Nonlinear vibration analysis of laminated shallow shells with clamped cutouts by the R-functions method

Kurpa, Lidiya; Timchenko, Galina; Osetrov, Andrey; Shmatko, Tetyana

doi:10.1007/s11071-017-3930-2

Cited by 9 publications

(5 citation statements)

References 23 publications

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“…where � i , i ¼ 1, 5 are undetermined components of the solution structure. 23,[30][31][32][33][34][35][36][37] They are expanded in a power series. In formula (27), !…”

Section: Investigation Of Laminated Fgm Shallow Shells With Complex Planformmentioning

confidence: 99%

“…[26][27][28] The problems of free vibration of laminated shallow shells with complex planform by RFM were considered in Kurpa et al 28 The algorithms of solving geometrically nonlinear vibrations of plates and shallow shells were proposed in various studies. [29][30][31][32][33][34] In the papers, [35][36][37] the method of the R-functions was extended to the problems of vibration of functionally graded shallow shells of arbitrary planforms.…”

Section: Introductionmentioning

confidence: 99%

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Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Kurpa

Shmatko

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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The purpose of the paper is to study stability and free vibrations of laminated plates and shallow shells composed of functionally graded materials. The approach proposed incorporates the Ritz method and the R-functions theory. It is assumed that the shell consists of three layers and is loaded in the middle plane. The both cases of uniform as well as non-uniform load are possible. The power-law distribution in terms of volume fractions is applied to get effective material properties for the layers. These properties are calculated for different arrangements and thicknesses of the layers by the analytical formulae obtained in the paper. The mathematical formulation is carried out in framework of the first-order shear deformation theory. The proposed approach consists of two steps. The first step is to define the pre-buckling state by solving the respective elasticity problem. The critical buckling load and frequencies of functionally graded material shallow shells are determined in the second step. The highlight of the method proposed is that it can be used for vibration and buckling analysis of plates and shallow shells of complex shape. The numerical results for frequencies and buckling load of plates and shallow shells of complex shape and different curvatures are presented to demonstrate the potential of the method developed. Different functionally graded material plates and shallow shells composed of a mixture of metal and ceramics are studied. The effects of the power law index, boundary conditions, thickness of the core, and face sheet layers on the fundamental frequencies and critical loads are discussed in this paper. The main advantage of the method is that it provides an analytical representation of the unknown solution, which is important when solving nonlinear problems.

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“…where � i , i ¼ 1, 5 are undetermined components of the solution structure. 23,[30][31][32][33][34][35][36][37] They are expanded in a power series. In formula (27), !…”

Section: Investigation Of Laminated Fgm Shallow Shells With Complex Planformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Kurpa

Shmatko

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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show abstract

“…They have to satisfy at least the geometrical boundary conditions as well. Construction of these functions is carried out by the R‐functions theory [24–30, 32] in case of complicated planform of the shell or complex boundary condition.…”

Section: Solution Methods Of the Vibration Problemmentioning

confidence: 99%

“…Therefore, combined application of the variational Ritz's method and the R‐functions theory allows to find an approximate solution in an analytical form. This approach was employed earlier for investigation of vibration of plates and shallow shells including laminated and FGM objects [14, 27–30]. Yet, in previous works there were considered temperature independent characteristics of the materials.…”

Section: Introductionmentioning

confidence: 99%

Application of the R‐functions in free vibration analysis of FGM plates and shallow shells with temperature dependent properties

2020

Self Cite

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The free vibration of plates and shallow shells with/without cutouts made of functionally graded materials (FGM) is investigated using variational FG shallow shells with temperature dependent mechanical characteristics of the constituent materials. First‐order shear deformation theory of shallow shells is employed. It is supposed that material properties vary through thickness according to a power‐law distribution of the constituent's volume fraction. They depend on both the temperature and the thickness. Temperature field is modeled by one‐dimensional heat transfer equation, since the temperature is varied only in thickness direction. Solution of this equation is determined by a polynomial power series expansion. Corresponding software was developed to implement the proposed approach. The vibration analysis was carried out for FG plates and shallow shells with cutout and various boundary conditions.

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“…R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells was studied by Shmatko et al [6]. Nonlinear vibration analysis of laminated shallow shells with clamped cutouts was considered by Kurpa et al [7] using the R-functions method. Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening was researched by Ko et al [8].…”

Section: Introductionmentioning

confidence: 99%

An analytical method for shallow spherical shell free vibration on two-parameter foundation

Gan

Yuan

et al. 2021

Heliyon

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The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of Rfunction, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method.

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Nonlinear vibration analysis of laminated shallow shells with clamped cutouts by the R-functions method

Cited by 9 publications

References 23 publications

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Application of the R‐functions in free vibration analysis of FGM plates and shallow shells with temperature dependent properties

An analytical method for shallow spherical shell free vibration on two-parameter foundation

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