Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory

Kurpa, L. V.; Shmatko, Tetyana; Awrejcewicz, Jan

doi:10.1590/1679-78254911

Cited by 12 publications

(16 citation statements)

References 42 publications

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“…The numerical results of the QGFM demonstrate its feasibility, efficiency and rationality by comparing with the FEM solution. The R-function theory can also be applied to effectively solve various boundary value problems of the plates and shells by constructing a trial function that satisfies the complicated boundary shape and by combining with the other method of weighted residuals such as the variational method [22], the spline-approximation [23] and the Ritz method [24].…”

Section: Resultsmentioning

confidence: 99%

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

Yuan

Yang

et al. 2020

Civ Eng J

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The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green's function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green's function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green's formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

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Section: Resultsmentioning

confidence: 99%

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

Yuan

Yang

et al. 2020

Civ Eng J

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“…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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“…Let w = 0 be the first-order normalized equation of boundary Γ . Satisfy the following expression 33–35…”

Section: Solution Of Two-dimensional Helmholtz Operatormentioning

confidence: 99%

“…, and k = 6 3 10 4 , respectively, Shear modulus of foundation G P = 1 3 10 4 , and uniform load subjected Z = 1000. According to R-function theory, [33][34][35] the following is obtained…”

Section: Boundary Constraintsmentioning

confidence: 99%

A computing method for bending problem of thin plate on Pasternak foundation

Gan

Yuan

et al. 2020

Advances in Mechanical Engineering

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The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

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Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory

Cited by 12 publications

References 42 publications

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

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

A computing method for bending problem of thin plate on Pasternak foundation

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