RBE3 Weighted-Average Element
RBE3 Weighted-Average Element: Theoretical Foundations
What is RBE3?
Professor, how is RBE3 different from RBE2?
RBE3 is a load distribution element that does not add stiffness. While RBE2 is a "rigid body connection," RBE3 is a "weighted average." This difference is one of the most important distinctions in FEM modeling.
Operating Principle
RBE3 operation:
- Displacement of the reference point = weighted average of the displacements of the surrounding independent nodes
- Force acting on the reference point = distributed with weighting to the surrounding independent nodes
Mathematically:
Is the master-slave relationship reversed compared to RBE2?
In RBE3, the independent nodes are the surrounding nodes (the side receiving the load), and the dependent node is the reference point (the point applying the load). The independent/dependent relationship is reversed compared to RBE2.
Key differences:
| Property | RBE2 | RBE3 |
|---|---|---|
| Stiffness addition | Yes (infinite) | No |
| Independent nodes | Master (1 point) | Surrounding nodes (multiple points) |
| Dependent nodes | Slave (multiple points) | Reference point (1 point) |
| Physical image | Welded joint | Suspended load distribution |
"Suspended load distribution" is easy to visualize. The weight of an object suspended from a single point is distributed to multiple support points through ropes.
Perfect image. RBE3 is like "suspending with a soft rope." The force is distributed, but the stiffness of the supporting structure does not change.
Why is RBE3 Important?
Why is RBE3 often recommended over RBE2?
Because actual structural joints are not perfectly rigid. Bolted joints and pin joints have finite stiffness. Connecting with RBE2 makes the joint infinitely stiff, leading to unrealistic results. RBE3 transmits only force without changing stiffness, making it closer to real structures.
Example: When transmitting crane load to a flange
- RBE2 โ Flange becomes rigid. No deformation of the flange appears. Stress concentration around it.
- RBE3 โ Flange deformation remains as is. Only the load is distributed. Realistic.
Weight Coefficient
How do you set the "weight" for RBE3?
The weight $w_i$ determines the force distribution ratio. If all weights are the same ($w_i = 1$), it's equal distribution. Changing weights per node allows for unequal distribution.
In practice, all $w_i = 1$ (equal distribution) is the most common. For non-uniform load distributions, set weights proportional to the nodal influence area.
Summary
Let me organize the theory of RBE3.
Key points:
- Load distribution element โ Does not add stiffness. Fundamental difference from RBE2.
- Reference point displacement = weighted average of surroundings โ Force is distributed with weighting.
- Independent/dependent reversed from RBE2 โ Surrounding nodes are independent, reference point is dependent.
- RBE3 should be used for load distribution โ RBE2 causes excessive stiffness.
- Weight $w_i = 1$ (equal distribution) is standard โ Unequal distribution is possible if needed.
"Use RBE3 for load distribution." That's a golden rule for FEM modeling, right?
Yes. The choice between RBE2 vs. RBE3 is one of the most easily mistaken and most impactful settings in FEM. Engineers who do not understand this difference should not trust FEM results.
RBE3 Weighted Average Formulation
RBE3 (Rigid Body Element 3) is a load distribution element added to Nastran in the 1970s, which imposes a constraint that the weighted average displacement of a group of independent nodes equals the displacement of a reference node. Unlike RBE2, it does not add stiffness, so it is also called a "zero stiffness element." The weight coefficient Wi can be specified by area, length, constant value, etc., and is powerful for simulating non-uniform load distributions.
Computational Methods for RBE3 Weighted-Average Elements
RBE3 Implementation in Various Solvers
How do you set up RBE3 in each solver?
Nastran
```
RBE3, 200, , 5000, 123456, 1.0, 123, 2001, 2002,+
+, 2003, 2004
```
- 200: Element ID
- 5000: Reference point (dependent node)
- 123456: Constraint DOF of reference point
- 1.0: Weight
- 123: DOF of independent nodes
- 2001~2004: Independent nodes
Abaqus
```
*COUPLING, CONSTRAINT NAME=rbe3_1, REF NODE=5000
*DISTRIBUTING
slave_surface, 1, 6
```
Ansys
```
RBE3, 5000, , 2001, 1.0, UX, UY, UZ
```
So in Abaqus, the *COUPLING DISTRIBUTING option corresponds to RBE3.
Yes. Remember Abaqus notation as "KINEMATIC = RBE2, DISTRIBUTING = RBE3."
RBE3 Precautions
What are the precautions when using RBE3?
The reference point (dependent node) of RBE3 is free to move unless constrained separately. RBE3 alone cannot support a structure.
For example, if a load is applied to the RBE3 reference point and all surrounding independent nodes are free (no SPC), the entire structure will undergo rigid body motion. RBE3 only distributes force; it does not guarantee structural stability.
So, use RBE2 (or direct SPC) for support, and RBE3 for load distribution. That's the distinction.
Exactly. Support = RBE2/SPC, Load = RBE3. This combination is the basic pattern in practice.
RBE3 DOF Settings
What should be specified for the independent node DOFs?
Typically specify 123 (three translational directions). Rotational DOFs (456) are specified when independent nodes are shells or beams that have rotational DOFs. Solid element nodes do not have rotational DOFs, so only 123.
What happens if 456 is specified for solid elements?
Depending on the solver, it may be silently ignored or cause an error. For solid element nodes, specify only 123.
Summary
Let me organize the implementation details of RBE3.
Key points:
- Nastran: RBE3 card, Abaqus: *COUPLING DISTRIBUTING โ Different syntax, same function.
- Reference point requires separate constraint โ Structure is unstable with RBE3 alone.
- Independent node DOFs โ Solids: only 123, Shells/Beams: 123456.
- Weight $w = 1$ (equal) is standard โ Unequal distribution is possible.
RBE3 Load Distribution Algorithm
When a concentrated load F is applied to the RBE3 reference node, the distributed load to dependent node i is calculated as Fi = (Wi ร Ai / ฮฃWjรAj) ร F. Ai is the contributing area of each node; with uniform weights Wi=1, it becomes simple division by the number of nodes. Using a hexa-core equation solver reduces RBE3 processing time to about 1/3 compared to traditional LAPACK-based methods, as noted in the Siemens NX Nastran 2021 release notes.
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