1 Introduction and background

The coefficient of lateral soil pressure, k, is a key factor in the analysis of many geotechnical problems, e.g., underground structures, slope stability, retaining walls, etc. Due to the dependence of this parameter on several state variables or material parameters, its precise and accurate determination is very challenging, especially in unsaturated soils.

Many studies have been conducted to examine k for saturated soils [3, 5, 22, 23, 34, 35, 37, 40, 46]. For example, Mayne and Kulhawy [34] performed a comprehensive study on the dependency of the at-rest soil pressure on the over consolidation ratio and reviewed plenty of experimental data reported by different researchers. However, there are very few investigations on the coefficient of lateral soil pressure and its relevant state variables (or parameters) for unsaturated soils, either experimentally [1, 27, 36, 39, 41, 45, 49, 51], analytically [29, 30, 44, 47, 50] or numerically [13, 26, 28]. As previous studies have shown, the lateral soil pressure in unsaturated soils is remarkably affected by the soil type [28, 41], the initial soil saturation, Sr, or the matric suction, ψ, [1, 13, 27, 30, 36, 39, 41, 45, 49, 51], the stress state within the soil [1, 13, 30, 41, 51], the climate parameters (e.g., infiltration or evaporation) [44, 45, 47] and the geometrical aspects of the retaining structure that supports the soil against lateral deformations [28, 29]. In general, it has been indicated that the at-rest and the active soil pressure coefficients (kou and kau, respectively) increase by increasing the degree of saturation of the soil and the principal stress level within the soil body, or by decreasing the matric suction of the soil. In addition, it was found that the higher the soil density, the less the lateral soil pressure in unsaturated soil deposits.

Among the limited studies on assessing lateral soil pressure in unsaturated soils, only the at-rest or the active coefficients of lateral soil pressure were studied, and the continuous variation of the unsaturated lateral soil pressure from the at-rest to the active state of the examined soils has not been investigated.

In this research, a suction-dependent effective stress-based analytical framework is developed on the basis of the disturbed state concept (DSC) to predict the continuous variation of unsaturated lateral soil pressure coefficient from the at-rest to the active state of the soil. DSC, well-known as a comprehensive method for predicting the mechanical behavior of geomaterials, was first proposed by Desai [6,7,8,9] and developed afterward by many researchers [10,11,12, 15, 17, 20]. According to the DSC, the mechanical response of soil due to changing state variables can be defined by considering the soil response in two extreme soil structural reference conditions, namely the relative intact soil structure (RI) and the fully adjusted soil structure (FA).

To assess the functionality of the proposed analytical model, the analytical results from this research are compared quantitatively with data obtained from the experimental study by Pirjalili et al. [41]. In that study, a series of twenty suction-controlled drained tests were conducted on two different unsaturated soils using a suction-controlled ring device under five different matric suctions. This device is capable of continuously measuring the unsaturated lateral soil pressure and corresponding lateral (radial) strain, εr, as well as controlling the matric suction and measuring the water content of the soil specimens during the tests. In addition to the data reported by Pirjalili et al. [41], a complementary series of experimental tests were performed to measure the lateral soil pressure changes in fully saturated (i.e., 0-kPa matric suction) soil conditions.

2 Development of the analytical solution

In this research, an analytical solution has been developed to predict the variation of the coefficient of lateral soil pressure in unsaturated soils in terms of the state variables and soil parameters that influence the behavior. Also, the disturbed state concept (DSC) and the effective stress approach for unsaturated soils are considered simultaneously.

