Keywords

1 Introduction

Bearing capacity and settlement are two key parameters required in the design of foundations. There are several techniques available today to determine or estimate both the bearing capacity and settlement behavior of foundations based on experimental methods, in situ tests, and numerical models including finite element analysis. In addition, there are different ground improvement methods to increase the bearing capacity and reduce the settlements. However, in spite of these advancements, various types of damages still can be caused to the superstructures placed on shallow foundations (hereafter referred to as SFs) due to the problems associated with the settlements leading to cracks, tilts, differential settlements, or displacements. This is particularly true for coarse-grained soils such as sands in which foundation settlements occur quickly after construction. Due to this reason, the settlement behavior is regarded as a governing parameter in the design of SFs in coarse-grained soils [25, 26, 34]. Foundation design codes suggest restricting the settlement of SFs placed in coarse-grained soils to 25 mm and also limit their differential settlements (e.g., [13]). Such design code guidelines suggest that the rational design of SFs can be achieved by estimating the applied stress versus surface settlement (hereafter referred to as SVS) behavior of SFs reliably instead of estimating the bearing capacity and settlement separately.

The most reliable testing method to estimate the SVS behaviors of SFs is in situ plate load tests (hereafter referred to as PLTs). In situ PLTs are commonly performed on the soils that are typically in a state of unsaturated condition. This is particularly true in arid or semiarid regions where the natural groundwater table is deep. Hence, the stresses associated with the constructed infrastructures such as SFs are distributed in the zone above the groundwater table, where the pore water pressures are negative with respect to the atmospheric pressure (i.e., matric suction). Several researchers showed that the SVS behaviors from model footings [35, 40, 42, 45] or in situ PLTs [16, 39] are significantly influenced by matric suction. However, in most cases, the in situ PLT results are interpreted without taking account of the negative pore water pressure above groundwater table. In other words, the influence of capillary stress or matric suction toward the SVS behavior is ignored in engineering practice. Moreover, the PLTs are generally conducted with small sizes of plates (either steel or concrete) in comparison to the real sizes of foundations. Due to this reason, the scale effect has been a controversial issue in implementing the PLT results into the design of SFs. These details suggest that the reliability of the design of SFs based on the PLT results can be improved by taking account of the influence of not only matric suction but also plate size on the SVS behaviors.

In this present study, two sets of in situ plate and footing load test results in unsaturated sandy and clayey soils available in the literature are revisited. Based on the results of these studies, an approach is presented such that the uncertainties associated with the scale effects are reduced or eliminated. In addition, discussions on how to interpret the in situ PLT results taking account of matric suction are also presented and discussed. Moreover, a methodology to estimate the variation of SVS behavior with respect to matric suction using finite element analysis (hereafter referred as FEA) is introduced.

2 Plate Load Test

In situ PLTs are generally conducted while designing SFs [3] or pavement structures [4, 5, 11] to estimate the reliable design parameters (i.e., bearing capacity and displacement) or to confirm the design assumptions. Figure 1 shows typical “applied stress” versus “surface settlement” (SVS) behavior from a PLT. The peak stress is defined as ultimate bearing capacity, q ult, for general failure; however, in the case where well-defined failure is not observed (i.e., local or punching failure), the stress corresponding to the 10% of the width of a foundation (ASTM D1194-94) or the stress corresponding to the intersection of elastic and plastic lines of the SVS behavior is regarded as q ult [16, 42, 48].

Fig. 1
figure 1

Typical stress versus displacement behavior from plate load test

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The elastic modulus can be estimated based on the modulus of subgrade reaction, k, that is a slope of SVS behavior (i.e., δ versus q) using the theory of elasticity as shown in Eq. (1). The maximum elastic modulus (i.e., initial tangent elastic modulus, E i ) can be computed using the k i value in the elastic range of SVS behavior (initial tangent modulus of subgrade reaction):

$$ E = \frac{{\left( {1 - {{\nu }^2}} \right)}}{{\left( {\delta /q} \right)}}B{{I}_{\rm{w}}} = k\left( {1 - {{\nu }^2}} \right){{I}_{\rm{w}}}B $$
(1)

where E = elastic modulus, ν = Poisson’s ratio, δ, q = settlement and corresponding stress, B = width (or diameter) of bearing plate, I w = factor involving the influence of shape and flexibility of loaded area, and k = modulus of subgrade reaction.

