Environmental & Engineering Geoscience
 © 2012 Association of Environmental & Engineering Geologists
Abstract
Many methods are available in literature for the prediction of immediate settlement of shallow foundations resting on cohesive soils. Of these, eight popular methods that are commonly used in practice are summarized briefly and compared using both hypothetical and real cases. In hypothetical cases, various scenarios with respect to the foundation geometry and embedment depth (i.e., different L/B and D_{f}/B ratios) under fixed loading and soil conditions were considered. Real cases, including all parameters required for the immediate settlement computations, were derived from literature after intensive effort. The results obtained from the comparisons of the hypothetical cases showed that the immediate settlements computed by the methods evaluated in this study are considerably different depending on the ratios of L/B and D_{f}/B. When the real cases were considered, for the majority of the cases, the best settlement estimates were obtained from the Bowles method. However, for mat foundations, the best settlement estimates were obtained from the Mayne method. In addition, immediate settlement values of foundations founded on multilayer soil profiles were also compared for both hypothetical and real cases. Two soil layers having different elastic modulus and thickness under fixed loading conditions, embedment depth, and aspect ratio for the foundation were considered in the hypothetical cases. Immediate settlements were calculated by weighted mean, harmonic mean, and principles of superposition in conjunction with the Mayne method, as well as with the full Mayne method with Gibson modulus. Results obtained from the hypothetical cases demonstrated that the immediate settlements calculated by these procedures are remarkably different depending on the variation in the elastic modulus and thickness of the soil layers. For the evaluation of real cases, immediate settlements were calculated by both Mayne and Bowles methods, as well as with the full Mayne method itself. As a result, the full Mayne method with Gibson modulus was determined to produce perfect settlement estimates for real cases compared with the conventional procedures such as weighted mean, harmonic mean, and principles of superposition. In addition, among conventional procedures, principles of superposition in conjunction with the Bowles method provide comparable settlement estimates to the full Mayne method.
INTRODUCTION
Shallow foundations founded on cohesive soils must be safe against shear failure of the soil, and settlement must be within tolerable limits associated with the superstructure under consideration. However, in general, the design of shallow foundations is often governed by settlement considerations rather than shear failure of the soil, and some design guidelines restrict the total settlement as well as differential settlements of the foundations to a reasonable extent according to the type of structure to prevent serviceability problems. Hence, settlement prediction is a major concern and an essential criterion in the design of shallow foundations. For this reason, the settlement of the foundation has to be predicted with reasonable accuracy before construction.
Total settlement of a foundation can be divided into two major categories: the immediate (or elastic) settlement, and the consolidation settlement. Immediate settlement can be defined as the elastic deformation or volume distortion of the area influenced by the foundation load without significant dissipation of excess porewater pressures. In many cases, the immediate settlement of a foundation on clay is small when compared with consolidation settlement (Foye et al., 2008); however, case histories have shown that it can be significant (as high as 0.3 m), particularly for highly plastic (plasticity index ≥ 50) or organic soils (Foot and Ladd, 1981). In addition, evaluation of the immediate settlement of structures on clay is important for a number of reasons (D'Appolonia et al., 1971): (1) Immediate settlement may constitute a large portion of the total final settlement, depending on the nature of the soil, the loading geometry, and the thickness of the compressible layer; (2) analysis of immediate settlement is an integral part of the analysis of the overall timesettlement behavior of foundations; and (3) immediate settlement is closely related to the undrained stability of a foundation, and excessive immediate settlement may be a warning of impending failure.
Conventionally, immediate settlement has been computed based on the theory of elasticity, which assumes that the soil is a linear elastic material that deforms according to Hooke's law. For this reason, immediate settlement is sometimes called “elastic settlement” in geotechnical literature. The general form of the immediate settlement formula based on the theory of elasticity is shown in Eq. 1 (Davis and Poulos, 1968; D'Appolonia et al., 1971; and Mayne and Poulos, 1999).
In some versions of Eq. 1, a reduction factor of (1 − ν^{2}) may be inserted on the righthand side of the equation. This is especially valid for surface foundations (Christian and Carrier, 1978).
