Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

Gorder, Robert A. Van

doi:10.1016/j.ijnonlinmec.2017.02.004

Cited by 10 publications

(2 citation statements)

References 54 publications

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“…Combining the series method and perturbation technique, Eipakchi and Shariati calculated elastic buckling stresses of a cylindrical panel under axial stress, in which the stress was selected as perturbation parameter. Van Gorder obtained asymptotic solutions for the Föppl‐von Kármán equations of thin axisymmetric annular plates under different boundary conditions, where the perturbation solutions agreed well with numerical solutions, even for relatively large values of the perturbation parameter. More recently, Fallah et al .…”

Section: Introductionmentioning

confidence: 67%

Application of biparametric perturbation method to functionally graded thin plates with different moduli in tension and compression

Yang

et al. 2019

Z Angew Math Mech

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In this study, a biparametric perturbation method is used for the solution of the large‐deflection bending problem of a functionally graded thin plate with different moduli in tension and compression. First, the Föppl‐von Kármán equations for the bimodular functionally graded thin plate are established in rectangular coordinates system, thus obtaining the axisymmetric simplified form in polar coordinates system. By adopting two groups of perturbation parameters, one group is gradient index and central deflection, another is gradient index and load, the biparametric perturbation solution for the established governing equations are obtained, respectively, under different boundary constrains. The result indicates that the two groups of parameter selections are valid, and the biparametric perturbation solution is also consistent to single‐parameter perturbation solution. The bimodular effects on stiffness and deformation of thin plates are also discussed via biparametric perturbation solutions obtained. The study indicates that the dominant factor influencing the stiffness is the relative magnitudes relation among the tensile modulus, the neutral layer modulus and the compressive modulus.

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Section: Introductionmentioning

confidence: 67%

Application of biparametric perturbation method to functionally graded thin plates with different moduli in tension and compression

Yang

et al. 2019

Z Angew Math Mech

View full text Add to dashboard Cite

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“…As indicated above, another issue with which we have to encounter is the corresponding solving method for thin plates in the case of large rotation angle. It is known that the classical large-deflection bending problem is often too challenging for analytical methods because the Föppl-von Kármán equations consist of two sets of high-order partial differential equations along with two kinds of deformation [27,28]. Generally, these analytical methods include the series expansion method in the form of various functions, the variation method based on energy principles, and the perturbation method of parameters.…”

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confidence: 99%

Nonlinear large deformation problem of rectangular thin plates and its perturbation solution under cylindrical bending: Transform from plate/membrane to beam/cable

et al. 2022

Z Angew Math Mech

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The Föppl‐von Kármán equations describing deformation of flexible thin plates are established on the basis of moderately large‐deflection and small rotation angle. For many years, the equations have been regarded as a classical and effective model used for the analysis of flexible thin plate problems, and at the same time, this model has also been constantly evolving and improving. The improvements for the model, however, come mainly from properties of materials perspective, but seldom from the deformation of plate perspective. In this study, we revisit Föppl‐von Kármán equations from the viewpoint of deformation concerning rotation angle. A new form for the classical equations without a small‐rotation‐angle assumption is derived, for the first time, by giving up the basic assumption that the sine function of the rotation angle equals to the first‐order derivative of the corresponding displacement, thus improving the governing equations while enhancing the nonlinearity. The abandonment of the small‐rotation‐angle assumption reveals such a fact that the second‐order derivative of displacement in classical equations originates from the curvature and twist of plates. Due to the complexity of the equations derived, its perturbation solution is obtained under cylindrical bending with two opposites fully fixed. Results indicate that via cylindrical bending, a two‐dimensional plate or membrane problem is easily associated with a one‐dimensional beam or cable problem. Results also show that the abandonment of the small‐rotation‐angle assumption will contribute to the free development of the deflection curve rotation of the plate, while at the same time, the deflection value will tend to decrease to agree with the deformation of the plate.

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Nonlinear vibration analysis of functionally graded flow pipelines under generalized boundary conditions based on homotopy analysis

Zhou

Chang

2022

Acta Mech

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Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

Abstract: Cite this article as: Robert A. Van Gorder, Asymptotic solutions for the Föpplvon Kármán equations governing deflections of thin axisymmetric annular plates,

Cited by 10 publications

References 54 publications

Application of biparametric perturbation method to functionally graded thin plates with different moduli in tension and compression

Application of biparametric perturbation method to functionally graded thin plates with different moduli in tension and compression

Nonlinear large deformation problem of rectangular thin plates and its perturbation solution under cylindrical bending: Transform from plate/membrane to beam/cable

Nonlinear vibration analysis of functionally graded flow pipelines under generalized boundary conditions based on homotopy analysis

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