RBE3加重平均要素
Theory and Physics
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 → Flange becomes rigid. No deformation of the flange appears. Stress concentration around it.
- RBE3 → 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 — Does not add stiffness. Fundamental difference from RBE2.
- Reference point displacement = weighted average of surroundings — Force is distributed with weighting.
- Independent/dependent reversed from RBE2 — Surrounding nodes are independent, reference point is dependent.
- RBE3 should be used for load distribution — RBE2 causes excessive stiffness.
- Weight $w_i = 1$ (equal distribution) is standard — 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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored." It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is not correct. Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | In mm system: N, in m system: N, keep unified. |
Numerical Methods and Implementation
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 — Different syntax, same function.
- Reference point requires separate constraint — Structure is unstable with RBE3 alone.
- Independent node DOFs — Solids: only 123, Shells/Beams: 123456.
- Weight $w = 1$ (equal) is standard — 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 × Ai / ΣWj×Aj) × 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.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommendation: For cases where stress evaluation is critical.
Full integration vs Reduced integration
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