3. Integrals to be calculated. Shell kinematics.#

For each node, the program calculates the coefficients of the 6 linear relationships [éq 2-8] that connect:

  • the 6 degrees of freedom of the \(\text{P}\) beam node (geometrically confused with the center of gravity \(G\) of the cross section of the shell mesh)

with the degrees of freedom of**all* the nodes in the shell edge mesh list.

These linear relationships are dualized, like all linear relationships that come from, for example, the LIAISON_DDL keyword in AFFE_CHAR_MECA. They are built as for the 3D-beam connection from the assembly of elementary terms.

_images/10002F0C000069D500002796DC6D90EBCCC47F81.svg

Linear shell or plate kinematics in thickness:

\(u(M)=u(Q)+(\theta (Q)\wedge n)\text{.}{y}_{3}\)

  • \(u\) is the displacement vector of the mean surface in \(Q\),

  • \(n\) is the normal vector to the mean surface of the shell in \(Q\),

  • \(\theta\) is the rotation vector in \(Q\) of the normal in the directions \({t}_{1}\text{et}{t}_{2}\) of the tangential plane

  • \({y}_{3}\) is the coordinate in the thickness (\({y}_{3}\mathrm{\in }\left]\mathrm{-}\frac{h}{2},\frac{h}{2}\right[\)).

3.1. Calculation of the average displacement on the S section#

The aim is to calculate the integral \({\int }_{S}u\text{dS}\), where \(u\) is the shell displacement (comprising 6 ddl per node), \(S\) is the shell edge of the connecting cross section.

The average displacement on section \(S\) is written as:

\({\int }_{s}u(M)\text{ds}=h{\int }_{l}u(Q)\text{ds}+{\int }_{l}(\theta (Q)\wedge n)({\int }_{-h/2}^{h/2}{y}_{3}{\text{dy}}_{3})\text{ds}\)

Be \({\int }_{s}u(M)\text{ds}=h{\int }_{l}u(Q)\text{ds}\)

In this expression, metric variations in the thickness of the shell are overlooked.

3.2. Calculation of the average rotation of the S section#

\(\begin{array}{}{\int }_{s}\mathrm{GM}\wedge u(M)\text{ds}={\int }_{l}{\int }_{-h/2}^{h/2}(\mathrm{GQ}+{y}_{3}n(Q))\wedge (u(Q)+\theta (Q)\wedge n(Q)\text{.}{y}_{3}){\text{dsdy}}_{3}\\ =h{\int }_{l}\mathrm{GQ}\wedge u(Q)\text{ds}+{\int }_{l}\mathrm{GQ}\wedge (\theta (Q)\wedge n(Q))\text{ds}{\int }_{-h/2}^{h/2}{y}_{3}{\text{dy}}_{3}\\ +{\int }_{l}n(Q)\wedge u(Q)({\int }_{-h/2}^{h/2}{y}_{3}{\text{dy}}_{3})\text{ds}+{\int }_{l}n(Q)\wedge (\theta (Q)\wedge n(Q)){\int }_{-\frac{h}{2}}^{\frac{h}{2}}{y}_{3}^{2}{\text{dy}}_{3}\text{.}\text{ds}\end{array}\)

Be \({\int }_{s}\mathrm{GM}\wedge u(M)\text{ds}=h{\int }_{l}\mathrm{GQ}\wedge u(Q)\text{ds}+\frac{{h}^{3}}{\text{12}}{\int }_{l}n(Q)\wedge (\theta (Q)\wedge n(Q))\text{ds}\text{.}\)

3.3. Calculation of the inertia tensor#

The inertia tensor is defined by [R3.03.03]:

\(I(\Omega )={\int }_{s}\mathrm{GM}\wedge (\Omega \wedge \mathrm{GM})\text{ds}\)

asking: \(\mathrm{GM}=\mathrm{GQ}+n(Q)\text{.}{y}_{3}\text{.}\)

We get: \(I(\Omega )=h{\int }_{l}\mathrm{GQ}\wedge (\Omega \wedge \mathrm{GQ})\text{ds}+\frac{{h}^{3}}{\text{12}}{\int }_{l}n(Q)\wedge (\Omega \wedge n(Q))\text{ds}\)

