The elastoplastic state of a thin spherical shell weakened by an elliptic hole is analyzed. Finite deflections are considered. The hole is reinforced with a thin ring. The shell is made of an isotropic homogeneous material. The load is internal pressure. A relevant problem is formulated and solved numerically with allowance for physical and geometrical nonlinearities. The distribution of stresses, strains, and displacements along the elliptic boundary and in the zone of their concentration is studied. The stress-strain state of the shell near the hole is analyzed Introduction. The stress distribution in thin-walled shells of revolution made of isotropic (metal) and anisotropic (composite) materials and weakened by a curvilinear (circular) hole is studied in [2, 3, 5, 7, 20, etc.]. The major results were obtained by solving boundary-value problems in the linear elastic axisymmetric case. Structural members (plates, shells) with circular holes either free or reinforced with a rigid inclusion were considered in most cases. Some boundary-value problems for shells with a curvilinear (circular, elliptic) nonreinforced hole were solved in physically nonlinear [1,7,15,16] or geometrical nonlinear [7,19] formulations. Numerical results on stress concentration in thin-walled shells were obtained in [7, 11, 16-18, 21, etc.] making joint allowance for nonlinear factors (elastoplastic strains and finite deflections). The elastoplastic state of shells of various shapes with circular holes reinforced with a thin elastic element (ring) is studied in [4,10,13,14]. Of considerable interest is the nonaxisymmetric deformation of shells weakened by an elliptic hole. Solutions for spherical and cylindrical shells with an elliptic free hole are presented in [1,8,11,12].We will analyze the stress-strain state of an elastoplastic flexible spherical shell with an elliptic hole reinforced with a thin curvilinear element (a rod or a ring) of given stiffness. We will use the method developed in [11,21] and numerical results to examine the effect of the nonlinearity and stiffness of the reinforcement on the distribution of stresses (strains, displacements) around a reinforced hole in a shell subjected to surface pressure.1. Consider a thin-walled isotropic spherical shell of radius R and thickness h weakened by an elliptic (circular) hole and subjected, in the general case, to surface ({p} = {p 1 , p 2 , p 3 } Ò ) and boundary ({m b } = {T b , S b , Q b , Ì b } Ò ) static loads. Let high levels of loading cause the shell to deform plastically near the hole and undergo large (finite) deflections comparable to the thickness of the shell, but smaller than all the other linear dimensions. We also assume that the properties and stress-strain curve of the shell's material are known from experiments.The midsurface of the shell is described [11] in a geographical coordinate system (y, v) with the origin at the center of the hole, and its geometry is described in a global Cartesian coordinate system (X, Y, Z) with the origin at the center of the sph...
The stress-strain state of thin flexible spherical shells weakened by an eccentric circular hole is analyzed. The shells are made of an isotropic homogeneous material and subjected to internal pressure. A problem formulation is given, and a method of numerical solution with allowance for geometrical nonlinearity is outlined. The distribution of displacements, strains, and stresses along the hole boundary and in the region of their concentration is examined. The data obtained are compared with numerical solutions of the linear problem. The stress-strain state around the eccentric circular hole is analyzed with allowance for geometrical nonlinearity Introduction. The major theoretical and experimental results on the stress distribution around curvilinear holes in spherical shells were obtained on the assumption of axisymmetric deformation. Elastic shells made of homogeneous isotropic or orthotropic (composite) materials and weakened by a central circular hole are numerically analyzed in [5-7, 10, 11]. In the case of doubly connected domains, the nonaxisymmetric deformation of spherical shells with a curvilinear (circular, elliptic) hole was analyzed in [1,4,9,10] considering the linear elastic, nonlinear elastic, and elastoplastic ranges. Two-dimensional boundary-value problems for a spherical shell with an eccentric circular hole were solved in linear elastic and nonlinear elastic (plastic strains) formulations using the theory of thin shells [5,10,14].Also of interest is to study the effect of large (finite) deflections within stress concentration regions in spherical doubly connected shells with an eccentric hole under loading of high level [8,12,13].The geometrically nonlinear problem for a shell with an external edge and a curvilinear (circular) hole reduces to a two-dimensional boundary-value problem for a domain with internal and external boundaries. In support of the method proposed earlier in [16] and of its application to the numerical solution of some two-dimensional linear elastic and nonlinear problems [15,17,18], we will present specific results on the stress-strain state of a flexible spherical shell weakened by an eccentric hole. We will analyze the effect of geometrical nonlinearity (large and finite deflections) on the distribution of stresses in regions of their concentration in a shell under a surface load.1. Consider a thin spherical shell of radius R and thickness h. Its planform is an eccentric ring (Fig. 1). The deep isotropic shell of constant thickness with a curvilinear hole (internal boundary) and external edges (external boundary) is generally under a surface load of given level (internal pressure q = q 0 ×10 5 Pa) and edge forces (moments) set at its boundaries. Assume that loads of high level cause normal displacements (deflections) comparable with or exceeding the thickness of the shell, strains remaining small. Finite (large) deflections in a shell with an eccentric circular hole are allowed for in solving a geometrically nonlinear two-dimensional boundary-value problem. Its solutio...
Концентрація напружень в пружнопластичній сферичній оболонці з рядом однакових кругових отворівПредставлено членом-кореспондентом НАН України І.С. Чернишенком Дано постановку періодичних задач статики для пружнопластичної сферичної оболонки з рядом однакових кругових отворів. Розроблено методику чисельного розв'язання даного класу нелінійних задач, яка базується на використанні методу додаткових напружень і варіаційного векторно-різницевого методу. Досліджено вплив пластичних деформацій і геометричних параметрів на напружено-деформований стан сферичної оболонки з рядом отворів при дії рівномірного внутрішнього тиску.
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