scholarly journals THERMAL POST-BUCKLING OF SLENDER ELASTIC RODS WITH DIFFERENT BOUNDARY CONDITIONS

2006 ◽  
Vol 5 (2) ◽  
pp. 50
Author(s):  
R. F. Solano ◽  
M. A. Vaz
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Mathematical Formulation ◽  
Different Boundary Conditions ◽  
Linear Ordinary Differential Equations ◽  
Linear Elastic ◽  
Buckling Behavior ◽  
Non Linear ◽  
Post Buckling ◽  
Complex Boundary

This paper presents mathematical formulation, critical buckling temperature and analytical and numerical solutions for the thermal post-buckling behavior of slender rods subjected to uniform thermal load. The material is assumed to be linear elastic, homogeneous and isotropic. Furthermore, large displacements are considered hence the formulation is geometrically non-linear. Three different boundary conditions are assumed: (i) double-hinged non-movable, (ii) hinged non-movable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring, and (iii) double-fixed non-movable. The governing equations are derived from geometrical compatibility, equilibrium of forces and moments, constitutive equations and strain-displacement relation, yielding a set of six first-order non-linear ordinary differential equations with boundary conditions specified at both ends, which constitutes a complex boundary value problem. The buckling and post-buckling solutions are respectively accomplished assuming infinitesimal and finite rotations. The results are presented in non-dimensional graphs for a range of temperature gradients and different values of slenderness ratios, and it is shown that this parameter governs the rod post-buckling response. The influence of the boundary conditions is evaluated through graphic results for deformed configuration, maximum deflection, maximum inclination angle and maximum curvature in the rod.

Geometrical and Material Non-Linear Formulation for Bend Stiffeners

2004 ◽  
Author(s):  
Murilo Augusto Vaz ◽  
Carlos Alberto Duarte de Lemos
Keyword(s):  
Differential Equations ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Power Series Expansion ◽  
Constitutive Models ◽  
Bending Moment ◽  
Accurate Analysis ◽  
Linear Ordinary Differential Equations ◽  
Linear Elastic ◽  
Non Linear

A mathematical formulation and a numerical solution for the geometrical and material non-linear analysis of bend stiffeners — employed to protect the upper terminations of flexible risers and subsea umbilical cables — are presented in this paper. The differential equations governing the problem result from geometrical compatibility, equilibrium of forces and moments and material constitutive relations, which can be linear elastic symmetric or non-linear elastic asymmetric. In this latter case, the bending moment versus curvature for each cross-section is calculated and then expressed by a polynomial power series expansion. Hence, a set of four first order non-linear ordinary differential equations is written and boundary conditions are defined at both ends. A one-parameter shooting method is employed and results are presented for a case study where linear elastic symmetric and non-linear elastic asymmetric constitutive models are compared and discussed. It is shown that an accurate analysis of bend stiffeners depends on a precise assessment of the material constitutive property.

Post-Buckling Analysis of Heavy Columns with Application to Marine Risers

1985 ◽  
Vol 29 (03) ◽  
pp. 162-169
Author(s):  
Theodore Kokkinis ◽  
Michael M. Bernitsas
Keyword(s):  
Boundary Conditions ◽  
Small Strain ◽  
Equilibrium Path ◽  
Drilling Mud ◽  
Tubular Columns ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Combined Action ◽  
Raphson Iteration ◽  
Good Agreement

The post-buckling behavior of heavy tubular columns following static instability under the combined action of weight, tension/compression at the top, and fluid static pressure forces in the gravity field is studied. A two-dimensional nonlinear small-strain large-deflection model of the column is derived, consisting of an integrodifferential equilibrium equation and two end rotation conditions. The equation of equilibrium is discretized using a finite-element method. An approximate solution valid in the neighborhood of the bifurcation point and an incremental solution are used to determine the secondary equilibrium path. The results of both methods are corrected by Newton-Raphson iteration. Conditions for unstable initial post-buckling behavior and existence of limit points on the secondary equilibrium path are presented. The numerical solution is applied to the problem of the elastica and is found to be in good agreement with the analytical solution. The secondary equilibrium path for a 500-m-long (1640 ft) marine drilling riser is calculated for two sets of boundary conditions and various values of the drilling mud density. The effect of the drilling mud density and the boundary conditions on the riser's post-buckling behavior is discussed.

