2. The 3D-beam connection

2.1. Objectives and solutions excluded

When one wishes to finely analyze a part of a complex slender structure [Figure 2.1-a], one may, in order to minimize the size of the mesh to be manipulated, one may want to represent the structure by a beam « far away » from the part to be analyzed. The aim of schematization by a beam is to bring realistic boundary conditions to the edges of the modelled and meshed part in a 3D continuous medium. The 3D‑Beam connection must therefore meet the following requirements:

P1	To be able to transmit the beam forces (twister) to the 3D mesh
P2	Do not generate “parasitic” constraints (or even a concentration of constraints), because the connection should then be placed far enough away from the area to be analyzed so that these disturbances are attenuated in the study area.
P3	Do not favor kinematic conditions or static connection conditions over each other. It must be equivalent to bringing an effort or displacement twister to the limits of the 3D domain.
P4	Admit any behaviors on either side of the connection (elasticity, plasticity…) and also allow dynamic analysis.

Figure 2.1-a

If these objectives are achieved, connection rules can also be used to deal with the problem of embedding a beam in a 3D mass. However, the distribution of stresses in the massif around the embedment will remain quite rough and must be used with care. It is preferable to mesh the joint in 3D and then to extend the start of the 3D meshing of the beam section by one of the beam elements with 3D connection/Beam [Figure 2.1-b].

Figure 2.1-b

In view of objectives 1 to 4, we can already eliminate two common connection techniques:

1. the first which brings the whole connection back to the treatment of connection conditions between the points facing each other at the intersection of the neutral axis of the beam and the 3D solid. Apart from the difficulty of correctly defining the « point » rotation of the material point belonging to the 3D solid, we concentrate the efforts (concentrated reaction, torque) at this point and we break the kinematic/static symmetry by giving priority to a particular kinematics.

2. the second solution which completely imposes a beam displacement (NAVIER - BERNOULLI) at the points of the 3D massif located at the intersection of the 3D solid and the section of the beam. In elasticity, we know that the hypothesis of the undeformability of sections in their plane is only an approximation. Correct from an energetic point of view for the beam, it leads to stress concentrations in the vicinity of the limits of the junction section for the 3D solid.

Note:

It goes without saying that everything presented here is only valid in the hypothesis of small disturbances (small displacements) .

2.2. Orientation

We will start from the mechanical elements of the connection:

• the constraint vector field \(\sigma \text{.}n\) defined on the trace of the section \(S\) of the beam on the 3D massif, \(n\) being the normal to the plane of \(S\),

• and the \({u}^{\text{3D}}\) movement field defined on this same domain,

for the three-dimensional solid, as well as:

• the \(T\) torsor of the beam forces at the geometric center of inertia \(G\) of \(S\),

• and the \(D\) twist of the beam movements at the same point,

for the beam.

These mechanical quantities are linked by:

• the conditions of kinematic continuity,

• the equilibrium conditions of the connection.

The first conditions are the connection conditions to be imposed in a « displacement » approach, the second are deduced from the weak formulation of balance via the virtual work of contact actions between the beam and the massif (which is none other than the expression of the « principle » of the action and the reaction written for the interface \(S\)). In fact, on surface \(S\), for any legal virtual movement, we have \((v,T,\Omega )\):

\({\int }_{}s\text{.}n\text{.}v\text{dS}=F\text{.}T+M\text{.}\Omega\) eq 2.2-1

where:

• \(T\text{et}\Omega\) are the infinitesimal translation and rotation of the beam respectively: \(D=(T,\Omega )\)

• \(F\text{et}M\) are respectively the resultant and the moment in the beam at the connection point: \(T=(F,M)\)

The first member of this equality will provide the dot product by which we will define the « beam component » of a 3D displacement field set to \(S\). By using this scalar product, the symmetry of the approach between kinematic and static connection conditions (P3) will be ensured as well as the possibility of dealing with any behaviors on either side of the connection (P4) since no aspect of behavior appears in the equality of balance used.

The process:

We will break down the 3D displacement field into a « beam » part and a « complementary » part. This will lead us to define quite naturally the conditions of kinematic connection between beam and 3D solid as the equality of the displacement (torsor) of the beam and of the beam part of the 3D displacement field [(§ 2.3)]. Once this is done, equality [éq 2.2-1] will allow us to interpret the connection conditions in static terms and thus access the static connection conditions [(§2.4)].

2.3. Breakdown of 3D movement on the interface

The junction between the three-dimensional solid \(B\) and the beam of section \(S\) is assumed to be plane and of normal \(n\) parallel to the tangent \(\tau\) to the beam at the point of contact \(G\), the geometric center of inertia of the section \(S\) [Figure 2.3-a].

Figure 2.3-a

We therefore exclude the case (b) where the beam does not « come out » by perpendicular to the surface of the solid. It should be understood that this restriction is necessary for the coherence of the connection as envisaged here since the theory of beams only knows of cuts normal to the average fiber: the equilibrium condition [éq 2.2-1] makes no sense if \(S\) is not the straight section of the beam. In the event that this condition is violated, the mesh can be modified to achieve it as shown in the diagram below.