Ultimate bearing capacity, q ult, and elastic modulus, E, estimated based on the SVS behavior from a PLT are representative of soils within a depth zone which is approximately 1.5B–2.0B [38]. Agarwal and Rana [1] performed model footing tests in sands to study the influence of groundwater table on settlement. The results of the study showed that the settlement behavior of relatively dry sand is similar to that of sand with a groundwater table at a depth of 1.5B below the model footing. These results indirectly support that the increment of stress due to the load applied on the model footing is predominant in the range of 0–1.5B below model footing. These observations are also consistent with the modeling studies results by Oh and Vanapalli [31]. This fact also indicates that the SVS behavior from PLT is influenced by plate (or footing) size since different plate sizes result in different sizes of stress bulbs and mean stresses in soils. This phenomenon which is conventionally defined as “scale effect” needs to be investigated more rigorously to rationally design the shallow foundations.

3 Scale Effect in Plate Load Test

3.1 Scale Effect and Critical State Line

The Terzaghi’s bearing capacity factor, N γ , decreases with an increase in the width of footings [18]. Various attempts have been made by several researchers to understand the causes of scale effects. Three main explanations for the scale effect that generally accepted are as follows:

  1. 1.

    Reduction in the internal friction angle, φ′, with increasing footing size (i.e., nonlinearity of the Mohr–Coulomb failure envelop) [7, 18, 21]

  2. 2.

    Progressive failure (i.e., different φ′ along the slip surfaces below a footing) [43, 49]

  3. 3.

    Particle size effect (i.e., the ratio of soil particles to footing size) [41, 43]

According to Hettler and Gudehus [21], there is lack of consistent theory to explain the progressive failure mechanism in the soils below different sizes of footings. In addition, the particle size effect for the in situ plate (or footing) load test (hereafter the term “plate” is used to indicate both steel plate and concrete footing) can be neglected since the ratio of plate size, B to d 50 (i.e., grain size corresponding to 50% finer from the grain size distribution curve), for in situ PLTs are mostly greater than 50–100 [24]. The focus of the present study is to better understand the scale effects of PLT results and suggest some guidelines of how they can be used in practice.

The reduction in φ′ with an increasing footing size is attributed to the fact that the larger footing size contributes to a higher mean stress in the soils. In other words, the larger footing induces higher mean stress that contributes to lower φ′ due to the nonlinearity of the Mohr–Coulomb failure envelop when tested rigorously over a large stress range. This phenomenon can be better explained using the critical state concept ([19]; Fig. 2).

Fig. 2
figure 2

Relationship between the initial states (i.e., void ratio and mean stress) of soils below footings and critical state line (After Fellenius and Altaee [19])

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In Fig. 2, the points plotted on the lines a–b, c–d, and e–f simulate the following scenarios:

  1. 1.

    Line a–b: Different sizes of footings placed at different depths in sand that have the same initial void ratio value, but the distances to the critical state line are different.

  2. 2.

    Line c–d: Same sizes of footings places at the same depth in sand that have different initial void ratio values, but the distances to the critical state line are the same.

  3. 3.

    Line e–f: Different sizes of footings placed at different depths in sands that have different void ratio values, but the distances to the critical state line are the same.

The main concept shown in Fig. 2 is that the behavior of sand below a footing is governed by a distance from the initial state to the critical state line. In other words, the initial states plotted on the line e–f will show the same SVS behaviors regardless of footing size since the distance to the critical state line for each initial state is the same. On the other hand, the sand below a larger footing (e.g., S1 in Fig. 2) will have larger displacement at a certain applied stress in comparison to a smaller footing (e.g., S2 in Fig. 2) due to the greater mean stress (i.e., closer to the critical state line) even though the initial void ratio is the same.

3.2 Plate Load Test Results

In this present study, two sets of in situ PLTs in sandy and clayey soils available in the literature are revisited to discuss scale effect of PLTs.

The Federal Highway Administration (FHWA) has encouraged investigators to study the performance of SFs by providing research funding. As part of this research project, several series of in situ footing (i.e., 1, 2, 2.5, and 3 m) load tests were conducted on sandy soils. These studies were summarized in a symposium held at the Texas A&M University in 1994 [9] (Fig. 3). Consoli et al. [15] conducted in situ PLTs in unsaturated clayey soils using three steel circular plates (i.e., 0.3, 0.45, and 0.6 m; PLT) and three concrete square footings (i.e., 0.4, 0.7, and 1.0 m; FLT) (Fig. 4).