Large-deflection and post-buckling behavior of slender beam-columns with non-linear end-restraints

2011 ◽  
Vol 46 (1) ◽  
pp. 79-95 ◽  
Author(s):  
Carlos Vega-Posada ◽  
Mauricio Areiza-Hurtado ◽  
J. Dario Aristizabal-Ochoa
Keyword(s):  
Large Deflection ◽  
Beam Columns ◽  
Buckling Behavior ◽  
Non Linear ◽  
Post Buckling

scholarly journals The Impact of 3D Printing Parameters on the Post-Buckling Behavior of Thin-Walled Structures

Materials ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 4742
Author(s):  
Tomasz Kopecki ◽  
Przemysław Mazurek ◽  
Łukasz Święch
Keyword(s):  
3D Printing ◽  
Numerical Solutions ◽  
Structure Mapping ◽  
Thin Walled Structures ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Thin Walled ◽  
The Impact ◽  
Printing Direction ◽  
Bearing Structure

This study presents the results of experimental research and numerical calculations regarding models of a typical torsion box fragment, which is a common thin-walled load-bearing structure used in aviation technology. A fragment of this structure corresponding to the spar wall was made using 3D printing. The examined system was subjected to twisting and underwent post-critical deformation. The research was aimed at determining the influence of the printing direction of the structure’s individual layers on the system stiffness. The experimental phase was supplemented by nonlinear numerical analyses of the models of the studied systems, taking into account the details of the structure mapping using the laminate concept. The purpose of the calculations was to determine the usefulness of the adopted method for modeling the examined structures by assessing the compliance of numerical solutions with the results of the experiment.

Effects of Friction on Post-Buckling Behavior and Axial Load Transfer in a Horizontal Well

SPE Journal ◽  
2010 ◽  
Vol 15 (04) ◽  
pp. 1104-1118 ◽  
Author(s):  
Guohua Gao ◽  
Stefan Miska
Keyword(s):  
Boundary Conditions ◽  
Critical Load ◽  
Axial Load ◽  
Load Transfer ◽  
Experimental Studies ◽  
Proper Solution ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Helical Buckling ◽  
Lateral Friction

Summary In this paper, the buckling equation and natural boundary conditions are derived with the aid of calculus of variations. The natural and geometric boundary conditions are used to determine the proper solution that represents the post-buckling configuration. Effects of friction and boundary conditions on the critical load of helical buckling are investigated. Theoretical results show that the effect of boundary conditions on helical buckling becomes negligible for a long pipe with dimensionless length greater than 5π Velocity analysis shows that lateral friction becomes dominant at the instant of buckling initiation. Thus, friction can increase the critical load of helical buckling significantly. However, once buckling is initiated, axial velocity becomes dominant again and lateral friction becomes negligible for post-buckling behavior and axial-load-transfer analysis. Consequently, it is possible to seek an analytical solution for the buckling equation. Analytical solutions for both sinusoidal and helical post-buckling configurations are derived, and a practical procedure for modeling of axial load transfer is proposed. To verify the proposed model and analytical results, the authors also conducted experimental studies. Experimental results support the proposed solutions.

Radiative Unsteady Rarefied Gaseous Flow Over a Stretching Sheet with Velocity Slip and Temperature Jump Effects

2020 ◽  
Vol 87 (3-4) ◽  
pp. 261
Author(s):  
Ram Prakash Sharma ◽  
N. Indumathi ◽  
S. Saranya ◽  
B. Ganga ◽  
A. K. Abdul Hakeem
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Stretching Sheet ◽  
Velocity Slip ◽  
Skin Friction Coefficient ◽  
Gaseous Flow ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In this study a mathematical analysis has been carried out to scrutinize the unsteady boundary layer flow of an incompressible, rarefied gaseous flow over a vertical stretching sheet with velocity slip and thermal jump boundary conditions in the presence of thermal radiation. Using boundary layer approach and suitable similarity transformations, the governing partial differential equations with the boundary conditions are reduced to a system of non-linear ordinary differential equations. The resulting non-linear ordinary differential equations are solved with the help of fourth order Runge-Kutta method with shooting technique. The results obtained for the velocity profile, temperature profile, skin friction coefficient and the reduced Nusselt number are described through graphs. It is predicted that the velocity and temperature profiles are lower for unsteady flow and has an opposite effect for steady flow.

Analytical Approximate Prediction of Thermal Post-Buckling Behavior of the Spring-Hinged Beam

2016 ◽  
Vol 08 (03) ◽  
pp. 1650028
Author(s):  
Youhong Sun ◽  
Baisheng Wu ◽  
Yongping Yu
Keyword(s):  
Numerical Solutions ◽  
Lateral Displacement ◽  
Admissible Function ◽  
Buckling Behavior ◽  
Practical Support ◽  
Differential Integral Equation ◽  
Post Buckling ◽  
Approximate Formulation ◽  
Support Conditions ◽  
The Galerkin Method

This paper is concerned with thermal post-buckling of uniform isotropic beams with axially immovable spring-hinged ends. The ends of the beam with elastic rotational restraints represent the actual practical support conditions and the classical hinged and clamped conditions can be achieved as the limiting cases of the rotational spring stiffness. The governing differential–integral equation is solved by assuming suitable admissible function for lateral displacement and by employing the Galerkin method. A brief and explicit analytical approximate formulation is established to predict the thermal post-buckling behavior of the beam. The present analytical approximate expressions show excellent agreement with the corresponding numerical solutions based on the shooting method. This confirms the effectiveness and verifies the accuracy of the formulas established.

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