Poulos 2018 Subgrade PDF
Poulos 2018 Subgrade PDF
Poulos 2018 Subgrade PDF
Harry G. Poulos1
1
Coffey Geotechnics, and the University of Sydney
E-mail: harry_poulos@coffey.com
ABSTRACT: The concept of modulus of subgrade reaction has been employed within the engineering world for almost 150 years. It has
been especially embraced by structural engineers who have found it convenient to represent the behaviour of the ground supporting their
structures by elastic springs. Despite the best efforts of the geotechnical profession to dissuade our structural colleagues from using this
flawed concept in foundation design, requests to provide a modulus of subgrade reaction continue almost unabated. Given this situation, a
suitable response is to provide such values via a rational process of evaluation, rather than by empirical correlations which have little
theoretical basis and which may not be applicable to the foundation being considered.
This paper sets out an approach to the estimation of values of modulus of subgrade reaction for various types of foundation. The key points
made are that the modulus of subgrade reaction (k) is not a fundamental soil property, but varies with the foundation type, foundation
dimension, and type of loading. k can be related to the Young’s modulus of the supporting soil and to the foundation dimensions, but for pile
groups, account must be taken of the reduction in k because of group effects arising from pile-soil-pile interaction. It is also emphasized that
careful distinction must be made between the modulus of subgrade reaction, k, and the spring stiffness K.
Keywords: Foundation, lateral loading, modulus of subgrade reaction, piles, pile group, settlement.
1. INTRODUCTION
2. BASIC APPROACH
where p = foundation pressure, and S = foundation deflection. The modulus of subgrade reaction for vertical loading, kv, can then
be obtained from eq. 2, as follows:
The units are typically MN/m3 or kN/m3.
kv = Es/BI (3)
Different values of k will apply for vertical and lateral loading, and
k will also vary with the type of foundation and its dimensions. The displacement influence factor can be obtained in the following
Initial consideration will be given to vertical loading on a raft or ways:
spread foundation, with the corresponding modulus of subgrade 1. From existing charts, such as those in Poulos and Davis
reaction being denoted as kv. Subsequently, pile foundations (1974) and Mayne and Poulos (1999). Such charts
subjected to vertical and lateral loading will be considered. usually require the soil profile to be simplified to have
Use can be made of elastic theory to compute kv. For example, either an equivalent uniform Young’s modulus with
assuming that, as shown in Figure 1, the foundation can be idealized depth (see Section 3.1.1), or one that increases linearly
as an equivalent circular footing of equivalent diameter d resting on with depth (see Sections 3.1.2 and 3.1.3).
a uniform layer of thickness h, elastic theory gives the following
general form of the expression for the settlement S of the footing 2. From a numerical analysis, such as the program FLEA
when subjected to uniform vertical load p: (Small, 1984). This approach is very versatile as it can
take account of such factors as soil layering, foundation
S = pBI / Es (2) shape and the presence of a slab or raft beneath the
loaded area.
where I = displacement influence factor, B = foundation
dimension, Es = Young’s modulus of soil. These options are discussed in more detail below.
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Geotechnical Engineering Journal of the SEAGS & AGSSEA Vol. 49 No.1 March 2018 ISSN 0046-5828
Solutions have been provided by Mayne and Poulos (1999) for the
cases set out below.
The solutions for this case are shown in Figure 2, where a = radius
of foundation, h = layer depth, q=applied uniform loading, Es =
Young’s modulus of soil layer.
It should be noted that the displacement is for the centre of the
loaded area, and that the displacements away from the centre will be
smaller. Thus, the modulus of subgrade reaction will NOT be a
Figure 3. Influence factors for circular foundation on a deep Gibson
constant over the entire area, but will increase from a minimum at
soil (b > 0.01)
the centre of the circle to a maximum at the edge of the circle.
For practical purposes, it may be preferable to compute an average
settlement of the loaded area, and for a circular area, the following
rough approximation may be used: 3.1.3. Flexible circular foundation on a “Gibson soil” of
finite thickness.
