Co-rotational Formulation
Co-rotational Formulation: Theoretical Foundations
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.
Computational Methods for Co-rotational Formulation
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.
Co-rotational Formulation in Practice
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.
Co-rotational Formulation: Software & Solver 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).
Advanced Co-rotational Formulation: Modern Research & Trends
Advanced Research in Co-rotational
Recent research directions:
- Machine learning-accelerated convergence prediction for large rotation problems
- Hybrid co-rotational formulation for structures with local buckling
- Real-time co-rotational analysis for digital twins of flexible structures