Case 4: In-Plane Bending of a Cantilever Beam Subjected to Concentrated Moment

Video Demonstration

Overview

This benchmark test investigates the in-plane pure flexural bending of a cantilever beam subjected to a concentrated moment at its free end.
The problem has been widely studied in the literature [Simo & Vu-Quoc (1986), Ibrahimbegovic (1995)] and serves to validate beam formulations for large in-plane rotations.

Geometry and Setup

Beam length: L = 10 m
Initial configuration: straight cantilever, clamped at the left end
Discretisation: 10 CVs (tested also with 5, 20, 40 CVs for convergence)

Material Properties
Axial rigidity: EA = 1 × 10⁴ N
Shear rigidity: GA₂ = GA₃ = 5000 N
Flexural rigidity: EI₂ = EI₃ = 100 Nm²
Torsional rigidity: GJ = 100 Nm²
Cross-section radius: r = 0.2 m
Young's modulus: E = 7.95 × 10⁴ Pa
Poisson's ratio: ν = 0

Boundary Conditions

Left end: clamped (fixed displacements and rotations).
Right end (free end): subjected to a concentrated moment about the z-axis:
- Example: M_z = 20π Nm applied in 4 increments.
Analytical solution (Euler formula):
[ ψ_z = \frac{M_z L}{EI}, \quad w_x = L - \frac{L}{ψ_z/2} \sin(ψ_z/2) \cos(ψ_z/2), \quad w_y = \frac{L}{ψ_z/2} \left( \sin(ψ_z/2) \right)^2 ]

Numerical Setup

Discretisation: 10 CVs (refined up to 40 CVs for convergence).
Newton–Raphson solver with average of 6 iterations per increment.
Execution time: < 0.5 s for 10 CVs.

Results

Deformation Pattern

For M_z = 20π Nm (ψ_z = 2π), the beam bends into a full circle.

How to Run

Execute:
```
./Allclean
./Allrun
```

Post-process in ParaView (WarpByVector using pointW).

References

Simo, J.C., & Vu-Quoc, L. (1986). A three-dimensional finite-strain rod model. Part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering, 58, 79–116.
Ibrahimbegovic, A. (1995). Computational aspects of vector-like parametrization of three-dimensional finite rotations. International Journal for Numerical Methods in Engineering, 38(21), 3653–3673.