1 Introduction

Since 1949, according to the best knowledge of the author (Pantelidis 2019), eighteen methods calculating the equivalent modulus of elasticity (\(E_{eq}\)) of stratified mediums have been proposed; indeed, eight of them after 2000 indicating that this research topic remains open. Parallelly, it makes sense that these methods refer exclusively to stratified soil mediums, although soil mediums with linearly increasing modulus with depth are often met in practice. In brief, Odemark’s (1949), de Barros’s (1966), Sridharan et al.’s (1990), Hirai and Kamei’s (2003, 2004), Hirai’s (2008), Abu-Farsakh and Chen’s (2012) methods are based on the concept of the flexural rigidity of thin slabs, Egorov and Nichiporovich (1961), Barden (1962), Gorbunov-Possadov and Malikova (1973), Sadrekarimi and Akbarzad (2009), Brahma and Mukherjee (2010), Budhu (2011) approached the problem from the mathematical mean point of view, Salamon (1968), Wardle and Gerrard (1972) and Gerrard (1982) suggested methods based on stress–strain relationships of the theory of elasticity while Ueshita and Meyerhof (1967), Fraser and Wardle (1985) and HariBharghav et al. (2017) proposed methods purely based on the theory of elastic settlement analysis. The comprehensive review of these methods offered recently by the author (Pantelidis 2019) clearly shows that there is still not a reliable method.

In the present paper a method for calculating both \(E_{eq}\) and \(\nu_{eq}\) of stratified soil mediums or homogenous mediums with linearly increasing modulus with depth is proposed. This method is based on the theory of elasticity.

2 Derivation of the Proposed Equivalent Elastic Moduli

2.1 Derivation of the Proposed Equivalent Elastic Modulus for Flexible Rectangular Footings

According to Boussinesq (1885), the normal stress increase at any point in a homogenous, elastic and isotropic semi-infinite medium due to a concentrated load \(P\) on its surface is:

$$\Delta \sigma_{x} = \frac{P}{2\pi }\left( {\frac{{3x^{2} z}}{{L_{1}^{5} }} - \left( {1 - 2\nu } \right)\left( {\frac{{x^{2} - y^{2} }}{{L_{1} r^{2} \left( {L_{1} + z} \right)}} + \frac{{y^{2} z}}{{L_{1}^{3} r^{2} }}} \right)} \right)$$
(1)
$$\Delta \sigma_{y} = \frac{P}{2\pi }\left( {\frac{{3y^{2} z}}{{L_{1}^{5} }} - \left( {1 - 2\nu } \right)\left( {\frac{{y^{2} - x^{2} }}{{L_{1} r^{2} \left( {L_{1} + z} \right)}} + \frac{{x^{2} z}}{{L_{1}^{3} r^{2} }}} \right)} \right)$$
(2)
$$\Delta \sigma_{z} = \frac{P}{2\pi }\frac{{z^{3} }}{{\left( {r^{2} + z^{2} } \right)^{\frac{5}{2}} }}$$
(3)

where \(L_{1} = \sqrt {x^{2} + y^{2} + z^{2} } ,\;r = \sqrt {x^{2} + y^{2} }\) and \(\nu\) is the Poisson’s ratio (see Das 2007). The notation followed is shown in Fig. 1.

Fig. 1
figure 1

Stresses in an elastic medium caused by a point load acting on the surface of a semi-infinite mass

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Considering a rectangular, uniformly loaded area which extends from \(x_{1}\) to \(x_{2}\) and from \(y_{1}\) to \(y_{2}\) (in the x- and y- direction respectively) on the surface of the elastic medium, the increase in the normal stress at any point in the medium in question will be:

$$\Delta \sigma_{i,S} = \int\limits_{{x_{1} }}^{{x_{2} }} {\int\limits_{{y_{1} }}^{{y_{2} }} {\Delta \sigma_{i} \left( {x,y,z} \right)} }$$
(4)