Different influence factors are available in literature to estimate immediate settlement of shallow foundations, including for perfectly flexible and/or rigid foundations. As known from the theory of elasticity, the magnitudes of settlement at the corner, center, edges, and average settlements of perfectly flexible foundations are different, whereas magnitudes of settlements of rigid foundations are almost equal at all points beneath the foundation. Davis and Poulos (1968) proposed the following relationships to obtain average settlement from flexible settlement beneath the center, corner, and edge of the foundation:
Similarly, Mayne and Poulos (1999) proposed the following approximate relationships between settlements at the center, corner, and edge of flexible foundations:
where K_{F} is the foundation flexibility factor and is defined as follows
Foundations of domestic buildings are generally designed as rectangular spread footing and constructed from reinforced concrete, which is almost rigid. Coduto (2001) proposed some reduction factors to obtain rigid settlement from flexible settlement beneath the center of the foundation. These reduction factors are 0.85 for perfectly rigid foundations (i.e., spread footings), and between 0.85 and 1.00 for intermediate rigid foundations (i.e., mat foundations).
In addition, Bowles (1996) argued that most foundations are flexible in practice, and even very thick ones deflect when loaded by superstructure loads. According to Bowles (1996), flexible settlement beneath the center of the foundation should be reduced by about 7 percent to obtain rigid settlement of the foundation. That is,
In geotechnical literature, many different methods are available for the prediction of immediate settlement of shallow foundations, including perfectly flexible and rigid foundations resting on ground surface or embedded into a soil mass having infinite thickness or underlain by an incompressible layer (i.e., bedrock). While the consolidation settlement tends to be more important to the overall design, it was not investigated here, and the comparisons here are only for methods used to estimate the immediate settlement. For this purpose, eight available methods that are commonly used in practice for the estimation of immediate settlement were compared both with a series of hypothetical cases involving different foundation geometries and embedment depths (i.e., different L/B and D_{f}/B ratios) under fixed loading and soil conditions and real cases derived from literature.
Because naturally deposited soils generally consist of soil layers having different elastic properties, the presence of soil layers with different elastic modulus values under the foundation is a common situation in practice. In this situation, several procedures can be performed for the computation of immediate settlements accounting for the stratification of the soil (i.e., variation of the elastic modulus with depth); these are the principle of superposition and the equivalent elastic modulus of the soil profile, which may be calculated by weighted or harmonic means. In this study, immediate settlements were computed using these procedures for a series of hypothetical cases including two representative soil layers having different elastic modulus and thicknesses under fixed loading conditions, foundation geometry, and embedment depth. These procedures were also compared for real cases collected from literature. Next, the available methods that are commonly used for the prediction of immediate settlement are briefly summarized; methods that are similar are presented together, rather than in chronological order.
A BRIEF REVIEW OF THE METHODS
The methods available in literature for the calculation of immediate settlement of shallow foundations may be grouped into two major categories with respect to the embedment depth: (1) methods that consider the embedment depth of the foundation, and (2) methods that assume foundations founded on ground surface (i.e., surface footing). In practice, reinforced concrete foundations are generally embedded within the soil to a reasonable depth by regarding various factors such as environmental conditions, field topography, and project requirements. In an immediate settlement equation, the embedment depth of the foundation can be taken into account by using an influence factor. A popular influence factor accounting for the embedment depth of the foundation is Fox's (1948) influence factor. In addition, a second influence factor must be present in the immediate settlement equations to take into account the dimension of the foundation and thickness of the compressible stratum. A popular influence factor for this is Steinbrenner's (1934) influence factor. These two influence factors are sometimes symbolized differently in literature, such as μ_{o}, μ_{1}; I_{s}, I_{F}; or I_{1}, I_{2}, etc., even though originally they were the same. In this study, the symbols are used as they were used in the cited studies.
The Method Proposed by Janbu et al. (1956)
A popular equation for the calculation of immediate settlement of shallow foundations was proposed by Janbu et al. (1956). This equation is as follows
In Eq. 6, the influence factors μ_{o} and μ_{1} are based on Fox's (1948) and Steinbrenner's (1934) elastic solutions, respectively. Janbu et al. (1956) presented two charts (Figure 1a) for determining the influence factors μ_{o} and μ_{1}, based on the values of Fox's (1948) and Steinbrenner's (1934) elastic solutions, respectively.
Since published in 1956, the charts and the equation proposed by Janbu et al. (1956) have become very popular among geotechnical engineers because of their considerable usefulness and simplicity. However, it should be noted that there are some assumptions in the equation and charts proposed by Janbu et al. (1956). These are: (1) Poisson's ratio of the soil is 0.5, (2) the foundation is perfectly flexible, and (3) the computed settlement is the average settlement beneath the foundation.
The Method Proposed by Christian and Carrier (1978)
Christian and Carrier (1978) made a critical evaluation of the charts proposed by Janbu et al. (1956) and concluded that the μ_{1} chart is inaccurate for H/B less than 5, and the μ_{o} chart overestimates the effect of embedment depth of the foundation, and so they recommended new charts for determining the influence factors μ_{o} and μ_{1}, provided that the equation proposed by Janbu et al. (1956) remains the same. The new charts suggested by Christian and Carrier (1978) are illustrated here sidebyside with Janbu's charts for easier comparison (Figure 1b).