3.4. Implementation of the method#

The coefficients of linear relationships are calculated in two stages:

  • calculation of elementary quantities on the elements in the list of shell edge cells (mesh of the type SEG2 or SEG3):

  • we calculate the 9 terms:

  • \({\int }_{\text{elt}}\text{ds};{\int }_{\text{elt}}\text{xds};{\int }_{\text{elt}}\text{yds};{\int }_{\text{elt}}{x}^{2}\text{ds};{\int }_{\text{elt}}{y}^{2}\text{ds};{\int }_{\text{elt}}{z}^{2}\text{ds};{\int }_{\text{elt}}\text{xyds};{\int }_{\text{elt}}\text{xzds};{\int }_{\text{elt}}\text{yzds}\)

as well as terms from \(I(\Omega )\text{:}\frac{{h}^{3}}{\text{12}}{\int }_{l}n\wedge (\Omega \wedge n)\text{ds}\)

This allows us to calculate: \(\frac{{h}^{3}}{\text{12}}{\int }_{l}({n}_{y}^{2}+{n}_{z}^{2})\text{ds},\frac{{h}^{3}}{\text{12}}{\int }_{l}{n}_{x}{n}_{y}\text{ds},\text{etc}\text{.}\text{.}\text{.}\)

  • summation of these quantities on \((S)\), hence the calculation of:

  • \(A=\mid S\mid\)

  • \(G\) position

  • inertia tensor \(I\)

  • knowing \(G\), elementary calculation on the elements in the list of shell edge cells of:

\({\mathrm{\int }}_{\text{elt}}{N}_{i}\text{ds};{\mathrm{\int }}_{\text{elt}}{\text{xN}}_{i}\text{ds};{\mathrm{\int }}_{\text{elt}}{\text{yN}}_{i}\text{ds};{\mathrm{\int }}_{\text{elt}}{\text{zN}}_{i}\text{ds}\) where \(\begin{array}{c}\text{GM}\mathrm{=}\left\{x,y,z\right\}\\ {N}_{i}\mathrm{=}\text{fonctions de forme de l'élément}\end{array}\)

(It should simply be noted that in this case, the integrals on the edge elements are to be multiplied by the thickness of the shell: \(\underset{\text{elt}}{\int }{N}_{i}\text{ds}=h\underset{l}{\int }{N}_{i}\text{dl}\) where \(l\) represents the curvilinear abscissa of the mean fiber of the shell edge element).

In addition, we add additional terms from: \(\frac{{h}^{3}}{\text{12}}{\int }_{l}n(Q)\wedge (\Omega \wedge n(Q))\text{ds}\)

By noting \(n=\mid \begin{array}{c}{n}_{x}\\ {n}_{y}\\ {n}_{z}\end{array}\) and \(\theta =\mid \begin{array}{c}{\theta }_{x}\\ {\theta }_{y}\\ {\theta }_{z}\end{array}\) in the global frame of reference we get:

\(n(Q)\wedge (\theta \wedge n(Q))=\mid \begin{array}{c}({n}_{y}^{2}+{n}_{z}^{2}){\theta }_{x}-{n}_{x}{n}_{y}{\theta }_{y}-{n}_{x}{n}_{z}{\theta }_{z}\\ -{n}_{x}{n}_{y}{\theta }_{x}+({n}_{x}^{2}+{n}_{z}^{2}){\theta }_{y}-{n}_{y}{n}_{z}{\theta }_{z}=A\theta \\ -{n}_{x}{n}_{z}{\theta }_{x}-{n}_{y}{n}_{z}{\theta }_{y}+({n}_{x}^{2}+{n}_{y}^{2}){\theta }_{z}\end{array}\)

so:

\(\frac{{h}^{3}}{\text{12}}{\int }_{l}n(Q)\wedge (\Omega \wedge n(Q))\text{ds}=\frac{{h}^{3}}{\text{12}}\sum _{\text{el}}({\int }_{\text{el}}A(s){N}_{j}(s)\text{ds}){\theta }_{j}\)

  • « assembly » of the terms calculated above to obtain, for each of the nodes of the edge cells, the coefficients of the terms of the linear relationships.