The differentials dx and dy of the variables \(x\) and \(y\) are included in \(\Delta \sigma_{i}\) as \(P = q{\text{d}}y{\text{d}}x\) (recall Eqs.  1 to 3; q is the uniform loading). Consequently, from Hooke’s law, the unit vertical strain is:

$$\varepsilon_{z} = \frac{1}{{E_{s} }}\left[ {\Delta \sigma_{z,S} - \nu \left( {\Delta \sigma_{x,S} + \Delta \sigma_{y,S} } \right)} \right]$$
(5)

Using Eqs. 1 to 3, the latter can be rewritten as follows:

$$\varepsilon_{z} = \frac{q}{{E_{s} }}\int\limits_{{x_{1} }}^{{x_{2} }} {\int\limits_{{y_{1} }}^{{y_{2} }} {\frac{{z\left( {1 + \nu } \right)\left( {3z^{2} - 2\left( {x^{2} + y^{2} + z^{2} } \right)\nu } \right)}}{{2\pi \left( {x^{2} + y^{2} + z^{2} } \right)^{5/2} }}dxdy} }$$
(6)

The settlement, \(\rho\), at the foundation level corresponding to a layer extending from \(z = 0\) (level of foundation) to \(z = H\), derives from the integration of Eq. 6 between these limits, that is:

$$\rho = \int\limits_{0}^{H} {\varepsilon_{z} dz}$$
(7)

The problem has been solved by Steinbrenner (1934) for a general \(B \times L\) footing. The solution can be found in any geotechnical engineering book, usually, combined with Fox’s (1948) embedment depth factor \(I_{E}\). More specifically,

$$\rho = q\left( {a^{\prime}B^{\prime}} \right)\frac{{1 - \nu^{2} }}{{E_{s} }}I_{s} I_{E}$$
(8)

where,

$$I_{s} = F_{1} + \frac{1 - 2\nu }{{1 - \nu }}F_{2}$$
(9)
$$F_{1} = \frac{1}{\pi }\left( {A_{0} + A_{1} } \right)$$
(10)
$$F_{2} = \frac{{n_{1} }}{2\pi }\tan^{ - 1} A_{2}$$
(11)
$$A_{0} = m_{1} \ln \frac{{\left( {1 + \sqrt {m_{1}^{2} + 1} } \right)\sqrt {m_{1}^{2} + n_{1}^{2} } }}{{m_{1} \left( {1 + \sqrt {m_{1}^{2} + n_{1}^{2} + 1} } \right)}}$$
(12)
$$A_{1} = \ln \frac{{\left( {m_{1} + \sqrt {m_{1}^{2} + 1} } \right)\sqrt {1 + n_{1}^{2} } }}{{m_{1} + \sqrt {m_{1}^{2} + n_{1}^{2} + 1} }}$$
(13)
$$A_{2} = \frac{{m_{1} }}{{n_{1} \sqrt {m_{1}^{2} + n_{1}^{2} + 1} }}$$
(14)

Also, \(a^{\prime} = 4,\;B^{\prime} = {B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2},\;m_{1} = {L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) and \(n_{1} = {H \mathord{\left/ {\vphantom {H {({B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2})}}} \right. \kern-\nulldelimiterspace} {({B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2})}}\) for the settlement at the center of foundation and \(a^{\prime} = 1,\;B^{\prime} = B,\;m_{1} = {L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) and \(n_{1} = {H \mathord{\left/ {\vphantom {H B}} \right. \kern-\nulldelimiterspace} B}\) for the settlement at the corner of foundation.

