the paper took about plate load test & sub-grade reaction

1. CASE STUDY ON THE ESTIMATION OF SUBGRADE REACTION FOR THE DESIGN OF RAFT
FOUNDATION ON AEOLIAN SOIL
Omar Hamza 1
, Abdul-Hakim Mawas 2
, Alaa Kourdey 3
1
College of Engineering and Technology, University of Derby, Derby DE22 3AW, UK (o.hamza@derby.ac.uk)
2
Structural Engineering Consultant, Dammam, KSA
3
Geotechnical Engineering Consultant, Dammam, KSA
Abstract
Subgrade reaction is an approximate representation of load-displacement behaviour of the supporting soil. Reliable
estimation of the modulus of subgrade reaction (k) is essential for the design of large and multi-column foundations
when flexible analysis is used to achieve economical and safe design. This paper presents a case study on the
estimation of the modulus of subgrade reaction (k) for the design of different sizes of raft foundations of multi-
story buildings constructed on Aeolian soil in Dammam, KSA. A site-specific ground investigation was conducted
for this project where boreholes and laboratory classification testing indicated that the soil is mainly loose to
medium dens Silty Sand. Plate Load Tests (PLT) were carried out at the foundation level and used to estimates the
modulus of subgrade reaction. The load-settlement response from five plate load tests (600 mm diameter) was
back analysed using 3D numerical modelling to verify the ground properties. Then, the validated ground model
was used to numerically simulate the raft foundations and estimate the k-values. As the size of the foundations is
larger than the plat, the modulus of subgrade reaction (k) decreased. The results also showed that the modulus of
subgrade reaction obtained from Terzaghi's equation (1955) for the prototype foundation is not always smaller
than the k-value obtained from the numerical analysis. The paper presents simple procedures that can be useful for
structural and geotechnical engineers who are involved with raft foundation design on sandy soils.
Keywords: Subgrade reaction, Raft foundation, Aeolian soil, Plate Load Test, Numerical Analysis
1. Introduction
In the design analysis of substructures (flexible raft foundation) or flexible pavement slab supported by the ground,
the soil interaction and tendency to deform are commonly represented by a parameter called coefficient or modulus
of subgrade reaction (k). The current practice in the selection of subgrade reaction tends to vary, depending upon
the availability of laboratory and field testing data as well as the analytical tools adopted in the design. The local
tradition and practice and the experience of the structural and geotechnical designers might also influence the
choice of the method selected for evaluating the k-value.
The earliest method was originally developed by Winkler (1867) assuming the soil medium as a system of identical
but mutually independent, closely spaced, discrete and linearly elastic springs (Marto et al. 2012). Accordingly,
the modulus of subgrade reaction, k, can be defined as follow:
k = p/ (1)
Equation (1) defines k-value as the ratio of contact pressure (p) at any given point to settlement () produced by
load application at that point. The unit of k-value is typically MN/m3
or kN/m3
. Figure 1 shows the analytical
model of concrete slab based on Winkler spring assumptions.
The common method to estimate the (vertical) modulus of subgrade reaction for the design of shallow foundations
is Plate Load Test (PLT) which is typically carried out using 300-600 mm diameter circular rigid steel plate
centrally loaded by vertical force (BS1377- Part 9, 1990). The output of PLT is a relationship between the plate
pressure (p) and average measured deflection across the plate () as shown in Figure 2. Accordingly, the modulus