Sav ≈ 0.78 [Scentre] (4)
where Sav = average settlement, and Scentre = settlement at centre. Figure 4 shows this case. These results apply for the centre of a
. uniformly loaded (flexible) footing. Eq 4 may be used to
approximate the average settlement.
3.1.2. Rigid or flexible circular foundation on a deep soil 3.2. Layered soils
layer whose modulus increases linearly with depth
(a “Gibson soil”) A layered soil profile can be transformed, approximately, into an
equivalent uniform layer of the same depth, via the procedure set out
The results for this case are plotted in Figure 3, where d=foundation in Figure 5 (Poulos, 1994). Here, Esbe is the equivalent Young’s
diameter, q=applied uniform loading, E0 = Young’s modulus at soil modulus of the layered profile, d is the foundation diameter, h i is the
surface, kE = rate of increase of Young’s modulus with depth. This thickness of layer i, Esi is the Young’s modulus of a layer i, and Wi
figure gives both the centre settlement of a uniformly loaded is the weighting factor for layer i.
(flexible) area and the settlement of a rigid footing. The average It should be noted that, when the thickness of the upper layer is
settlement of the uniformly loaded area is closely approximated by greater than about 4d, Esbe can be taken as the modulus of the upper
the settlement of the rigid footing. layer.
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3.5. Summary
For footing shapes other than circular, they can generally be 5. EFFECTS OF EXCAVATION
transformed into a circle of equal area. The equivalent diameter, d e,
of the circle is then:
In many cases, excavation, for example for a basement, will be
carried out prior to construction of a building or structure. In this
de = 2.(A/) 0.5 (5)
case, there are at least two important issues to consider when
assessing the modulus of subgrade reaction and the consequent
where A = foundation area.
foundation behaviour:
1. Because of the unloading arising from excavation, soils
3.4. Limiting value of modulus of subgrade reaction – one- that were in a normally consolidated or lightly
dimensional compression overconsolidated state will be subjected to recompression
upon the application of the building load, and thus will
It should be recognised that there is an important situation that give tend to be stiffer than under initial loading. Once the
rise to a limiting value of the modulus of subgrade reaction, and that previous vertical stress has been reinstated, the soil will
needs to be recognised when assessing kv. again exhibit the initial loading stiffness. Thus, the overall
If the loaded area is large, and/or the soil layer thickness is small soil behaviour will tend to be more stiff than if there was
(see Figure 6), then there will be essentially one-dimensional no excavation.
compression below the area. In this case, the value of kv, denoted as
kv1D, can be calculated via the following expression: 2. If the excavation extends below the water table, there will
be a resulting hydrostatic uplift on the base of the
kv1D = Ds/h (6) foundation, which will reduce the net loading on the
foundation.
where Ds = constrained modulus of soil, and h = soil layer
thickness. Considering first a single soil layer, the effects of excavation and
The constrained modulus Ds is related to Young’s modulus Es and soil unloading can be estimated approximately as set out below.
Poisson’s ratio s, and for a typical value of s of 0.3, Ds = 1.35Es. The foundation settlement in the layer can be estimated via the
Thus, eq. 6 can be re-expressed as follows: following expression:
kv1D = 1.35 Es/h (7) S = I.B { ex / Esr +(p - ex) / Es } (9)
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where = Esr / Es = ratio of recompression to initial loading of subgrade reaction modulus, taking account of the following
modulus values. factors:
is typically 5-10 for soft clays, but close to 1 for very stiff soils
and rocks. Thus, for a single soil layer, the corresponding value of
modulus of subgrade reaction, kvex, taking into account the effects
of unloading and reloading, can be simply related to the normal 1. The type of loading, i.e. axial or lateral;
value without excavation, kv, as:
2. The effects of interaction among the piles within the
kvex = kv / [1 – ex /p (1-1/)] (12) group; such effects will tend to “soften” the soil springs.
