In-Plane Cantilever Subjected to an End Follower Load

Tutorial Aims

This tutorial demonstrates how to:

• simulate a geometrically nonlinear quasi-static beam problem
• apply a non-conservative circulatory follower force
• use pseudo-time to increment a static load
• compare free-end displacements with literature reference results
• perform a mesh sensitivity study

The case is the second benchmark presented in the beamFoam paper.

Prerequisites

• A compiled beamFoam installation
• A sourced, supported OpenFOAM.com environment
• gnuplot with the pdfcairo terminal for generating the supplied plots
• ParaView for visualising the deformed beam

Problem Description

A straight cantilever is clamped at its left end and subjected to a circulatory follower force at its right end. Unlike a force with a fixed global direction, the follower force rotates with the instantaneous end configuration. It is therefore configuration-dependent and non-conservative.

The load is increased linearly from zero to 134 kN over 1000 pseudo-time increments. Both time-derivative schemes are set to steadyState, so pseudo-time controls load incrementation rather than physical dynamics.

The corresponding rigidities are approximately:

• flexural rigidity: EI = 3.5e7 Nm^2
• shear rigidity: GA = 1.61538e8 N

Case Setup

Beam Geometry and Nonlinear Controls

constant/beamProperties defines the circular beam and the nonlinear solution controls:

beams
(
    beam_0
    {
        crossSectionModel circle;

        circleCrossSectionModelDict
        {
            radius 0.65828153;
        }

        length      100;
        nSegments   40;
        E           2.3731844e8;
        G           1.1865922e8;
    }
);

beamModel coupledTotalLagNewtonRaphsonBeam;

The model uses:

• nCorrectors 1000
• solutionTol 1e-10
• residualTol 1e-8

Boundary Conditions and Loading

The left patch is clamped:

• 0/W: zero fixed displacement
• 0/Theta: zero fixed rotation

At the right patch:

• 0/W uses followerForceBeamDisplacementNR
• 0/Theta uses momentBeamRotationNR with a zero applied moment

The follower force is read from constant/timeVsFollowerForce:

(
    (0 (0 0 0))
    (1 (0 0 1.34e5))
)

The force direction is updated as the free-end configuration changes. This boundary condition is the defining feature of the benchmark.

Quasi-Static Solution and Output

system/fvSchemes selects steadyState for both time-derivative schemes. system/controlDict advances from pseudo-time 0 to 1 with deltaT = 0.001.

The enabled function objects write:

• right-end displacement history
• left-end resultant force and moment history
• nonlinear convergence information

Running the Tutorial

From this tutorial directory:

./Allclean
./Allrun

The script creates the beam mesh, runs beamFoam and generates displacementPlot.pdf.

The principal output files are:

• log.createBeamMesh
• log.beamFoam
• displacementPlot.pdf
• postProcessing/0/beamDisplacements_right.dat

The default plot compares the calculated free-end displacement components against referenceResultsSimo.dat.

Post-Processing

To inspect the changing beam configuration:

touch case.foam
paraview case.foam

Apply Warp By Vector using pointW. Inspect multiple pseudo-time directories to see how the beam direction and follower-force direction evolve as the load increases.

The displacement history contains the free-end components. allPlots.gnuplot combines this history with AppliedLoad.dat to plot displacement against load.

Expected Results

The paper reports:

• good agreement with the Simo and Vu-Quoc reference solution
• approximately four to six Newton iterations per load increment
• closer agreement with the reference solution as the mesh is refined
• second-order spatial convergence of the final free-end displacement

The important validation quantities are the in-plane free-end displacement components w_x and w_z as the load increases to 134 kN.

Mesh Sensitivity Study

Run:

./runSweep.sh

The script runs meshes with 5, 10, 20 and 40 segments. It writes:

• timing_summary.txt
• displacement histories under dispResults/
• AppliedLoad.dat
• dispComparison_Wx_MeshSizes.pdf
• dispComparison_Wz_MeshSizes.pdf

The script generates AppliedLoad.dat from the 20-segment displacement history and compares the mesh results with referenceResultsSimo.dat.

The sweep modifies nSegments in constant/beamProperties and leaves the last tested value, 40, in place.

The beamFoam paper additionally reports a convergence study using 10, 20, 40 and 80 control volumes. Extend the mesh_sizes list in runSweep.sh if the 80-segment result is required.

Troubleshooting

• If the force behaves as a fixed global load, confirm that 0/W uses followerForceBeamDisplacementNR, not forceBeamDisplacementNR.
• If the load-displacement plot fails, confirm that postProcessing/0/beamDisplacements_right.dat exists and that gnuplot can find AppliedLoad.dat and referenceResultsSimo.dat.
• If nonlinear convergence deteriorates near the final load, reduce deltaT to use smaller load increments.
• Run ./Allclean before repeating the standard case after changing the mesh.

References

• Bali, S., Taran, A., Tuković, Ž., Pakrashi, V., and Cardiff, P. (2025). beamFoam: A Cell-Centred Finite Volume Solver for Nonlinear Geometrically-Exact Beams in OpenFOAM. OpenFOAM Journal, 5, 180-210.
• Argyris, J., et al. (1981). Nonlinear Finite Element Analysis of Elastic Systems under Nonconservative Loading: Natural Formulation. Part I: Quasistatic Problems.
• Simo, J. C., and 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.