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Modulus of subgrade reaction or coefficient of subgrade reaction is the reaction pressure sustained by the soil sample under a rigid plate of standard diameter per unit settlement measured at a specified pressure or settlement.

It is a stiffness parameter that is commonly used to define the support conditions of footings and mat foundations. This parameter explains a linear elastic response. Hence during the design, the pressure generated by the modulus of subgrade reaction is limited to the allowable bearing pressure of the soil.

The coefficient of sub-grade reaction (k) is the primary parameter used in the design of pavements, foundation, and soil-structure interaction studies.

This article elaborates on the basic concept and determination of the coefficient of sub-grade reaction.

## Modulus of Subgrade Reaction

Modulus of subgrade reaction is defined as the pressure per unit deformation of the subgrade at specific pressure or deformation. It is expressed as:

**k = p/s**

Here, 'k' is the modulus or coefficient of subgrade reaction, 'p' is the pressure, and 's' is the deformation of soil settlement. The coefficient of subgrade reaction is measured and expressed as load intensity per unit of displacement. In the SI unit system, it is expressed as kN/m^{2}/m.

## Determination of Modulus of Sub-Grade Reaction

The modulus of subgrade reaction is determined by the field plate load test. In this test, compressive stress is applied to the soil layer through rigid plates, and the deflections are measured for different values of stress.

The test setup consists of a loading frame that has a hydraulic jack, a reaction beam, and a proving ring. The settlement of the plate is measured using a dial gauge. The dial gauges are attached to a separate datum frame as shown in Figure-1. The standard size of the plate is 750 mm in diameter.

The plate is properly placed on the test site. The test site must be levelled and prepared before placing the plate. As shown in the figure, the stiffening plates are placed on the bottom plate in the order of decreasing diameter.

The test procedure is commenced by applying a seating load of 7 kN/m^{2}. This load is released after a few seconds. The load is then applied to cause a settlement of s= 0.25 mm.

When the settlement does not increase, or it is less than 0.025 mm per minute, then the dial gauge reading is taken, and the average settlement is determined. This procedure is repeated till the settlement goes to 0.175 cm.

As shown in Figure-2, a graph is plotted between the bearing pressure (p) and the corresponding settlement caused (s) or delta.

The pressure corresponding to the settlement 0.125 cm is read from the plot, and the value of 'k' is determined as:

**k = p/0.125**

The soil, after construction, is likely to soften due to external vibrations, water penetration, and other imposed loads . This effect is considered by providing certain corrections to the value of 'k'. Here, 'k' is corrected for full saturation. For this, two specimens from the site are subjected to consolidation test in soaked and unsoaked condition. The consolidation test is conducted, and the pressure corresponding to 0.125 cm is determined for both the specimens. Then the corrected modulus of rupture is given by:

**k' = k x [p/p _{s}]**

Where p and p_{s} are the pressures for unsoaked and soaked specimen as shown in Figure-3.

In practice, it is used by geotechnical and structural engineers for the analysis of structure. The analysis provides insights into the settlement of the structure and settlement-induced stresses within the structure.

## FAQs

**What is modulus of subgrade reaction?**

Modulus of subgrade reaction is defined as the pressure per unit deformation of the subgrade at specific pressure or deformation. It is expressed as k=p/s.

Here, k = modulus or coefficient of subgrade reaction; p = applied pressure and s = deformation of soil settlement. The coefficient of subgrade reaction is expressed in load intensity per unit of displacement.

**How to determine modulus of subgrade reaction?**

The field plate load test is used to determine the modulus of subgrade reaction. In this test, compressive stress is applied to the soil layer through rigid plates, and the deflections are measured for different values of stress.

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