple mechanical models.

NONLINEAR GENERALIZED BEAM THEORY FOR COLD-FORMED STEEL MEMBERS

2003 ◽  
Vol 03 (04) ◽  
pp. 461-490 ◽  
Author(s):  
N. SILVESTRE ◽  
D. CAMOTIM
Keyword(s):  
Numerical Technique ◽  
Beam Theory ◽  
Equilibrium Equations ◽  
Steel Members ◽  
Mode Decomposition ◽  
Nonlinear Equilibrium ◽  
Geometrical Imperfections ◽  
Post Buckling ◽  
Cold Formed Steel ◽  
Generalized Beam Theory

A geometrically nonlinear Generalized Beam Theory (GBT) is formulated and its application leads to a system of equilibrium equations which are valid in the large deformation range but still retain and take advantage of the unique GBT mode decomposition feature. The proposed GBT formulation, for the elastic post-buckling analysis of isotropic thin-walled members, is able to handle various types of loading and arbitrary initial geometrical imperfections and, in particular, it can be used to perform "exact" or "approximate" (i.e., including only a few deformation modes) analyses. Concerning the solution of the system of GBT nonlinear equilibrium equations, the finite element method (FEM) constitutes the most efficient and versatile numerical technique and, thus, a beam FE is specifically developed for this purpose. The FEM implementation of the GBT post-buckling formulation is reported in some detail and then employed to obtain numerical results, which validate and illustrate the application and capabilities of the theory.

scholarly journals Optimal Shape and First Integrals for Inverted Compressed Column

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 334
Author(s):  
Enes Kacapor ◽  
Teodor M. Atanackovic ◽  
Cemal Dolicanin
Keyword(s):  
Variational Principle ◽  
First Integrals ◽  
Arbitrary Cross Section ◽  
Optimal Shape ◽  
Equilibrium Equations ◽  
Concentrated Force ◽  
Cross Sectional ◽  
Nonlinear Equilibrium ◽  
Elastic Column ◽  
Special Case

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.

Synthesis of Compliant Mechanisms With Specified Equilibrium Positions

2005 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy
Keyword(s):  
Compliant Mechanisms ◽  
Equilibrium Constraints ◽  
Equilibrium Equations ◽  
Equilibrium Configurations ◽  
Form Design ◽  
Design Requirements ◽  
Synthesis Procedure ◽  
Position Synthesis ◽  
Four Bar Linkage ◽  
Sine And Cosine Functions

This paper presents a synthesis procedure for a compliant four-bar linkage with three specified equilibrium configurations. The finite position synthesis equations are combined with equilibrium constraints at the flexure pivots to form design equations. These equations are simplified by modeling the joint angle variables in the equilibrium equations using sine and cosine functions. Solutions to these design equations were computed using a polynomial homotopy solver. In order to provide a design specification, we first compute the six equilibrium configurations of a known compliant four-bar mechanism. We use these results as design requirements to synthesize a compliant four-bar. The solver obtained eight real solutions which we refined using a Newton-Raphson technique. A numerical example is provided to verify the design methodology.

Sign in / Sign up

Export Citation Format

Share Document