Case 09 — Cantilever beam with rotational spring under tip load
A cantilever beam of finite length is subjected to a concentrated vertical load applied at its free end. The fixed end of the beam is supported by a rotational spring with a specified rotational stiffness. Vertical displacement and rotation at the free end are evaluated. This verification case validates the correct implementation of rotational spring supports and their influence on displacements and rotations in beam elements.
Description
A cantilever beam of length 10 inches is subjected to a vertical concentrated load of P = 10 lbf at its free end. The fixed end of the beam is supported by a rotational spring with stiffness kr = 10 000 lbf-in/rad.
Determine:
- The displacements at the free end (node 2)
Structural scheme
Geometry, boundary conditions, and load applications used in the verification model.
Model parameters
| Parameter | Value |
|---|---|
| Element type | Beam element |
| Material | E = 30 000 ksi |
| Section properties | I = 1 000 in⁴ |
| Boundary conditions | Left Node – constrain Ux and Uy, Rotational Spring constant about Z-axis kr = 10 000 lbf-in/rad |
| Loads | Nodal load −10 lbf |
Numerical results
Displacements
Table results from RodX
Comparison
| Parameter | Analytical solution | RodX | Midas/Civil |
|---|---|---|---|
| Δx(2) | 0.00 | 0.00 | 0.00 |
| Δy(2) | −0.10 | −0.10 | −0.10 |
| θz(2) | −0.01 | −0.01 | −0.01 |
The numerical results obtained with RodX are in agreement with the analytical solution and reference FEA results.
Reference
- Midas verification examples, Midas/Civil Ltd., 2022, example – Static-8.