Fig. 3
figure 3

Stress versus displacement behavior from in situ footing load tests (After Briaud and Gibbens [9])

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Fig. 4
figure 4

Stress versus displacement behavior from in situ plate (PLT) and footing (FLT) load tests (After Consoli et al. [15])

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As can be seen in Figs. 3 and 4, the bearing capacity increases with decreasing plate size, and different displacement values are observed under different stresses. The SVS behaviors clearly show that the SVS behavior is dependent of plate size (i.e., scale effect). These observations are consistent with the SVS behaviors along the line a–b shown in Fig. 2. In other words, the soil below a larger footing induces higher mean stress; therefore, the initial state is closer to the critical state. This phenomenon makes the soil below a larger footing behave as if it is loose soil compared to a smaller footing [12].

3.3 Elimination of Scale Effect of Shallow Foundations

Briaud [8] suggested that scale effect (Fig. 3) can be eliminated by plotting the SVS behaviors as “applied stress” versus “settlement/width of footing” (i.e., δ/B) curves (i.e., normalized settlement; Eq. (2)). Similar trends of results were reported by Osterberg [36] and Palmer [37]:

$$ \frac{\delta }{B} = \frac{{q\left( {1 - {{\nu }^2}} \right)}}{E}{{I}_{\rm{w}}} $$
(2)

According to the report published by FHWA [10], this behavior can be explained using triaxial test analogy (Fig. 5). If triaxial tests are conducted for identical sand samples under the same confining pressure where the top platens are different sizes of footings, the stress versus strain behaviors for the samples are unique regardless of the diameter of the samples (i.e., the same stress for the same strain). This concept is similar to relationship between q and δ/B from PLTs since the term δ/B can be regarded as strain.

Fig. 5
figure 5

Triaxial test/shallow foundation analogy [10]

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Consoli et al. [15] suggested that the scale effect of PLTs (Fig. 4) can be eliminated when the applied stress and displacement are normalized with unconfined compressive strength, q u, and footing width, B, respectively, as shown in Eq. (3). They also analyzed PLTs results available in literature [17, 22] and showed that the concept in Eq. (3) can be extended to the PLT results in sandy soils as well:

$$ \begin{array}{llllllllll} \left( {\frac{q}{{{{q}_{\rm{u}}}}}} \right) = \left( {\frac{1}{{{{q}_{\rm{u}}}}}} \right)\left( {\frac{E}{{1 - {{\nu }^2}}}} \right)\left( {\frac{1}{{{{C}_{\rm{s}}}}}} \right)\left( {\frac{\delta }{B}} \right) \\ = \left\{ {\frac{{{{C}_{\rm{d}}}}}{{{{q}_{\rm{u}}}{{C}_{\rm{s}}}}}} \right\}\left( {\frac{\delta }{B}} \right) \\ \end{array} $$
(3)

where q = applied stress, q u = unconfined compressive strength at the depth of embedment, δ = surface settlement, B = width of footing, C s = coefficient involving shape and stiffness of loaded area (I w in Eq. 1), and C d = coefficient of deformation (=E/(1 − ν 2)).

As can be seen in Figs. 6 and 7, the curves (δ/B versus q) fall in a narrow range. From an engineering practice point of view, these curves can be considered to be unique. Consoli et al. [15] suggested that uniqueness of the normalized curves can be observed at sites where the soils are homogeneous and isotropic in nature.

Fig. 6
figure 6

Normalized in situ footing load test results [9]

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Fig. 7
figure 7

Normalized in situ plate and footing load test results [15]

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4 Scale Effect of Plate Size in Unsaturated Soils

The critical state concept discussed above can be effectively used to explain the scale effect of SFs in saturated or dry sands. However, this concept may not be applicable to interpret the scale effect of plate size in unsaturated soils since the SVS behaviors in unsaturated soils are influenced not only by footing size but also by matric suction value. The influence of matric suction however is typically ignored in conventional engineering practice.

4.1 Average Matric Suction Value

Matric suction distribution profile is mostly not uniform with depth in fields. In this case, the concept of “average matric suction” [45] can be used as a representative matric suction value to interpret mechanical properties of a soil at a certain matric suction distribution profile. The average matric suction value, Ψ, is defined as a matric suction value corresponding to the centroid of the suction distribution diagram from 0 to 1.5B depth (Fig. 8).

