RBE2剛体要素
Theory and Physics
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 — Slaves move rigidly according to the master's motion
- Adds Stiffness — Connection becomes infinitely stiff
- Eliminates DOFs — Slave DOFs are reduced
- Connection Stresses are Inaccurate — Evaluate one or two elements away
- Use with Caution — 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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? 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, assuming "forces are applied slowly enough that acceleration can be ignored". It 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. So a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall 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 for a smoother ride. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
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 stress-strain relationship is linear
- Isotropic material (unless specified otherwise): 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 equilibrium 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. Beware of unit 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) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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 line—very 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 — Constrain all DOFs. Complete rigid connection
- 123 — Constrain only translations. Rotations free (pin-like connection)
- 12 — 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 — Concise notation
- Abaqus uses *COUPLING with KINEMATIC option — Equivalent functionality
- Selecting DOFs to constrain is crucial — Don't constrain unnecessary DOFs
- Beware of over-constraint — 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 + θ_ind × 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.
Linear Elements (1st Order)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but DOFs increase by about 2-3x. Recommended: when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Quadratic convergence within convergence radius, but computationally expensive.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying full load at once, apply in small increments. The arc-length method (Riks method) can trace beyond limit points on load-displacement curves.
Analogy: Direct Method vs Iterative Method
The direct method is "solving simultaneous equations by long division correctly...
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