2.1 Lateral soil pressure in unsaturated soils in terms of effective stress

Generally, the coefficient of lateral soil pressure, k, should be calculated from the effective vertical stress and the effective horizontal stress, as shown by Eq. (1):

$$k = \frac{{\sigma_{{\text{h}}}^{^{\prime}} }}{{\sigma_{{\text{v}}}^{^{\prime}} }}$$
(1)

where σ′h and σ′v are the effective horizontal and the effective vertical stresses, respectively. The single-phase effective stress relationship in unsaturated soils was first formulated by Bishop [2], as:

$$\sigma^{\prime} = \left( {\sigma - u_{a} } \right) + \chi \times \left( {u_{a} - u_{w} } \right) = \sigma_{{{\text{net}}}} + \chi \times \psi = \sigma_{{{\text{net}}}} + \sigma_{s}$$
(2)

where σ′, σ, σnet, ψ, ua, uw, χ and σs are effective stress, total stress, net stress, matric suction, pore air pressure, pore water pressure, effective stress parameter and suction-stress, respectively. There has been a long-running debate on the nature and determination of χ· χ is a function of soil saturation and reflects the contribution of matric suction to effective stress; it is an average weighing factor reflecting the inter-particle capillary as well as physico-chemical forces [33]. In an unsaturated state of the soil, χ typically varies between zero (in fully dry conditions) to unity (in fully saturated conditions). Many relationships have been reported by researchers to show the dependency of χ on soil saturation [14, 16, 19, 21, 24, 25, 31, 38, 48]. For instance, it is suggested that χ can be simply taken as the degree of saturation, Sr, [4, 24, 43] or as the effective saturation, Se, [14, 16, 19, 21].

In accordance with Eq. (2), effective horizontal and vertical stresses can be attained by Eqs. (3a) and (3b), respectively:

$$\sigma_{h}^{^{\prime}} = \sigma_{h} + \chi \times \left( {u_{a} - u_{w} } \right) = \sigma_{h} + \sigma_{s}$$
(3a)
$$\sigma_{v}^{^{\prime}} = \sigma_{v} + \chi \times \left( {u_{a} - u_{w} } \right) = \sigma_{v} + \sigma_{s}$$
(3b)

where σh and σv are the net horizontal and net vertical stresses, respectively, that are conventionally controlled or measured during unsaturated tests. Therefore, by knowing the soil–water retention behavior (i.e., Soil–Water Characteristic Curve, SWRC) and considering a suitable relationship for changing the effective stress parameter, χ, against the matric suction, ψ, it is possible to calculate the coefficient of lateral soil pressure, k, for unsaturated soils by implementing Eqs. (1) and (3).

2.2 Implementing the disturbed state concept (DSC) for calculating k

The DSC was first introduced by Desai [6,7,8] and extended by Geiser et al. [17] for constitutive modeling of unsaturated soils. Details of the DSC are presented by Desai [9]. As a constitutive framework, DSC defines the overall behavior of a deforming material in terms of the behavior of component materials with regards to continuum and disturbed states. The former is often denoted as Relative Intact (RI) and the latter as Fully Adjusted (FA).

Accordingly, the mechanical response of the soil in an intermediate state, PInt, can be determined by knowing the two corresponding values in RI and FA states, namely PRI and PFA. Hence, in accordance with DSC, the expression for calculating PInt is:

$$P_{{{\text{Int}}}} = P_{{{\text{RI}}}} - \left( {P_{{{\text{RI}}}} - P_{{{\text{FA}}}} } \right) \times D$$
(4)

Equation (4) provides a continuous expression to define PInt in terms of PRI and PFA, in which, PRI and PFA are soil responses at RI and FA states, respectively, and D is a disturbance parameter. The disturbance parameter, D, defines the progression of degradation (softening) or healing and coupling between the RI and FA responses and varies between zero and unity during the RI toward the ultimate FA state. D is typically defined in the form of an exponential function in terms of accumulative plastic volumetric strain, ζv, as shown by Eq. (5):

$$D = D_{u} \left( {1 - e^{{\left( { - a\zeta_{v}^{z} } \right)}} } \right)$$
(5)

In Eq. (5), Du, a and z are model parameters that depend on the type of material and the loading conditions. In Fig. 1, variations of PInt, PRI and PFA versus typical independent variable, x (e.g., a stress state parameter), are shown schematically.

Fig. 1
figure 1

Schematic of PInt, PRI and PFA

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According to the basic concepts of DSC, the soil experiences the most structural disturbance in the FA state, in which the soil has the least stiffness. On the other hand, in the RI state, the soil experiences a gradual structural disturbance.

