Exact parametric analytic solutions of the elastica ODEs for bars including effects of the transverse deformation

Sotiropoulou, Anastasia; Panayotounakos, D.E.

doi:10.1016/j.ijnonlinmec.2003.09.004

Cited by 13 publications

(6 citation statements)

References 4 publications

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“…(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: 99%

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

2006

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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.

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Section: Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2006

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“…By now, using transformation (3.9) as well as relations (3.11) one derives 4) or, by way of (3.7) and (3.5), one also derives…”

Section: Approximate Analytic Solutionsmentioning

confidence: 99%

“…[3] a thorough parametric analysis of the above problem was presented in the form of elliptic integrals, while in Ref. [4] an exact parametric solution of the governing nonlinear ordinary differential equations (ODEs) of the elastica buckling analysis for cantilevers due to a generalized terminal loading by taking into account the influence of transverse deformation was developed.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic solutions to the non-linear cantilever elastica

Stampouloglou

Theotokoglou

Andriotaki

2005

International Journal of Non-Linear Mechanics

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“…Special planar problems can be seen at [11] which includes the effects of transverse deformation; in [8] a variety of loads (terminal, transverse, continuous) is tackled; whereas the effect of large loads can be seen at [19]. Unextendibility and circular shape of undeformed rod are (numerically) analyzed in [10] where the load is a uniform centrally directed force.…”

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confidence: 99%

Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces

Scarpello

Ritelli²

2006

Meccanica

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In this article we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod's centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears.

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Exact parametric analytic solutions of the elastica ODEs for bars including effects of the transverse deformation

Cited by 13 publications

References 4 publications

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

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

Asymptotic solutions to the non-linear cantilever elastica

Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces

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