# On second order asymptotic solutions of axial symmetrical problems of*r*>0 thin uniform circular toroidal shells with a large parameter*a* ^{2}/R_{0}h

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## Abstract

According to the classical shell theory based on the Love-Kirchhoff assumptions, the basic differential equations for the axial symmetrical problems of r>0 thin uniform circular toroidal shells in bending are derived, and the second order asymptotic solutions are given for r>0 thin uniform circular toroidal shells with a large parameter a^{2}/R_{0}h. In the resent paper, the second order asymptotic solutions of the edge problems far from the apex of toroidal shells are given, too. Their errors are within the margins allowed in the classical theory based on the Love-Kirchhoff assumptions.

## Keywords

Asymptotic Solution Large Parameter Axisymmetric Problem Comparative Equation Ring Shell## Nomenclature

*a*Radius of curvature in meridional direction of toroidal shell

- \(\tilde C_1 \tilde C_2 \)
Arbitrary complex constants

*E*Modulus of elasticity

*H,V*Horizontal and Vertical forces

*h*Wall thickness

*M*_{φ},*M*_{0}Meridional and circumferential moments per unit length

*N*_{φ},*N*_{0}Meridional and circumferential forces per unit length

*Q*_{φ}Transverse shear force per unit circumferential width

*q*_{H}, q_{V}components of external loading forces per unit middle surface area of toroidal shell

*R*_{0}Radius of whole toroidal shell

*r*_{2}Radius of curvature in the circumferential direction of toroidal shell

*r**r*_{2}sinφ*N*_{φ},*M*_{θ}Meridional and circumferential strains

- ν
Poisson's ratio

- ϑ
Rotation of tangent to meridian

- φ
Coordinate defining angular position on meridian of toroidal shell

*V*^{*},*r*^{#},*ψ*^{*}Values of

*V, r*, φ at upper edge of toroidal shell, respectively

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