Subgrade Modulus Calculator - k from Pressure & Settlement

Calculate modulus of subgrade reaction k, applied pressure, or settlement from plate-load data in kN/m³, MN/m³, pci, kPa, MPa, psi, mm, and inches.

Related Calculators

Subgrade Modulus Formula

The following formula is used to calculate the subgrade modulus:

K = P / δ

Variables:

K is the subgrade modulus (pci)
P is the pressure applied to the soil (psi)
δ is the deflection of the soil (inches)

To calculate the subgrade modulus, divide the pressure applied to the soil by the deflection of the soil.

What is Subgrade Modulus?

Subgrade modulus, formally called the modulus of subgrade reaction and symbolized as k, is the ratio of contact pressure to settlement at the surface of a soil mass. It was introduced by Winkler in 1867 and later formalized by Terzaghi in 1955 as a practical way to represent soil-structure interaction without modeling the full elastic continuum of the ground beneath a structure. The parameter is used across two distinct engineering domains: rigid pavement design (where k characterizes support beneath concrete slabs) and mat/spread foundation design (where k defines spring stiffness in structural analysis software).

A critical distinction: the k-value used in pavement design and the k-value used in foundation design are not interchangeable. The cited FAA static pavement procedure uses a 30-inch (762 mm) diameter plate; an approximately 12-inch FWD plate is a different, dynamic measurement. A 1-foot-square reference plate used in foundation size correlations is different again. The applicable procedure must address loaded geometry, stiffness, soil profile and test conditions.

Illustrative k Magnitudes by Soil Group

The numerical ranges and single values below are illustrative magnitudes only. They are not verified USCS-specific design ranges, recommended defaults or a substitute for appropriate project data. Actual k depends on loaded geometry, site conditions, moisture, density, soil profile and the applicable test or design procedure.

USCS Group	Soil Description	Illustrative range (pci)	Illustrative value (pci)
OL, OH, Pt	Organic soils, peat	25 to 100	25
CH, MH	High-plasticity clays and silts	50 to 150	50
CL, ML	Low-plasticity clays and silts	50 to 200	100
SM, SC	Silty and clayey sands	50 to 250	150
SW, SP	Well-graded and poorly graded sands	150 to 400	200
GM, GC	Silty and clayey gravels	200 to 500	250
GW, GP	Well-graded and poorly graded gravels	300 to 500+	350

For unit-conversion context only, 50 pci is approximately 13.57 MN/m³ and 1,000 pci is approximately 271.45 MN/m³. These examples do not define weak or strong support, a universal design threshold or a maximum usable k. Use the requirements and limits of the applicable project procedure.

Measurement Methods

Plate Load Test (ASTM D1196 / AASHTO T 222)

A static plate load test measures pressure and displacement under a specified method. FAA AC 150/5320-6G §2.3.9.12.4 calls for a 30-inch (762 mm) diameter plate. In the cited historical nonrepetitive static elastic-k interpretation, pressure at 0.05 inch (1.27 mm) deflection is divided by that deflection; this is not a universal definition for every test method. Plate seating, displacement reference, load increments, moisture and density conditions matter.

Plate-size adjustments depend on soil behavior, loaded geometry, stress range, layering and the prescribed test or design model. FHWA GEC 6 Figure 8-4 gives different size relations for firm cohesive and cohesionless soils at the same contact pressure. A single diameter multiplier is not a universal correction. This calculator applies no plate-size adjustment.

CBR and Resilient Modulus Correlations

Some design procedures estimate support from California Bearing Ratio (CBR) or resilient modulus (M_r) using empirical relationships. A relationship from CBR to M_r or from M_r to k depends on soil, stress range, moisture, density and the procedure; it is not a universal unit conversion.

NCHRP 1-37A (2004) correlation: M_r (psi) = 2,555 × CBR^0.64, as summarized by FHWA. CBR is the percent number; use only where soil and test conditions support the relationship.
Historical fine-grained, low-CBR estimate: M_r (psi) = 1,500 × CBR was used for CBR ≤ 10 in older guidance. It requires comparable moisture and density and is not a general current recommendation.
Granular soils: A validated project-specific relationship is needed for M_r; do not assume 3,000 × CBR^0.65 is universally applicable.
Older empirical M_r-to-static-k approximation: k (pci) = M_r (psi) / 19.4 appears in a historical AASHTO analysis context. It is not a unit conversion, a measured result or a universal current design rule.