2. of subgrade reaction can be estimated as secant modulus at a selected value of deflection (i.e. typically taken at a
deflection value 1 of 1.25mm).
Figure 1. Analytical model of slab based on Winkler springs
Figure 2. Modulus of subgrade reaction
According to Winkler’s formula, a larger value of k means the ground is stiffer and therefore larger pressure is
required to produce a unit settlement. Despite its simplicity, this formula does not consider several factors that
might influence the k-value, including the geometry and depth of foundation, the magnitude and direction of
loading, as well as the non-linear stress-strain behaviour of the soil and the variation of groundwater conditions.
To improve the accuracy of the evaluation of the modulus of subgrade reaction, several formulas have been
developed by researchers, including Biot (1937), Terzaghi (1955), Vesic (1961), Meyerhof and Baike (1963),
Selvadurai (1979), and Bowles (1996). Table 1 provides a summary of these studies, where the estimation of k-
value is correlated with several parameters related to the foundation and the soil, including the modulus of elasticity
(Es) and the Poisson ratio of the soil (s), in addition to the moment of inertia of the foundation cross section (I)
and modulus of elasticity of the foundation (E).
Empirical equations (Widjaja 2008) have been also suggested to estimate the k-value in clay and sand in correlation
with undrained shear strength (Cu) and Standard Penetration Test N-value (SPTN). For large foundation where
failure is governed by settlement, an approximate relationship between the subgrade reaction and the allowable
soil bearing pressure has been suggested by Bowles (1996).
Table 1. Methods used in the literature to estimate the (vertical) modulus of subgrade reaction (k) for foundation
Method Reference Suggested formula Comment
1 Terzaghi (1955) Clay: 𝑘 = 𝑘 𝑝 [
𝐷
𝐵
]
Sand: 𝑘 = 𝑘 𝑝 [
𝐵+𝐷
2𝐵
]
2
B = Width of foundation (≤ 3D)
D = Diameter of plate in the load test
(PLT)
kp = Modulus of subgrade reaction from
plate load test = Pressure divided by
Settlement.
2 Biot (1937)
𝑘 =
0.95 𝐸𝑠
𝐵(1 − 𝜐𝑠
2)
× [
𝐵4
𝐸𝑠
𝐸𝐼(1 − 𝜐𝑠
2)
]
0.18
Es = modulus of elasticity of soil
(layered soil should be simplified)

3. Several recent studies have pointed out 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 (Poulos 2018). Avci and Gurbuz
(2018) have considered varying value of k depending on the displacement magnitude of soil.
From the literature, it may be concluded that k can be applied to assess structural actions (moments and shears) in
a raft or slab. However, using a constant k-value can give misleading estimates of the distribution of settlement
across a foundation as well as the magnitude of displacement at different stress level.
The focus of this paper is to evaluate the (vertical) modulus of subgrade reaction for shallow foundations with
different sizes constructed on sandy soil using various methods proposed in the literature. The procedures were
exemplified using a case study of residential projects with several buildings constructed on Aeolian soil.
2. Geology of the area
The Eastern Coast of Saudi Arabia has been intermittently submerged during the Tertiary Age, with the movement
of the Arabian Gulf resulting in the sedimentation of marine deposits. A marine deposit, known as the Bahr
Formation and consisting of calcarenite and quartz sand inter-bedded with dune sand, forms low cliffs along much
of shoreline (Al-Refeai and Al-Ghamdy 1994).
A recent deposit consisting of marine terraces raised beach, Sabha and Aeolian sand, which confirm the type of
soil found on the project site. The project site is located in medium dense light olive brown to olive brown, poorly
graded sand with silt (SP-SM) trace of gravel and in some other location traces of broken seashell fragment and
decayed wood noticed at the time of the investigation.
In general, Aeolian deposit of silts and dune sand usually has high dry strength but a low wet strength (Shehata
and Amin 1997). It is also composed of one particle size and a large void ratio due to honeycomb structure. When
Aeolian sand gets wet its structure may collapse leading to settlement of structures built on it. Therefore ground
improvement and/ or suitable foundation solution may be required to reduce the risk of any differential settlement.
3. Description of the project and the ground investigation
The case study considered in this paper was a residential project constructed in Dammam, KSA in 2018. The
project area extends over 25000 m2
and consists of 4 buildings as shown in Figure 3. Each building has a projection
area of approximately 5285 m2
and consists of ground floor which is dedicated for parking or showroom followed
by 4-6 floors depending on the plot. Several sizes of shallow foundations were used to support the structural
loadings of the columns and walls. The maximum foundation size was approximately 10x40m with 0.6m thickness.
νs = the Poisson ratio of the soil.
3 Vesic (1961)
𝑘 =
0.65 𝐸𝑠
𝐵(1 − 𝜐𝑠
2)
× [
𝐵4
𝐸𝑠
𝐸𝐼
]
0.083
E= modulus of elasticity of foundation.
B = foundation dimension.
If = displacement influence factor which
can be determined from charts (e.g.
Poulos and Davis 1974; Mayne and
Poulos 1999) depending on foundation
flexibility and soil layers.
4 Meyerhof and
Baike (1963); 𝑘 =
𝐸𝑠
𝐵(1 − 𝜐𝑠
2)
5 Selvadurai
(1979) 𝑘 =
0.65 𝐸𝑠
𝐵(1 − 𝜐𝑠
2)
6 Poulos (2018)
𝑘 =
𝐸𝑠
𝐵𝐼𝑓
7 Widjaja (2008) Clay: k = 40 to 50 Cu (t/m3
)
Sand: k = 70 to 100 SPTN (t/m3
)
Cu = undrained shear strength
SPTN = Standard Penetration Test
value of the soil.
8 Bowles (1996) 𝑘 =
𝑞 𝑢
𝑠 𝑎
qu=ultimate bearing capacity,
sa= allowable foundation settlement
(large foundation with B ≥ 3m where
failure is governed by settlement)

