Modulus of Subgrade Reaction (k): Values and Plate Test

The modulus of subgrade reaction, usually just called k or the k-value, is how concrete pavement designers describe the support a slab gets from the ground below it. It answers a simple question: push down on the soil with a known pressure, and how far does it sink? Firm soil sinks a little and has a high k. Soft soil sinks more and has a low k. The number goes straight into slab thickness design for roads, parking lots, warehouse floors and large driveways.

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The short version

Definition: k = pressure on the soil ÷ deflection.
Units: pci (psi per inch) or MPa/m.
Typical range: about 50 pci (weak) to over 1,000 pci (strong).
Measured by: the plate load test, or estimated from other soil tests.

Where the idea came from

In the 1920s, H. M. Westergaard worked out stresses in concrete slabs by treating the ground as a bed of closely spaced springs. Each spring pushes back in proportion to how much it is squeezed, and k is the stiffness of those springs. It is a simplification. Real soil spreads load sideways and does not act like separate springs. But the model works well enough for rigid slabs, which spread a wheel load over a wide area, that it is still in use.

Typical values

The full range runs from roughly 50 pci for weak support to more than 1,000 pci for very strong support. Soft, wet clays and silts sit at the low end. Dense, well-graded granular soils and stabilized layers sit toward the high end. Moisture and compaction matter as much as soil type, so the same clay can give quite different values dry and well compacted versus wet and loose. Getting compaction right, checked against a Proctor test, is one of the most practical ways to raise support.

The plate load test

The plate load test is the field method for measuring k directly. AASHTO T 222 and ASTM D1196 describe the nonrepetitive static version. The U.S. Army Corps of Engineers method is a common example of how it is run:

Prepare the spot. Strip the area down to subgrade elevation, at least twice the plate width, and level it with a thin bed of sand or plaster.
Stack the plates. Seat a 30 inch steel plate, then center 24 inch and 18 inch plates on top so the stack acts stiff.
Set up the reaction. A hydraulic jack pushes up against something heavy, often a steel beam between two loaded trucks, with supports at least 8 feet from the plate.
Mount the gauges. Three dial gauges rest on the 30 inch plate near its edge, 120 degrees apart, on a frame supported well away from the plate.
Seat the plate. A small seating load of 1 psi (about 707 lb on a 30 inch plate) is applied and the gauges are zeroed.
Load in steps. Load goes on in 5 psi steps, each held until the plate stops moving, reading all three gauges each time.
Plot and correct. Pressure is plotted against average deflection. If the line does not pass through zero, it is shifted to correct for seating.

In the Corps method, the uncorrected value is the pressure of 10 psi divided by the deflection read at 10 psi. That value is then adjusted for bending of the plates and for the soil being wetter later in its life than on test day. Some other procedures instead take the pressure at a deflection of 0.05 inch and divide by 0.05.

The math

Example (made-up numbers): at 10 psi on a 30 inch plate, the corrected average deflection is 0.065 inch. k = 10 ÷ 0.065 = about 154 pci before any plate-bending or saturation correction. That would count as moderate support.

Plate size matters. A 30 inch plate is about 707 square inches, so 10 psi is roughly 7,070 lb of jack force. Smaller plates give higher apparent k values on the same soil, which is why results are tied to the plate size used.

Estimating k from other tests

Plate tests are slow and need heavy equipment, so k is often estimated. The 1993 AASHTO pavement guide offers a relationship between plate-test k and the subgrade's resilient modulus (MR): k = MR ÷ 19.4, with MR in psi and k in pci.

Example (made-up numbers): a subgrade with an MR of 7,000 psi gives k = 7,000 ÷ 19.4 = about 361 pci. Designers often treat values from this kind of conversion with caution and may adjust them for seasonal moisture, loss of support at the slab edges and any stiff layer near the surface.

Charts that relate k to the California bearing ratio and to soil classification are also widely used for smaller projects.

How k is used in design

Concrete slab design takes k along with traffic, concrete strength and joint details to set thickness. Concrete is stiff enough that it spreads loads widely, so moderate changes in k usually move the required thickness less than changes in concrete strength or traffic. Uniform support matters more than very high support. Soft spots, voids and pumping under joints cause more cracking than an evenly modest subgrade.

A granular or stabilized base on top of the subgrade raises the effective k the slab feels. Overlays such as whitetopping treat the old asphalt below as a very stiff support layer.

Driveways and patios

Homeowners will not run a plate load test, but the same ideas apply. Strip soft topsoil, use good structural fill where you need to build up grade, compact in thin lifts and keep water away from the edges. A uniform, well-drained base under a properly jointed slab, as covered in our control joint spacing guide, does more for a driveway than extra thickness on a soft spot. The concrete calculator turns your slab size into cubic yards.

Questions people ask

What is the modulus of subgrade reaction?

It is a number, called k, that describes how stiff the soil under a slab acts. It is the pressure on the soil divided by how far the soil sinks under that pressure. It treats the ground like a bed of springs supporting the slab.

What units is k in?

Pounds per square inch for each inch of deflection, written as psi/in or pci, which works out to pounds per cubic inch. Metric values are in MPa/m.

What are typical k values?

They range from about 50 pci for weak support to over 1,000 pci for very strong support, roughly 13.5 to 270 MPa/m. Most soils fall between those ends depending on soil type, moisture and density.

How is k measured in the field?

With a plate load test, often using a 30 inch steel plate pushed down by a hydraulic jack against a heavy reaction such as a loaded truck. Dial gauges near the plate edge read the deflection.

Can k be found in a lab?

There is no direct lab test for k. It is usually estimated from other tests, such as resilient modulus or California bearing ratio, using published relationships.

Where is k used?

Mainly in designing concrete pavements and slabs on grade, which spread loads over a wide area. Asphalt pavement design usually relies on resilient modulus or CBR instead.

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