Investigation of the stress‐strain state of the laminated shallow shells by R‐functions method combined with spline‐approximation

Awrejcewicz, Jan; Kurpa, L. V.; Osetrov, Andrey

doi:10.1002/zamm.201000164

Cited by 20 publications

(19 citation statements)

References 11 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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show abstract

“…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%

R-function theory method for bending problem of slip clamped trapezoidal shallow spherical shell

Li¹

2017

Proceedings of the 2017 6th International Conference on Energy and Environmental Protection (ICEEP 2017)

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Abstract. In this paper, the R-function theory(RFT) is applied to solve the bending problem of slip clamped trapezoidal 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.

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“…We set a = b = 1, Poisson’s ratio v = 0.3, thickness t = 0.1, Elastic modulus E = 4 × 10 6 , the coefficient of elastic foundation k = 4 × 10 4 , k = 5 × 10 4 , and k = 6 × 10 4 , respectively, Shear modulus of foundation G P = 1 × 10 4 , and uniform load subjected Z = 1000. According to R-function theory, 33–35 the following is obtained…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Gan

Yuan

et al. 2014

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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Investigation of the stress‐strain state of the laminated shallow shells by R‐functions method combined with spline‐approximation

Cited by 20 publications

References 11 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

R-function theory method for bending problem of slip clamped trapezoidal shallow spherical shell

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

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