For multiple soil layers, eq. (11) has to be applied to each layer If The discussion below considers both axial and lateral loading, for
FLEA is used, then these modified Young’s modulus values are single piles and for piles within a group.
input into the program. If a hand calculation is carried out, then an
equivalent profile can be developed via Figure 5. 8.2. Axially loaded piles – single piles
kv = 1.4Es/d (13) where Esv = Young’s modulus of the soil along the shaft for
vertical loading, d = pile shaft diameter.
This expression has been derived from the elastic solution for a For the pile base, the corresponding modulus of subgrade reaction,
rigid footing on a semi-infinite homogeneous mass with Poisson’s kb, is:
ratio of 0.3, and is only correct if:
kb ≈ 1.4Esb /db (15)
1. The soil stiffness is uniform with depth;
2. The diameter or width, d, of the foundation loading is not where Esb = average soil modulus below pile base, for vertical
very large in relation to the overall depth of the loading, db = pile base diameter.
compressible soil profile;
8.3. Axially loaded piles – a pile within a group
3. The load is not applied to, or through, a concrete slab;
4. There is no excavation involved. Pile-soil-pile interaction within a group will tend to reduce the
equivalent stiffness of the soil and hence the modulus of subgrade
In other cases, it is possible that the use of eq 13 will give a reaction. A simple approach to take group effects into account is to
conservative (and sometimes extremely conservative) assessment of multiply the single pile values in eqs 14 and 15 by a group reduction
kv. Such conservatism can have significant cost implications for factor, RG, so that the group values of modulus of subgrade reaction,
clients. ksG for the shaft and kbG for the base, can be approximated as:
It is therefore strongly recommended that the approach outlined in
this paper be followed to avoid the provision of misleading values of ksG ≈ RG.ks (16a)
kv.
It should also be noted that kv, if it is to be used, should be applied kbG ≈ RG.kb (16b)
only to assess structural actions (moments and shears) in a raft or
slab. When applied to estimate settlements, the modulus of subgrade RG can be approximated as follows (Poulos, 1989):
reaction can give misleading estimates of the distribution of
settlement across a foundation. RG ≈ n-w (17)
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More accurate values of RG can be obtained via a computer 9. SPRING STIFFNESS VALUES
analysis of the pile group behaviour, for example, via programs such
as DEFPIG (Poulos, 1990), REPUTE (Geocentix, 2014) or PIGLET It is not uncommon for the structural engineer to request values of
(Randolph, 2005). the spring stiffness for the foundation elements, rather than the
modulus of subgrade reaction. It is critical to distinguish between
these two values.
If the foundation width or diameter is d, and an elemental length of
the foundation of L is considered, then the spring stiffness K for
that element can be calculated from the relevant modulus of
8.4. Laterally loaded piles – single piles
subgrade reaction, k, as follows:
K = k.d.L (23)
A reasonable method for evaluating the modulus of subgrade
reaction for lateral loading, kh, is to equate the solutions for K will then have the units of stiffness (force per unit length, for
deflection of a rigid fixed head pile from elastic continuum theory example MN/m), whereas k has the units of force per length cubed,
and from subgrade reaction theory. On this basis, the following for example MPa/m or MN/m3.
approximate relationships can be obtained: For the overall foundation, or for individual piles with a group, the
spring stiffness can be computed as the ratio of the applied load to
kh ≈ X1. Esh / d (18) the deflection. In general, different values will be obtained for each
pile within a group, and for each component of load (i.e. vertical,
where X1 is typically 0.8 to 1.0, depending on the length to lateral and moment).
diameter ratio of the pile, Esh = Young’s modulus of soil, for lateral It has been found that the most reliable approach to estimating the
loading, d = pile diameter, or width in the direction of loading. stiffness of individual piles within a group is to carry out analyses in
For lateral loading, it is common practice to adopt Young’s which an equal load (generally equal to the working load) is applied
modulus values which are less than those for vertical loading, simultaneously to each pile within the group. The resulting
because of the greater strain levels in the soil under lateral loading. deflection of each pile can then be used to compute the pile head
A reduction factor of 0.7 is commonly applied to E sv values to stiffness. Vertical, lateral and moment loads should be considered
obtain Esh. A more complex expression has been derived by Vesic separately. This procedure avoids unrealistic computed stiffness
(1961) in which the relative stiffness of the pile and soil is included. values that can arise under some combinations of vertical, lateral
The factor X1 in eq 17 can then be reduced to the following and moment loading.
expression:
The FLEA analysis gives the following result for the central The first step is to estimate revised values of the “operative”
deflection Sc of the foundation: modulus which allow for the effects of the excavation and the
subsequent reloading of the soil profile.
