Elastoplastic state of flexible spherical shells with a reinforced elliptic hole

Abstract

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...

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

“…The stress-strain states of a thin spherical shell of radius R and thickness h with a reinforced elliptic hole under uniform internal pressure q = q 0 ×10 5 Pà was analyzed in [88].…”

“…The majority of publications that take into account nonlineas factor deal with axisymmetric problems [8,13,48,72]. Nonlinear two-dimensional problems for shells with reinforced holes were solved in a few papers [46,61,63,88].…”

Stress concentration in a transversely isotropic spherical shell with two circular rigid inclusions

“…Despite their long history [1][2][3], studies of stress concentration in plates and shells with holes and inclusions (rings, washers, covers) are still relevant [5][6][7][8]. A similar situation exists with other problems in the mechanics for multiply connected and structurally inhomogeneous bodies [4,9].…”

“…The perturbation of the stress state caused by the inclusions can be found from the homogeneous system of governing differential equations of thin transversely isotropic shells proposed in [3,5] and based on the refined Timoshenko-type theory that accounts for the transverse-shear strains. Let us represent them as…”

“…Problem-Solving Method. The solutions of the homogeneous system of differential equations (2) decreasing in absolute magnitude with distance from G q , according to [3,5], have three different analytic forms, depending on the range of variation in the parameter d. The functions U and w are defined to have cylindrical and polyharmonic (power) components: …”