The charts suggested by Christian and Carrier (1978) (Figure 1b) are composed of Burland's (1970) and Giroud's (1972) results for μ_{o} and μ_{1}, respectively. The shape correction factors μ_{1} are composed of Giroud's (1972) results, which are based on the numerical integration of Taylor's results (Giroud, 1972), while embedment correction factors μ_{o} are the values computed by Burland (1970), which are based on the results of finite element analysis conducted for circular foundations taking the Poisson's ratio of the soil equal to 0.49. In fact, Burland (1970) computed μ_{o} factors for different Poisson's ratios of the soil varying from 0 to 0.49. However, because the value of 0.49 (about 0.5) is a usual design value for saturated clay soil and often employed in practice, only the μ_{o} factors for Poisson's ratio equal to 0.49 were proposed by Christian and Carrier (1978).
Originally, Giroud (1972) published tabulated results for the correction factor μ_{1} for L/B ratios equal to 1, 1.5, 2, 2.5, 3, 4, 5, 10, and ∞, in conjunction with H/B ratios ranging from 0 to 20, and ∞, including Poisson's ratios of 0, 0.3, and 0.5. From these tabulated results, Christian and Carrier (1978) selected only the values for L/B ratios equal to 1, 2, 5, 10, and ∞, and for Poisson's ratio equal to 0.50 for simplicity, and they transformed these values into a chart as shown in Figure 1b. In the present paper, because these possibilities are encountered in practice, the remaining values for the L/B ratio (that is, 1.5, 2.5, 3, and 4) published by Giroud (1972) were added into the μ_{1} chart and are reproduced in Figure 2. Unlike the chart published by Christian and Carrier (1978), the chart presented in Figure 2 does not contain a curve for circular foundations due to the lack of any values for circular foundations in the tabulated results published by Giroud (1972).
The assumptions in the charts and the equations proposed by Janbu et al. (1956) and Christian and Carrier (1978) are the same; that is, the foundation is assumed to be perfectly flexible, the Poisson's ratio of the soil is equal to about 0.5, and the settlement is the average immediate settlement beneath the foundation. However, it should be noted that the charts recommended by Christian and Carrier (1978) give lower values for μ_{1} and higher values for μ_{o} when compared with the charts proposed by Janbu et al. (1956).
The Method Proposed by Harr (1966)
Harr (1966) published the following equation to compute the immediate settlement of flexible and rigid foundations having rectangular and circular shapes resting on the ground surface.
Harr (1966) published various tabulated values assembled by different investigators (e.g., Egorov, 1958; Tsytovich, 1963) for the influence factor I_{1}, which is based on various assumptions, such as foundation flexibility and thickness of the compressible layer. Three different charts were prepared from the tabulated values published by Harr (1966) and are presented in Figure 3.
The influence factors shown in Figure 3a are based on the values assembled by Tsytovich (1963) for flexible and rigid foundations ranging from circular to rectangular shapes resting on the surface of an infinitely deep homogeneous compressible clay layer (i.e., H = ∞). To calculate the immediate settlement under the corner of rectangular foundations by using the influence factors given in Figure 3a, I_{cor} can be taken to be ½I_{cen}. The influence factors given in Figure 3b and c are based on the values assembled by Egorov (1958) for flexible and rigid foundations, respectively, resting on the surface of a homogeneous compressible layer underlain by a rigid base (i.e., H < ∞). Since the equation published by Harr (1966) is valid for surface foundations, it does not include any correction factors for the embedment depth of the foundation. However, it is possible to use this equation by applying an embedment correction factor given by Fox (1948) or Burland (1970).
The Method Proposed by Giroud (1968)
Giroud (1968) proposed the following equation to compute the immediate settlement of rectangular flexible foundations resting on the surface of a soil mass having semiinfinite thickness that behaves as an isotropic homogeneous continuum, which obeys Hooke's law.
If the Poisson's ratio ν is taken to be equal to 0.5 in Eq. 8, the second term of the equation will be equal to zero. Hence, the second term may be omitted, and the equation is identical to Harr's (that is Eq. 7).
Giroud (1968) calculated the influence factor I_{1} by integrating the formulas by Boussinesq (1885) and Cerruti (1882), which give the settlement due to a single load, and obtained the following equations to compute the influence factor I_{1}, which depends on the point where the settlement has to be computed.
