Subgrade Reaction Modulus k Calculator — Winkler, Plate Load Test and Correlations
The subgrade reaction modulus k (or coefficient of subgrade reaction) is the parameter that characterises the soil as a "spring" in the Winkler model, used for designing slabs, ground slabs, rigid pavements and foundations on elastic ground. This calculator provides k from the plate load test (BS 1377-9), from correlations with CBR (Terzaghi 1955) and from the conversion ks1 → ks(B) due to scale effect according to Terzaghi-Peck. Essential in the design of industrial structures with raft foundations, transmission towers, airport pavements and machine foundations under cyclic loading.
What is it and when is it applied?
The Winkler 1867 model idealises the soil as a set of independent springs with stiffness k (force per unit displacement and area, kN/m³). It is a useful simplification for rapid calculation of stresses in slabs and pavements when the interaction between soil points is secondary (dense granular soils and relatively uniform sites). For foundations on soft clays or stratified soils the model loses accuracy; it is replaced by continuum models (Boussinesq, Mindlin) or by finite elements with the soil as an elastic continuum. The k value is measured directly with the 760 mm circular plate load test, which gives k₁ (modulus for a 1 ft = 300 mm plate).
Applied Formulas
Subgrade reaction modulus from plate load test (BS 1377-9):
k₁ = σ (at settlement 0.05 in = 1.25 mm) / 0.00125 m [kN/m³]
Correction for the size of the actual foundation B ≠ 0.30 m:
Sands: ks(B) = k₁ · ((B + 0.30) / (2·B))² (Terzaghi 1955)
Clays: ks(B) = k₁ · (0.30 / B)
where B is in metres and k₁ is the value for the 0.30 m plate
Correlation with elastic modulus E:
ks ≈ E / (B · (1 − ν²)) for average flexible foundation (Vesic 1961)
Correlation with CBR (Ullidtz 1987 / PCA):
ks(MPa/m) ≈ 10 · CBR^0.66 (CBR of the subgrade, for rigid pavements)
Correlation with N_SPT (sands, Scott 1981):
ks ≈ 1.8·N_SPT MPa/m (in subgrade with small B) — useful only as a preliminary estimate
Calculation Example
| Parameter | Value |
|---|---|
| Plate load test 300 mm at 6 m depth | σ = 250 kPa @ δ = 1.25 mm |
| k₁ calculated from test | 250 / 0.00125 = 200,000 kN/m³ = 200 MPa/m |
| Actual ground slab width B | 6.0 m (equivalent effective section) |
| Soil type | Compacted coarse sand, γ = 20 kN/m³ |
| Measured CBR (subgrade) | 25 % (well-compacted sand) |
| Elastic modulus E (correlation) | 80 MPa |
Size conversion (Terzaghi sand formula): ks(B = 6 m) = 200 · ((6 + 0.30) / (2·6))² = 200 · (6.30/12)² = 200 · 0.525² = 200 · 0.276 = 55.2 MPa/m = 55,200 kN/m³. The value drops 3.6 times when going from a small plate to a large ground slab — reason why k₁ cannot be used directly. Verification by Vesic: ks(Vesic) ≈ E / (B·(1 − ν²)) = 80,000 / (6 · (1 − 0.3²)) = 80,000 / (6 · 0.91) = 14,652 kN/m³ = 14.7 MPa/m. Verification by CBR (Ullidtz): ks ≈ 10·25^0.66 = 10·8.41 = 84.1 MPa/m. The three methods give comparable orders of magnitude but a typical 3× scatter. Design value: take the lower conservative value or average → 55 MPa/m is reasonable for a well-compacted gravel ground slab. Verify with in-situ load test if the project is critical (silos, buildings with dynamic loads).
Result: k₁ = 200 MPa/m (plate) · ks(B=6m) = 55 MPa/m · Use 55 MPa/m in slab software
Interpretation of Results
The k value controls the stresses in the slab: a high k concentrates stresses near the loads (high moments at supports, shear at perimeters); a low k distributes the load over a wider area (smooth moments, greater deflection). Typical ks values (B=1 m): soft clays 5-15 MPa/m; firm clays 20-50; loose sands 15-30; dense sands 50-100; compacted gravels 80-150. For the design of slabs on heterogeneous soils it is recommended to calculate with the minimum and maximum expected ks and take the envelope of stresses, since k is neither linear nor unique in practice.
Reference Standards
- BS 1377-9:1990 — Methods of test for soils for civil engineering purposes — In-situ tests (static plate load test for determination of k)
- BS EN 1997-1:2004 (Eurocode 7) — Geotechnical design — General rules (cites subgrade reaction modulus as an accepted parameter)
- PCA (1984) Thickness design for concrete highway and street pavements — k tables from CBR
- Terzaghi, K. (1955). Evaluation of coefficients of subgrade reaction
- Vesic, A.S. (1961). Beams on elastic subgrade and the Winkler's hypothesis
- BS 1377-9:1990 — Standard test method for nonrepetitive static plate load tests of soils
- Highways Agency Design Manual for Roads and Bridges — Plate load test and correlation with CBR
Frequently Asked Questions
Why does ks(B) depend on size?
Because the depth of influence of a foundation is proportional to its width B. A small plate (0.30 m) compresses a soil bulb down to ~0.60 m; a large slab (6 m) compresses down to 12 m and integrates more strata. If the deeper strata are softer, ks(B) decreases; if they are stiffer, ks(B) may increase. Terzaghi's formula assumes a relatively uniform profile.
When not to use Winkler?
Uniform soft clays, very rigid foundations, slabs with highly concentrated loads, or sites with variable strata with depth. Alternatives: Pasternak model (two parameters: k and Gp for coupling between springs), Boussinesq elastic layer model, or 3D finite elements with soil as a continuum (PLAXIS, SAP2000 with meshing). In tall buildings with piled raft foundations, FEM modelling is always used.
How do I use k in slab software?
In SAP2000, ETABS, ANSYS the k is entered as an area spring stiffness (kN/m³ in area spring flag). In SAFE it is assigned as "soil k" directly to each slab panel. Caution: some programs require k to be converted to force per area units (kN/m²) by multiplying by the finite element size, others take it directly. Always verify with a simple uniformly loaded slab example (expected deflection = q/k).
Is k different for static and dynamic loads?
Yes. Under dynamic loading (machines, vibration, earthquake), the soil responds more stiffly (k increases by 20-50 %). The dynamic modulus Ed from geophysical tests (MASW, cross-hole) which gives Vs is used, then Ed = 2·ρ·Vs²·(1+ν). For anti-vibration design, the dynamic k is used and amplitudes are verified according to Richart-Hall 1970 for machine foundations.