Sc = 1.069 m. Assuming that the unit weight of the upper two layers is 20 kN/m3,
the stress relief due to excavation will be ex = 2.5x20 +2.3x(20-
Thus, the modulus of subgrade reaction for the centre of the 10) = 75kPa.
foundation is: Assuming that ex is reasonably constant with depth, and that
the ratio of reload to initial loading modulus,, is 5 for Soil 3 and
kv = p/Sc = 1.0/1.069 = 0.94 MPa/m. 2.5 for Soil 4, application of eq 10 gives the following equivalent
values of Young’s modulus:
The same problem has been evaluated by hand calculations, using Soil 3: Eseq = 15.0/[1-{75/125(1-1/5)}] = 28.8 MPa
Figure 5 to obtain an equivalent modulus for the layered profile,
Figure 2 to obtain the displacement influence factor I ( denoted as Ih Soil 4: Eseq = 40.0/[1-{75/125(1-1/2.5)}] = 62.5 MPa
in this figure), and then eq 3 to evaluate kv. In these calculations, the
equivalent modulus was calculated as 12.6 MPa, while interpolation
from Figure 2 gave a value of I of 1.25. From eq 3, kv was found to
be 1.01 MPa/m, which was similar to, but somewhat larger than, the
value of 0.94 computed from FLEA.
Figure 8. Example 1: loaded area on the surface of a layered soil For the two-layer system consisting of Soils 3 and 4, the program
profile FLEA gives a central deflection of 0.302m/MPa applied pressure.
Thus, the modulus of subgrade reaction is:
Had the simplified expression (eq 13) been applied in this case,
with a modulus of 10 MPa used (i.e. the modulus for the layer kv = 1/0.302 = 3.31 MPa/m.
directly below the foundation), the value of kv would have been
computed to be 1.4.10/20 = 0.70 MPa/m, about 25% less than the Using hand calculation methods with Figures 2 and 5, and
value derived from the FLEA analysis. assuming a rough footing and a soil Poisson’s ratio of 0.3, the
For the case of one dimensional compression, which would give a computed central deflection is 0.279m/MPa, thus giving:
lower limit value, the use of eq 8 leads to a value of kv1D = 0.90
MPa/m, which is slightly less than the values obtained from FLEA kv = 1/0.279 = 3.58 MPa/m.
and the hand calculation method. It would thus appear that, in this
case, the geometry for this example is approaching that of a one- This is similar to, but slightly larger than, the value obtained from
dimensional situation. the FLEA analysis.
It should be noted that, comparing Cases 1 and 2, the effect of the
10.2. Case 2 – foundation within an excavation excavation in Case 2 is to increase kv by a factor in excess of 3. This
arises both because the upper two soft layers are removed, and also
because the Young’s modulus of the lower two layers is increased
This case is illustrated in Figure 9. The soil profile is similar to that
due to the effects of the larger recompression modulus over a part of
for Case 1, but now it is assumed that a 5m deep excavation will be
the range of foundation loading.
made and the foundation will be constructed at the base of the
excavation.
The following assumptions are made: 10.3. Case 3 – group of 30 piles
• The water table is located 2.5m below the surface;
Figure 10 shows a group of 30 piles within a layered soil profile
• The unit weight of the upper two layers is 20 kN/m3 consisting of stiff clays extending to considerable depth. Values of
the modulus of subgrade reaction will be developed for both axial
• The ratio of the reload to unload moduli is 5 for the upper and lateral loading.
three layers, and 2.5 for the lower (and stiffer) layer.