Apparently, there is an equivalent homogenous soil medium with elastic constants \(\left( {E_{eq} ,v_{eq} } \right)\) producing under identical loading conditions the same settlement with the original soil medium. Equating these settlements (note: for the stratified medium the principle of superposition is necessary to be applied; see Barnes 2016)

$$\begin{aligned} & q\left( {a^{\prime}B^{\prime}} \right)\frac{{1 - \nu_{eq}^{2} }}{{E_{eq} }}I_{s} (h_{n} ,v_{eq} )I_{E} (v_{eq} ) \\ & = q\left( {a^{\prime}B^{\prime}} \right)\sum\limits_{i = 1}^{n} {\frac{{1 - \nu_{i}^{2} }}{{E_{s,i} }}\left( {I_{s} (h_{i} ,v_{i} ) - I_{s} (h_{i - 1} ,v_{i} )} \right)I_{E} (v_{i} )} \\ \end{aligned}$$
(15)

and solving the latter as for \(E_{eq}\), the following expression is obtained:

$$E_{eq} = \frac{{I_{s} (h_{n} ,v_{eq} ) \cdot I_{E} (v_{eq} ) \cdot \left( {1 - \nu_{eq}^{2} } \right)}}{{\sum\nolimits_{i = 1}^{n} {\left[ {\frac{{I_{s} (h_{i} ,v_{i} ) - I_{s} (h_{i - 1} ,v_{i} )}}{{E_{s,i} }}I_{E} (v_{i} ) \cdot \left( {1 - \nu_{i}^{2} } \right)} \right]} }}$$
(16)

where, \(h_{i}\) is the vertical distance extending from the level of foundation to the bottom of the \(i\)-th layer (that is, it is not the thickness of the \(i\)-th layer); there are \(n\) such layers. Also, it is \(I_{s} \left( {h_{0} ,v_{1} } \right) = 0\). Equation 16 has two unknowns, namely, \(E_{eq}\) and \(\nu_{eq}\); however as shown later, the two unknowns can easily be reduced to one, the equivalent modulus \(E_{eq}\). Finally, from Eq. 16 it is inferred that, \(E_{eq}\) depends on the point considered on the plan-view of footing, as the parameter \(n_{1}\) in the \(I_{s}\) factor depends on the location of this point (recall Eqs. 12–14). By suitably applying the principle of superposition on the foundation level (see Barnes 2016), Eq. 16 can be used so that the \(E_{eq}\) value to be calculated at any point on the plan-view of footing.

It is noted that, Eq. 16 stands for any \(I_{s}\) factor derived from a settlement equation of the form:

$$\rho = q\left( {{\text{Width}}\;{\text{or}}\;{\text{Diameter}}\;{\text{of}}\;{\text{footing}}} \right)\frac{{1 - \nu^{2} }}{{E_{s} }}I_{s} I_{E}$$
(17)

2.2 Derivation of the Proposed \(E_{eq}\) for Circular Footings

2.2.1 Circular Flexible Footings

In a similar way, the settlement at the center of a flexible circular footing is:

$$\rho = q\left( {2r_{1} } \right)\frac{{1 - \nu^{2} }}{{E_{s} }}I_{s} I_{E}$$
(18)

where

$$I_{s} = \left( {1 - \frac{{r_{1} }}{{\sqrt {r_{1}^{2} + H^{2} } }}} \right) + \frac{1 - 2\nu }{{2\left( {1 - \nu } \right)}}\left( {1 - \frac{H}{{\sqrt {r_{1}^{2} + H^{2} } }}} \right)\frac{H}{{r_{1} }}$$
(19)

and \(r_{1}\) is the radius of footing and \(H\) is the thickness of the compressible medium measured from the foundation level.

Equations 18 and 19 (ignoring the \(I_{E}\) factor) derives from the following single integral:

$$\varepsilon_{z} = \frac{q}{{E_{s} }}\int\limits_{0}^{{r_{1} }} {\frac{{z\left( {1 + \nu } \right)\left( {3z^{2} - 2\left( {r^{2} + z^{2} } \right)\nu } \right)}}{{2\pi \left( {r^{2} + z^{2} } \right)^{5/2} }}\left( {2\pi r_{1} } \right)dr}$$
(20)

where, the term \(x^{2} + y^{2}\) in Eq. 6 has been replaced by \(r^{2}\). \(E_{eq}\) is then obtained from Eq. 16 using the \(I_{s}\) factor of Eq. 19.

