2. Benchmark solution#

2.1. Calculation method#

The reference solution is a digital solution [1].

If \(a\mathrm{-}b\mathrm{\le }r<a+b\)

  • Displacement: \({\delta }_{r}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2Eh}}(r\mathrm{-}\nu (r+a))\)

  • Constraints: \({\sigma }_{11}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2h}}\mathrm{\times }\frac{r+a}{r}\) \({\sigma }_{22}\mathrm{=}\frac{\mathit{pb}}{\mathrm{2h}}\)

These formulas are only applicable to thin tori, such as \(\frac{b}{h}>10\) and with a radius of curvature such as \(r\mathrm{\times }\pi <100\sqrt{\frac{{I}_{x}}{A}}\) with \({I}_{x}\) =moment of inertia and \(A\) = area of the geometric cross section of the torus

2.2. Reference quantities and results#

  • Travel

Dot

\(\mathit{DX}(m)\)

\(r\mathrm{=}a\mathrm{-}b\)

\({\delta }_{r}\mathrm{=}1.19\mathrm{\times }{10}^{\mathrm{-}7}\)

\(r\mathrm{=}a+b\)

\({\delta }_{r}\mathrm{=}1.79\mathrm{\times }{10}^{\mathrm{-}6}\)

  • Constraints

Point

Stresses (Pa)

\(\mathrm{\forall }r\)

\({\sigma }_{22}\mathrm{=}\mathrm{2,5}\mathrm{\times }{10}^{5}\)

\(r\mathrm{=}a\mathrm{-}b\)

\({\sigma }_{11}\mathrm{=}7.5\mathrm{\times }{10}^{5}\)

\(r\mathrm{=}a+b\)

\({\sigma }_{11}\mathrm{=}4.17\mathrm{\times }{10}^{5}\)

2.3. Uncertainties about the solution#

Analytical solution

2.4. Bibliographical references#

  1. Guide VPCS - 1990 edition.