To facilitate the immediate settlement computations, representative influence factors were computed by using Eq. 8a and Eq. 8b, and a chart was prepared from the values obtained. As a result, the same chart given in Figure 3a is obtained. In other words, the values calculated by Eq. 8a and Eq. 8b are the same as those presented by Harr (1966). In that case, the influence factors I_{cen} and I_{av} can be obtained directly from Figure 3a for computation of immediate settlement by the Harr method. Although the method originally did not involve any correction factors for the embedment depth, an embedment correction factor given by Fox (1948) or Burland (1970) may be applied to the equation.
The Method Proposed by Bowles (1996)
Bowles (1996) suggested the following equations to compute the immediate settlement beneath a corner of a rectangular flexible foundation.
The influence factors I_{1} and I_{2} shown in Eq. 9 are the Steinbrenner's (1934) influence factors and can be computed by using Eq. 9a and Eq. 9b.
where the tan^{−1} term is in radians, M = L/B, and N = H/B.
The embedment correction factor I_{F} (Fox, 1948) may be computed by using the computer program FFACTOR enclosed in the textbook of Bowles (1996). In the present study, representative values for the embedment correction factor I_{F} were computed by using the FFACTOR, including L/B ratios of 1, 1.5, 2, 2.5, 3, 4, 5, 10, 20, 50, and 100, and D_{f}/B ratios ranging from 0.1 to 1000. In these computations, Poisson's ratio of the soil was taken as to be equal to 0.5. When a chart was prepared from the values of I_{F} obtained from these computations, the same μ_{o} chart as the one presented by Janbu et al. (1956) and given in Figure 1a is obtained. Therefore, the embedment correction factor I_{F} given in Eq. 9 can be obtained from the μ_{o} chart depicted in Figure 1a for ν = 0.5.
To facilitate the immediate settlement computations, representative values for the influence factor I_{1} were also computed by using Eq. 9a with an Excel spreadsheet, and the chart given in Figure 4 was prepared from the values obtained. Therefore, the influence factor I_{1} can be calculated from either Eq. 9a or can be obtained from Figure 4.
Comparing the charts μ_{1} and I_{1} (Figures 1a and 4, respectively), it can be seen that they are similar in form. However, it should be noted that the I_{1} chart gives lower values than the μ_{1} chart. The reason is that the influence factor I_{1} is used to compute the settlement beneath the corner of the foundation, whereas the influence factor μ_{1} is used to compute the average settlement of the foundation.
If Poisson's ratio of the soil is taken to be equal to 0.5, which is the usual design value for saturated clay in immediate settlement computations, the influence factor I_{2} shown in Eq. 9 would be ineffective because the second term is equal to zero. Therefore, this correction factor is not considered further.
As stated previously, Eq. 9 is valid for the immediate settlement beneath the corner of a rectangular flexible foundation. In order to calculate the immediate settlement beneath the center of a rectangular flexible foundation from Eq. 9, the foundation base should be divided by 4, and the settlement beneath the corner of each contributing rectangle should be summed. Alternatively, settlement of one of the contributing rectangles should be multiplied by 4 to calculate the settlement beneath the center of the whole base. In accordance with these considerations, to compute the immediate settlement beneath the center of a rectangular flexible foundation, Eq. 10 can be used.
The influence factor I_{2} was neglected in Eq. 10 because the Poisson's ratio of the soil was assumed as 0.5. In order to compute the immediate settlement under the center of the foundation by Eq. 10, the value of B/2 (i.e., B′) instead of B should be used during computations of both I_{F} and I_{1} and for use with the charts given in Figures 1a and 4, respectively. As a note, Bowles (1996) suggested that the equations can also be used for a circular foundation by converting it to an equivalent square.
The Method Proposed by Gazetas et al. (1985)
Gazetas et al. (1985) proposed an equation for the estimation of immediate settlement of arbitrarily shaped rigid foundations resting on an elastic deep homogeneous soil (it was not defined clearly what signifies “deep”). The equation is applicable to a large range of embedment depths and a variety of solid base shapes, ranging from circular to strip and including rectangles of any aspect ratio as well as odd shapes differing substantially from rectangular or circular. The equation also regards the effect of contact between vertical sidewalls of the embedded foundation and the surrounding soil. Originally, while the method is valid for rigid foundations, Gazetas et al. (1985) suggested that the proposed equation may also be applicable to compute the average immediate settlement of flexible foundations with sufficient accuracy. The equation and the geometry proposed by Gazetas et al. (1985) are given in Eq. 11.
The shape, embedment, and sidewall friction factors, μ_{s}, μ_{emb}, and μ_{wall} can be determined by the following equations.
The dimensionless shape parameter A_{b}/4(L′)^{2} shown in Eq. 11a has the typical values for common footing shapes as listed in Table 1.
