Green quasifunction method for free vibration of clamped thin plates

Li, Shanqing; Yuan, Hong

doi:10.1016/s0894-9166(12)60004-4

Cited by 15 publications

(22 citation statements)

References 30 publications

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“…Thus, the problem can always be reduced to two simultaneous Fredholm integral equations of the second kind without the singularity. Using the R-function theory, Li and Yuan described successfully the rectangular, trapezoidal, triangular and parallelogrammic domains of plates [5][6][7] and shallow spherical shells [8,9]. For the first time, the Rfunction theory is applied to describe the dodecagon domain of the shallow spherical shells with concave boundary.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

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

Yuan

Yang

et al. 2020

Civ Eng J

Self Cite

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“…Thus, the problem can always be reduced to the Fredholm integral equation of the second kind without singularity. Using the present method, Li and Yuan solved successfully the free vibration of clamped thin plates [2], the simply-supported thin plate [3,4] and shallow spherical shells [5,6]. For the first time, the proposed R-function theory method is applied to study the free vibration problem of slip clamped trapezoidal shallow spherical shell.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for bending problem of slip clamped shallow spherical shell

Li¹

2017

Proceedings of the 2017 6th International Conference on Measurement, Instrumentation and Automation (ICMIA 2017)

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Abstract. In this paper, the R-function theory(RFT) is applied to solve the f bending problem of slip clamped shallow spherical shell. Firstly the fundamental solution of the biharmonic operator, the boundary equation and the R-function are used to construct the quasi-Green's function. Then the model governing differential equation of the problem is reduced to the Fredholm integral equation of the second kind by Green's formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation by the R-function. A numerical example shows that this method is an effective numerical method. IntroductionAs a kind of structural forms, a shell is widely used in various fields, such as, in the large-span roof, the underground foundation engineering, the hydraulic engineering, the large container manufacturing, the aviation, the shipbuilding, the missiles, the space technology, the chemical industry, and so on.In the analysis and calculation of various physical and mechanical problems in engineering, the governing differential equation describing their physical state and process needs firstly to be built. Only few problems with a regular geometric boundary and a simple differential equation can be solved with an analytical or a half analytical method. For most complex engineering problems, it is difficult to find an analytical solution so that an approximate method is used to analyze and calculate the problems. In many calculation problems of engineering, although geometry of arbitrary shapes, complex boundary conditions, various properties and inhomogeneous of materials, and so on, but a numerical solution can be obtained directly by using a numerical method from a mathematical model. The main numerical methods are the boundary element method, the finite element method, the finite difference method and the coupling method.In the paper, the R-function theory proposed by Rvachev[1] are utilized. The bending problem of slip clamped shallow spherical shell is studied. A quasi-Green function is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem, but it does not satisfy the fundamental differential equation. The key point of establishing the quasi-Green function consists in describing the boundary of the problem by normalized equation 0 = ω and the domain of the problem by inequality 0 > ω . There are multiple choices for the normalized boundary equation. Based on a suitably chosen form of the normalized boundary equation, a new normalized boundary equation can be established such that the singularity of the kernel of the integral equation is overcome. For any complicated area, a normalized boundary equation can always be found according to the R-function theory. Thus, the problem can always be reduced to the Fredholm integral equation of the second kind without singularity. Using the present method, Li and Yuan solved successfully the free vibration o...

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Section: Introductionmentioning

confidence: 99%

R-function theory method for free vibration of slip clamped trapezoidal shallow spherical shell on Winkler foundation

Li¹

2017

Proceedings of the 2017 2nd International Conference on Civil, Transportation and Environmental Engineering (ICCTE 2017)

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Abstract. In this paper, the R-function theory(RFT) is applied to solve the free vibration of slip clamped trapezoidal shallow spherical shell on Winkler foundation. Firstly the fundamental solution of the biharmonic operator, the boundary equation and the R-function are used to construct the quasi-Green's function. Then the model governing differential equation of the problem is reduced to the Fredholm integral equation of the second kind by Green's formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation by the R-function. A numerical example shows that this method is an effective numerical method.

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Green quasifunction method for free vibration of clamped thin plates

Cited by 15 publications

References 30 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

A numerical method for bending problem of slip clamped shallow spherical shell

R-function theory method for free vibration of slip clamped trapezoidal shallow spherical shell on Winkler foundation

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