(1) Axial Loading
• The final applied loading on the foundation will be 150
kPa. Considering first a single isolated pile, the modulus of subgrade
reaction for the shaft is given by eq 13. Thus, from the surface to a
depth of 12m, ks ≈ 0.6x30/0.8 = 22.5 MPa/m.
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Geotechnical Engineering Journal of the SEAGS & AGSSEA Vol. 49 No.1 March 2018 ISSN 0046-5828
From 12m to the pile base at 20m, ks ≈ 0.6x100/0.8 = 75 MPa/m. (3) K can be related to the Young’s modulus of the supporting
For the pile base, from eq 14, kb ≈ 1.4x100/0.8 = 175 MPa/m. soil and to the foundation dimensions.
Considering now the group effects via eq 16, a value of w of
between 0.25 (for a hard stratum) and 0.5 (for a uniform stratum) (4) For pile groups, account must be taken of the reduction in
would be appropriate and a value of 0.375 is therefore chosen. The k because of group effects arising from pile-soil-pile
corresponding group reduction factor from eq 17 is thus 30 -0.375 = interaction.
0.279.
Accordingly, the values of ksG and kbG for the piles in a group (5) Careful distinction must be made between the modulus of
environment would be, from eqs 16a and 16b: subgrade reaction, k, and the spring stiffness K. K is often
ksG ≈ 0.279x22.5 = 6.3 MPa/m for 0 to 12m depth, best obtained via a foundation analysis, as the ratio of the
0.279x75 = 20.9 MPa/m for 12 to 20m depth, and applied load to the computed displacement of the
0.279x175 = 48.8 MPa/m for the base. oundation.
REFERENCES
Fleming, K., Weltman, A., Randolph, M. and Elson, K. (2009).
“Piling Engineering”. 3rd Edition. Taylor and Francis,
London.
Mayne, P.W. and Poulos, H.G. (1999). “Approximate Displacement
Influence Factors for Elastic Shallow Foundations”. Jnl.
Geot. and Geoenvironmental Eng., ASCE, 125(6): 453-460.
Poulos, H.G. (1989). “Pile Behaviour – Theory and Application”.
29th Rankine Lecture. Geotechnique, Vol. 39, No. 3, pp. 365-
415.
Poulos, H.G. (1994). "Settlement Prediction for Driven Piles and
Pile Groups". Vert. and Horizl. Deformns. of Foundns. and
Embankments, Geotech. Spec. Publ. No. 40, ASCE, New
York, Vol. 2, 1629-1649.
Poulos, H.G. (2001). “Pile Foundations”. Chapter 10 of
Geotechnical and Geoenvironmental Engineering Handbook,
Ed. R.K. Rowe, Kluwer Publishers.
Poulos, H.G. and Davis, E.H. (1974). “Elastic Solutions for Soil and
Figure 10. Case 3: 30-Pile group Rock Mechanics”. John Wiley, New York.
Poulos, H.G. and Davis, E.H. (1980). “Pile Foundation Analysis and
Design”. John Wiley, New York.
(2) Lateral Loading
Small, J.C. (1984). “FLEA User’s Manual”. Centre for Geotechnical
A reduction factor of 0.7 will be applied to Esv values to obtain Research, University of Sydney.
Young’s modulus for lateral loading, Esh. Vesic, A.S. (1961). “Bending of Beam Resting on Isotropic Elastic
For a single isolated pile, from eq 18, taking X 1 = 0.9, the values of Solid”. Jnl. Eng. Mechanics Divn., ASCE, 87(EM2): 35-53.
horizontal modulus of subgrade reaction would be as follows:
0-12m: kh ≈ 0.9x(0.7x30)/0.8 = 23.6 MPa/m
12-20m: kh ≈ 0.9x(0.7x100)/0.8 = 78.7 MPa/m.
Allowing now for group effects, and adopting the case of constant
stiffness with depth (rather than a linearly increasing stiffness with
depth), Lc/d is about 12.8 m (for Ep = 30000 MPa), and for s/d = 5,
wl = 0.3 from Figure 7. Thus, from eq 21, RGh ≈ 30 -0.3 = 0.36.
11. CONCLUSIONS