Does generalized elastica lead to bimodal optimal solutions?

Spasic, D. T.; Glavardanov, V.B.

doi:10.1016/j.ijsolstr.2009.03.019

Cited by 4 publications

(5 citation statements)

References 36 publications

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“…First we note that when j ¼ 0 and k 1 ¼ 0, we recover the optimality condition for the Pflüger beam posed in the sense of the classical Bernoulli-Euler elastica theory (see Atanackovic and Simic, 1999). Secondly, the influence of nonlocal theory as well as the generalization of the classical elastica theory shown in Spasic and Glavardanov (2009), lead to a complicated two point boundary value problem since the optimality conditions are given in complex form. Namely, in both cases the posed boundary value problems require the solution of a cubic or quadratic equation.…”

Section: Optimization Problem and Its Solutionmentioning

confidence: 65%

“…Note that generalizations of the classical elastica theory, like the one at hand, lead to more complicated two point boundary value problems since optimality conditions are given in the form of polynomials. For example, see Spasic and Glavardanov (2009) where the posed boundary value problem requires a solution of the cubic equation and where its explicit solution, that determines optimal shape, was shown in terms of the Chebishev radicals. Thus, in obtaining a numerical solution that determines the optimal shape of a micro/nano beam for given arc length, in each iteration one solve the depressed quartic equation explicitly by the use of Ferrari's method.…”

Section: Introductionmentioning

confidence: 99%

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Stability and optimal shape of Pflüger micro/nano beam

Glavardanov

Spasic

Atanacković

2012

International Journal of Solids and Structures

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a b s t r a c tThis paper deals with optimal shapes against buckling of an elastic nonlocal small-scale Pflüger beams with Eringen's model for constitutive bending curvature relationship. By use of the Pontryagin's maximum principle the optimality condition in form of a depressed quartic equation is obtained. The shape of the lightest (having the smallest volume) simply supported beam that will support given uniformly distributed follower type of load and axial compressive force of constant intensity without buckling, is determined numerically. A special attention is paid to the influence of the characteristic small length scale parameter of the nonlocal constitutive law to both critical load and optimal shape of the analyzed beams. For the case when distributed follower type of load is zero, our results reduce to those obtained recently for compressed nonlocal beam. Also the post buckling shape of the optimally shaped rod is studied numerically.

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Section: Optimization Problem and Its Solutionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Stability and optimal shape of Pflüger micro/nano beam

Glavardanov

Spasic

Atanacković

2012

International Journal of Solids and Structures

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“…From ( 30), (31) and [16], p.4, we conclude that 𝑓 is contact equivalent to 𝑓 given below, that is, the terms of the order 𝑂(𝑎 3 Δ𝜆, 𝑎Δ𝜆 2 , 𝑎 5 ) may be neglected in the analysis of number of solution of (29). Therefore, we analyze only…”

Section: 𝔽(𝜆 𝜑) = 𝔹(𝜆)𝜑 + ℕ(𝜆 𝜑)mentioning

confidence: 92%

“…This theory represents one of two approaches to the mathematical rod theory and it identifies the rod with the axis of the rod. For details of derivation see [4] p. 45, or [5]. Assume that the rod axis in the undeformed state coincides with the x axis of the rectangular coordinate system

x-y

.…”

Section: Introductionmentioning

confidence: 99%

On the generalized Clausen problem

2022

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We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load 𝐹 of the column and the length 𝐿, find crosssectional area 𝐴, such that the volume 𝑊 of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.

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“…In order to solve more complex problems of plane elastica i.e. problems with different constitutive axioms and different load, engineering communities are still interested in efficient methods for solving both linear and nonlinear differential equations, see [2][3][4][5][6][7][8]. Recently, due to an enormous and wide-spread availability of computational power one more efficient method was added to the list, see [9] where the Laplace transform and the method of successive approximations (LT&MSA for short), was used in finding the analytical approximative solutions describing Toda oscillators …”

Section: Introductionmentioning

confidence: 99%