Gazetas et al. (1985) argued that the friction resistance (i.e., adhesion force) between vertical sides of the wall and the surrounding soil leads to an additional reduction in the settlement of the embedded foundation. In addition, they also stated that for cases where there is doubt about the quality of the contact between sidewall and the surrounding soil, a reduction factor between zero and unity can be applied to the sidewall friction factor μ_{wall}. However, Budhu (2000) has argued that the full wall resistance will only be mobilized if sufficient settlement occurs, and it is difficult to ascertain the quality of the soilwall adhesion. Consequently, Budhu (2000) has recommended that caution should be exercised in relying on the reduction of settlement resulting from the sidewall friction factor. If the effect of wall friction is neglected, then sidewall friction factor μ_{wall} should be taken as unity (that is μ_{wall} = 1) in the computations.
The Method Proposed by Mayne and Poulos (1999)
Mayne and Poulos (1999) presented more comprehensive design charts and approximation formula for the immediate settlement of shallow foundations, which account for the dimension, rigidity, and embedment depth of the foundation, the variation of the elastic modulus with depth, and the Poison's ratio of the soil. With this method, the immediate settlement under the center of a flexible circular foundation can be computed. The equation proposed by Mayne and Poulos (1999) is given by
The correction factors I_{G} and I_{F} can be obtained from Figure 5. The embedment correction factor I_{E} can be taken from the μ_{o} chart presented in Figure2b because they are the same correction factors as those suggested by Burland (1970). Actually, although the method was developed for circular foundations, according to developers, other geometries can be accommodated by setting the foundation plan area equal to the area of an equivalent circle.
Additional terms, such as normalized Gibson modulus β and foundation flexibility factor K_{F}, need to be known to determine the displacement influence and rigidity correction factors I_{G} and I_{F}. Foundation flexibility factor K_{F} can be determined from Eq. 3c given previously. Normalized Gibson modulus β assumes the variation (or increase) of the elastic modulus of the soil varies with depth linearly and can be expressed by (Gibson, 1967)
The elastic modulus of the foundation material E_{found} (i.e., reinforced concrete) varies depending on the compressive strength and unit weight of the concrete. According to ACI 318 Code (1989), the elastic modulus of a reinforced concrete foundation is given by (Bowles, 1996).
In the ACI 318 Code, the unit weight of concrete γ_{c} is given as values varying between 14 and 25 kN/m^{3} (Bowles, 1996); therefore, it was taken to be 23 kN/m^{3} for this study. Compressive strength of concrete, , in fact, is a design value related to the superstructure, but 25 MPa may be suitable for generalized dimensioning of the foundation; therefore, the elastic modulus of the foundation may be taken as about 25,000 MPa for the calculation of immediate settlement for preliminary design purposes. As for thickness of the foundation, t = 0.5 m may be convenient for immediate settlement calculations of rectangular spread footings.
Other Methods Available for the Estimation of Immediate Settlement
In addition to the previously described methods, a few more methods are available for the estimation of immediate settlement of shallow foundations, such as the methods suggested by D'Appolonia et al. (1971), Han et al. (2007), and Foye et al. (2008). These methods have not been incorporated into the comparisons conducted for the present study because either they require extra parameters other than the usual parameters (such as initial shear stress ratio f, which is a function of both the coefficient of lateral earth stress at rest, K_{o}, and undrained shear strength of the soil, s_{u}, and initial vertical effective stress ), or they do not include the correction factors for all possible conditions (e.g., L/B, H/B, and D_{f}/B) required for the immediate settlement calculations.
COMPARISON OF THE METHODS
The methods were compared first on a series of hypothetical cases including all possible conditions and then on real cases derived from literature.
Comparison of the Methods Using Hypothetical Cases
In the hypothetical cases, the undrained elastic modulus of the soil E_{u} was taken as 15 MPa (no change with depth was assumed), which corresponds to a medium to soft clay (Bowles, 1996), and undrained Poisson's ratio ν was taken as 0.5, which is the usual design value for saturated clay in undrained conditions (Mayne and Poulos, 1999; Coduto 2001). Vertical uniform pressure applied by footing to the soil q_{o} was assumed as 150 kPa, which corresponds to an approximately tenstory building. A schematic illustration of the mentioned cases is depicted in Figure 6, and other assumptions related to these cases are given on this figure. For the calculation of immediate settlements in conjunction with the Bowles method, Eq. 10 was used; therefore, immediate settlements under the center of the foundation were computed. For the Gazetas method, wall friction effect μ_{wall} was neglected for simplicity; therefore, μ_{wall} was taken to be equal to 1. For computations of Mayne method, the elastic modulus and thickness of the foundation were taken as 25,000 MPa and 0.5 m, respectively. In order to calculate the normalized Gibson modulus β, the rate of increase of elastic modulus of the soil with depth k_{E} was taken as 0.01. This k_{E} value is so small that can be assumed no change occurs in the elastic modulus with depth. The embedment correction factor of Burland (1970) was applied to the both Harr and Giroud methods to be able to compare them with the other methods. The main characteristics of the methods that were used for comparison are given in Table 2. The immediate settlements computed by these methods are presented in Figure 7.
