A Parametric Solution to the Elastic Pole-Vaulting Pole Problem

Griner, Gary M.

doi:10.1115/1.3167633

Cited by 33 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2.4) was first presented in [2] and since then has been solved in exact form only for special cases where the bar buckles under the action of concentrated forces and couples at its free end [1,4,10]. Taking into consideration Eq.…”

Section: Formulationmentioning

confidence: 99%

“…On the other hand, the problem of nonlinear buckling analysis (elastica problem) of a straight bar due to terminal loads was first examined by Euler and Lagrange, and since then by many other researchers [1,2,4,10]. In particular, Griner [1] succeeded in constructing a parametric solution of the elastica problem concerning straight bars subjected to compressive forces and couples, while in [10] exact parametric analytic solutions of the same problem were constructed including effects of the transverse deformation. In [4] the authors presented a closed-form solution of the exact third-order nonlinear differential equation (ODE), concerning the elastica analysis of a cantilever due to its own weight, by approximating the slope θ of the deflected elastica up to O θ 2 ; cot θ ∼ = 1/θ .…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

2006

View full text Add to dashboard Cite

It is shown that by a series of admissible functional transformations the already derived (third-order) strongly nonlinear ordinary differential equation (ODE), describing the elastica buckling analysis of a straight bar under its own weight [Int. J. Solids Struct. 24(12), 1179-1192, 1988, The Theory of Elastic Stability, McGraw-Hill, New York, 1961], is reduced to a first-order nonlinear integrodifferential equation. The absence of exact analytic solutions of the reduced equation leads to the conclusion that there are no exact analytic solutions in terms of known (tabulated) functions of this elastica buckling problem. In the limits of large or small values of the slope of the deflected elastica, we expand asymptotically the above integrodifferential equation to nonlinear ODEs of the Emden-Fowler or Abel nonlinear type. In these cases, using the solution methodology recently developed in Panayotounakos [Appl. Math. Lett. 18:155-162, 2005] and Panayotounakos and Kravvaritis [Nonlin. Anal. Real World Appl., 7(2):634-650, 2006], we construct exact implicit analytic solutions in parametric form of these types of equations and thus approximate implicit analytic solutions of the original elastica buckling nonlinear ODE.

show abstract

Section: Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

2006

View full text Add to dashboard Cite

show abstract

“…Many models were studied in order to investigate the stability of the beam models under variety of load cases. Focusing on the simple beam model, there are a number of research works dealing with the postbuckling of the beams under several load cases such as transverse point load [1], axial point load [2][3][4], follower point load [5], uniform load [6], follower uniform load [7], and concentrated moment [8,9]. Majority of previous studies focuses on the postbuckling behaviors of beams under only two specific load cases: point and uniform loads.…”

Section: Introductionmentioning

confidence: 99%

“…The concentrated moments occurring in previous examples may apply at ends or within the span length of the beam depending on design purpose. Griner [8] and Seide [9] have studied a case of beam with the concentrated moments acting at the end. However as aforementioned, in the case of the concentrated moment acting within the span length of the beam, the research regarding to such problem has never been found.…”

Section: Introductionmentioning

confidence: 99%

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

Phungpaingam

Chucheepsakul

2007

Acta Mech Sin

View full text Add to dashboard Cite

This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The nonlinear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance (10 −5 ). The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.

show abstract

“…Large flexural deflections of straight bars (elastica problem) can be solved in terms of elliptic functions and integrals when the loads consist of concentrated end forces and couple [1][2][3]. In Ref.…”

Section: Introductionmentioning

confidence: 99%