Free vibration analysis of laminated shallow shells with complex shape using the R-functions method

Kurpa, Lidiya; Shmatko, Tetyana; Timchenko, Galina

doi:10.1016/j.compstruct.2010.05.016

Cited by 33 publications

(32 citation statements)

References 19 publications

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“…Compared with ANSYS finite element solution, it shows good agreement. R-function theory can also be used to effectively solve various boundary value problems in engineering by constructing a trial function that satisfies the boundary conditions and by combining with the method of weighted residuals such as the variational method and the spline-approximation [7][8].…”

Section: Discussionmentioning

confidence: 99%

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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: Discussionmentioning

confidence: 99%

“…The governing differential equations of the bending problem of slip clamped shallow spherical shell [7] can be expressed as follow…”

Section: Fundamental Equationsmentioning

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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“…The obtained results of the DQM were compared with the results obtained using the finite element method. Kurpa et al [6] studied vibration of composite laminated shallow shells resting on an arbitrary planform using the FSDT. The proposed method involves the use of the R-functions theory and variation methods.…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

Irani-Rahaghi

Karroubi

Arefi

2016

Appl. Math. Mech.-Engl. Ed.

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An analytical method for the three-dimensional vibration analysis of a functionally graded cylindrical shell integrated by two thin functionally graded piezoelectric (FGP) layers is presented. The first-order shear deformation theory is used to model the electromechanical system. Nonlinear equations of motion are derived by considering the von Karman nonlinear strain-displacement relations using Hamilton's principle. The piezoelectric layers on the inner and outer surfaces of the core can be considered as a sensor and an actuator for controlling characteristic vibration of the system. The equations of motion are derived as partial differential equations and then discretized by the Navier method. Numerical simulation is performed to investigate the effect of different parameters of material and geometry on characteristic vibration of the cylinder. The results of this study show that the natural frequency of the system decreases by increasing the non-homogeneous index of FGP layers and decreases by increasing the non-homogeneous index of the functionally graded core. Furthermore, it is concluded that by increasing the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increase considerably.

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“…The governing differential equations of the free problem of slip clamped trapezoidal shallow spherical shell on Winkler foundation [7] can be expressed as follow …”

Section: Fundamental Equationsmentioning

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.

show abstract

Free vibration analysis of laminated shallow shells with complex shape using the R-functions method

Cited by 33 publications

References 19 publications

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

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

Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer

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

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