Physically and Geometrically Nonlinear Static Problems for Thin-Walled Multiply Connected Shells

Guz, A. N.; Storozhuk, E. À.; Chernyshenko, I. S.

doi:10.1023/a:1025793808397

Cited by 20 publications

(78 citation statements)

References 5 publications

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“…This system is obtained using the virtual-displacement principle [1,91] and iterative methods (modified Newton-Kantorovich method, method of additional stresses) and the finite-element method (FEM).…”

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

“…where dA W , 0 , dW r is the work done by the external and internal forces through virtual displacements, The linearization by these methods leads to the following functional [91]: We will use the FEM to solve two-dimensional linear elastic boundary-value problems at each approximation. To this end, we divide the mid-surface ( ) S into E two-dimensional finite elements (FEs) and the reinforcement axis G r into E r one-dimensional finite elements.…”

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

“…To this end, we divide the mid-surface ( ) S into E two-dimensional finite elements (FEs) and the reinforcement axis G r into E r one-dimensional finite elements. Within each FE, the components of the vector of increments of displacements of the mid-surface, { } DU , are approximated by polynomials of two variables [36]: Let us describe the finite elements for a shell and curvilinear rod developed in [91] to design shells of complex geometry with reinforced holes. This FE modification does not define the angles of rotation j 1 , j 2 in the expressions for flexural strains by formulas (2.1), as is the case in the classical FEM, but rather approximates them by biquadratic serendipity polynomials [36] with the Kirchhoff-Love hypotheses being valid only at the FE nodes.…”

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

“…At the nth step of loading, this system has the following matrix form: Comparing the linear and nonlinear solutions for shells of various shapes obtained using the developed method and software programs with the exact solutions or solutions obtained by other methods suggests that the method is quite efficient [91] and can be used to solve doubly nonlinear two-dimensional problems for thin shells with free or reinforced curvilinear holes.…”

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

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Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

2009

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Results on stress concentration in thin shells with curvilinear holes subject to plastic deformation and finite deflections are reviewed. The holes (circular, elliptical) are reinforced with thin-walled elements (rings, rods) of different stiffness. A numerical method of solving doubly nonlinear problems of statics for shells of complex geometry is outlined. The stress distribution near curvilinear holes in spherical, cylindrical, and conical shells under statical loading is studied. The numerical results are analyzed Introduction. Thin shells and plates are used as structural members in various fields of modern engineering. In most cases, they have complex geometry (geometry variations, holes, notches, and inclusions of various shapes) for the purposes of design or technology. High loads on structural members made of homogeneous isotropic materials (metals, alloys) may cause structural changes (plastic or creep strains) near stress concentrators and microcracks, which may lead to failure of structural members and then the whole structure. There are also large or finite displacements or strains in the zone of high stresses. One of the basic and most important tasks in the mechanics of shell structures is to analyze the distribution of stresses and strains in structural members of complex form. Therefore, in designing and manufacturing load-bearing structures or their elements with high strength and minimum weight, the need arises to take into account the real operation conditions of structural members and the real properties of structural materials (plastic strains) and their deformation (large or finite displacements).Results of theoretical and experimental analysis of the stress distribution in shells (plates) obtained by solving linear elastic boundary-value problems (linear Hooke's law, small displacements, strains) are discussed in a great many publications most fully generalized in the monographs [2,21,22,25,26,58,65]. They mainly discuss solutions of linear (elastic) problems for thin (spherical, cylindrical, conical, etc.) shells weakened by curvilinear holes of various shapes and made of advances metallic materials.The available theoretical and applied results on the stress distribution in thin and nonthin anisotropic (composite) shells made of materials obtained by solving linear elastic problems (generalized Hooke's law; Kirchhoff-Love or Timoshenko hypotheses) are presented in [27,50].To solve this class of problems in linear elastic formulation, analytic, variational, and numerical methods are used. Methods and results of solving specific classes of physically and geometrically nonlinear problems for some metal shells are discussed in the monographs [20,32,38,42,67,70]. The issue of stress concentration in shells (plates) involving the development of methods of solving nonlinear problems is analyzed in the reviews [24,89,93,94].Note that most studies on nonlinear stress concentration in shells, including problem formulation, development of methods for solving certain classes of nonlinear problems, a...

