Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

Storozhuk, E. À.; Chernyshenko, I. S.

doi:10.1007/s10778-005-0134-0

Cited by 18 publications

(30 citation statements)

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“…As the rigidity of fixation is increased, the points smear Introduction. The asymmetric stress-strain state (SSS) near curvilinear holes (cutouts) in isotropic [4,6,9,10] and composite [4,5,8,11] plates and shells is of theoretical and practical interest: the results of a stress-strain analysis would allow us to evaluate the stress and strain of various thin-walled structural members under various kinds of loads.Numerical stress-strain analyses were performed mainly for thin-walled spherical caps with a central elliptical hole [4,5,8,10,11].As pointed out in [1,7], in the off-center case, numerical results are mainly available only for a circular hole, with the emphasis being on the stress distribution around the bridge between the outer and inner edges, which does not give an accurate account of the maximum stresses. Most publications employ special orthogonal coordinate frames and address simply connected stress-concentration problems for elastic isotropic shells.…”

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On the stress distribution in a spherical shell with an off-center curvilinear hole

Mulyar

2006

Int Appl Mech

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A numerical analysis is made of the joint effect of two factors of asymmetry-ellipticity and eccentricity-on the stress distribution near a free hole in a spherical shell. The nature of deformation is determined by the predominant factor. Whether there are "fixed" points on the graphs of stress distribution around a small hole at which they intersect depends on how rigidly the outer edge is fixed. As the rigidity of fixation is increased, the points smear Introduction. The asymmetric stress-strain state (SSS) near curvilinear holes (cutouts) in isotropic [4,6,9,10] and composite [4,5,8,11] plates and shells is of theoretical and practical interest: the results of a stress-strain analysis would allow us to evaluate the stress and strain of various thin-walled structural members under various kinds of loads.Numerical stress-strain analyses were performed mainly for thin-walled spherical caps with a central elliptical hole [4,5,8,10,11].As pointed out in [1,7], in the off-center case, numerical results are mainly available only for a circular hole, with the emphasis being on the stress distribution around the bridge between the outer and inner edges, which does not give an accurate account of the maximum stresses. Most publications employ special orthogonal coordinate frames and address simply connected stress-concentration problems for elastic isotropic shells. Such problems for elastoplastic shells were analyzed in [6, 10] and for shells made of nonlinear elastic composites in [8].Note that the paper [5], where the SSS of elastic orthotropic laminates with an elliptic hole was analyzed, gives specific results only for a special case (a circular hole).Specific simply and doubly connected problems for spherical shells with a central circular hole or with two circular holes were solved in [6].We will examine the joint effect of two factors of asymmetry (ellipticity and eccentricity) on the SSS only for a rigidly fixed shell with finite mechanical and geometrical parameters.Nowadays, stress-strain studies of doubly and multiply connected thin-walled structural members are of interest. The same applies to the stress-strain analysis of spherical shells (bottoms). We will discuss below results from a stress-strain analysis of elastic isotropic spherical shells with an off-center curvilinear (elliptical or circular) hole.1. Problem Formulation. Let us consider a thin isotropic nonclosed spherical shell of radius R and thickness h with an off-center [7] curvilinear (elliptical, circular) hole. The shell is a doubly connected domain (Fig. 1); its boundaries are the outer and inner edges of the shell. The shell is subjected to surface forces { } { , , } p p p p T = 1 2 3 . The inner boundary can be free, fixed,

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On the stress distribution in a spherical shell with an off-center curvilinear hole

Mulyar

2006

Int Appl Mech

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“…We used a 24´16 FE mesh and pressure q 0 = 5. Table 6 collects the values of relative deflection (w * = w/h) along the boundary of the reinforced hole (r = r 0 , 0 £ £ q p/2) The results obtained for a nonreinforced elliptic hole are reported in [97]. Figure 8 shows the variation of the concentration factor for membrane hoop stress, k h q R q q s = 2 / , along the meridional section q = 0 (l a = (yR -a)/r 0 ) for q 0 = 5.…”

Section: Spherical Shell With a Reinforced Ellipticmentioning

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“…Compared with [97], the reinforcement (ring) reduces the maximum stresses s q by 65, 51, 52, and 15% and the maximum deflection by 65,95,44, and 71% in LP, PNP, GNP, and PGNP, respectively. Allowing for both plastic strains and large deflections leads to equalization of stresses throughout the thickness of the shell and decrease in the maximum stresses by 39, 18, and 42% in PNP, GNP, and PGNP, respectively, compared with the LP.…”

Section: Spherical Shell With a Reinforced Ellipticmentioning

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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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“…The papers [12,15,18,19] outline methods of solving geometrically nonlinear problems and specific results on the nonlinear deformation of spherical and cylindrical shells.…”

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Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells

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Gromov

2006

Int Appl Mech

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A nonlinear boundary-value problem for shells is formulated. The problem statement permits us to analyze the behavior of the postbuckling solutions in a finite-dimensional space with the dimension independent of the discretization technique. A branching pattern is established for the solutions of the problem in the case of a uniform external pressure. Four characteristic types of solution are recognized in the case of an arbitrary external pressure. The associated buckling loads are determined. It is found out that five significant parameters are sufficient for a qualitative analysis of a cylindrical shell Keywords: nonlinear boundary-value problem, shell theory, internal and external pressure, buckling loads Introduction. The behavior of thin-walled shells under predominantly compressive stress was addressed in a great many studies, which is due to the wide use of such thin-walled structures in aircraft and mechanical engineering. The majority of these studies involve an analysis of the bifurcation points on the prebuckling solution branch of the corresponding nonlinear boundary-value problem. Secondary branching in local modes discovered in problems of uniform hoop [9] and axial [11] compression and the experimentally observed variety of deformation modes of shells [1,7] call for a more careful study into the general properties of the solutions of nonlinear systems. This would make it possible to predict and justify real buckling loads.A stability analysis of continuous conservative systems usually reduces to an analysis of infinite-dimensional discrete systems based on numerical procedures (the finite-element method, finite-difference methods, and direct variational methods) [4]. The development of computational methods of finding postbuckling solutions of nonlinear boundary-value problems (the parameter continuation method and a technique of moving to a postbuckling branch in the neighborhood of a bifurcation point) made it possible to analyze the postbuckling behavior of cylindrical shells. Noteworthy are the studies [16,21], where the dynamics of buckling is modeled. In [16], a group-theoretical analysis of a nonlinear boundary-value problem for shells was performed, and the structure of postbuckling branches was determined. Of special importance are the studies [10, 11], which set forth a "jump" method for finding the postbuckling branches of a nonlinear boundary-value problem. This method made it possible to obtain secondary postbuckling branches in the axial-compression problem for a cylindrical shell. The jump occurs in a Euclidean space obtained from a finite-element approximation of the governing partial differential equations.Also worthy of mention are the paper [13], where the Mushtari-Donnell-Vlasov equations were solved analytically, and the papers [5, 6], which used the solution to study the prebuckling and postbuckling stress-strain states of noncircular cylindrical shells. The papers [12,15,18,19] outline methods of solving geometrically nonlinear problems and specific results on the nonlinear ...

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Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

Cited by 18 publications

References 12 publications

On the stress distribution in a spherical shell with an off-center curvilinear hole

On the stress distribution in a spherical shell with an off-center curvilinear hole

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

Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells

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