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Research Papers

Application of Continuation Methods to Uniaxially Loaded Postbuckled Plates

[+] Author and Article Information
Theodore C. Lyman

e-mail: theodore.lyman@duke.edu

Lawrence N. Virgin

Professor
e-mail: l.virgin@duke.edu

Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708

R. Benjamin Davis

Aerospace Engineer
Structural Dynamics and Analysis
Branch (ER41),
NASA Marshall Space Flight Center,
Huntsville, AL 35812
e-mail: robert.b.davis@nasa.gov

Manuscript received October 22, 2012; final manuscript received May 6, 2013; accepted manuscript posted May 29, 2013; published online September 19, 2013. Assoc. Editor: Anthony Waas.

J. Appl. Mech 81(3), 031010 (Sep 19, 2013) (10 pages) Paper No: JAM-12-1486; doi: 10.1115/1.4024672 History: Received October 22, 2012; Revised May 06, 2013; Accepted May 29, 2013

Continuation methods are used to examine the static and dynamic postbuckled behavior of a uniaxially loaded, simply supported plate. Continuation methods have been extensively used to study problems in mathematics and physics; however, they have not been as widely applied to problems in engineering. When paired with a Galerkin approximation, continuation methods are shown to be well suited to solving nonlinear buckling problems. In addition to providing a robust solution method for nonlinear equations, the linearized Jacobians from the continuation steps will contain natural frequency and mode shape information for mechanical systems (provided inertia terms are included). Results for the primary buckling branch are compared to previously published results. Using the open-source continuation package Auto, stable, remote secondary buckling branches were discovered. These secondary stable equilibrium persist even in the presence of geometric imperfections and their existence is confirmed by experiment.

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Figures

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Fig. 1

Simple parameterization in (x, λ) space

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Fig. 2

Schematic diagram of the uniaxially compressed square plate

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Fig. 3

Bifurcation diagram for the uniaxially compressed square plate with and without imperfections: solid, stable; dashed, unstable

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Fig. 4

Static deflection for the primary branch of the uniaxially loaded square plate: Px/Pcritical = 4.0

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Fig. 5

Variation of four lowest frequencies about the primary buckling branch for the uniaxially compressed square plate: solid, perfect; dashed, imperfect with a011 = h

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Fig. 6

Linearized vibration mode shapes for the uniaxially compressed square plate corresponding to the four lowest natural frequencies about the primary buckling branch

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Fig. 7

Bifurcation diagram for the uniaxially compressed square plate including secondary stability branch: solid, stable; dashed, unstable

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Fig. 8

Static deflection for the stable secondary buckling branch of the uniaxially loaded square plate: Px/Pcritical = 4.0, L2 Norm = 6.6834

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Fig. 9

Bifurcation diagram for the imperfect, uniaxially compressed square plate including secondary stability branch: solid, stable; dashed, unstable

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Fig. 10

Static deflection for the stable secondary buckling branch of the uniaxially loaded square plate: Px/Pcritical = 4.0, L2 Norm = 6.6834

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Fig. 11

Variation of the four lowest frequencies about the secondary buckling branch for the uniaxially compressed square plate: perfect, solid; imperfect, dashed

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Fig. 12

Linearized vibration mode shapes for the uniaxially compressed square plate corresponding to the four lowest natural frequencies about the secondary branch

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Fig. 13

Experimental plate setup

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Fig. 14

Experimental measurement points

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Fig. 15

Experimental deflection with Auto stable solution: solid lines, Auto; markers, experimental

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Fig. 16

Centerline profiles (y = 0.5) for four different deflected states at Px/Pcritical = 4.35: solid, Auto; dashed and °, experimental

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