Simply-supported beam on a Winkler elastic foundation (CSI 1-013)

• elasticBeamColumn — Euler-Bernoulli (bending only)
• Point Springs (zeroLength) ⇒ Winkler foundation
• k_node = k · tributary length (one per interior node)
• Simply supported, central point load
• Deflection & moment vs exact continuum (Timoshenko)

A prismatic beam spanning between a pin and a roller rests on a continuous elastic (Winkler) foundation and carries a single point load at midspan. The foundation reaction is proportional to the local deflection, so the governing equation is EI y⁗ + k y = 0 away from the load — the classic beam-on-elastic-foundation problem with a closed-form continuum solution. STRIX represents the distributed foundation with production Point Springs (a vertical zeroLength spring at every interior node, stiffness k times the nodal tributary length) and runs the ordinary linear-static pipeline. The lumped-spring discrete model converges to the exact continuum center deflection and moment as the mesh is refined — the same problem, and the same independent reference, documented in CSI SAP2000 verification example 1-013.

1. Center deflection Uzexact continuum: EI y'''' + k y = 0, pinned-pinned, central P (Timoshenko 1956 Prob 3 p23)= −0.089333 in
2. Center moment Myexact continuum center moment (sagging), Timoshenko 1956 Prob 3 p23= 17697.995034 kip-in

CSI SAP2000 Software Verification Example 1-013 'Simply Supported Beam on Elastic Foundation'; independent reference = Timoshenko, Strength of Materials Part II, 1956, Problem 3, p.23 (exact continuum solution of EI y'''' + k y = 0, pinned-pinned, central load). Reproduced from scratch by sb7Theory() to −0.0893327 in / 17697.995 k-in, matching CSI's published −0.08933 in / 17698 k-in.

Unlike the exact cubic beam element, the lumped nodal foundation is a first-order approximation of the continuous Winkler support, so the discrete model approaches the exact continuum solution as the mesh is refined. The center deflection is essentially converged by a few elements per half-span; the center moment tightens monotonically toward the reference. At the finest mesh (100 elements per half, matching the reference discretization) STRIX reproduces both the exact continuum deflection and moment to under 0.001% — the same independent Timoshenko reference that CSI 1-013 uses.

1. Timoshenko, S. — Strength of Materials, Part II. Beam on elastic foundation, Problem 3, p.23 (exact continuum solution).
2. Hetényi, M. — Beams on Elastic Foundation. University of Michigan Press, 1946. Winkler-foundation theory, EI y⁗ + k y = q.
3. CSI — SAP2000 Software Verification, Example 1-013: Simply Supported Beam on Elastic Foundation (independent reference, Timoshenko 1956).
4. OpenSees — element zeroLength + uniaxialMaterial Elastic (nodal foundation spring); elasticBeamColumn (Euler-Bernoulli).

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SB7.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SB7/

sha256

(pending publish)