2. Reference solution#

2.1. Calculation method#

We use the theory of LOVE - KIRCHHOFF to calculate the analytical solution of this problem.

The arrow in the axisymmetric coordinate system is written:

\({u}_{x}=\{\begin{array}{cc}\frac{P}{8{\mathrm{\alpha }}^{4}D}(2-{\mathrm{e}}^{\mathrm{\alpha }y}\mathrm{cos}(\mathrm{\alpha }y))& \forall y\le 0\\ \frac{P}{8{\mathrm{\alpha }}^{4}D}{\mathrm{e}}^{-\mathrm{\alpha }y}\mathrm{cos}(\mathrm{\alpha }y)& \forall y\ge 0\end{array}\)

with \(D=\frac{E{t}^{3}}{12(1-{\mathrm{\nu }}^{2})}\) and \(4{\mathrm{\alpha }}^{4}=\frac{\mathit{Et}}{{\mathit{DR}}^{2}}\).

The rotation is written as:

\({\mathrm{\beta }}_{s}=\{\begin{array}{cc}\frac{P}{8{\mathrm{\alpha }}^{3}D}{\mathrm{e}}^{\mathrm{\alpha }y}(\mathrm{cos}(\mathrm{\alpha }y)-\mathrm{sin}(\mathrm{\alpha }y))& \forall y\le 0\\ \frac{P}{8{\mathrm{\alpha }}^{3}D}{\mathrm{e}}^{-\mathrm{\alpha }y}(\mathrm{cos}(\mathrm{\alpha }y)+\mathrm{sin}(\mathrm{\alpha }y))& \forall y\ge 0\end{array}\)

The widespread efforts are:

\({N}_{\mathrm{\theta }\mathrm{\theta }}=\frac{\mathit{Et}}{R}{u}_{x}\left(y\right)\),

math:

{M} _ {mathit {ss}} = {mathit {ss}} = {mathit {ss}}} = {mathit {ss}} =frac {p} {4 {mathrm {alpha}} {4 {mathrm {alpha}}} ^ {alpha}} ^ {-|y|} {4 {mathrm {alpha}}} ^ {-|y|} {4 {mathrm {alpha}}} ^ {-|y|} {4 {mathrm {alpha}}} ^ {-|y|} {4 {mathrm {alpha}}} ^ {-|y|}mathrm {sin}}} (mathrm {alpha} y)

The three-dimensional constraints are:

\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=\frac{{N}_{\mathrm{\theta }\mathrm{\theta }}}{t}+12\frac{{M}_{\mathrm{\theta }\mathrm{\theta }}(x-R)}{{t}^{3}}\),

\({\mathrm{\sigma }}_{\mathit{ss}}=12\frac{{M}_{\mathit{ss}}(x-R)}{{t}^{3}}\),

either:

\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(y,x)=\{\begin{array}{cc}\frac{\mathit{pR}}{t}\left(1-\frac{{\mathrm{e}}^{\mathrm{\alpha }y}}{2}\left(\mathrm{cos}(\mathrm{\alpha }y)+2\mathrm{\nu }\frac{R-x}{t}\sqrt{\frac{3}{1-{\nu }^{2}}}\mathrm{sin}(\mathrm{\alpha }y)\right)\right)& \forall y\le 0\\ \frac{\mathit{pR}}{t}\cdot \frac{{\mathrm{e}}^{-\mathrm{\alpha }y}}{2}\left(\mathrm{cos}(\mathrm{\alpha }y)-2\mathrm{\nu }\frac{R-x}{t}\sqrt{\frac{3}{1-{\nu }^{2}}}\mathrm{sin}(\mathrm{\alpha }y)\right)& \forall y\ge 0\end{array}\),

and:

\({\mathrm{\sigma }}_{\mathit{ss}}(y,x)=\{\begin{array}{cc}\frac{\mathit{pR}}{t}\cdot \frac{x-R}{t}\sqrt{\frac{3}{1-{\mathrm{\nu }}^{2}}}{\mathrm{e}}^{\mathrm{\alpha }y}\mathrm{sin}(\mathrm{\alpha }y)& \forall y\le 0\\ \frac{\mathit{pR}}{t}\cdot \frac{x-R}{t}\sqrt{\frac{3}{1-{\mathrm{\nu }}^{2}}}{\mathrm{e}}^{-\mathrm{\alpha }y}\mathrm{sin}(\mathrm{\alpha }y)& \forall y\ge 0\end{array}\).

2.2. Reference quantities and results#

The following values are tested:

  • the arrow \(\mathit{DX}\) at points \(A\), \(B\) and \(C\),

  • rotation \(\mathit{DRZ}\), at points \(A\) and \(B\),

  • the generalized effort \(\mathit{NYY}\) at points \(B\) and \({B}_{1}\),

  • the generalized moment \(\mathit{MXX}\) at point \({B}_{1}\)

2.3. Bibliographical references#

    1. ANDRIEUX - F. VOLDOIRE: Shell models. Linear static applications. Summer School CEA - EDF - INRIA of Numerical Analysis 1992.

2.4. Uncertainty about the solution#

There is no step