The equilibrium state of a hollow transversally-isotropic thick-walled sphere, which is fixed on the external surfaces and is subject to a uniform internal lateral pressure and weight forces

We use the decomposition of the components of the displacement vector along the hoop and radial coordinates in series in Legendre polynomials and generalized power series to obtain an exact analytical solution to the equilibrium problem of a thick-walled transversely isotropic centrally symmetric hollow body, which is rigidly fixed on the external surface and is subject to a uniform internal pressure and weight forces.
As an example of using the obtained analytical solution, we analyzed the influence of weight forces on distribution of independent invariants of the stress tensor in the cross section of a heavy reinforced concrete sphere, which internal surface is free from pressure. Based on the multicriteria approach describing various loss of strength mechanisms (from tension or compression in the radial and hoop direction and interlayer shear), we found the regions of a heavy reinforced concrete sphere, in which damage can be initiated.
A qualitative and quantitative comparison of the stress fields at the points of the cross-sections of the thick-walled heavy spheres with the results of the numerical solution of the same problem in the axisymmetric and 3D formulations in the FEM packages ANSYS 13.0 and ABAQUS 6.11 is carried out. Both packages demonstrated the minimum deviation of the numerically determined values of the stress invariants from the exact analytical solution in the axisymmetric formulation. Also the difference with a comparable error in the 3D setting was found. In the latter case, the presentation of the FEM results for stresses and strains in the component form led to an unexpected result, i.e. significant errors in comparison with the exact analytical solution. To eliminate the errors found in determining the stress-strain state, which are caused only by features of determining the spherical coordinate system in the FEM packages ANSYS 13.0 and ABAQUS 6.11, it is necessary to use the presentation of the results obtained in the invariant form.