Case 03 — Beam on elastic supports
Verification of vertical displacement of a steel beam supported by two identical vertical elastic springs. A concentrated vertical load is applied at an intermediate node along the beam span. This verification case validates the correct implementation of vertical elastic supports (spring elements) and their interaction with beam elements in the RodX finite element solver.
Description
A steel beam is supported at its ends, A and B, by two identical vertical springs. Each spring has a stiffness of 45 kN/m and is initially unstressed. A vertical concentrated load of 3 kN is applied at node C, located 1.0 m from support A and 2.0 m from support B. The self-weight of the beam is neglected, and linear elastic behavior is assumed.
Determine:
- Vertical displacement at node C.
Structural scheme
Geometry, boundary conditions, and load applications used in the verification model.
Model parameters
| Parameter | Value |
|---|---|
| Element type | Beam element |
| Material | Steel |
| Young's modulus | 2.0 × 10⁸ kN/m² |
| Section properties | I = 4.6875×10⁻⁶ m⁴ |
| Boundary conditions | Ux restrained, Uy – spring with ky = 45 kN/m |
| Loads | Concentrated load −3 kN applied at node C in the Y direction |
Numerical results
Displacements
Table results from RodX
Comparison
| Node | Analytical solution | RodX | Midas/Civil |
|---|---|---|---|
| Δy(C) | -38.5 | -38.5 | -38.5 |
The numerical results obtained with RodX are in agreement with the analytical solution and reference FEA results.
Reference
- Hibbeler, R. C. Mechanics of Materials, 10th Edition (SI Units), Pearson Education, 2017, Example 12.16, p. 648.