RBE2 Rigid Element
RBE2 Rigid Elements: Theoretical Foundations
What is RBE2?
Professor, RBE2 is the most commonly used rigid body element in FEM, right?
Yes. RBE2 (Rigid Body Element, Type 2) is a Nastran term, but the concept exists in all FEM solvers. It rigidly connects multiple dependent nodes (slaves) to one independent node (master).
Operating Principle
Given the displacement of the independent node $\{u_m\} = (u, v, w, \theta_x, \theta_y, \theta_z)$, the displacement of dependent node $i$ is:
Here, $\Delta x_i = x_i - x_m$, etc., are the distances from the independent node.
That's the equation of rigid body kinematics. "When the master node moves, the slave nodes also move rigidly along with it."
Exactly. RBE2 enforces a rigid body motion constraint. Since the slave node displacements are uniquely determined from the independent node's displacement and rotation, the specified DOFs of the slave nodes are eliminated from the global equations.
Effects of RBE2
RBE2 adds stiffness to the structure. The connected region becomes undeformable.
| Effect | Explanation |
|---|---|
| Stiffness Addition | Connection becomes infinitely stiff |
| DOF Reduction | Slave DOFs are eliminated |
| Force Transmission | Forces acting on the master are rigidly distributed to slaves |
| Displacement Unification | Slave displacements become dependent on the master |
Typical Use Cases
What does it mean to impose a "surface remains planar" condition with RBE2?
For example, if you want to impose a uniform axial displacement condition on the end face of a pressure vessel, connect all nodes on that end face with an RBE2. If you leave the master node's axial displacement free, the end face moves as a single body in the axial direction, but the displacement is not constrained.
Cautions for RBE2
Most important caution: RBE2 increases structural stiffness.
Since real structural joints are not perfectly rigid, connecting them with RBE2 will give overly stiff results, right?
Exactly. At RBE2 connections:
- Stress concentration occurs in the surrounding deformed elements
- Overall stiffness is overestimated
- Deflection is underestimated
"Do not trust stresses near RBE2s" is a golden rule of FEM.
Summary
Let me summarize the theory of RBE2.
Key points:
- Rigid Connection โ Slaves move rigidly according to the master's motion
- Adds Stiffness โ Connection becomes infinitely stiff
- Eliminates DOFs โ Slave DOFs are reduced
- Connection Stresses are Inaccurate โ Evaluate one or two elements away
- Use with Caution โ Use only where absolutely necessary
So RBE2 is an element that's "convenient but can distort results if overused."
Exactly. RBE2 is a tool like a "hammer". Don't hit everything with RBE2; use it only where rigid body connection is truly needed. For load distribution, RBE3 should be used.
RBE2 Multi-Point Constraint Equations
RBE2 (Rigid Body Element 2) is a multi-point constraint (MPC) element introduced in NASA's Nastran in the late 1960s. It transmits displacement from an independent node (independent DOF) to dependent nodes (dependent DOFs) via rigid body transformation. Mathematically, the constraint equations are incorporated into the global stiffness matrix using the Lagrange multiplier method or penalty method, enforcing displacement compatibility across all degrees of freedom.
Computational Methods for RBE2 Rigid Elements
RBE2 Implementation in Various Solvers
How is RBE2 set up in different solvers?
Nastran
```
RBE2, 100, 1000, 123456, 2001, 2002, 2003, 2004
```
- 100: Element ID
- 1000: Master node
- 123456: DOFs to constrain
- 2001~2004: Slave nodes
Abaqus
```
*RIGID BODY, REF NODE=1000, ELSET=slave_nodes
```
or
```
*COUPLING, CONSTRAINT NAME=rbe2_1, REF NODE=1000
*KINEMATIC
slave_surface, 1, 6
```
Ansys
```
CERIG, 1000, 2001, ALL
CERIG, 1000, 2002, ALL
```
Nastran's RBE2 can specify many slaves in one lineโvery concise.
Nastran's card format is optimized for aerospace workflows that use many RBE2s. Abaqus's *COUPLING and Ansys's CERIG can achieve the same, but the notation differs.
Selecting DOFs to Constrain
Can we constrain only some DOFs instead of "123456"?
Yes. For example, using "123" (constrain only translations, rotations free) allows slave nodes to rotate relative to the master.
Typical choices:
- 123456 โ Constrain all DOFs. Complete rigid connection
- 123 โ Constrain only translations. Rotations free (pin-like connection)
- 12 โ Constrain only in-plane translations
Choosing the wrong DOFs can lead to over-constraint, right?
Using 123456 where not all DOFs need constraining will lock even rotational DOFs. For example, if you only want "in-plane displacements of slaves to follow the master", constrain only 12.
Summary
Let me organize the implementation details of RBE2.
Key points:
- Nastran allows specifying many slaves in one line โ Concise notation
- Abaqus uses *COUPLING with KINEMATIC option โ Equivalent functionality
- Selecting DOFs to constrain is crucial โ Don't constrain unnecessary DOFs
- Beware of over-constraint โ 123456 is not always correct
RBE2 Rigid Displacement Matrix
The RBE2 transformation matrix T is composed of a 6-column matrix that calculates dependent node translational displacements from the independent node's 3 translational and 3 rotational components. Given the arm length r=[rx,ry,rz], the dependent translational displacement is u_dep = u_ind + ฮธ_ind ร r. This linear transformation introduces errors in large deformation problems, so in practical nonlinear analysis, replacing RBE2 with RBE3 or contact conditions is standard practice.
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