Beam on elastic foundation

A reinforced-concrete grade beam (or continuous footing) of finite length resting on the soil, analysed as a beam on a Winkler foundation and designed to ACI 318-19 with ASCE 7-22 loads. The soil is a bed of independent springs: the contact pressure at a point is the modulus of subgrade reaction ks times the settlement there, so a stiff beam spreads its loads and a flexible one concentrates the pressure under them. The calculation gives the settlement, soil pressure, shear and moment along the beam for every load combination, checks the peak service pressure against the allowable bearing pressure and the settlement against your limit, and designs the concrete section for flexure and shear.

Inputs of the beam on elastic foundation #

• Beam: length L, width b in contact with the soil, depth h, and whether the self-weight is added to the dead load. The beam is free at both ends: the soil alone holds it up (lengths up to 80 ft, 24 m).
• Loads, unfactored and per load type (D, L, Lr, S, R, W, E): point loads P (downward positive) with an optional couple M (clockwise positive) at any position x from the left end, and line loads, uniform or linearly varying, over the full length or any part of it (From = To = 0 loads the full length). The ASCE 7-22 2.4.1 combinations are used for the soil pressure and the settlement, 2.3.1 for the concrete; wind and earthquake act in both directions.
• Modulus of subgrade reaction ks. It is not a soil constant: it depends on the loaded width and shape and on the soil, so it is your input, with the value the geotechnical engineer gives for this beam. When you only have a 1 ft (0.3 m) plate-load test value ks1, choose "From a 1 ft plate-load test": the calculation converts it to the beam width with Terzaghi's (1955) expressions, ks = ks1 ((B + 1)/2B)² for granular soil and ks = ks1 (1/B)(m + 0.5)/(1.5 m) for cohesive soil (B = b in ft, m = L/B), and shows the steps. Nothing is converted unless you choose it.
• Soil springs: compression only (the default: the soil cannot pull the beam down, so the springs that would be in tension are removed and the beam re-analysed until the contact zone settles, per combination) or linear (the classical Hetényi solution; any tension in the springs is then reported as a failed contact check).
• Allowable bearing pressure (gross, from the geotechnical report) and the settlement limit (your limit, commonly 1 in. total for buildings).
• Beam stiffness: Ec by ACI 318-19 19.2.2.1(b) (57,000√f'c), 19.2.2.1(a) (wc^1.5 × 33√f'c) or entered; the moment of inertia gross, a fraction of Ig, or entered. A stiffer beam attracts more moment and spreads the pressure; a cracked (softer) beam concentrates the pressure under the loads: Table 6.6.3.1.1(a) gives 0.35Ig for beams at factored loads and 6.6.3.2.2 lets you use 1.4 times that (0.49Ig) for service loads. Consider both; the calculation tells you when the service moment exceeds the cracking moment.
• Concrete and reinforcement: f'c, fy, λ, clear covers (3 in. cast against earth at the bottom), top and bottom bars (size and number, continuous over the length), stirrups (size, legs, spacing) or none.

Auto-design (editor toolbar) varies the width (4-in. steps) and depth (2-in. steps), bottom and top bars (size and count), stirrup size, spacing and legs; the length, loads, soil and limits are kept. Objectives: lowest cost index (concrete, formwork, reinforcement); least concrete, then cost. Every other input is kept; see Auto-design.

Checks performed #

The drawings (screen, report, DXF and PDF sheet) show the beam elevation on its soil springs with the loads by type (point loads keyed P1, P2, ... with their values in the sheet notes when there are more than three positions), the load positions and the length, the service soil pressure envelope (with the settlement at the labelled peaks) and the factored shear and moment envelopes under it, and the typical section with the bars, stirrups and cover. Bars are drawn continuous over the length as designed; cut-offs, hooks, splices and column dowels are not part of the calculation and the sheet notes say so.

Limitations #

• Winkler model: the springs are independent, so the soil next to a loaded area does not settle with it (no shear coupling, no elastic half-space); the results are only as good as ks. Vary ks (for example half and twice the value) when it is uncertain.
• Constant section and constant ks along the beam; no axial load, no torsion, no supports other than the soil (piles or walls under the beam are not modelled); the length is limited to 80 ft (24 m).
• Shear is checked at its largest value anywhere, conservatively; punching and bearing under the columns are not checked (use the spread or combined footing calculations where they govern).
• The plate-load conversion follows Terzaghi as given in the textbooks (Bowles, Foundation Analysis and Design, 9-6); the original paper is not on file and the value should be confirmed by the geotechnical engineer.

Verification examples #

Verification status: 20 values compared in 7 examples (5 published, 2 hand calculations from the cited equations): 18 within tolerance, 2 recorded deviations. Code basis: partial. Only some provisions were read in the target edition; the rest rests on older editions, examples or summaries.

backend/tests/test_verify_elastic_foundation_beam.py; the finite elements are also compared with Hetényi's closed forms for the infinite beam (force and couple), the semi-infinite beam and the finite beam with free ends in backend/tests/test_elastic_foundation_beam.py (within 0.1 %).