As can be seen from Figure 7, the settlement curves obtained by the methods that utilize the same embedment correction factor are almost parallel for all L/B and D_{f}/B ratios. Among the methods, the greatest settlement values were obtained from Giroud's method for all L/B and D_{f}/B ratios because this method does not consider an incompressible layer in reasonable depth under the foundation (i.e., H = ∞). Differences between immediate settlements computed by the Mayne and the other methods also increase with increasing L/B ratio. Comparing the flexible and rigid settlements obtained by Harr methods, the flexible settlement is always greater than the rigid settlement, as can be expected. As highlighted also by Christian and Carrier (1978), the immediate settlements estimated by the Janbu method are more sensitive to the embedment depth of the foundation than that of the Christian method. Because the Bowles and Janbu methods use the same embedment and shape correction factors (Table 2), the immediate settlements computed by these methods are very close to each other for all L/B and D_{f}/B ratios. A remarkable result that can be observed in Figure 7 is that the immediate settlements computed by the Gazetas method decrease linearly toward zero as D_{f}/B ratio increases. The reason is that the embedment correction factor μ_{emb} proposed by Gazetas et al. (1985) is directly proportional to the D_{f}/B ratio, and as D_{f}/B ratio increases, μ_{emb} approaches to zero. This is in agreement with the findings obtained by Mei et al. (2005) about the Gazetas method.
Comparison of the Methods on Real Cases
Indeed, the performance of the various methods of the immediate settlement analysis should be evaluated by applying them to real case histories in which measurements of immediate settlement are available. However, although a large number of case histories is available in literature in which total settlements were monitored, unfortunately the case histories where immediate settlement was measured, including all parameters required for the immediate settlement analysis, are very scarce. Nevertheless, 13 settlement observations were collected from literature. Structure, soil, and foundation properties of these cases are summarized in Table 3. For all cases, immediate settlement was assessed as the settlements that occurred at the end of the construction period. Measured and computed immediate settlements from these case histories are presented in Table 4.
As can be seen from Table 4, the best settlement estimates were obtained from the Bowles method for majority of the cases. The Mayne method also produced quite good settlement predictions in many cases. If dimensions of the footing are relatively small, and it can be assumed to be almost rigid, the Harr “rigid” method also gives good settlement estimates with Burland's embedment correction factors. However, if the L/B ratio of the footing is relatively large, the Harr “flexible” method provides better settlement prediction with Burland's embedment correction factors than the Harr “rigid” method. The Janbu and the Christian methods yielded moderate settlement predictions (generally underestimated) compared with the other methods. The best settlement estimates were obtained from the Mayne method for mat foundations. The Giroud method generally overestimated the settlement because it does not consider an incompressible layer under the foundation. The Gazetas method produced very poor settlement predictions for some cases (especially for mat foundations). However, this is not surprising because, as stated by the developers of the method, the accuracy of the settlement computed from it may not be better than 10 to 20 percent, since it is obtained by curve fitting. Similar findings for the Gazetas method were reported by other investigators (e.g., Bowles, 1996; Mei et al., 2005).
THE CASES WHERE A MULTILAYER SOIL PROFILE EXISTS UNDER THE FOUNDATION
Naturally deposited soils generally consist of soil layers having different elastic properties and strengths. Hence, the elastic moduli of these layers generally increase with depth. In multilayer cases, immediate settlement may be computed either by using the principle of superposition or equivalent elastic modulus, which represents the soil profile. Equivalent elastic modulus of the soil layers may be calculated by two procedures, weighted mean and harmonic mean. Each of these procedures has a different calculation scheme: Weighted mean and harmonic mean of the elastic modulus can be computed by using Eq. 13 and Eq. 14, respectively.
where E_{u(w)} and E_{u(h)} are the equivalent elastic modulus of the soil layers calculated by weighted and harmonic mean, respectively, (E_{u})_{n} and H_{n} are the elastic modulus and thickness of the nth soil layer, respectively, and n is the layer number.
