Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review)

Maksimyuk, V. A.; Storozhuk, E. À.; Chernyshenko, I. S.

doi:10.1007/s10778-012-0544-8

Cited by 33 publications

(18 citation statements)

References 52 publications

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“…In paper, the variational-difference method (VDM) [2,4,[7][8][9][10][11][12][13] is used for analysis of the stress-strain state of the shell. The variational-difference method is based on the Lagrange principle, the principle of minimum total potential energy [14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Thin Walled Wavy Shell of Monge Type Surface with Parabola and Sinusoid Curves by Variational-Difference Method

Иванов

Rynkovskaya

2017

MATEC Web Conf.

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Abstract. The paper is about the stress-strain state of the thin shell in the form of Monge surface with parabola generatrix and sinusoid guide. Coordinate system of the Monge surface is a system of coordinate lines of the principal curvatures of the surface. The variational-difference method is used for analysis. Variational-difference method allows using the geometric characteristics of the middle surface of the shell, which is important in the calculation of shells of complex shape. In the finite element method, which is often used in the shells of complex shape analysis, the equation of the middle surface of the shell is used only for finite element mesh.

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Section: Introductionmentioning

confidence: 99%

Analysis of Thin Walled Wavy Shell of Monge Type Surface with Parabola and Sinusoid Curves by Variational-Difference Method

Иванов

Rynkovskaya

2017

MATEC Web Conf.

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“…It is established that the stresses in the shell increase substantially with decrease in the inclusion radius and the distance between the two inclusions Keywords: shallow isotropic spherical shell, stress state, circular rigid inclusion, bridge between inclusions Introduction. The analysis of the stress state of shells and plates with various stress concentrators such as holes and inclusions [2][3][4][5][6][7][8][9][10][11][12], concentrated [1] and local [3] loads is still of theoretic and practical interest.Analytic and numerical solutions for a spherical shell loaded by a force or a moment through a rigid ring were obtained in [3] for a shell with one perfectly rigid inclusion. However, studies of shells and plates with two circular holes or rigid inclusions showed that the stress concentration can be very high if the stress concentrators are close to each other [2,[4][5][6][7][8][9][10][11][12].…”

mentioning

confidence: 99%

“…The analysis of the stress state of shells and plates with various stress concentrators such as holes and inclusions [2][3][4][5][6][7][8][9][10][11][12], concentrated [1] and local [3] loads is still of theoretic and practical interest.…”

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confidence: 99%

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confidence: 99%

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Stresses in a Spherical Shell Loaded Through Rigid Inclusions

Шевченко

Zakora

2015

Int Appl Mech

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The stress state of a shallow isotropic spherical shell with circular rigid inclusions subject to a force or a moment is determined. The case of two inclusions of unequal radii is analyzed numerically. It is established that the stresses in the shell increase substantially with decrease in the inclusion radius and the distance between the two inclusions Keywords: shallow isotropic spherical shell, stress state, circular rigid inclusion, bridge between inclusions Introduction. The analysis of the stress state of shells and plates with various stress concentrators such as holes and inclusions [2][3][4][5][6][7][8][9][10][11][12], concentrated [1] and local [3] loads is still of theoretic and practical interest.Analytic and numerical solutions for a spherical shell loaded by a force or a moment through a rigid ring were obtained in [3] for a shell with one perfectly rigid inclusion. However, studies of shells and plates with two circular holes or rigid inclusions showed that the stress concentration can be very high if the stress concentrators are close to each other [2,[4][5][6][7][8][9][10][11][12]. In view of this, we will consider a spherical shell with two circular perfectly rigid inclusions loaded by a force or a moment and will analyze in detail the stress state of a shell with two unequal rigid rings, including the case where they are very close to each other.1. Problem Formulation. Consider a shallow isotropic spherical shell with m circular perfectly rigid inclusions with centers located on the Ox-axis. The following boundary conditions are defined on the rigid boundaries G q of the inclusions [2]:

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“…The lower frequencies of square plates of different thickness made of materials with different types of anisotropy were analyzed in [3]. Various combinations of boundary conditions on the lateral faces, except for the cantilever case, were considered.To solve the relevant two-and three-dimensional eigenvalue problems, use is often made of variational difference and finite-element methods as well as discrete-continuous and Ritz methods [3,12,14,[17][18][19]21]. The frequencies of an isotropic parallelepiped obtained by the Ritz method using various basis functions (algebraic and Chebyshev polynomials, trigonometric functions, etc.)…”

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confidence: 99%

Three-Dimensional Analysis of the Lower Frequencies of a Cantilevered Anisotropic Parallelepiped

Bespalova

Urusova

2014

Int Appl Mech

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An approach to determining the lower frequencies of a cantilevered elastic parallelepiped using a three-dimensional problem statement and allowing for material anisotropy is developed. The approach combines the inverse-iteration and extended Kantorovich-Vlasov methods and is validated against the combination of the finite-element and Ritz methods. The influence of the anisotropy of the material on the lower frequencies of the parallelepiped is analyzed. It is shown that the variation in the frequencies of the parallelepiped with the boundary conditions being considered follows the variation in the predominant stiffness characteristics Keywords: elastic parallelepiped, cantilever, different types of anisotropy, three-dimensional problem statement, natural frequencies, extended Kantorovich-Vlasov methodIntroduction. Cantilevered elastic members of various shape and structure are widely used in various fields of engineering and construction. They are used to model wings of aircraft and blades of gas-turbine engines in mechanical engineering; elements of floors of buildings, balcony girders, and multistorey structures in construction; elements of scanning electron microscopes in nanoelectromechanical systems. No wonder that cantilever restraint is deemed to be a technological masterpiece. An analysis of the vibrations of such members is an important initial stage in their dynamic design, which may prevent emergencies under real service conditions. By now, the dynamic characteristics of cantilevered elastic members have been most extensively studied for beams and thin plates either of simple structure or with some complicating factors. For example, the influence of lamination of beams on their vibrations was studied in [9], the dependence of the lower frequencies on the elastic properties of composites was analyzed in [20], and discrete inclusions such as lumped masses were allowed for in [8]. We will use beam models, the classical Kirchhoff-Love model, and shear models of the first and higher orders.The vibrations of cantilevered bodies have been studied, using a three-dimensional problem statement, mainly for thick plates of various shapes such as an isosceles triangle, a rectangular parallelepiped, a prism with skew edge opposite to fixed one, etc. [18,19,21,22]. The three-dimensional problem statement in elasticity was compared in [18] with different two-dimensional problem statements for plates depending on their geometrical parameters. Particular attention was given to the boundary conditions on the lateral faces, and the frequency analysis was usually restricted to the case of an isotropic material [19]. The free vibrations of thick anisotropic plates were studied in few publications. The lower frequencies of square plates of different thickness made of materials with different types of anisotropy were analyzed in [3]. Various combinations of boundary conditions on the lateral faces, except for the cantilever case, were considered.To solve the relevant two-and three-dimensional eigenvalue problems, use is often...

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Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review)

Cited by 33 publications

References 52 publications

Analysis of Thin Walled Wavy Shell of Monge Type Surface with Parabola and Sinusoid Curves by Variational-Difference Method

Analysis of Thin Walled Wavy Shell of Monge Type Surface with Parabola and Sinusoid Curves by Variational-Difference Method

Stresses in a Spherical Shell Loaded Through Rigid Inclusions

Three-Dimensional Analysis of the Lower Frequencies of a Cantilevered Anisotropic Parallelepiped

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