Follower Force
Follower Force: Theoretical Foundations
What is a Follower Force?
Professor, what is a "follower force"?
A load whose direction changes following deformation. A typical example is internal pressure: when a container deforms, the orientation (normal direction) of the pressure-acting surface changes, and thus the direction of the pressure also changes.
Examples of Follower Forces
| Load | Follower? | Reason |
|---|---|---|
| Internal/External Pressure | Yes | Follows the surface normal direction |
| Gravity | No | Always downward |
| Rocket Thrust | Yes | Follows the vehicle's orientation |
| Wind Load | No (usually) | Fixed direction |
| Contact Force | Yes | Follows the contact surface normal |
Internal pressure is a follower force... When it expands, the area changes and the magnitude of the force changes too.
Both area change and direction change contribute to nonlinear effects. Follower forces are automatically considered when NLGEOM=YES.
Summary
Ziegler's Paradox and the Instability of Follower Forces
The stability of follower forces (forces where the external load follows deformation) is famous for the paradox shown by Wilhelm Ziegler (ETH Zurich) in 1952. There exists "Ziegler's Paradox" where a continuum that should become stable with added viscous damping instead becomes unstable. This demonstrates the peculiarity of follower force systems where static and dynamic stability do not coincide, and is deeply involved in the analysis of flutter-type instability (Beck's column problem).
Computational Methods for Follower Force
Follower Forces in FEM
In nonlinear analysis with NLGEOM=YES, for each iteration:
1. Recalculate the surface normal direction based on the deformed shape
2. Pressure × deformed area = Follower force
3. Reflect in the right-hand side of the global equation
Abaqus: *DLOAD, P (pressure) automatically becomes follower with NLGEOM=YES.
Comparison with Non-Follower Forces
Results can differ by 10-20% depending on whether internal pressure is treated as follower in large deformation (e.g., balloon expansion). With NLGEOM=NO, internal pressure acts fixed to the initial surface.
Summary
Finite Element Formulation of Follower Forces and Load Stiffness Matrix
To incorporate follower forces into FEM, it is necessary to add a "load stiffness matrix" (Kσ_load) that depends on deformation to the tangent stiffness. For uniform pressure (normal direction follower force), the load stiffness matrix becomes anti-symmetric. While the standard full Newton method (Newton-Raphson) can automatically consider follower forces by updating the load stiffness each analysis, note that errors may arise depending on the update frequency in the Modified Newton method.
Follower Force in Practice
Follower Forces in Practice
Problems where follower forces are important:
- Large deformation (expansion) of pressure vessels
- Balloon/airbag inflation
- Rocket thrust direction
- Brake friction force (direction changes with deformation)
Practical Checklist
Thrust Follower Effect in Solid Rockets
The thrust nozzle of a solid rocket is a typical example of a follower force that always generates a rearward resultant force relative to the deformed vehicle body. The Japan Aerospace Exploration Agency (JAXA) applied nonlinear dynamic analysis including follower forces to the flight load analysis of the Epsilon rocket (first launched in 2013) and incorporated into the design standard the conservative evaluation of structural loads during roll programs as 3-5% higher.
Follower Force: Software & Solver Comparison
Tools
All solvers support follower forces with NLGEOM=YES. No difference.
Discovery of Follower Forces: Euler's Compression Buckling Research
The problem of follower forces (follower loads) emerged in nascent form when Leonhard Euler studied rod compression buckling in 1744. Since ABAQUS 6.14, the `FOLLOWER FORCE` option allows explicit specification of follower loads; there is a case where buckling load decreased by up to 23% when the follower nature of nozzle combustion gas pressure was not considered in rocket engine thrust analysis.
Advanced Technology
Advanced Research
Variational Principle for Non-Conservative Force Systems: Extension of Hamilton's Principle
Since follower forces are non-conservative, the usual potential energy minimization principle cannot be used. It is necessary to use an extended version of Lagrange's equations, Hamilton's variational principle (principle of virtual work), to explicitly incorporate the work of non-conservative forces. Research by Bolotin (Moscow) in the 1970s introduced probability theory into the stability analysis of non-conservative forces, laying the foundation for Dynamic stability theory.
Follower Force: Common Issues & Debugging
Troubles
When Stiffness Appears to Decrease with Load Increase in Follower Force Analysis
If Newton-Raphson method convergence worsens in follower force analysis and apparent stiffness decreases as load increments accumulate, the sign or magnitude of the load stiffness matrix may be inappropriate. When using Abaqus pressure follower force (FOLLOWER FORCE TYPE=PRESSURE), set the iterative convergence criterion strictly to 1e-6 or lower and check the residual at each load step. It is good to verify stability by making load steps finer (1/5 to 1/10).
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