There is no general formula for the principle of superposition because it varies depending on the number of the soil layers. For two soil layers, total immediate settlement can be computed by using Eq. 15.
where H1 and H2 refer to the first and second soil layers, respectively, whereas E1 and E2 represent the elastic modulus of the first and second soil layers, respectively. The meaning of these subscripts can be seen more clearly from Figure 8. Because each procedure is based on a different calculation scheme, the resulting immediate settlement obtained from these procedures will also be different. In order to observe the possible differences between the immediate settlements computed by these three procedures, they were first compared on hypothetical cases including two soil layers, and then on real cases derived from literature.
Comparison of MultiLayer Procedures on Hypothetical Cases
In the hypothetical cases, a square foundation having base dimensions of 1.5 × 1.5 m carrying a uniform pressure of 150 kPa and embedded in a 1.5 m depth from the surface was considered. It was assumed that there are two soil layers having different elastic moduli. The elastic modulus of the first layer was set at 15 MPa, while the elastic modulus of the second layer varied from 15 to 75 MPa, which provides an E_{2}/E_{1} ratio equal to 1, 2, 3, 4, and 5. The depth of the first layer was also varied at 0.5B, 1B, 2B, and 3B. The depth of the incompressible layer was set at 5B (that is 7.5 m) from the bottom of the foundation by considering the effective influence depth of the stresses due to foundation load. A schematic illustration of the hypothetical cases is depicted in Figure 8.
In order to compare the procedures, immediate settlements were computed by the Mayne method, because this method takes into account the variation of the elastic modulus with depth. The immediate settlements computed by this method and other procedures are presented in Figure 9.
As can be seen from Figure 9, weighted mean calculations always yielded less settlement than the other calculation procedures for all conditions. Because Mayne and harmonic mean calculations are not affected by the thickness of the soil layers, settlements calculated by these procedures were not changed with respect to the thickness of the soil layers; therefore, they are the same for all depths of H_{1} (Figure 9). Because principles of superposition calculations are dependent on the thickness of the soil layers, settlements calculated by this procedure increased with increasing thickness of the first layer of the soil. For cases where thickness of the first layer is smaller than about 2B, the immediate settlements computed by the principles of superposition were quite affected by the elastic modulus variations within the layers. However, for cases where thickness of the first layer is greater than about 2B, effects of elastic modulus variations within the layers are less pronounced on the immediate settlement values computed by the principles of superposition.
According to the Boussinesq solution, the stress increment caused by the foundation load attenuates substantially at a depth of about 2B for a uniformly loaded rectangular area (Coduto, 2001). Therefore, for cases where the thickness of the first layer is greater than about 2B, overall immediate settlement is controlled by the elastic modulus of the first layer, and hence immediate settlement is not significantly affected by the ratio of E_{2}/E_{1}. In this respect, results obtained from the principles of superposition (Figure 9) seem to be more logical than the other methods.
In general, relatively large differences, depending on the E_{2}/E_{1} ratio and thickness of the soil layers, are observed between the immediate settlements obtained from these procedures. However, the procedures that give reliable settlements cannot be evaluated by comparing hypothetical cases. For this reason, immediate settlements obtained from these procedures are compared with real settlements in the following section.
Comparison of MultiLayer Procedures Using Real Cases
For comparison purposes, only four cases could be derived from literature due to the lack of case studies that include all parameters required for immediate settlement calculations. A summary of these cases is presented in Table 5. Elastic moduli of the soil profiles in the cases were obtained by reinterpreting the in situ test results such as CPT and SPT reported in the cited studies given in Table 5. The soil profile was divided into three layers by considering the variation of the elastic modulus of the soil layers with depth. The weighted mean and harmonic mean of the elastic modulus of the soils, as well as elastic modulus of the top and bottom soil layers are given in Table 5. Immediate settlements were calculated by both Mayne and Bowles methods because best prediction performance was obtained from these methods in the previous sections. The results obtained from these calculations are presented in Table 6.
According to Table 6, perfect settlement estimates for all cases evaluated in this study were obtained from the full Mayne method with Gibson modulus. Among the classical procedures, the best prediction performance was obtained from the principles of superposition in conjunction with the Bowles method, while weighted and harmonic mean provided poor settlement estimates, as shown in Table 6. This is not surprising because principles of superposition gave more logical results on the hypothetical cases evaluated in the previous section.
CONCLUSIONS
While consolidation settlement is usually considered critical to design of structures on clay soils, the immediate settlement is also an integral part of the total settlement of a foundation resting on cohesive soils. For this reason, it should also be estimated accurately. In geotechnical literature, many methods are available for the prediction of immediate settlement of shallow foundations on clay. In this study, eight of them that are widely used in practice were evaluated and compared both using a series of hypothetical and real cases. Comparisons were made for cases where both a single and a multilayer soil profile were present underneath the foundation. Based on the results obtained, the following conclusions can be drawn:

According to the results obtained from the comparisons of the hypothetical cases designed for a single soil layer, almost all methods produced relatively large different immediate settlements depending on the L/B and D_{f}/B ratios.