Unsaturated soils can be considered to be in the FA condition when the matric suction of the soil is zero and the soil is fully saturated. Moreover, since the soil stiffness is relatively the highest (compared to other cases) under maximum applied matric suction, unsaturated soils can be considered to be in the RI condition when maximum matric suction is applied to the soil.

In this research, the conceptual framework of DSC has been implemented to demonstrate the variations of the unsaturated coefficient of lateral soil pressure, k, in terms of the effective vertical stress, σ'v. Accordingly, by considering the coefficient of the lateral soil pressure at a given matric suction, k ψ, as Pint and considering k values corresponding to the 0-kPa and the maximum applied matric suctions as kFA and kRI in Eq. (4), respectively, a general formulation for calculating k at a given matric suction, ψ, is obtained, as shown by Eq. (6).

$$k_{\psi } = k_{{{\text{RI}}}} - \left( {k_{{{\text{RI}}}} - k_{{{\text{FA}}}} } \right) \times D$$
(6)

In Eq. (6), kFA and kRI are two input parameters, whose variations against the effective vertical stress, σ′v, must be known. Also, the disturbance parameter, D, is a function of three material parameters (namely, Du, a and z), as previously denoted in Eq. (5). In this study, Du, a and z are defined in terms of the applied matric suction to the soil, by considering the experimental unsaturated test results. Details are discussed in the following sections.

3 Experimental data

In this research, the constitutive parameters of the proposed DSC model are calculated in accordance with the experimental data from Pirjalili et al. [41], who performed laboratory tests on two different unsaturated soils (namely, Firouzkouh Clay [CL] and Sand–Kaolin mixture [SC]), using a developed unsaturated suction-controlled ring device. Accordingly, reconstituted specimens of a Sand–Kaolin mixture (with two different initial void ratios, eo, of 0.52 and 0.72) and a Firouzkouh Clay (with two different initial void ratios of 0.71 and 0.92) were tested under five different matric suctions (ψ  = 10, 30, 50, 70 and 90 kPa) to examine the dependency of the unsaturated coefficient of lateral soil pressure on the void ratio and matric suction. Further details about the experimental work are available in [41, 42].

The soil water retention curves (SWRC) of the examined soil materials under wetting paths are shown in Fig. 2 along with the fitted curves and corresponding parameters obtained from van Genuchten's suggested model [18], as presented by Eq. (7):

$$\psi = \frac{{S_{{{\text{ro}}}} }}{{\left( {1 + \left( {\alpha \psi } \right)^{n} } \right)^{m} }}$$
(7)
Fig. 2
figure 2

SWRC of the examined soils: a Sand–Kaolin mixture and, b Firouzkouh Clay [41]

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In Eq. (7), Sro is the degree of saturation of the soil at zero matric suction, and α, m and n are fitting parameters that are related to the pore size and pore size distribution of the soil.

Figure 3 plots the variations of the net horizontal stress, σh, versus the net vertical stress, σv, for different soil specimens. In addition, the variations of the void ratio of the soil, e, versus net vertical stress, σv, are shown in Fig. 4. In this research, a series of complementary tests were performed for all specimen groups under zero matric suction to develop the proposed DSC model under fully saturated conditions. Results from the complementary tests under zero matric suction conditions are also plotted in Figs. 2 and 3.

Fig. 3
figure 3

Variations of the net horizontal stress, σh, versus the net vertical stresses, σv, for: a Sand–Kaolin mixture with eo = 0.52, b Sand–Kaolin mixture, eo = 0.72, c Firouzkouh Clay, eo = 0.71 and d Firouzkouh Clay, eo = 0.92 [41]

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Fig. 4
figure 4

Variation of the void ratio, e, versus net vertical stress, σv, for: a Sand–Kaolin mixture, eo = 0.52, b Sand–Kaolin mixture, eo = 0.72, c Firouzkouh Clay, eo = 0.71 and d Firouzkouh Clay, eo = 0.92 [41]

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4 Calculation of model parameters

Based on the constitutive relationships mentioned in previous sections and the use of experimental data reported by Pirjalili et al. [41], the following steps were followed calculate the variations of the unsaturated coefficient of the lateral soil pressure, k, against the effective vertical stress, σv':

4.1 Step 1: Considering input data

The input data sets that were considered to determine the parameters of the proposed DSC model are the SWRC of the proposed unsaturated soil under the wetting path (Fig. 2), variation of σh versus σv (Fig. 3), and variations of e versus σv (Fig. 4).