Figure 2.3-b

We note:

:math:`(G,{e}_{1}{e}_{2})` a**main geometric inertia coordinate system* of \(S\) whose origin is the center of inertia \(G\), and \(({x}_{1},{x}_{2})\) the associated coordinates,

• \(n\) or \({\mathrm{e}}_{3}\) the normal one at plane \(S\), outgoing to the 3D massif,

• \({\varepsilon }^{\text{\alpha \beta }3}=({e}_{\mathrm{\alpha }},{e}_{\mathrm{\beta }},{e}_{3})\) the alternating form of the mixed product of the base vectors,

finally \(I\) the geometric inertia tensor of \(S\) (diagonal in the coordinate system \(({e}_{1}{e}_{2})\)) and \(A=\mid S\mid\) the area of the section \(S\).

Recall that the inertia tensor \(I\) can be defined equivalently by a linear application (mixed representative):

\(I(U)={\int }_{S}\text{GM}(x)\wedge (U\wedge \text{GM}(x))\text{dx}\)

or a symmetric bilinear application (covariant representative):

\(I(U,V)={\int }_{S}(U\wedge \text{GM}(x))\text{.}(V\wedge \text{GM}(x))\text{dx}\)

These two expressions will be useful, in coordinate system \((G,{e}_{1}{e}_{2}{e}_{3})\) the matrix representing \(I\) is:

\(\left[I\right]=\left[\begin{array}{ccc}{I}_{1}& 0& 0\\ 0& {I}_{2}& 0\\ 0& 0& {I}_{1}+{I}_{2}\end{array}\right]\)

with \({I}_{\mathrm{\alpha }}\) geometric moment of inertia of \(S\) with respect to the \((G,{e}_{\alpha })\) axis. By convention Greek indices take the values 1 or 2.

The useful space for the displacement fields and constrained vectors defined in \(S\) is \(V\mathrm{=}{L}^{2}{(S)}^{3}\). We introduce the \(T\) space for the fields associated with a torsor (defined by two vectors):

\(T=\left\{v\in V/\exists (T,\Omega )\text{tel que}v(M)=T+\Omega \wedge \text{GM}\right\}\) eq 2.3-1

For the displacement fields of \(S,T\) is the translation of the section (or of the point \(G\)), \(\Omega\) the infinitesimal rotation and the fields \(v\) are the displacements maintaining the section \(S\) plane and not deformed (Assumptions of NAVIER - BERNOULLI).

For constrained vector fields, \(\mid S\mid T\) is the \(F\) resultant of the actions applied to \(S\), and \(I(\Omega )\) is the moment resulting \(M\) in \(G\). Fields \(v\) then correspond to affine stress distributions in the section. In fact, we have:

\(\begin{array}{}F(\sigma )\equiv {\int }_{S}s\text{.}n\text{dS}={\int }_{S}T\text{dS}=\mid S\mid T\\ M(\sigma )\equiv {\int }_{S}\text{GM}(x)\wedge \sigma \text{.}n\text{dS}={\int }_{S}\text{GM}(x)\wedge (\Omega \wedge \text{GM})\text{dS}=I(\Omega )\end{array}\)

We used here the fact that \(G\) is the center of geometric inertia so: \({\int }_{S}{x}_{\mathrm{\alpha }}\text{dS}=0\). The vector subspace \(T\) being of finite dimension (equal to 6) has an orthogonal additional for the dot product defined on \(V\):

\({T}^{\perp }=\left\{v\in V/{\int }_{S}v\text{.}w\text{dS}=0\forall w\in T\right\}\) eq 2.3-2

Or, more explicitly:

\({T}^{\perp }=\left\{v\in V/{\int }_{S}v\text{dS}=0\text{et}{\int }_{S}\text{GM}\wedge v\text{dS}=0\right\}\) eq 2.3-3

Any \(V\) field is uniquely broken down into the sum of one element of \(T\) and one element of \({T}^{\perp }\).

\(\begin{array}{ccc}u={u}^{p}+{u}^{s}& {u}^{p}\in T,& {u}^{s}\in {T}^{\perp }\end{array}\) eq 2.3-4

In addition, we have the following property:

For any pair of 3D fields \((u,v)\) defined to \(S\),

\(\begin{array}{ccc}u={u}^{p}+{u}^{s}& & \\ & & \Rightarrow {\int }_{S}v\text{.}w\text{dS}={\int }_{S}{v}^{p}\text{.}{w}^{p}\text{dS}+{\int }_{S}{v}^{s}\text{.}{w}^{s}\text{dS}\\ v={v}^{p}+{v}^{s}& & \end{array}\) eq 2.3-5

The following definition is therefore natural:

Definition:

We call the beam displacement component of a field \(u\) defined on the section the component \({u}^{p}\) of \(u\) on the subspace.