When considering the real cases, both the Bowles and the Mayne methods provided good immediate settlement estimates for a single soil layer compared with the other methods evaluated in this study. However, for mat foundations, the Mayne method provided the best immediate settlement estimates.

Weighted mean, harmonic mean, principles of superposition, and the Mayne method produced quite different immediate settlement values for foundations resting on a multilayer soil profile depending on the variation of the elastic modulus and thickness of the soil layers. However, among these procedures, the principle of superposition gives more logical results than the other methods.

By considering the real cases, the best settlement estimates for foundations resting on multilayer soil profiles were obtained by the full Mayne method with Gibson modulus, compared with the conventional procedures such as principles of superposition, weighted mean, and harmonic mean. However, among these classical procedures, principles of superposition in conjunction with the Bowles method produced quite good settlement estimates for multilayer soil profiles.
In addition to above conclusions, it should be noted that the reliability of any method is based on the accuracy of the elastic parameters of soil, even if it has produced good settlement estimates for some cases. The most significant parameter for the resulting immediate settlement is the elastic modulus of the soil. Several methods based on laboratory or in situ testing are available for determining (actually estimating) the elastic modulus of the soil. Because laboratory testing such as unconfined compression and triaxial tests are very sensitive to sampling disturbance, in situ testing such as standard penetration test (SPT) and cone penetration test (CPT) have been widely used to obtain the elastic modulus of soils in conjunction with empirical equations and/or correlations. Several equations for possible use can be found in some textbooks (e.g., Bowles, 1996; Look, 2007). However, the constants in the equations for determining the elastic modulus of the soil may vary from region to region. For this reason, as suggested also by Bowles (1996), the value to be used should be based on local experience with that equation giving the best fit for the locality.
LIST OF SYMBOLS
ν: Poisson's ratio of the soil.
μ_{1}: Shape factor.
μ_{o} and I_{F}: Fox's correction factor for the embedment depth.
μ_{s}, μ_{emb}, and μ_{wall}: Shape, embedment, and sidewall friction factors, respectively.
α: Ratio of L/B.
β: Normalized Gibson modulus.
γ_{c}: Unit weight of concrete in kN/m^{3}.
: 28day compressive strength of concrete in MPa.
A_{b}: Plan area of the foundation in m^{2}.
A_{w}: Area of the sidewallsoil interface in m^{2}.
B: Width of the foundation in m.
B′: Halfwidth of the rectangular foundation or halfwidth of the rectangle circumscribed to the actual base of an arbitrarily shaped foundation in m.
d: Diameter of the circular foundation or equivalent diameter of the rectangular foundation.
D_{f}: Embedment depth of the foundation in m.
E_{c}: Elastic modulus of the reinforced concrete in MPa.
E_{found}: Elastic modulus of the foundation material (i.e., reinforced concrete).
E_{o}: Elastic modulus of the soil just beneath the foundation in MPa.
E_{soilAV}: Representative (or average) elastic modulus of the soil located beneath the foundation base.
E_{u}: Undrained elastic modulus of the soil in MPa.
H: Thickness of the compressible layer beneath the foundation in m.
I_{1} and I_{2}: Settlement influence factors depending on the point where the settlement has to be computed and Steinbrenner's (1934) influence factors for Bowles method.
I_{cen}, I_{cor}, and I_{av}: Influence factors for under the center, corner, and average settlement of the foundation, respectively.
I_{F} and μ_{o}: Fox's (1948) correction factor for the embedment depth.
I_{G}, I_{F}, and I_{E}: Displacement, rigidity, and embedment correction factors, respectively, for Mayne method.
I_{s}: Settlement influence factor.
k_{E}: Rate of increase of elastic modulus of the soil with depth (depicted in Figure 5a).
K_{F}: Foundation flexibility factor.
L: Length of the foundation in m.
L′: Halflength of the rectangular foundation or halflength of the rectangle circumscribed to the actual base of an arbitrarily shaped foundation in m.
P: Vertical force carried by the foundation in kN.
q_{o}: Vertical uniform pressure applied by the foundation to the soil in kPa.
S_{i}: Immediate settlement of the foundation in mm.
t: Thickness of the foundation in m.
Acknowledgments
The author would like to thank the anonymous reviewers for invaluable comments that improved the quality of the manuscript.