Effective pavement k is procedure-specific. FHWA §5.4.6 describes effects from base stiffness and thickness, shallow rock, loss of support and seasonal conditions. Layered design procedures can derive effective k internally; dividing M_r by a single constant does not perform these adjustments or establish a universal current modulus conversion.

Terzaghi's Size and Shape Corrections

The following historical foundation relations use a 1-foot-square reference plate, foundation width B in feet, the same contact pressure and the stated soil-model assumptions. They distinguish cohesive and cohesionless behavior; they are not universal plate corrections and must not be confused with the FAA 30-inch circular pavement reference.

For cohesionless (granular) soil:

k f = k₁ \times [(B + 1) / (2B)]²

Here k₁ is the 1-foot-square plate value, B is foundation width in feet and k_f is the model-adjusted value. Under these assumptions, B = 10 gives k_f = k₁ × 0.3025, or about 30% of the reference value.

For cohesive (clay) soil:

k f = k₁ / B

Here B is in feet and the same 1-foot-square, same-pressure reference applies. Under this cohesive-soil relation, B = 10 gives 0.1 times the reference k; it is a model result, not a general field correction.

For rectangular foundations (L > B):

kᵣect = kₛquare \times [(m + 0.5) / (1.5m)]

Here m = L/B is length-to-width ratio. This empirical shape illustration approaches 2/3 of the square-foundation value as m increases, within its assumptions. It is not a universal rectangular-footing correction.

A plate-test value should not be transferred to a mat or footing without assessing geometry, structural stiffness, soil profile, load range and the applicable model. The size and direction of an adjustment are project-specific; no fixed stiffness-error multiplier is assumed here.

Composite k-Value: Effect of Subbase Layers

In rigid pavement design, effective k can differ from raw subgrade k because the base, subbase and underlying profile form a layered support system. Layer thickness and stiffness, load distribution, interface conditions and the selected procedure affect the result.

Thickness alone does not determine a numerical increase in k. Base and subbase stiffness, layering, support conditions, drainage and climate all matter; stabilized layers also require appropriate erosion and loss-of-support assessment. Use a project-specific layered or prescribed design procedure rather than fixed thickness-to-k increments or doubling/tripling rules.

Pavement vs. Foundation k: Why They Differ

Pavement and foundation models both use pressure per unit displacement, but their k-values are calibrated to different procedures, geometries and response definitions. FAA static pavement testing uses a 30-inch circular plate; approximately 12-inch FWD dynamic backcalculation and composite support values are different quantities. They are not directly interchangeable, and a small wheel footprint does not by itself eliminate geometry or model adjustments.

A footing or mat can engage a different soil volume and depth range from a small plate. The response depends on geometry, structural stiffness, compressible-layer thickness, layering and load distribution. Evaluate those differences using site data and the chosen model instead of assuming a universal influence-depth multiplier.

Relationship Between k and Elastic Modulus (E)

For a uniformly pressured flexible circular area on a homogeneous, isotropic, linear-elastic half-space, the center deflection gives the relation below. Here E is Young’s modulus, v is Poisson’s ratio and r is the loaded radius, all in consistent units. This center-response idealization is not a universal rigid-plate or layered-foundation conversion.

k = E / [2r \times (1 - v²)]

Within this idealization, doubling radius halves k for the same E and v. With E = 10,000 psi and v = 0.3, r = 6 inches gives approximately 916 pci; r = 60 inches gives approximately 91.6 pci. Real rigid plates, layered ground and footings can require different response models.

Limitations and Practical Considerations

The Winkler model represents support with independent springs and does not explicitly model shear coupling between adjacent springs. Its accuracy depends on calibration and the problem; no universal direction of stiffness error follows. Coupled or continuum models can represent additional interactions but require appropriate parameters and assumptions.

Moisture, freeze-thaw and seasonal changes can affect support by amounts that depend on the site and test conditions. Some pavement procedures derive effective annual support from seasonal response and performance relationships. Evaluate representative project moisture, density and seasonal conditions; do not assume a universal percentage reduction or one moisture state for every soil.

Unit Conversions

From	To	Multiply by
pci (lbf/in³)	kN/m³	≈ 271.447
pci (lbf/in³)	MN/m³	≈ 0.271447
kN/m³	pci	≈ 0.00368396
MN/m³	pci	≈ 3.68396
pci	kPa/mm	≈ 0.271447
kPa/mm	pci	≈ 3.68396