Fig. 8
figure 8

Estimation of average matric suction value using the centroid of suction distribution diagram

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As discussed earlier, the stress increment in a soil due to a load or a stress act on a SF is predominant in the range of 0–1.5B. Hence, when loads are applied on two different sizes of footings, the sizes of stress bulbs (in the depth zone of 0–1.5B) are different (Fig. 9). In other words, the stress bulb for the smaller footing (i.e., B 1) is shallower in comparison to that of the larger footing (i.e., B 2). These facts indicate that the SVS behaviors from PLTs are governed by E and ν values within the stress bulb. If a matric suction distribution profile is uniform with depth, the average matric suction value is the same regardless of footing size. However, if the matric suction distribution profile is nonuniform, the average matric suction value is dependent on the footing size. For example, the average matric suction value for the smaller plate, B 1, (i.e., Ψ 1) is greater than that of larger plate, B 2, (i.e., Ψ 2). In this case, the concept shown in Eqs. (2) and (3) cannot be used to eliminate the scale effect of plate since q u [29], E i [34], and ν [30, 32, 33] are not constant but vary with respect to matric suction. More discussions are summarized in later sections.

Fig. 9
figure 9

Average matric suction values for different footing sizes under nonconstant matric suction distribution profile

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4.2 Variation of Ei with Respect to Matric Suction for Coarse-Grained Soils

Oh et al. [34] analyzed three sets of model footing test results in unsaturated sands [28, 42] and showed that the initial tangent elastic modulus, E i , is significantly influenced by matric suction. Based on the analyses, they proposed a semiempirical model to estimate the variation of E i with respect to matric suction using the soil–water characteristic curve (SWCC) and the E i for saturated condition along with two fitting parameters, α and β:

$$ {{E}_{{i({\rm{unsat}})}}} = {{E}_{{i({\rm{sat}})}}}\left[ {1 + \alpha \frac{{\left( {{{u}_{\rm{a}}} - {{u}_{\rm{w}}}} \right)}}{{\left( {{{P}_{\rm{a}}}/101.3} \right)}}\left( {{{S}^{\beta }}} \right)} \right] $$
(4)

where E i(sat) and E i(unsat) = initial tangent elastic modulus for saturated and unsaturated conditions, respectively, P a = atmospheric pressure (i.e., 101.3 kPa), and α, β = fitting parameters.

They suggested that the fitting parameter, β = 1, is required for coarse-grained soils (i.e., I p = 0%; NP). The fitting parameter, α, is a function of footing size, and the values between 1.5 and 2 were recommended for large sizes of footings in field conditions to reliably estimate E i (Fig. 10) and elastic settlement (Fig. 11). Vanapalli and Oh [46] analyzed model footing [47], and in situ PLT [16, 39] results in unsaturated fine-grained soils and suggested that the fitting parameter, β = 2, is required for fine-grained soils. The analyses results also showed that the inverse of α (i.e., 1/α) nonlinearly increases with increasing I p and the upper and the lower boundary relationship can be used for low and high matric suction values, respectively, at a certain I p (Fig. 12).

Fig. 10
figure 10

Variation of modulus of elasticity with the parameter, α, for the 150 mm × 150 mm (Using data from Mohamed and Vanapalli [28])

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Fig. 11
figure 11

Variation of elastic settlement with the parameter, α, for the 150 mm × 150 mm footing (Using data from Mohamed and Vanapalli [28])

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Fig. 12
figure 12

Relationship between (1/α) and plasticity index, I p

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4.3 Variation of qu with Respect to Matric Suction for Fine-Grained Soils

Oh and Vanapalli [29] analyzed six sets of unconfined compression test results and showed that the q u value is a function of matric suction (Figs. 13 and 14). Based on the analyses, they proposed a semiempirical model to estimate the variation of undrained shear strength of unsaturated soils using the SWCC and undrained shear strength under saturated condition along with two fitting parameters, ν and μ (Eq. 5). Equation (5) is the same in form as Eq. (4):

$$ {{c}_{{{\rm{u(unsat)}}}}} = {{c}_{{{\rm{u(sat)}}}}}\left[ {1 + \frac{{\left( {{{u}_{\rm{a}}} - {{u}_{\rm{w}}}} \right)}}{{\left( {{{P}_{\rm{a}}}/101.3} \right)}}\frac{{\left( {{{S}^{\nu }}} \right)}}{\mu }} \right] $$
(5)

where c u(sat), c u(unsat) = shear strength under saturated and unsaturated condition, respectively, P a = atmospheric pressure (i.e., 101.3 kPa) and ν, μ = fitting parameters.

