4. Use#

4.1. Modeling#

For each connection, the user must define under the keyword factor LIAISON_ELEM of AFFE_CHAR_MECA:

\(S\):

the trace of the right section of the beam on the shell: it does it with the keywords MAILLE_1et /or GROUP_MA_1c “in other words, it gives the list of line cells (assigned to shell modeling “edge” elements) that represent this section geometrically.

\(\text{P}\):

a node (keyword NOEUD_1ou GROUP_NO_1) carrying the 6 classical beam degrees of freedom: DX, DY, DZ, DRX, DRY, DRZ

\(V\):

the vector defining the axis of the beam, oriented from the shell to the beam, and defined by its coordinates using the AXE_POUTRE keyword: (v1, v2, v3)

Note:

**the node* \(\text{P}\) can be a beam element node or a discrete element node,

  • the list of shell edge meshes, defined by MAILLEou * GROUP_MAdoit represent **exactly the right section of the beam. This is an important constraint for meshing.*

4.2. Examples and tests#

4.2.1. Quiz SSLX101#

It is a straight beam subjected to unit forces in \(\text{B}\) (traction, bending and torsional moments). A section of thin tube with a thickness of \(\text{h}\ll \text{R}\) is taken.

_images/Object_80.svg

The embedding in \(\text{O}\) is achieved using a connection between the edge of the shell and a point element located in \(\text{O}\). This element is embedded (translations and rotations zero).

This makes it possible to obtain a stress state in the shell that is very similar to a « beam » solution: there is no disturbance in the stress field. The solution differs from the analytical solution (solution RDM) by 3%, this being solely due to the fineness of the mesh in shell elements.

4.2.2. Flexion of a plate#

Let us consider a sufficiently long thin plate, of length \(\mathrm{2L}\), of width \(\text{b}\), of width, of thickness \(\text{h}\), modeled by a shell element \(\text{OA}\) and a beam element on \(\text{AB}\):

_images/Object_81.svg

  • The 1st link condition is written as:

\(bhU(A)=h{\int }_{\text{CD}}U(y)\text{dy}\)

the displacement of point \(\text{A}\) (belonging to the beam) is the average of the movements of the edge \(\text{CD}\) of the plate.

  • The 2nd connection condition is written as:

\(I(\Omega )=h{\int }_{\text{CD}}\mathrm{AQ}\wedge U(Q)\text{ds}+\frac{{h}^{3}}{\text{12}}{\int }_{\text{CD}}\theta (Q)\text{ds}\)

In the case of a bend around \(y\), the only non-zero term is: \(\frac{{h}^{3}}{\text{12}}{\int }_{-\frac{b}{2}}^{b/2}\theta (y)\text{dy}\)

Indeed, \(h{\int }_{\text{CD}}\mathrm{AQ}\wedge U(Q)\text{ds}=h({\int }_{-\frac{b}{2}}^{\frac{b}{2}}{U}_{z}\text{ydy})\text{.}x=0\)

For a bend around \(y\), the link is therefore written:

\({I}_{y}{\theta }_{y}(A)=\frac{b{h}^{3}}{\text{12}}{\theta }_{y}\) because \({\theta }_{y}\) is constant on CD.

This application is implemented in test SSLX100B: 3D_shell_beam mix.