4. Figure 3. Architectural view of the residential project
Site investigation was carried out to explore the ground conditions using five boreholes with in-situ and laboratory
testing. The boreholes were about 10-15m deep and indicated that the ground consists of three different types of
soils as shown in Figure 4. The soil layer below the foundation level (1.5m) was classified as poorly graded
Sand with silt (SP-SM) according to the Unified Soil Classification System (USCS). This layer has an average
thickness of 3m and is underlain by moderately weathered Limestone with minimum Rock-quality designation
(RQD) value of 11 increasing with depth.
Standard Penetration Test (SPT) was conducted at three different depths. A small SPTN-value of 7-9 was measured
near ground surface, but the SPTN-values increased with depth to reach 29 at 3m below ground surface. The
ground condition was assessed to obtain the geotechnical parameters of the soils as shown in Table 2. The
assessment of the geotechnical parameters was based on the SPTN and published data about similar types of soil.
Figure 4. Typical borehole log from the site investigation carried out for the project

5. Groundwater table was found at 2.2m below ground surface, which is consistent with the sea water level as the
site is located within a coastal city.
4. Plate load testing – result and analysis
As part of the site investigation, 5 Plate Load Tests were conducted according to BS1377- Part 9, 1990. The tests
used a circular plate with a diameter of 600mm and the pressure was applied gradually up to 6.75 kg/cm2
(675
kPa) and then reduced back to 0 kPa creating a loading cycle as shown in Figure 5.
The experimental results of the plate load tests (PLT) were analysed to obtain the variation of k-values at different
displacements as presented in Figure 6. The k-value for each test did not show any significant variation with the
increase of displacement. This might be explained on the basis that the soil was mostly in the elastic condition
under the small plate settlement. The average k-value from all PLTs was approximately 124 MN/m3
, which is
significantly larger than expected for similar types of soils.
The soil elastic stiffness (Modulus of elasticity, Es) obtained from the PLTs results indicated an average value of
63 MPa. This is much larger than the Es value (20 MPa) based on SPTN (as shown in Table 2). This uncertainty
in soil parameters led the design team to consider a multi-craft foundation option for the buildings.
0
1
2
3
4
5
0 100 200 300 400 500 600 700
Settlement(mm)
Pressure (kPa)
0
1
2
3
4
5
0 100 200 300 400 500 600 700
Settlement(mm)
Pressure (kPa)
Test 2
0
1
2
3
4
5
0 100 200 300 400 500 600 700
Settlement(mm)
Pressure (kPa)
Test 3
0
1
2
3
4
5
0 100 200 300 400 500 600 700
Settlement(mm)
Pressure (kPa)
Test 4
0
1
2
3
4
5
0 100 200 300 400 500 600 700
Settlement(mm)
Pressure (kPa)
Test 5
Table 2. Geotechnical parameters
Parameter description Method/ equation used Parameter value
Modulus of elasticity of soil (Es) Schultze and Menzenbach (1961) 2400+530SPTN±2100 (kPa)
for silty Sands
Poisson ratio of the soil (νs) Bowles (1996) 0.3
Effective friction angle (’) Peck et al. (1974) 35o
(based on SPTN value)
Effective Cohesion (C’) Minimum value representing the
effect of silt within Sand matrix
5 kPa
Ultimate bearing capacity (qu) Brinch Hansen (1970) 260 kPa (B=10m and L/B=3)
367 kPa (B=3m and L/B=3)
Settlement (S) Schultze and Menzenbach (1961) Settlement caused by qu
Figure 5. The results of
Plate Load Test (PLT)
showing the relationship
between plat settlement and
applied pressure.