2.2.2 Circular rigid footings

The settlement of a rigid circular footing is (see Kézdi and Rétháti 1988):

$$\rho = \left( {1 + \nu } \right)\frac{{qr_{1} }}{{2E_{s} }}\left[ {2\left( {1 - \nu } \right)\left( {\frac{\pi }{2} - \alpha } \right) - \sin \alpha \cos \alpha } \right]$$
(21)

where \(\alpha = \arctan ({{r_{1} } \mathord{\left/ {\vphantom {{r_{1} } H}} \right. \kern-\nulldelimiterspace} H})\) and \(r_{1}\) and \(H\) as before. Equation 21 is rewritten in the form of Eq. 17 as follows:

$$\rho = q\left( {2r_{1} } \right)\frac{{1 - \nu^{2} }}{{E_{s} }}I_{s} I_{E}$$
(22)

where,

$$I_{s} = \frac{1}{2}\left[ {\left( {\frac{\pi }{2} - \alpha } \right) - \frac{\sin \alpha \cos \alpha }{{1 - \nu }}} \right]$$
(23)

\(E_{eq}\) is then obtained from Eq. 16 using the \(I_{s}\) factor of Eq. 23.

2.3 Derivation of the Proposed \(E_{eq}\) for Flexible Rectangular Footings Founded on Soil Medium with Linearly Increasing Elastic Modulus

Considering now that, a flexible rectangular footing rests on “Gibson’s soil” (Gibson 1967), that is, on a soil with elastic modulus linearly increasing with depth as follows:

$$E_{s,z} = E_{s,0} + k \cdot z$$
(24)

Equation 6 then becomes:

$$\varepsilon_{z,G} = \frac{q}{{E_{s,0} }}\int\limits_{{ - {B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2}}} {\int\limits_{{{{ - L} \mathord{\left/ {\vphantom {{ - L} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z\left( {1 + \nu } \right)\left( {3z^{2} - 2\left( {x^{2} + y^{2} + z^{2} } \right)\nu } \right)}}{{\left( {1 + \frac{k}{{E_{s,0} }}z} \right)2\pi \left( {x^{2} + y^{2} + z^{2} } \right)^{5/2} }}dxdy} }$$
(25)

where, \(E_{s,0}\) is the soil modulus immediately below the footing, \(k\) is the rate of increase of the modulus per unit depth and \(z\) is the depth measured from the foundation level.

On the other hand, the strain at depth \(z\) for the equivalent homogenous medium is:

$$\varepsilon_{z,eq} = \frac{q}{{E_{eq} }}\int\limits_{{ - {B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{B \mathord{\left/ {\vphantom {B 2}} \right. \kern-\nulldelimiterspace} 2}}} {\int\limits_{{{{ - L} \mathord{\left/ {\vphantom {{ - L} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z\left( {1 + \nu_{eq} } \right)\left( {3z^{2} - 2\left( {x^{2} + y^{2} + z^{2} } \right)\nu_{eq} } \right)}}{{2\pi \left( {x^{2} + y^{2} + z^{2} } \right)^{5/2} }}dxdy} }$$
(26)

Assuming, now, that the soil medium has thickness \(H\), the equivalent elastic modulus, \(E_{eq}\), derives from the numerical solution of the equation below:

$$\int\limits_{0}^{H} {\varepsilon_{z,eq} dz} = \int\limits_{0}^{H} {\varepsilon_{z,G} dz}$$
(27)

The results are given in chart form in Fig. 2 for various aspect ratio of footing \({L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) and \(\nu_{eq} =\)\(\nu\) (assuming that all layers have the same Poisson’s ratio value) or \(\nu_{eq} =\) 0. \(H\) in Fig. 2 is the influence depth of footing as proposed by Terzaghi et al. (1996), i.e.:

$$z_{l} = 2B\left( {1 + \log \frac{L}{B}} \right)$$
(28)

