A simple approach to bifurcation and limit point calculations

Fujikake, Masahisa

doi:10.1002/nme.1620210115

Cited by 19 publications

(4 citation statements)

References 4 publications

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“…Within ABAQUS TM the incremental formulation of the finite element approach to non-linear analysis (as shown in Fig. 3) is (Bathe, 1982;Fujikake, 1995) …”

Section: Cool-down and Equilibrium Shapesmentioning

confidence: 99%

Analysis of thermally induced multistable composites

Mattioni

Weaver

Potter

et al. 2008

International Journal of Solids and Structures

119

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This paper models the non-linear flexural response of laminates that have piecewise variation of lay-up in the planform, using finite element analysis. Attention is focused on the effects that thermal stresses have on the potential multiple shapes of a composite structure. Unsymmetric laminates may possess more than a single equilibrium configuration, and during the cool-down the solution thus bifurcates at a critical temperature. In static analyses, numerical solutions are often coaxed to converge into one or the other branch of the solution. A methodology to overcome this problem is presented. Such modelling is necessary to allow application of multistable composite within morphing aircraft structures as multistable composites could provide a viable solution for the realisation of shape-adaptable structures.

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“…Within ABAQUS TM the incremental formulation of the finite element approach to non-linear analysis (as shown in Fig. 3) is (Bathe, 1982;Fujikake, 1995) …”

Section: Cool-down and Equilibrium Shapesmentioning

confidence: 99%

Analysis of thermally induced multistable composites

Mattioni

Weaver

Potter

et al. 2008

International Journal of Solids and Structures

119

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“…To this line of approach to the solution of the extension problem belong the papers of Ya. M. Grigorenko and A. P. Mukoed [8], V. I. Gulyaev, V. A. Bazhenov and E. A. Gotsulyak [9], M. Fujikake [14], G. Hunt and K. Williams [15], W. Koiter [16], I. Kubor [17], P. Samuels [18], J. Thompson and G. Hunt [19,20], and G. Thurston [21]. The basis for practically all of the approaches used in these papers is a method that consists in the final analysis of a power-series representation of the solution of systems of nonlinear equations in the neighborhood of the bifurcation point under consideration.…”

Section: The Methods Is Based On the Fact That The Eigenvectors Of Thementioning

confidence: 99%

“…The basis for practically all of the approaches used in these papers is a method that consists in the final analysis of a power-series representation of the solution of systems of nonlinear equations in the neighborhood of the bifurcation point under consideration. Exceptions are the papers [14], where the eigenvalue problem formed using tangent stiffness matrices is used to study the befiavior of the solution, and [17] and [20], where methods of catastrophe theory are used.…”

Section: The Methods Is Based On the Fact That The Eigenvectors Of Thementioning

confidence: 99%

Extension of the solution of nonlinear equations in the neighborhood of a bifurcation point

Grigolyuk¹,

Lopanitsyn²

1999

J Math Sci

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Introduction. There exists a rather large class of different mathematical models for describing the nonlinear behavior of elastic thin-walled structures. Among these are, first of all, the equations of the theory of rods, plates, and shells. Under static loads Lagrange's principle makes it possible to obtain the equations of equilibrium for such structures and the corresponding boundary conditions. Here the equations of equilibrium are usually nonlinear partial differential equations.Independently of the methods of solving the resulting boundary-value problems, such as the methods of shooting and orthogonal sweep, Rayleigh-Ritz and Bubnov, collocations and least squares, finite differences, finite and boundary elements, and so forth, the problem of determining the stress-strain state of the structure, in the final analysis, reduces to the need to solve systems of nonlinear algebraic equations. Their solutions for various values of the load are represented by points in a space of generalized variables. Here the geometric locus of these points is a continuous curve called the load trajectory (see the figure). Due to the nonlinear behavior of the structures the load trajectories are complicated curves, sometimes loop-shaped and consisting of several curves that may intersect. Two types of singular points may be present on them. The first type consists of limit points, at which stability is lost due to a jump in the deflection: cracking (see the points A, B, C, D, E, and F in the figure). The second type consists of bifurcation points: points I and H in the figure. They are points of intersection of the principal branch of the trajectory and its bifurcation branches. At these points there is Eulerian loss of stability. At all singular points the Jacobian of the system is degenerate, and the application of the traditional methods based on different recursive schemes is impossible.Nowadays the occurrence of limit points on a load trajectory is no hindrance to the application of the well-thought-out and algorithmized methods of extension [2-7, 10, 13], which implement the idea of equivalence of arguments. However, when there are bifurcation points on the load trajectory and it is necessary to pass through them, there methods are unusable, and it becomes necessary to apply methods that differ from those just mentioned, methods that divide into two basic groups.

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“…But, at bifurcation points (or branch points), two equilibrium path branches emanate. Limit and simple bifurcation points are broadly discussed in literatures (Fujikake, 1985;Fujii and Choong, 1992;Onate and Matias, 1996;Planinc and Saje, 1999;Vejdani-Noghreiyan, 2004;Wriggers and Simo, 1990). Among critical points, multiple bifurcation points are very difficult to be found.…”

Section: Introductionmentioning

confidence: 99%

Computation of multiple bifurcation point

Rezaiee‐Pajand

Vejdani-Noghreiyan

2006

Engineering Computations

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PurposeThe aim of this paper is to develop a new method for finding multiple bifurcation points in structures.Design/methodology/approachA brief review of nonlinear analysis is presented. A powerful method (called arc‐length) for tracing nonlinear equilibrium path is described. Techniques for monitoring critical points are discussed to find the rank deficiency of the stiffness matrix. Finally, by using eigenvalue perturbation of tangent stiffness matrix, load parameter associated with multiple bifurcation points is obtained.FindingsSince other methods of finding simple bifurcation points diverge in the neighborhood of critical points, this paper introduces a new method to find multiple bifurcation points. It should be remembered that a simple bifurcation point is a multiple bifurcation point with rank deficiency equal to one. Therefore, the method is applicable to simple critical points as well.Practical implicationsGlobal buckling of the structures should be considered in design. Many structures (specially symmetric space structures) have multiple bifurcation points, therefore, analyst and designer should be aware of these points and should control them (for example, by changing the geometry or other related factors) for obtaining a safe and optimum design.Originality/valueIn this paper a robust method to find multiple bifurcation points is introduced. By using this method, engineers can be aware of critical load of multiple bifurcation points to control global buckling of related structures.

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A simple approach to bifurcation and limit point calculations

Cited by 19 publications

References 4 publications

Analysis of thermally induced multistable composites

Analysis of thermally induced multistable composites

Extension of the solution of nonlinear equations in the neighborhood of a bifurcation point

Computation of multiple bifurcation point

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