Fig. 13
figure 13

Comparison between the measured and the estimated shear strength using the data by Babu et al. [6]

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Fig. 14
figure 14

Comparison between the measured and the estimated shear strength using the data by Vanapalli et al. [47]

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The fitting parameter, ν = 2, is required for unsaturated fine-grained soils. Figure 15 shows the relationship between the fitting parameter, μ, and plasticity index, I p, on semilogarithmic scale for the soils used for the analysis. The fitting parameter, μ, was found to be constant with a value of “9” for the soils that have I p values in the range of 8 and 15.5%. The value of μ however increases linearly on semilogarithmic scale with increasing I p.

Fig. 15
figure 15

Relationship between plasticity index, I p, and the fitting parameter, μ

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4.4 Variation of ν with Respect to Matric Suction

The Poisson’s ratio, ν, is typically considered to be a constant value in the elastic settlement analysis of soils. This section briefly highlights how ν varies with matric suction by revisiting published data from the literature. Mendoza et al. [27] and Alramahi et al. [2] conducted bender element tests to investigate the variation of small-strain elastic and shear modulus with respect to degree of saturation for kaolinite and mixture of glass beads and kaolin clay, respectively. Oh and Vanapalli [33] reanalyzed the results and back calculated the Poisson’s ratio, ν, with respect to degree of saturation. The analyses of the results show that ν is not constant but varies with the degree of saturation as shown in Fig. 16.

Fig. 16
figure 16

Variation of Poisson’s ratio, ν, with respect to degree saturation

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5 Reanalysis of Footing Load Test Results in Briaud and Gibbens [9]

The site selected for the in situ footing load tests was predominantly sand (mostly medium dense silty sand) from 0 to 11 m overlain by hard clay layer (Fig. 17). The groundwater table was observed at a depth of 4.9 m, and the soil above the groundwater table was in a state of unsaturated condition. In this case, different footing sizes may result in different average matric suction values. In other words, scale effect cannot be eliminated with normalized settlement since the soils at the site are not “homogenous and isotropic.” Despite this fact, as can be seen in Fig. 6, the SVS behaviors from different sizes of footing fall in a narrow range. This behavior can be explained by investigating the variation of matric suction with depth at the site as follows.

Fig. 17
figure 17

The average soil profile at the test site (After Briaud and Gibbens [9])

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Figure 18 shows the grain size distribution curves for the soil samples collected from three different depths (i.e., 1.4–1.8 m, 3.5–4.0 m, and 4.6–5.0 m). The grain size distribution curve the Sollerod sand shown in Fig. 18 is similar to the sand sample collected at the depth of 1.4–1.8 m. The reasons associated with showing the GSD curve of Sollerod sand will be discussed later in this chapter. The soil properties used in the analysis are summarized in Table 1.

Fig. 18
figure 18

Grain size distribution curves for the soil samples collected from three different depths [9] and Sollerod sand [42]

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Table 1 Summary of the soil properties (From Briaud and Gibbens [9])
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As shown in Table 1, the water content at the depths of 0.6 and 3.0 m is 5%. This implies that the matric suction value can be assumed to be constant up to the depth of approximately 3.0 m. The field matric suction distribution profile is consistent with the typical matric suction distribution profile above groundwater table for the coarse-grained soils. In other words, matric suction increases gradually (which is close to hydrostatic conditions) up to residual matric suction value and thereafter remains close to constant conditions (i.e., matric suction distribution (1) in Fig. 19). This matric suction distribution profile resulted in the same average matric suction value regardless of footing size (for this study). However, it also should be noted that the average matric suction value for each footing can be different if a nonuniform matric suction distribution profile is available below the footings (i.e., matric suction distribution (2) in Fig. 19).

Fig. 19
figure 19

Average matric suction for different sizes of footing under uniform and nonuniform matric suction distribution

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6 Variation of SVS Behaviors with Respect to Matric Suction

After construction of SFs, the soils below them typically experience wetting–drying cycles due to the reasons mostly associated with the climate (i.e., rain infiltration or evaporation). Hence, it is also important to estimate the variation of SVS behaviors with respect to matric suction.

Oh and Vanapalli [32, 33] conducted finite element analysis (FEA) using the commercial finite element software SIGMA/W (Geo-Slope 2007; [23]) to simulate SVS behavior of in situ footing (B × L = 1 m × 1 m) load test results ([9]; Fig. 3) on unsaturated sandy soils. The FEA was performed using elastic–perfectly plastic model [14] extending the approach proposed by Oh and Vanapalli [30]. The square footing was modeled as a circular footing with an equivalent area (i.e., 1.13 m in diameter, axisymmetric problem).