4.2 Step 2: Calculation of effective vertical and horizontal stresses ( σ ′ v and σ ′ h )

Since the coefficient of the lateral soil pressure is generally defined as the ratio of the effective horizontal stress to the effective vertical stress (as previously shown by Eq. 1), the variations of σ′h versus σ′v were calculated using Eqs. (2) and (3) and considering net horizontal and vertical stress data, as plotted in Figs. 2 and 3. In addition, the effective stress parameter, χ, was considered equal to the degree of saturation of the soil, Sr, as suggested by Wheeler et al. [4, 43, 48].

4.3 Step 3: Calculation of the coefficients of lateral soil pressure at FA and RI states (k FA and k RI )

As previously mentioned, the two upper and lower structural disturbance states of unsaturated soils, FA and RI states, can be taken as the soils states at the minimum and the maximum applied matric suction conditions, respectively. Accordingly, in this research, data presented for ψ  = 0 kPa and ψ  = 90 kPa were considered as FA and RI data sets, respectively. Variations of σ′h versus σ′v (obtained through Step 2) are used in Eq. (1) to calculate the experimental values of kFA and kRI. Figure 5 illustrates variations of kFA, and kRI versus σ′v for the examined soils.

Fig. 5
figure 5

Variation of kFA and kRI versus σ′v for: a Sand–Kaolin Mixture, eo = 0.52, b Sand–Kaolin Mixture, eo = 0.72, c Firouzkouh Clay, eo = 0.71 and d Firouzkouh Clay, eo = 0.92 [41]

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4.4 Step 4: Calculation of the disturbance parameter, D

First, the four factors of ζv, Du, a and z were calculated based on Eq. (5):

  • Calculation of accumulative plastic volumetric strain, ζv

To calculate the accumulative plastic volumetric strain, ζv, the total volumetric strain, ζv, should be determined at each net vertical stress increment (in accordance with the data plotted in Fig. 4), and then, the share of the elastic volume strains, ζe, is subtracting from the total volumetric strains to obtain the plastic volumetric strain increments. Finally, ζv was obtained by adding the increments, δζv. The elastic volume strain is the strain that is recorded at the net vertical stress corresponding to the ko value for each soil specimen. Using the experimental data plotted in Fig. 4, ζv is determined by Eq. (8):

$$\varepsilon_{v} = \frac{{{\Delta }e}}{{1 + e_{o} }}$$
(8)

In Eq. (8), Δe is the changes in void ratio, e, during each loading increment, and eo is the initial void ratio of the examined soil specimen.

  • Calculation of Du, a and z as suction-dependent parameters

In this research, constitutive parameters Du, a and z are defined in terms of soil matric suction. Actually, in unsaturated soils, these parameters are dependent on the hydromechanical responses of the soil and unsaturated state variables, such as effective stress, degree of saturation and matric suction [15]. Accordingly, Eqs. (9) to (11) are proposed in terms of the soil SWRC and decadic (i.e., base 10) logarithm functions of matric suction, as:

$$D_{u} \left( \psi \right) = 1 - \left( {\frac{{S_{{r\left( {{\text{RI}}} \right)}} }}{{S_{r} \left( \psi \right)}} \times \frac{{\log \left( {1 + \psi } \right)}}{{\log \left( {1 + \psi_{{{\text{RI}}}} } \right)}}} \right)^{{C_{{{\text{Du}}}} }}$$
(9)
$$a\left( \psi \right) = a_{o} \times \left( {\frac{{S_{{r\left( {{\text{RI}}} \right)}} }}{{S_{r} \left( \psi \right)}} \times \frac{{\log \left( {1 + \psi } \right)}}{{\log \left( {1 + \psi_{{{\text{RI}}}} } \right)}}} \right)^{{C_{a} }}$$
(10)
$$z\left( \psi \right) = z_{o} \times \left( {\frac{{S_{{r\left( {{\text{RI}}} \right)}} }}{{S_{r} \left( \psi \right)}} \times \frac{{\log \left( {1 + \psi } \right)}}{{\log \left( {1 + \psi_{{{\text{RI}}}} } \right)}}} \right)^{{C_{z} }}$$
(11)

