Mathematical models and algorithms for studying strength and stability of shell structures

Karpov, Vladimir; Semenov, Alexey

doi:10.1134/s1990478917010082

Cited by 13 publications

(3 citation statements)

References 8 publications

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“…Several mathematical models of deformation of reinforced shell structures, including those that account for various properties of the material are described in [8]. For the structures composed of orthotropic and isotropic materials, linearly elastic and physically nonlinear problems, as well as the problems of creep, are considered.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

et al. 2019

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The problem of parametric oscillations of an isotropic viscoelastic shallow shell of variable thickness under periodic load is considered. It is believed that under the influence of specified load, the shallow shell allows displacements (in particular, deflections), commensurate with its thickness. In a geometrically nonlinear statement, taking into account the viscoelastic properties of material, a mathematical model of the problem has been developed using the classical Kirchhoff-Love hypothesis. Using the Bubnov-Galerkin method based on the polynomial approximation of the deflections, the problem is reduced to the study of the system of integro-differential equations, where time is the independent variable. The solution of the system of integrodifferential equations is determined by the proposed numerical method. Based on this method, a numerical solution algorithm is described. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. At the same time, the effect of geometric nonlinearity, viscoelastic properties of material, as well as other physicomechanical and geometric parameters and factors (rheological parameters, thickness, initial shape imperfections, aspect ratios, boundary conditions, excitation coefficient) on the area of dynamic instability is taken into account. The results obtained in this study are in good agreement with the results and data obtained by other authors.

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Section: Introductionmentioning

confidence: 99%

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

et al. 2019

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“…If we account for physical non-linearity, then the functional of full potential deformation energy of the shell can be written in the following form [18]:…”

Section: Mathematical Model Of Shell Structure Deformationmentioning

confidence: 99%

Nonlinear Deformations of Stiffened Reinforced Concrete Shells

Panin

Semenov

2019

KEM

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The paper discusses the process of non-linear deformation of shell structures made of reinforced concrete. A mathematical model of deformation in the form of the functional of full potential deformation energy is provided. The model is based on the Kirchhoff–Love hypotheses, and allows accounting for structure reinforcement with stiffeners. An orthogonal network of stiffeners, located from the concave side, is considered as the structure support. Type of load — external, uniformly distributed. The Ritz method is applied to the functional to reduce the variational problem of the functional minimum to a system of nonlinear algebraic equations. Then, for each load value, the problem is solved using iterative methods. Analysis of strength and stability of shallow shells of double curvature and rectangular planform is performed. Values of critical loads, deflection and stress fields are obtained. Curves of deflection depending on load are provided. All results are given in dimensionless parameters. The Mohr–Coulomb criterion was used to analyze concrete strength, and the Lyapunov criterion was used for stability analysis. Influence of the number of stiffeners reinforcing the shell on the resulting stress values is shown. It has been revealed that with account for physical non-linearity of concrete, when the dependence of stresses and deformations is curvilinear, deformations (and deflections as well) of shells increase in comparison with the linear-elastic solution. It has been also found that when nonlinearity is taken into account, redistribution of stresses over the shell field occurs (the maximum stresses shift towards the shell contour).

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“…Also, shells are generally exposed to a wide range of working areas. This necessitates investigating the static deformation, dynamic behavior and stability conditions of shells [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Large elastic deformation of micromorphic shells. Part I: Variational formulation

Norouzzadeh

Ansari

Darvizeh

2019

Mathematics and Mechanics of Solids

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We aimed to study the static deformation of geometrically nonlinear shell-type structures on the basis of micromorphic theory. Employing the most comprehensive model in the micro-continuum field, shells in low-dimensions and made of inhomogeneous materials are precisely investigated. The seven-parameter two-dimensional (2D) kinematic model is used which satisfies three-dimensional (3D) constitutive relations and represents the macro-deformation components in mid-surface area of the shell. Also, in the framework of micromorphic continua with three deformable director vectors, nine micro-deformation degrees of freedom, including micro-scale rotations, shears and stretches, are taken into account. Utilizing the energy approach in the convected curvilinear coordinate system leads to the general derivation of the variational formulations in Lagrangian description. High-order stress–strain relations are obtained via introducing the size-dependent as well as size-independent elasticity tensors for the isotropic micromorphic solid. Finally, an equivalent matrix–vector form of representation is proposed to facilitate the solution procedure of the extracted tensor-based formulation. Determining the kinetic and kinematic fields in terms of 16 macro and micro-deformation components, provides the opportunity to directly implement the interpolation-based solution methodologies, such as the finite element isogeometric analysis presented in Part II of this study. Two parts of the article, that are organized to be independent, contribute to the literature respectively from theoretical and computational perspectives.

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Mathematical models and algorithms for studying strength and stability of shell structures

Cited by 13 publications

References 8 publications

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

Nonlinear Deformations of Stiffened Reinforced Concrete Shells

Large elastic deformation of micromorphic shells. Part I: Variational formulation

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