scholarly journals Dynamic Buckling of a Clamped Finite Column Resting on a Non – linear Elastic Foundation

2019 ◽  
pp. 1-47
Author(s):  
E. Julius, Bassey ◽  
M. Anthony, Ette ◽  
U. Joy, Chukwuchekwa ◽  
C. Atulegwu, Osuji
Keyword(s):  
Elastic Foundation ◽  
Governing Equation ◽  
Asymptotic Expansions ◽  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Linear Elastic ◽  
Non Linear ◽  
Relevant Variables ◽  
Independent Parameters

The analysis of the dynamic buckling of a clamped finite imperfect viscously damped column lying on a quadratic-cubic elastic foundation using the methods of asymptotic and perturbation technique is presented. The proposed governing equation contains two small independent parameters (δ and ϵ) which are used in asymptotic expansions of the relevant variables. The results of the analysis show that the dynamic buckling load of column decreases with its imperfections as well as with the increase in damping. The results obtained are strictly asymptotic and therefore valid as the parameters δ and ϵ become increasingly small relative to unity.

NON-LINEAR VIBRATIONS OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR ELASTIC FOUNDATION

1998 ◽  
Vol 212 (2) ◽  
pp. 295-309 ◽  
Author(s):  
H.R. Öz ◽  
M. Pakdemirli ◽  
E. Özkaya ◽  
M. Yilmaz
Keyword(s):  
Elastic Foundation ◽  
Curved Beam ◽  
Linear Elastic ◽  
Non Linear ◽  
Linear Vibrations

scholarly journals Perturbation Approach in the Dynamic Buckling of a Model Structure with a Cubic-quintic Nonlinearity Subjected to an Explicitly Time Dependent Slowly Varying Load

2019 ◽  
pp. 1-19
Author(s):  
A. M. Ette ◽  
I. U. Udo-Akpan ◽  
J. U. Chukwuchekwa ◽  
A. C. Osuji ◽  
M. F. Noah
Keyword(s):  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Time Dependent ◽  
Initial Time ◽  
Perturbation Approach ◽  
Elastic Model ◽  
Model Structure ◽  
Regular Perturbation ◽  
Long Run

This investigation is concerned with analytically determining the dynamic buckling load of an imperfect cubic-quintic nonlinear elastic model structure struck by an explicitly time-dependent but slowly varying load that is continuously decreasing in magnitude. A multi-timing regular perturbation technique in asymptotic procedures is utilized to analyze the problem. The result shows that the dynamic buckling load depends, among other things, on the first derivative of the load function evaluated at the initial time. In the long run, the dynamic buckling load is related to its static equivalent, and that relationship is independent of the imperfection parameter. Thus, once any of the two buckling loads is known, then the other can easily be evaluated using this relationship.

scholarly journals Oscillations of a Beam on a Non-Linear Elastic Foundation under Periodic Loads

2006 ◽  
Vol 13 (4-5) ◽  
pp. 273-284 ◽  
Author(s):  
Donald Mark Santee ◽  
Paulo Batista Gonçalves
Keyword(s):  
Mathematical Model ◽  
Elastic Foundation ◽  
Period Doubling ◽  
Algebraic Expression ◽  
Geometrical Parameters ◽  
Flexible Beam ◽  
Algebraic Relation ◽  
Structural Element ◽  
Linear Elastic ◽  
Non Linear

The complexity of the response of a beam resting on a nonlinear elastic foundation makes the design of this structural element rather challenging. Particularly because, apparently, there is no algebraic relation for its load bearing capacity as a function of the problem parameters. Such an algebraic relation would be desirable for design purposes. Our aim is to obtain this relation explicitly. Initially, a mathematical model of a flexible beam resting on a non-linear elastic foundation is presented, and its non-linear vibrations and instabilities are investigated using several numerical methods. At a second stage, a parametric study is carried out, using analytical and semi-analytical perturbation methods. So, the influence of the various physical and geometrical parameters of the mathematical model on the non-linear response of the beam is evaluated, in particular, the relation between the natural frequency and the vibration amplitude and the first period doubling and saddle-node bifurcations. These two instability phenomena are the two basic mechanisms associated with the loss of stability of the beam. Finally Melnikov's method is used to determine an algebraic expression for the boundary that separates a safe from an unsafe region in the force parameters space. It is shown that this can be used as a basis for a reliable engineering design criterion.

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