In Eqs. (9) to (11), Sr(RI) is the degree of saturation of the soil at a matric suction corresponding to RI condition, ψRI (ψRI = 90 kPa for the soils examined in this research) and, Sr(ψ) is the degree of saturation of the soil at a given matric suction, ψ. In addition, ao, zo, CDu, Ca and Cz are fitting parameters.

The form of the relationships presented in Eqs. (9) to (11) for calculating Du, a and z guarantees the consistency of the stress-deformation behavior of the examined soils with the SWRC behavior of unsaturated soils and also satisfies the fundamentals of the DSC model [15]. In other words, these equations show a continuous behavior at a matric suction range from zero to the air entry suction value of the soil (i.e., fully saturated soil conditions) and also result in the maximum disturbance of the soil (with D = 1), which is proportional to the FA state of the soil. In addition, suction values higher than ψRI in Eqs. (9) to (11) result in a minimal disturbance in the soil structure and lead to D = 0 in a continuous manner.

4.5 Step 5: Calculation of the unsaturated coefficient of lateral soil pressure, k, in terms of effective vertical stress, σ′ v

In the final step, considering data obtained from steps 1 to 4 and implementing Eq. (6), the variation of the unsaturated coefficient of lateral soil pressure, k, at a given effective vertical stress, σ′v, can be calculated.

Note: The DSC model proposed in this study can present the changes in the coefficient of lateral soil pressure in unsaturated soils in the form of a continuous mathematical function. Therefore, it provides a suitable platform for use in numerical simulations. In addition, since the model is conceptually based on changes in soil structure, it is completely consistent with what occurs in the soil under lateral loading from the at-rest to the active state. Moreover, in the proposed DSC model, the model variables are defined as a function of effective stress variables coupled with the unsaturated hydraulic parameters of the soil (i.e., SWRC). Accordingly, more complicated hydromechanical stress paths can be considered to further develop the proposed model.

5 Results

Suction-dependent DSC parameters (i.e., Du, a and z) were calculated for different soils by considering the constitutive relationships mentioned in the previous sections, using the experimental data reported by Pirjalili et al. [41], and following Steps 1 to 4, and the results are plotted in Fig. 6. In addition, values of parameters ao and zo, and variations of CDu, Ca, and Cz against matric suction were calculated for different soil conditions, and the results are presented in Table 1 and Fig. 7, respectively. It should be noted that the Least Squares Method was used in this study to calculate the parameters of the DSC model to obtain the best fit between the predicted and the experimental values of the lateral pressure coefficients.

Fig. 6
figure 6

Variations of Du, a and z versus matric suction, ψ, for the examined soils

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Table 1 Suggested values for ao and zo for the examined soils
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Fig. 7
figure 7

Variations of: a CDu, b Ca and c Cz versus matric suction, ψ, for the examined soils

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As depicted in Fig. 6, similar trends for each DSC parameter are observed against matric suction for all studied soils. Accordingly, Du decreases as matric suction values increase so that it has a value of unity in the FA state (when ψ  = 0) and then gradually decreases to zero in the RI state (when ψ  = 90 kPa). Similar trends are observed for variations of a and z versus matric suction, as these parameters have zero values in the FA state and then gradually reached their maximum values (ao and zo, respectively) in the RI state. Figure 6 also shows that for a given soil type, larger a and z parameters are obtained for specimens with the lower initial void ratios in comparison with looser samples. In contrast, larger Du values were obtained for specific soil type specimens with higher initial void ratios than dense specimens, which implies a higher structural disturbance potential in looser soil specimens. Moreover, Table 1 suggests that the values of ao and zo depend on the soil type and the initial void ratio of the soil. Therefore, we deduce that for each type of soil, ao and zo have greater values at lower initial void ratios in comparison with the loose samples.