Recalling Eq. 24, the depth where the elastic modulus of soil is equal to the equivalent one is:

$$z_{eq} = {{\left( {E_{{{\text{eq}}}} - E_{s,0} } \right)} \mathord{\left/ {\vphantom {{\left( {E_{{{\text{eq}}}} - E_{s,0} } \right)} k}} \right. \kern-\nulldelimiterspace} k}$$
(29)
Fig. 2
figure 2

\({{E_{eq} } \mathord{\left/ {\vphantom {{E_{eq} } {E_{0} }}} \right. \kern-\nulldelimiterspace} {E_{0} }}\) versus \({k \mathord{\left/ {\vphantom {k {E_{0} }}} \right. \kern-\nulldelimiterspace} {E_{0} }}\) charts for various \({L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) cases for \(\nu_{eq} =\)\(\nu\) (left) and \(\nu_{eq} =\) 0 (right). \(\nu\) ranges from 0.1 to 0.5 (from the lower to the upper curve) with 0.1 interval

Full size image

In this respect, \({{z_{eq} } \mathord{\left/ {\vphantom {{z_{eq} } {z_{l} }}} \right. \kern-\nulldelimiterspace} {z_{l} }}\) versus \({k \mathord{\left/ {\vphantom {k {E_{0} }}} \right. \kern-\nulldelimiterspace} {E_{0} }}\) charts for various \({L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) cases for \(\nu_{eq} =\)\(\nu\) (0.1 ≤ \(\nu\) ≤ 0.5) are given in Fig. 3. It is noted that, Eqs. 25 and 26 refer to the center of footing.

Fig. 3
figure 3

\({{z_{eq} } \mathord{\left/ {\vphantom {{z_{eq} } {z_{l} }}} \right. \kern-\nulldelimiterspace} {z_{l} }}\) versus \({k \mathord{\left/ {\vphantom {k {E_{0} }}} \right. \kern-\nulldelimiterspace} {E_{0} }}\) charts for various \({L \mathord{\left/ {\vphantom {L B}} \right. \kern-\nulldelimiterspace} B}\) cases for \(\nu_{eq} =\)\(\nu\). \(\nu\) ranges from 0.1 to 0.5 (from the lower to the higher curve) with 0.1 interval

Full size image

3 Application Examples

3.1 Schmertmann’s (1970) Stratified Medium

The multilayer soil profile used by Schmertmann’s (1970; “Figure 6”) is also used here. This consists of eleven (horizontal) soil strata over bedrock; the thickness and modulus of each soil stratum is given in Table 1. A perfectly flexible, a smooth rigid and a rough rigid circular footing of 2.6 m diameter resting on the surface of the stratified medium were considered. Each one of these footings carries uniform load of 200 kPa. The Poisson’s ratio value for all strata is the same and equal to 0.4. The derived homogenous equivalent soil mediums were compared against the original stratified medium in a finite element analysis framework using Rocscience’s RS2. The criterion used was that a homogenous soil medium is equivalent to a stratified one if under the same loading conditions, the two footings over them yield the same settlement. Prior to running the finite element models, the validity of the models was checked against analytical results (see Appendix).

Table 1 \(E_{s}\) values and vertical extend of each layer (based on \(q_{c}\) given in Schmertmann (1970); also, according to Schmertmann et al. (1978), \(E_{s} ({\text{kPa}}) = 2.5\;q_{c}\) for the axisymmetric case)
Full size table

Considering that \(\nu_{eq} =\) 0.4, Eq. 16 (with the use of the \(I_{s}\) factor of Eq. 19) gave \(E_{eq} =\) 9132 kPa; for \(\nu_{eq} =\) 0 it was found that \(E_{eq} =\) 11,108 kPa. The settlement values obtained are summarized in Table 2. As shown, the error due to the use of the equivalent elastic constants is negligible. It is very interesting that, as shown in Fig. 4, not only the maximum settlement values do not differ but also the whole settlement profiles of the equivalent mediums are almost similar to the original one. Indeed, the same profile is taken using any \((\nu_{eq} ,E_{eq} )\) pair of values derived from Eq. 16; these pair of values are shown in Fig. 5. In the same figure, the relative difference \((R_{d} )\) in \(E_{eq}\) is also given; the following formula for \(R_{d}\) was used:

$$R_{d} = 1 - \frac{{E_{eq} (\nu_{eq} = \nu )}}{{E_{eq} (\nu_{eq} = 0)}}$$
(30)
Table 2 Comparison of settlements derived from Schmertmann’s (1970) original example medium and its equivalent both for \(\nu_{eq} =\) 0.4 and 0
Full size table
Fig. 4
figure 4