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Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

Section: Methods Of Solving Physically and Geometrically Nonlinear Promentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

2009

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“…Many of the technology problems associated with the deformation of a thin shell, and this explains the development of a geometrically and physically nonlinear theory of shells, with the development of methods of research of stress strain state of shell structures, operating beyond the limits of elasticity [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

Research of elasto-plastic bending of thin shells and plates by the R-fanctions method

Morachkovska¹,

Timchenko²,

Lyubitskaya³

2016

Bulletin NTU "KhPI": JDSM

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Розглядається ефективний метод розв'язання нелінійних крайових задач пружно-пластичного згину тонких пологих оболо-нок який базується на теорії R-функцій. Задача зводиться до знаходження точок стаціонарності запропонованих змішаних варіаційних функціоналів, лінеаризованих за схемою методу послідовних навантажень і Ньютона-Канторовича спільно з методом змінних параметрів пружності. Чисельні дослідження виконані з використанням програмуючої системи «ПОЛЕ». Встановлено нові закони нелінійного деформування пологих оболонок і пластин складної форми в плані.Ключові слова: тонкі пологі оболонки, пружно-пластичні деформації, теорія R-функцій.Рассматривается эффективный метод решения нелинейных краевых задач упруго-пластического изгиба тонких пологих обо-лочек базирующийся на теории R-функций. Задача сводится к нахождению точек стационарности, предложенных смешан-ных вариационных функционалов, линеаризованных по схеме метода последовательных нагружений и Ньютона-Канторовича совместно с методом переменных параметров упругости. Численные исследования выполнены с использовани-ем программирующей системы «ПОЛЕ». Установлены новые законы нелинейного деформирования пологих оболочек и пла-стин сложной формы в плане. Ключевые слова: тонкие пологие оболочки, упруго-пластические деформации, теория R-функций.The effective method basing on theory of R-functions and variational structural method is developed for solving non-linear boundary problems. Elastic-plastic bending of thin shallow shells is considered. The problems are reduced to finding stationary points of suggested mixed variational functionals according to the initial linearization due to usage of subsequent loading and Newton-Kantorovich jointly with method of varying elastic parameters. The method is used for automatic calculations in «POLE» programming system for investigations of shell structural elements. The numerical justification of the method is given. New laws of non-linear deformation of shallow shells and plates with complex shapes in plane are established. Keywords: shallow shells, elasto-plastic deformations, R-functions theory.© I. Morachkovska, G. Timchenko, E. Lyubitskaya, 2016 IntroductionMany of the technology problems associated with the deformation of a thin shell, and this explains the development of a geometrically and physically nonlinear theory of shells, with the development of methods of research of stress strain state of shell structures, operating beyond the limits of elasticity [1][2][3].Mathematical problems of elasto-plastic deformation of flexible membranes are formulated for non-linear differential equations under certain boundary and initial conditions. Mathematical methods, allowing to explore and find solutions of nonlinear differential equations, quite complicated. This paper proposes algorithms and some results of the solution of such problems on the basis of the known variational-structural method and the theory of Rfunctions [4][5][6][7]. This method became widespread in the international scientific literature under the name abbrevi...

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Elastoplastic state of flexible cylindrical shells with two circular holes

2004

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The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Theoretical and experimental data on stress concentration in isotropic and anisotropic structural elements (plates and shells) with two or more curvilinear (circular, elliptic, or rectangular) holes have been obtained mainly based on the linear elastic theory of thin shells [1, 4, 6-9, 11, 12, 15, etc.].The distribution of stresses (strains or displacements) in multiply connected thin-walled shells of various shapes with physical (plastic strain or creep) or geometrical (finite or large deflections) nonlinearities was addressed in a few studies [1-3, 5-10, etc.]. Note that the unsteady creep of multiply connected shallow shells was studied in the publications [2,5], which analyzed the stress state of internally loaded spherical shells with two equal circular holes and with cyclically symmetrically located holes centered on a circle of given radius. Experimental data for a cylindrical shell with two longitudinally arranged nonreinforced circular holes are presented in the papers [3, 10], which analyzed the elastoplastic stability of shells and determined the critical axial compressive forces.An important task is to solve static and dynamic problems for cylindrical shells taking physical and geometrical nonlinearities into account. The paper [13] includes a generalized formulation of nonlinear stress-concentration problems for arbitrary thin shells with several holes, the governing equations, and a solution technique. Numerical data for spherical shells with two circular holes under a surface load are presented in the paper [14], which studies the influence of one and two nonlinearities on the stress distribution in shells for different distances between the holes.

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Physically and Geometrically Nonlinear Static Problems for Thin-Walled Multiply Connected Shells

Cited by 20 publications

References 5 publications

Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

Research of elasto-plastic bending of thin shells and plates by the R-fanctions method

Elastoplastic state of flexible cylindrical shells with two circular holes

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