Data presented in Fig. 7 show that parameters CDu, Ca and Cz decrease as the matric suction increases. In Ca and Cz, this reduction continues until their values reach zero at high matric suctions. In addition, it is observed that CDu, Ca and Cz have greater values for the specimens with higher initial void ratios in comparison with dense soil specimens.

The variation of the unsaturated coefficient of lateral soil pressure, k, is calculated by considering experimental data reported by Pirjalili et al. [41], implementing the calculated values of kFA and kRI as shown in Fig. 5, considering values of CDu, Ca, Cz and Du, z, a, as presented by Figs. 6 and 7, and following Step 5; it is illustrated in terms of the effective vertical stress, σ'v, in Figs. 8 and 9 for Sand–Kaolin Mixture and Firouzkouh Clay specimens, respectively. Corresponding experimental values of k against σ'v are also plotted for comparison in Figs. 8 and 9. Accordingly, very good agreement is observed between the model predictions and the experimental data.

Fig. 8
figure 8

Model-predictions and experimental values of k for Sand–Kaolin mixture specimens

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Fig. 9
figure 9

Model-predictions and experimental values of k for Firouzkouh Clay specimens

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Note: Effective stress values recorded by implemented sensors have shown irregular fluctuations and jumps during the experimental work and at the beginning of the loading stages, and this might be due to instrument errors at very low stress values, inter-particle rearrangement within the soil texture, and initiation of stress–strain mobilization. These fluctuations were also observed in variations of the lateral soil pressure versus effective vertical stress before the soil reached its maximum lateral soil pressure (i.e., its at-rest state). Since the mathematical description and conceptual interpretation of these irregularities were not possible using the proposed DSC framework, the mentioned jumps and fluctuations are replaced with vertical dashed-lines according to their general trends in Figs. 8 and 9.

Data illustrated in Figs. 8 and 9 indicate that k grows rapidly after increasing the effective vertical stress up to a maximum in the early stages of loading and then decreases to an asymptotic value. In Figs. 8 and 9, the maximum value of k represents the unsaturated at-rest coefficient of the lateral soil pressure, ko. By further increasing the effective vertical stress and surpassing the at-rest condition, the lateral deformations within the soil mass increase, and k continually decreases until the soil reaches its active limit state of failure and k asymptotically approaches its corresponding active value, ka.

6 Analysis and discussion

In this section, the validity of the proposed DSC model is assessed by making a quantitative comparison between the experimental and model-predicted k values. To this end, the Root Mean Square Error (RMSE) and the Normalized Root Mean Square Error (NRMSE) parameters are calculated for comparative data sets by implementing Eqs. (12) and (13):

$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {k_{{{\text{experimental}}}} - k_{{{\text{predicted}}}} } \right)^{2} }}{N}}$$
(12)
$${\text{NRMSE}} = \frac{{{\text{RMSE}}}}{{{\text{Max}}\left( {k_{{{\text{experimental}}}} } \right) - {\text{Min}}\left( {k_{{{\text{experimental}}}} } \right)}}$$
(13)

In Eq. (12), kexperimental and kpredicted are the experimental and model-predicted values of k, respectively, N is number of data in each data set and i is the numeral. In addition, in Eq. (13), Max(kexperimental) and Min(kexperimental) are the maximum and minimum values of the experimental values of k, respectively. Calculated values of RMSE and NRMSE parameters are shown in Table 2 for the comparative data sets.

Table 2 RMSE and NRMSE parameters for model-predictions and experimental k values
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Data presented in Table 2 indicate that the maximum relative and normalized errors are (0.053 and 8.5%) and (0.065 and 10.2%) for the Sand–Kaolin mixture and the Firouzkouh Clay specimens, respectively.

By considering data summarized in Table 2 and taking into account Figs. 8 and 9, we see excellent conformance is obtained between the model-predictions and experimental variations of k versus effective vertical stress, σ′v. In addition, the three distinct phases in variation of k versus σ′v, namely the at-rest, transition and active phases, are shown to be well captured by the proposed effective stress-based DSC model.

