Modulus of Subgrade Reaction
Modulus of Subgrade Reaction
Modulus of Subgrade Reaction
This term is measured and expressed as load intensity per unit of displacement. For the English unit
system, it is often expressed in kip/in2/in; in the SI system it is expressed as kN/m2/m. Some express this
term in kip/in3 (or kN/m3) which can be misleading. Numerically, kip/in3 is correct but does not properly
represent the physical significance of the measured value and could be mistaken as a density unit or a
volumetric measurement.
Mathematically, the coefficient of subgrade reaction is expressed as:
Ks = p/s (Eqn 1)
where p = contact pressure intensity and s = soil settlement
As Terzaghi mentioned, proper estimation of contact pressure for a flexible foundation could be very
cumbersome, so it is assumed that Ks remains constant for the entire footing. In other words, the ratio
between pressure and settlement at all locations of a footing will remain constant. So the displacement
diagram of a footing with a load at center will have a dishing effect. A point at the center of the footing will
experience the highest displacement. Displacement reduces as it moves away from the center. Figure 1a
shows a simple slab-on-grade foundation. It was modeled and analyzed in STAAD Foundation as Mat,
which is a flexible foundation; the soil was defined using coefficient of subgrade reaction. For this
exercise, the software default value for the modulus of subgrade reaction was used. The displacement
diagram shows a dishing effect as discussed earlier. Figure 1b shows the soil pressure contour. It is also
obvious that the pressure intensity at the center is maximum and reduces as the elements (or node
coordinates) move away from the center. So, it could be assumed that the ratio of pressure intensity and
settlement is constant.
Table 1: Soil pressure, node displacement and their ratio.
Consider some of the numbers from the same example. Soil pressure, corresponding displacement and the
ratio are listed in Table 1. The points are represented on a diagonal to illustrate the variation of pressure
and displacement as the points move away from the center to the most distant point in the corner of the
rectangular footing. Figure 2 shows the points on the mat slab.
This is hardly a surprise as, by definition, the modulus of subgrade reaction (Ks) is a constant for the entire
footing and the program used Ks as its soil property. It is also important to note that the software default
Ks value (10858 kN/m2/m) was exactly the same as the constant ratio calculated in Table 1.
Base pressure was calculated from the support reaction. One might think that the ratio of support reaction
and corresponding displacement will also be a constant. As shown in Table 2, the ratios are not constant for
all values. How is the Ks value used inside the program and how is the base pressure calculated?
Tributary Area
Often an assumption is made to calculate how much area of a plate can be attributed to a node or, in other
words, the influence of each node on the surface area of a plate. It depends on the shape of the plate. For a
perfect square or rectangular plate, each node will influence exactly of the plate surface area (Figure 3a).
But for a generalized quadrilateral, the best practice would be to calculate the center of the mass of the
plate and then draw lines from that center point to the middle points of each side. In Figure 3b, the shaded
area represents the influence surface area of the corresponding node.
Allowable Settlement
Bearing capacity is the measurement of the soil pressure a soil can safely bear. In other words, bearing
capacity is the pressure which soil can withstand before it fails. The two most important soil failure criteria
are:
Shear failure
Maximum allowable settlement
Among many factors, foundation width (B) can influence failure criteria. Normally, shear failure governs
for smaller foundations and settlement failure governs bigger foundations. Table 4 is a typical example
which shows the relationship among different foundation sizes and failure criteria.
Table 4: Final allowable bearing capacity for allowable settlement = 25 mm and a given embedment depth.
Foundations can be rigid or flexible. Bearing capacity is used to design rigid foundations, but subgrade
reaction is used for flexible foundations. The very assumption of a rigid foundation is that the distribution
of the subgrade reaction p over the base of the foundation must be planar, because a rigid foundation
remains plane when it settles. Consider a simply supported beam loaded at its center, as shown in the
Figure 5a. By statics, we can obtain R1 = P/2 and R2 = P/2. If the same beam is loaded eccentrically, the
reaction can be calculated as shown in Figure 5b.
Figure 5: Reactions for a simply supported beam.
The same concept is extended for rigid foundation design. But instead of the end supports, the whole
foundation is supported. It is also assumed that the relative stiffness of the concrete slab is much higher
than the soil stiffness. So, the slab is assumed to remain planar even after the application of load.
Figure 6a shows a footing loaded at the center. From a rigid wide beam analogy, P = R x L. Similarly, for
an eccentrically loaded footing, the reaction will vary linearly from one end to the other as shown in Figure
6c. Equations 3 and 4 can be solved to find end reactions. But none of the equations contain modulus of
subgrade reaction (Ks). So, the distribution of subgrade reaction on the base of a rigid footing is
independent of the degree of compressibility of the subgrade it is resting on. As many authors have
concluded, a rigid foundation can be safely designed using bearing capacity, as in most cases this method
yields more conservative results.
Again, the definition of Ks is the pressure per unit settlement. In other words, soil capacity to withstand
pressure for a given displacement. From earlier discussions, it is also clear that even bearing capacity has
an allowable settlement. It is therefore tempting to conclude that the modulus of subgrade reaction is the
bearing capacity per unit settlement.
Ks = stress/displacement (Eqn 7)
where
I = Safety factor
These equations clearly indicate that the appropriate safety factor must be used, and the Ks value can be
better compared with ultimate bearing capacity rather than the allowable bearing capacity. The safety
factor can vary depending on projects and geotechnical engineers. The other important factor is the
assumed allowable settlement for the calculated bearing capacity.
Similarly, it is to be noted that the base pressure values reported by FEA analysis cannot be directly
compared with the bearing capacity. Maximum base pressure should be multiplied by the safety factor and
then compared with the allowable bearing capacity of the soil.
However the above mentioned equations have limitations. They can be applied to footings where
settlement failure governs, but cannot be related to footings where shear failure occurs before reaching the
allowable settlement limit. So, engineers must exercise caution before using these equations.
Conclusion
The correlation between bearing capacity and modulus of subgrade reaction is at best an estimation. It can
be used for estimation, but a Ks value determined by a plate load test should always be used if available or
should be requested whenever possible. However, the above discussion gives insight into these values and
helps engineers to understand the physical significance of modulus of subgrade reaction. And, as always,
structural engineers should consult a geotechnical engineer professional prior to finalizing soil stiffness
and bearing values.
References
Soil Mechanics in Engineering Practice (Third Edition) Terzaghi, Peck, Mesri