6. Figure 6. The variation of k-values at different displacements – obtained form PLT
5. Estimation of k-value based on numerical modelling
Numerical modelling was used to back analyse the plate load test (PLT) to verify and validate ground properties.
The ground model was then used to simulate different sizes of foundation representing the range of foundations
used in the project to evaluate the k-value at different pressure. Midas-GTS (2009) was used to simulate the Plate
Load Test, where the plate was modelled as a rigid material with a diameter of 600mm and the load was applied
gradually using the construction stages command, starting from 0 kPa to 675 kPa during the loading phase, then
from 675 to 0 kPa during the unloading phase. The soils were modelled using Mohr-coulomb failure criterion.
Figure 7 shows the module outputs in comparison with the results of the PLT tests. To reduce the differences
between the model results and the PLT results it was necessary to increase the elastic stiffness of the soil (Modulus
of elasticity, Es) to approximately 52 MPa, while keeping the shear strength parameters fixed (C’=5 kPa, ’=35o
–
see Table 2). This adjusted Es value (obtained from the back analysis) is significantly larger than expected for a
similar type of soil.
Figure 7. The results of the numerical model of the plate load test in comparison with the actual tests. The model
was used to verify the soil properties
Using the validated ground model, different size of foundations were analysed to evaluate the pressure- settlement
behaviour and thus estimate the range of k-value. Several sizes of shallow foundations were modelled to reflect
the actual design of the project. The maximum foundation size was approximately 10x40m with 0.6m thickness.
The reinforced concrete of the foundation was modelled as elastic material with a modulus of elasticity (E) equal
to 30 GPa. The k-value of the foundations obtained from the numerical analysis are presented and compared with
other results as discussed in Section 6.
0
50
100
150
200
250
0 1 2 3 4 5 6 7
k-value(MN/m3)
Displacement (mm)
Test 1
Test 2
Test 3
Test 4
Test 5
0
1
2
3
4
5
6
7
0 200 400 600 800
Settlement(mm)
Pressure (kPa)
Test 3 Test 4
Test 5 Numerical Analysis

7. 6. Comparison between the values of subgrade reaction
Several methods were used to evaluate the k-value for the project, including the methods described in Table 1 as
well as the numerical analysis presented above. The calculation showed a variation of up to 35% between these
methods as shown in Figure 8.
Figure 8. Comparison between the k-value obtained by different methods
It is apparent from Figure 8 that the variation of k-value obtained from different methods is not consistent. In
particular, Bowles’s method provided the largest values for small foundation (i.e. width < 3m), however the same
method gave the smallest k-values for larger size of foundations. Interestingly, the numerical model showed a
consistent decrease in k-value with the increase of foundation width.
7. Conclusion
The aim of this paper is to provide an overview of the calculation methods of the subgrade reaction for the design
of different sizes of raft and multi-column foundations. The paper provided an insight into some of the factors
affecting the k-value particularly the width of the foundation and the methods of analysis. The study has shown a
significant variation of up to 35% between the methods commonly used for the evaluation of the subgrade reaction.
This significant variation would increase the uncertainty which has been already experienced by structural and
geotechnical engineers over the calculation of the subgrade reaction.
Bowles’s method provided the largest values for smaller foundation widths (i.e. less than 3m), however, the same
method gave the smallest k-value for larger size of foundations. On the contrary, the numerical model showed a
consistent drop in k-value with the increase of foundation width. The results also showed that the modulus of
subgrade reaction obtained from Terzaghi's equation (1955) for prototype footing is not always smaller than the
k-value obtained for the foundation using numerical analysis. Overall the results of the numerical analysis are
more moderate and closer to Terzaghi’s method.
The principal implication of this study is that subgrade reaction is not always constant value and its variation
depends on several factors including the shape and geometries of the foundation, the location across the foundation
(centre or edge of foundation), the magnitude of pressure applied on the foundation, the non-linear behaviour of
the soil, and the type of field test and method used for assessing this parameter.
8. References
Al-Refeai, T., Al-Ghamdy, D. (1994). Geological and geotechnical aspects of Saudi Arabia. Geotechnical &
Geological Engineering, 12(4): 253-276.
Avci, B., Gurbuz, A. (2018). Modulus of subgrade reaction that varies with magnitude of displacement of
cohesion-less soil. Arabian Journal of Geosciences, 11(13): 351.
Bowles, J.E. (1996). Foundation analysis and design (fifth edition), USA: McGraw-Hill, 219–270 and 501–588.
0
20
40
60
80
100
120
140
0 2 4 6 8 10
k-value(MN/m3)
Foundation Width (m)
PLTs (ave)
Bowles (1996)
Terzaghi (1955)
Numerical

8. Brinch Hansen, J. (1970). A revised and extended formula for bearing capacity, Danish Geotechnical Institute
Copenhagen (Bulletin 28): 5-11.
British Standards Institute (1990). BS 1377-2: Methods of test for Soils for Civil Engineering purposes. Part 2:
Classification Tests.
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situ tests.
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Winkler, E. (1867). Die Lehre von Elastizat and Festigkeit (on elasticity and fixity): 182.