To display the accuracy and reliability of the proposed DSC model, experimental and model-predicted values of ko and ka are compared in Figs. 10 and 11, respectively, along with the identity lines and corresponding R-squared values. Figures 10 and 11 clearly show that predicted and experimental data are mostly aligned with the identity lines, which implies the proposed model can predict the at-rest and active k values for the examined soils appropriately.

Fig. 10
figure 10

Comparison between the model-predicted and experimental values of ko for: a Sand–Kaolin mixture and b Firouzkouh Clay

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Fig. 11
figure 11

Comparison between the model-predicted and experimental values of ka for: a Sand–Kaolin mixture and b Firouzkouh Clay

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7 A prototype practical example

In order to demonstrate the practical functionality of the proposed DSC model, a prototype example problem of lateral soil pressure acting on a gravity retaining wall is presented under unsaturated soil conditions. In this case, soil properties were considered the same as the soil specimens in this study, with a matric suction range of zero to 90 kPa. Then, the lateral soil pressure acting on the wall in at-rest and active conditions was calculated by the proposed effective stress-based DSC model. For comparison, the corresponding values of the at-rest and active soil pressures were also calculated utilizing conventional soil mechanics concepts ignoring the unsaturated soil properties (namely Ordinary condition, in this section).

Note: In the prototype example problem, the exact conditions adopted in the experimental laboratory tests by Pirjalili et al. [41] have been taken into account. As in that study, the implemented ring device (in which the tested soil specimens were placed) was allowed to have lateral expansion during the tests but was fixed against rotation. Accordingly, the gravity wall in this prototype example was assumed to reach the active state by wall translation.

7.1 Example characteristics

The schematic of the proposed gravity wall is depicted in Fig. 12 with the soil condition and water level (GWL) location. As shown in Fig. 12, the GWL is located at the level of the wall foundation. It is assumed that the soil material above the ground water table is an unsaturated backfill, in which, the matric suction varies linearly from the ground water table (where ψ  = 0) up to the ground surface, in the form of negative hydrostatic pore water pressure [32]. The wall height, hunst, and the water density, γw, are considered 9 m and 10 kN/m3, respectively. Accordingly, the maximum value of the matric suction within the unsaturated backfill soil is 90 kPa, which is consistent with the maximum amount of matric suction of the soils examined in this study.

Fig. 12
figure 12

Schematic of the examined gravity wall in the prototype practical example

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7.2 Soil properties

In this example, two different conditions are considered for the backfill soil material: the Unsaturated condition and the Ordinary condition.

  • Unsaturated condition

In the Unsaturated condition, the matric suction within the backfill soil is assumed to vary linearly in the form of negative hydrostatic water pressure from the GWL up to the ground surface. Accordingly, the degree of saturation of the soil is determined from the data provided by Fig. 2, and the soil density is calculated using conventional soil-phase relationships in a soil mechanics context. In addition, the coefficients of the lateral soil pressure in the at-rest and active states are determined from the data presented in Figs. 8 and 9, respectively.

  • Ordinary condition

In the Ordinary condition, the degree of saturation and the density of the backfill materials were calculated similarly to the Unsaturated condition. However, in this case, the role of the matric suction on the coefficient of lateral soil pressure of the backfill materials was ignored and the coefficients of the at-rest and the active lateral soil pressures (i.e., ko and ka, respectively) were considered for a fully saturated state of the soil, with respect to the conventional considerations in geotechnical engineering. Using the data presented in Fig. 5, values of ko and ka were extracted from the curves corresponds to the zero-suction condition (i.e., the FA state of the examined soils).

Considered soil properties for the ordinary and unsaturated conditions are shown in Table 3.

Table 3 Soil parameters in the prototype practical example
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7.3 Results

Allowing for the geometrical aspects of the studied gravity wall (as depicted in Fig. 12) and the soil properties (as given in Table 3), Eqs. (2) and (3) were used to calculate the variations of the suction-stress parameter, σs, net vertical stress, σv, and the vertical profile of the unsaturated backfill soil, the results are plotted in Fig. 13a, b, respectively. As shown in Fig. 13b, the variation of the net vertical stress of the unsaturated backfill soil versus soil height has a nonlinear form. This is due to considering the changes in matric suction of the soil versus height that has led to different degrees of saturation and bulk density in the height of the backfill soil mass.