The soil–water characteristic curve (SWCC) can be used as a tool to estimate the variation of total cohesion, c, (Eq. 6; [44]) and initial tangent elastic modulus, E i , (Eq. 4) with respect to matric suction:

$$ c = c^{\prime} + \left( {{{u}_{\rm{a}}} - {{u}_{\rm{w}}}} \right)\left( {{{S}^{\kappa }}} \right)\tan \phi ^{\prime} $$
(6)

where c = total cohesion, c′ and φ′ = effective cohesion and internal friction angle for saturated condition, respectively, (u a − u w) = matric suction, S = degree of saturation, and κ = fitting parameter (κ = 1 for sandy soils (i.e. I p = 0%); [20]).

Information on the SWCC was not available in the literature for the site where the in situ footing load test was carried out. Hence, the SWCC for the Sollerod sand (Fig. 20) used for the analysis as an alternative based on the following justifications. Among the grain size distribution (hereafter referred as GSD) curves shown in Fig. 18, the grain size distribution curve for the range of depth 1.4–1.8 m can be chosen as a representative GSD curve since the stress below the footing 1 m × 1 m is predominant in the range of 0–1.5 m (i.e., 1.5B) below the footing. This GSD curve is similar to that of Sollerod sand (see Fig. 18) used by Steensen-Bach et al. [42] to conduct model footing tests in a sand to understand influence of matric suction on the load carrying capacity. In addition, the shear strength parameters for the Sollerod sand (c′ = 0.8 kPa and φ′ = 35.8°) are also similar to those of the sand where the in situ footing load tests were conducted (see Table 1). The influence of wetting–drying cycles (i.e., hysteresis) and external stresses on the SWCC is not taken into account in the analysis due to the limited information.

Fig. 20
figure 20

SWCC used for the analysis (Date from Steensen-Bach et al. [42])

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The variation of SVS behavior with respect to matric suction from the FEA is shown in Fig. 21. Figure 22a, b shows the variation of settlement under the same stress of 344 kPa and the variation of stress that can cause 25-mm settlement for different matric suction values, respectively. The stress 344 kPa is chosen since the settlement for saturated condition at this stress is 25 mm. The settlement at the matric suction of 10 kPa (i.e., field condition) is approximately 4 mm and then increases up to 25 mm (i.e., permissible settlement) as the soil approaches saturated conditions under the constant stress (i.e., 344 kPa). The permissible settlement, 25 mm, can be induced at 2.7 times less stress as the soil approaches saturated conditions (i.e., from 10 to 0 kPa). The results imply that settlements can increase due to decrease in matric suction. It is also of interest to note that such a problem can be alleviated if the matric suction of the soil is maintained at 2-kPa value.

Fig. 21
figure 21

Variation of SVS behavior with respect to matric suction

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Fig. 22
figure 22

Variation of (a) settlement under the applied stress of 344 kPa and (b) stress that can cause 25-mm settlement with respect to matric suction

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7 Summary and Conclusions

Plate load test (PLT) is regarded as the most reliable testing method to estimate the applied stress versus surface settlement (SVS) behavior of shallow foundations. However, there are uncertainties in interpreting the PLT results for soils that are in a state of unsaturated condition. This is mainly attributed to the fact that the SVS behavior from the PLTs is significantly influenced by both footing size and the capillary stresses (i.e., matric suction). Previous studies showed that the scale effect can be eliminated by normalizing settlement with footing size. This methodology is applicable to the soils that are homogeneous and isotropic with depth in nature such as saturated or dry soils. In case of unsaturated soils, matric suction distribution profile with depth should be taken into account to judge whether or not this methodology is applicable. This is because if the matric suction distribution profile is nonuniform with depth, different plate sizes lead to different average matric suction values. In other words, the soil below the plates cannot be regarded as homogeneous and isotropic since strength, initial tangent elastic modulus, and the Poisson’s ratio are function of matric suction. These facts indicate that the reliable deign of shallow foundations based on the PLT results can be obtained only when the results are interpreted taking account of the matric suction distribution profile with depth and influence of average matric suction value on the SVS behavior.

In case of the shallow foundations resting on unsaturated sandy soils, it is also important to estimate the variation of SVS behavior with respect to matric suction. This can be achieved by conducting finite element analysis using the methodology presented in this chapter. According to the finite element analysis for the in situ footing (1 m × 1 m) load test results discussed in this chapter [9], unexpected problems associated with settlement are likely due to decrease in matric suction. Such a problem can be alleviated if the matric suction of the soil is maintained at a low of 2 kPa.