The calculation of the beam part of a 3D \(u\) field is performed using the orthogonal projection property since \(\mathrm{T}\) and \({\mathrm{T}}^{\mathrm{\perp }}\) are orthogonal by definition.

If we write down \({u}^{p}={T}_{u}+{\Omega }_{u}\wedge \text{GM}\), then:

\(({T}_{u},{\Omega }_{u})=\begin{array}{c}\text{Argmin}\\ (T,\Omega )\end{array}{\int }_{S}{(u-T-\Omega \wedge \text{GM})}^{2}\) eq 2.3-6

Note in passing the interpretation of the beam component of \(u\): it is the beam displacement field closest to \(u\) in the sense of least squares. The calculation of the minimum immediately leads to the characterization of:

\(\begin{array}{ccc}{T}_{u}=\frac{1}{\mid S\mid }{\int }_{S}u\text{dS},& & {\Omega }_{u}={I}^{-1}({\int }_{S}\text{GM}\wedge u\text{dS})\end{array}\) eq 2.3-7

The kinematic connection condition sought is therefore the following linear connection between the 3D field on \(S\) and the elements of the displacement torsor of the beam in \(G\):

\(\begin{array}{ccc}\mid S\mid T-{\int }_{S}u\text{dS},& & I(\Omega )-{\int }_{S}\text{GM}\wedge u\text{dS}\end{array}=0\) eq 2.3-8

2.4. Expression of the static connection condition

Returning to the weak formulation of interface balance [éq 2.2-1], we can to derive/to establish the necessary and sufficient conditions for static connection. In fact, we have:

\(\begin{array}{ccc}{\int }_{S}\sigma \text{.}n\text{.}v\text{dS}=R\text{.}{T}_{v}+M\text{.}{\Omega }_{v}& & \forall v\in V\end{array}\) eq 2.4-1

Thanks to the expressions [éq 2.3-7] and to the decomposition of space \(V\), and to the property [éq2.3‑5], we immediately have the three equations:

\(\begin{array}{}F={\int }_{S}\sigma \text{.}n\text{dS}\\ M={\int }_{S}\text{GM}(x)\wedge \sigma \text{.}n\text{dS}\\ {(\sigma \text{.}n)}^{\sigma }=0\text{ou de manière Žéquivalente}{\int }_{S}\sigma \text{.}n\text{.}v\text{dS}=0\text{}\forall v\in {T}^{\perp }\end{array}\) eq 2.4-2

The static connection conditions are therefore:

• transmission of the torsor of the girder forces, (satisfies property P1),

• nullity of the complementary part (« non-beam ») of the 3D stress vector field on the 3D solid section (satisfies the P2 property).

The static and kinematic symmetry (P3 property) will also be noted since the connection conditions are also interpreted as:

• the equality in the sense of least squares between the 3D displacement and the displacement of the beam,

• the equality in the sense of least squares between the stress vector field and the elements of reduction of the torsor of the girder forces.

2.5. Implementation of the connection method

For each connection, the user must define:

S:	the trace of the section of the beam on the 3D massif: it does so with the keywords MAILLE_1et /or GROUP_MA_1; that is to say, it gives the list of surface meshes (\(\mathrm{lma}\)) (assigned with 3D modeling “edge” elements) that geometrically represent this section.
P:	a node (keyword NOEUD_1ou GROUP_NO_1) carrying the 6 classical beam degrees of freedom: \(\mathrm{DX}\), \(\mathrm{DY}\), \(\mathrm{DZ}\), \(\mathrm{DRX}\), \(\mathrm{DRY}\), \(\mathrm{DRZ}\)

Note:

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

• the list of meshes \(\mathrm{lma}\) must represent **exactly the section of the beam. This is an important constraint for meshing.*

For each node, the program calculates the coefficients of the 6 linear relationships [éq 2.3-8] that connect:

• the 6 degrees of freedom of node \(P\),

with the degrees of freedom of**all* the nodes of \(\mathrm{lma}\).

These linear relationships will be dualized, like all linear relationships derived for example from the LIAISON_DDL keyword in AFFE_CHAR_MECA.

The calculation of the coefficients of linear relationships is carried out in several steps:

• calculation of elementary quantities on the elements of \(\mathrm{lma}\): (OPTION: CARA_SECT_POUT3)

• area = \({\int }_{\text{elt}}1;{\int }_{\text{elt}}x;{\int }_{\text{elt}}y;{\int }_{\text{elt}}\mathrm{x2};\mathrm{...}\)

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

• \(A=\mid S\mid\)

• \(G\) position

• inertia tensor \(\Omega\)

• knowing \(G\), elementary calculation on the elements of \(\mathrm{lma}\) of: (OPTION: CARA_SECT_POUT4)

\({\int }_{\text{elt}}\text{Ni};{\int }_{\text{elt}}\text{xNi};{\int }_{\text{elt}}\text{yNi};{\int }_{\text{elt}}\text{zNi}\text{où :}\begin{array}{c}\text{GM}=\left\{x,y,z\right\}\\ \text{Ni}=\text{fonctions de forme de l'éléŽment}\end{array}\)

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