剛体要素
Theory and Physics
What is a Rigid Element?
Professor, what is a "rigid element" in FEM? An element that doesn't deform?
Yes. A rigid element does not deform. It is an element that "rigidly connects" two or more nodes so that the displacement/rotation of one is transmitted to the other.
What kind of physical structure does it represent?
Mathematical Formulation
Rigid elements are implemented as Multi-Point Constraints (MPC: Multi-Point Constraint) in FEM. The displacement of dependent nodes (slave nodes) relative to the displacement of the independent node (master node) $\{u_m\}$ is:
$[T]$ is the displacement transformation matrix, based on the equations of rigid body motion.
So the dependent node's displacement is "determined automatically" from the independent node's displacement.
Yes. The degrees of freedom (DOF) of the dependent nodes are eliminated (condensation). This effectively reduces the DOF count.
Types of Rigid Elements
| Type | Nastran | Abaqus | Features |
|---|---|---|---|
| Rigid Coupling (All DOF) | RBE2 | RIGID BODY / COUPLING | Couples all degrees of freedom rigidly |
| Weighted Average (Load Distribution) | RBE3 | *DISTRIBUTING COUPLING | Distributes forces. Does not add stiffness. |
| Rigid Surface | RBAR | *RIGID BODY | Rigid link between two nodes |
Are RBE2 and RBE3 completely different?
Fundamentally different. RBE2 is "rigid coupling" and adds stiffness to the structure. RBE3 is "load distribution" and does not add stiffness. Not understanding this difference leads to serious modeling errors. Details are explained on the RBE2 and RBE3 pages.
Precautions for Rigid Elements
Rigid elements are convenient, but misuse can significantly distort results:
1. Excessive local stiffness from rigid elements — Connection areas become unnaturally stiff
2. Creates stress singularities — Stress becomes inaccurate at the boundary between rigid and deformable elements
3. Over-constraint — Unnecessary degrees of freedom become constrained
So stress at the boundary of a rigid element is not reliable?
The stress in the first one or two layers of deformable elements connected to a rigid element is for reference only. Evaluate stress at a location away from the connection.
Summary
Let me organize the theory of rigid elements.
Key points:
- Non-deforming element — Rigidly connects nodes
- Implemented as MPC — DOF of dependent nodes are eliminated
- RBE2 (rigid coupling) and RBE3 (load distribution) are completely different — Must not be confused
- Beware of excessive local stiffness — Stress at connection points is inaccurate
- Convenient for connecting different element types — Beam-to-solid, shell-to-solid
Understanding RBE2 vs. RBE3 is one of the most important skills in FEM modeling, isn't it?
Exactly. Whether one can correctly distinguish and use these two reveals the skill level of an FEM engineer.
Mechanical Basis of Rigid Elements
Rigid elements are finite elements that behave as ideal rigid bodies with no internal deformation, introduced in the 1960s by Turner (M.J. Turner) et al. for NASA's space structure analysis. Mathematically, all six degrees of freedom of dependent nodes are determined deterministically from independent nodes via a rigid body transformation matrix T. Since actual materials deform elastically, rigid elements can be interpreted as simulating an elastic modulus at least 10 to 100 times greater than the true stiffness.
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, based on the assumption that "acceleration can be ignored because forces are applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ and $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under 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". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (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 typical mistake here: getting the load direction wrong. Intending "tension" but applying "compression" — it 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 and 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 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) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Implementation of MPC (Multi-Point Constraint)
How is the MPC for rigid elements implemented?
There are two methods:
1. Elimination of Dependent DOF (Transformation Method)
Express the DOF of dependent nodes in terms of the DOF of independent nodes and eliminate them from the global equations. The size of the simultaneous equations becomes smaller. Nastran's RBE2 uses this method.
2. Penalty Method
Approximates rigid constraints with "very stiff springs". Not perfectly rigid but practically accurate enough. Too large a penalty value worsens the condition number. Abaqus's KINEMATIC option uses the transformation method, the PENALTY option uses the penalty method.
Which is better?
The transformation method (KINEMATIC) is the basic recommendation. The penalty method is sometimes used when combined with contact (to align with the penalty method used for contact).
Implementation by Solver
How is the MPC for rigid elements implemented?
There are two methods:
Express the DOF of dependent nodes in terms of the DOF of independent nodes and eliminate them from the global equations. The size of the simultaneous equations becomes smaller. Nastran's RBE2 uses this method.
Approximates rigid constraints with "very stiff springs". Not perfectly rigid but practically accurate enough. Too large a penalty value worsens the condition number. Abaqus's KINEMATIC option uses the transformation method, the PENALTY option uses the penalty method.
Which is better?
The transformation method (KINEMATIC) is the basic recommendation. The penalty method is sometimes used when combined with contact (to align with the penalty method used for contact).
| Function | Nastran | Abaqus | Ansys |
|---|---|---|---|
| Rigid Coupling | RBE2 | RIGID BODY / COUPLING, KINEMATIC | CERIG |
| Load Distribution | RBE3 | *COUPLING, DISTRIBUTING | RBE3(MPC) |
| Rigid Link (2 nodes) | RBAR | *RIGID BODY | MPC184 |
| Rigid Surface | RBE2 (multi-node) | *RIGID BODY | TARGE170 |
Rigid Element Connection Patterns
Typical connection patterns:
Beam Element and Solid Element Connection
Beam elements have rotational DOF, but solid elements do not. Connect using RBE2/RBE3:
- RBE2 — Beam endpoint as master, nodes on solid surface as slaves. The cross-section moves rigidly (can be too stiff).
- RBE3 — Beam endpoint as slave (reference point), nodes on solid surface as masters (load distribution points). Does not add stiffness (more realistic).
RBE3 is a "gentler" connection, isn't it?
Exactly. RBE2 is like a "rigid weld" connection, RBE3 is like a "balanced load transfer" connection. Actual bolted joints in structures are often closer to RBE3.
Summary
Let me organize the numerical methods for rigid elements.
Key points:
- Transformation method (KINEMATIC) is the basic recommendation — Accurate via DOF elimination
- Penalty method is for use with contact — Approximate rigidity
- Beam-to-solid connection — RBE2 (stiff) vs. RBE3 (soft)
- Stress at connection points is inaccurate — Evaluate at a location 1-2 elements away
Comparison of Rigid Element Implementation Methods
There are three main methods for implementing rigid elements: ① Penalty method, ② Lagrange multiplier method, ③ Direct stiffness condensation. The penalty method is easy to implement but the choice of penalty coefficient affects solution accuracy. The Lagrange multiplier method is mathematically rigorous but increases the size of simultaneous equations. Nastran adopts the direct condensation method, which allows calculation while maintaining the system equation size by eliminating dependent DOF.
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 DOF increase by about 2-3 times. Recommendation: When stress evaluation is important.
Related Topics
なった
詳しく
報告