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Theory and Physics
What is Co-rotational Formulation?
Professor, what is "co-rotational formulation"?
Co-rotational formulation is an optimal method for large rotation problems of beams and shells. It makes the local coordinate system of each element follow the rotation, allowing the use of small deformation theory within the element.
Even if the element rotates, the deformation inside can be treated as "small deformation"?
Yes. Even for large overall rotations, if each element is sufficiently small, the deformation within the element is infinitesimal. Think of it as "large rotation = accumulation of many small rotations".
The Problem of Large Rotations
Finite rotations in 3D are non-commutative (the result changes depending on the order of rotations). This complicates the formulation for large rotations.
The order of rotations affects the result... This can be ignored for small rotations, but is important for large rotations.
Co-rotational formulation manages the rotation matrix $[R]$ for each element, correctly handling the non-commutativity of rotations. OpenSees' nonlinear beam elements and Abaqus' B31 elements are co-rotational based.
Summary
Key Points:
- The local coordinate system of each element follows the rotation — Small deformation within the element
- 3D large rotations are non-commutative — The order of rotations affects the result
- Optimal for large deformations of beams/shells — Collapse analysis of frame structures
- OpenSees, Abaqus B31 — Co-rotational based beam elements
The Invention of Co-rotational Coordinates and Argyris
Co-rotational formulation was independently developed by Argyris (University of Stuttgart) and Wempner (Georgia Tech) in the 1960s-70s. By having each element possess a "local coordinate system that rotates and translates with itself," it can handle large displacements/rotations while keeping local deformations in the linear range. Argyris described it as "slightly deformed rigid body motion," greatly simplifying the geometric nonlinear formulation for large deformation analysis.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "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, assuming "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. 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 forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). 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 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 ending up with "compression"—sounds like a joke, but it actually happens when coordinate systems rotate 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 away. That's because vibration 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 vibration energy to improve ride comfort. What if damping were zero? Buildings would keep shaking 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 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: Large deformations and large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity and creep require constitutive law extensions.
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 inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| 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 Co-rotational
Co-rotational formulation algorithm:
1. Set the initial local coordinate system for each element.
2. Extract the rigid body rotation from the deformed element's nodal coordinates (Polar decomposition).
3. Update the local coordinate system by the rigid body rotation amount.
4. Calculate the small deformation stiffness matrix in the updated local system.
5. Transform to the global system.
"Extraction of rigid body rotation" is the core, isn't it.
Use polar decomposition ($[F] = [R][U]$, where $[R]$ is rotation, $[U]$ is stretch) to separate the rigid body rotation.
Summary
Update Procedure of the Co-rotational Method
In the co-rotational method, each load step proceeds in the order: ① Update the element's local coordinate system in the current configuration, ② Calculate local displacements, ③ Transform the internal force vector to the global coordinate system, ④ Assemble the tangent stiffness matrix. Updating local displacements using spinor algebra instead of vectorial rotation reduces numerical error in large rotations to less than 1/10. Abaqus' beam and shell elements use this method.
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 deformations. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. 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 for the situation.
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 the tangent stiffness matrix every iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies the full load not all at once, but in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Co-rotational in Practice
Used for seismic collapse analysis of steel frames, large deformations of marine risers, and deformation tracking of flexible robots.
Practical Checklist
Springback Analysis of Thin Sheet Metal Pressing
"Springback" (shape change due to elastic recovery after press forming) is a typical application example of co-rotational large deformation analysis. In pressing high-strength steel (980MPa class), springback of 5-10mm can occur after punch release, directly affecting dimensional accuracy. Toyota, Honda, and Subaru all use co-rotational formulation-based sheet forming analysis (Autoform, PAM-STAMP, etc.) as standard in design for springback prediction.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If the mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
How to Think About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong, right? "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Co-rotational Tools
Selection Guide
Accuracy of ANSYS Beam188 Large Rotation Analysis
ANSYS's BEAM188 adopts Timoshenko beam theory + co-rotational formulation, enabling high-precision large deformation/large rotation analysis for beams with slenderness ratio L/D>10. With the full Newton method setting KEYOPT(2)=2, it converges even for problems involving rotations over 90°. It is also used for robot arm endpoint accuracy analysis and design verification of deployable space structures (unfolding of folded solar panels).
The Three Most Important Questions for Selection
- "What are you solving?": Does the necessary physical model/element type for co-rotational formulation have support? For example, in fluids, the presence of LES support; in structures, the ability to handle contact/large deformations makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) in cars.
- "How far will you expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Research in Co-rotational
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