Comparison of settlement profiles derived from Schmertmann’s (1970) original example medium and its equivalent both for \(\nu_{eq} =\) 0.4 (\(E_{eq} =\) 9132 kPa) and 0 (\(E_{eq} =\) 11,108 kPa)

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Fig. 5
figure 5

Black, solid line: \((\nu_{eq} ,E_{eq} )\) combinations giving the same settlement profile in the example considered (obtained from Eq. 16). Gray, dashed line: Relative difference of the \(E_{eq}\) value between the \(\nu_{eq} =\)\(\nu\) and the \(\nu_{eq} =\) 0 case (Eq. 30)

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It is a common practice, however, a study to be restricted to a specific influence zone depth (recall Eq. 28), due to the phenomenon of cementation of soils. Cementation refers to the additional bonding between particles through certain extraneous chemical substances present and hence exhibit higher resistance against deformation (Nagaraj et al. 1995); it contributes to the shear strength of a soil only where the shear strain is kept at low levels (0.001%; see Budhu 2011). Considering an influence zone depth equal to two times the footing diameter (value suggested for circular and square footings; see Schmertmann et al. 1978 and Terzaghi et al. 1996), the equivalent modulus of elasticity, \(E_{eq}\), for the example studied herein is 8336 kPa and 10,580 kPa for \(\nu_{eq} =\) 0.4 and 0 respectively; the variation of \(E_{eq}\) with the ‘(normalized) influence zone depth’ for the present example is also given (see Fig. 6) for both \(\nu_{eq} =\) 0.4 and 0.

Fig. 6
figure 6

Variation of \(E_{eq}\) with the normalized influence zone depth for the example considered for both the \(\nu_{eq} =\) 0.4 and 0 case

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Some more examples are given in Table 3 for comparison purposes. From the table in question it is inferred that footing rigidity also affects the \(E_{eq}\) value; in this respect, \(E_{eq}\) is greater for rigid footings. The effect of footing area on \(E_{eq}\) is also shown; the difference in \(E_{eq}\) between footings of different footprint area is attributed to the fact that, as known, bigger footings have greater influence zone depth (bigger stress bulb) and thus, the \(E_{eq}\) value is affected by the deeper strata. Moreover, as shown in Table 3, the effect of the footing point (on plan-view) considered on \(E_{eq}\) is significant for square (flexible) footings and negligible for (flexible) long footings. It is additionally noted that, the relative difference between any \(\nu_{eq} =\) \(v\) case and the \(\nu_{eq} =\) 0 case can be taken by the simple empirical formula \(R_{d} = \nu^{2}\); that is, the same \(R_{d}\) value is obtained for any \(BxL\) footing, footing width or location on footing’s plan-view (see Table 3). The \(E_{eq}\) value corresponding to \(\nu_{eq} =\) 0 and infinite influence zone depth can be taken by the following equation:

$$E_{eq} (\nu_{eq} = 0) = \frac{{E_{eq} (\nu_{eq} = \nu )}}{{1 - \nu^{2} }}$$
(31)
Table 3 The equivalent elastic modulus for various footing cases on Schmertmann’s (1970) stratified medium both for \(\nu_{eq} =\) 0.4 and 0 (the depth of influence zone was considered to be the whole thickness of Schmertmann’s (1970) medium)
Full size table

For comparison purposes “Bowles” (Egorov and Nichiporovich 1961; Bowles 1996) \(E_{eq}\) value is also given in Table 3:

$$E_{eq} = \frac{{\sum {H_{i} E_{i} } }}{{\sum {H_{i} } }}$$
(32)

where \(H_{i}\) and \(E_{i}\) are the thickness and modulus of the i-th layer.

“Bowles” method is probably the most widely used method among practitioners; it is noted however that, it does not consider the effect of the Poisson’s ratio of soils, the footing shape and size, the location on the plan-view of footing and more importantly, the vertical distance of each soil layer from the foundation level. In every case, for the example cases considered herein, Bowles’ \(E_{eq}\) value is very high and thus, on the non-conservative side (see Table 3). Finally, for circular flexible footings, the diameter seems to affect the relative difference in a neither significant nor negligible manner, whilst for rigid circular footings, the footing diameter plays less significant role.

3.2 Reducing the Elastic Modulus of a Homogenous Medium to a Value Suitable for a Winkler Spring Analysis

One of the important outcomes of the present paper is that, since the actual deformation pattern of soil mediums defers from the respective one assuming an array of Winkler springs, it is mandatory that, the elastic modulus of soil in a Winkler’s spring analysis be modified as to correspond to a \(\nu =\) 0 medium, even when the soil is homogenous. The \(\nu_{eq} = 0\) value corresponds to compression with no lateral expansion, that is, to the deformation pattern of springs. For example, if the soil below a circular foundation of diameter 2.6 m is homogenous with \(v\) = 0.2 and \(E_{s} =\) 5000 kPa, the correct \(E_{s}\) value for a Winkler type analysis, according to Eq. 16 (for \(n =\) 1), is 5332 kPa if \({{z_{l} } \mathord{\left/ {\vphantom {{z_{l} } {2r}}} \right. \kern-\nulldelimiterspace} {2r}} =\) 2, and 5208 kPa if \({{z_{l} } \mathord{\left/ {\vphantom {{z_{l} } {2r}}} \right. \kern-\nulldelimiterspace} {2r}} =\)∞ (both values stand for \(v\) = 0); the 5208 kPa value can alternatively be obtained from Eq. 31, i.e. 5000/(1–0.22) = 5208. If, now, the soil below the same foundation had \(v\) = 0.4 and \(E_{s} =\) 5000 kPa, the correct \(E_{s}\) value would be 6346 kPa for \({{z_{l} } \mathord{\left/ {\vphantom {{z_{l} } {2r}}} \right. \kern-\nulldelimiterspace} {2r}} =\) 2 and 5952 kPa for \({{z_{l} } \mathord{\left/ {\vphantom {{z_{l} } {2r}}} \right. \kern-\nulldelimiterspace} {2r}} =\)∞ (again both values stand for \(v\) = 0).

4 Summary and Conclusions

The two elastic constants of soil are essential parameters for assessing the rigidity of shallow foundations (e.g. Gupta 2007), calculating the critical column spacing (e.g. ACI Committee 336, Ulrich et al. 1988) and calculating the modulus of subgrade reaction of soil, \(k_{s}\) (e.g. Vesic 1961). The latter is used as the spring constant in a Winkler type analysis. These springs must effectively represent the deformability of the ground in every case, that is, in homogenous soils, stratified mediums or homogenous soils with linearly increasing modulus; an effective representation involves the use of the \(E_{eq}\) value corresponding to \(\nu_{eq}\) = 0 so that the equivalent soil to be consistent with the deformation pattern of Winkler’s springs (compression with no lateral expansion). This paper presents a method for the determination of the equivalent elastic constants, \(E_{eq}\) and \(\nu_{eq}\), of the above-mentioned cases. The method in question is based on the theory of elasticity combining the principle of vertical and horizontal superposition. Flexible rectangular footings over stratified mediums or over homogenous mediums with linearly varying modulus, as well as rigid and flexible circular footings over stratified mediums were considered. The results are presented either as closed-form expressions or in chart form. A number of application examples are also given. As shown, the derived \(E_{eq}\) values not only return reliable settlement results (reliable as for the “equivalency”), but also settlement profiles that are similar to those corresponding to the original soil mediums.