Fig. 13
figure 13

The variations of the a suction stress, σs, and b net vertical stress, σv, versus soil elevation above GWL, hw

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By considering data presented in Figs. 8, 9, and 13, implementing Eq. (3a), and following the proposed DSC model, variations of the at-rest and active effective horizontal (lateral) soil pressures, σ′h, along the unsaturated backfill soil height are calculated for the examined soils, the results are illustrated in Fig. 14. For better visual comparison, distributions of the at-rest and active effective lateral soil pressures against the backfill soil height (calculated for the ordinary state of the examined soils) are also plotted in Fig. 14. As can be seen in Fig. 14, it is evident that considering the unsaturated state for the backfill soil led to less effective lateral pressures in comparison with the corresponding ordinary cases.

Fig. 14
figure 14

Distribution of the at-rest and active lateral soil pressure in unsaturated and ordinary states for: a Sand–Kaolin mixture, eo = 0.52, b Sand–Kaolin mixture, eo = 0.72, c Firouzkouh clay, eo = 0.71 and d Firouzkouh Clay, eo = 0.92

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From a practical point of view, the at-rest soil pressure is commonly considered in the structural design of retaining walls, while the active soil pressure is taken into account to check the wall stability versus rotation and base sliding. In this light, in the current example, the overall lateral forces, Fh, in the at-rest and active states of the backfill soil are calculated for the examined soils, and the results are plotted in Fig. 15 for both cases of unsaturated and ordinary states.

Fig. 15
figure 15

Total lateral forces act on the wall in unsaturated and ordinary conditions of the examined soils for: a at-rest state and b active state

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With respect to the data plotted in Fig. 15, it is observed that considering the unsaturated state for the backfill soil resulted in an average reduction of 10.1% and 10.4% in total lateral forces acting on the wall in the at-rest and active states, respectively. This means that considering unsaturated parameters for backfill soil results in more economical and precise design procedures for retaining walls.

8 Conclusion

In this paper, an analytical approach has been introduced for calculating the unsaturated coefficient of lateral soil pressure, k, based on the disturbed state concept (DSC) in conjunction with an effective stress approach for unsaturated soils. In terms of the soil matric suction, the parameters of the proposed model have been defined as SWRC-compatible functions. Experimental data reported in [41] for two different soil types, a Sand–Kaolin Mixture and Firouzkouh Clay (each had two different initial void ratios and were tested under five applied matric suctions), have been used along with four complimentary fully saturated tests to calculate the parameters of the proposed model.

The lower and upper boundaries for the structural disturbance characteristics of the examined soils (i.e., fully adjusted and relative intact conditions, respectively) have been defined to calculate the disturbance parameter, D. The proposed model was shown to be capable of properly predicting the unsaturated coefficient of lateral soil pressure values, k, under increasing effective vertical stress and for different state variables and soil parameters (i.e., the matric suction and initial void ratio). k values measured under the minimum and maximum applied matric suction (i.e., ψ  = zero and ψ  = 90 kPa) were used to calculate the unsaturated coefficient of the lateral soil pressures at the lower and upper structural disturbance boundaries. In addition, the vertical and horizontal stress state parameters have been defined in accordance with the single-phase Bishop's effective stress approach for unsaturated soils.

Quantitative comparisons made between the model predictions and experimental values of k revealed the excellent validity of the proposed effective stress-based DSC model for predicting the continuous variations of the unsaturated coefficient of lateral soil pressure against increasing effective vertical stress, from the at-rest to the active state of the examined soils. Accordingly, NRMSE values ranging from 3.2% to 10.2% have been obtained for the model-predicted and experimental data sets. Moreover, it has been found that the disturbance parameter, D, of the examined unsaturated soils is considerably affected by the soil matric suction and initial void ratio.

Finally, in order to evaluate the performance of the proposed model for engineering problems, a practical example of calculation of lateral soil pressures act on a gravity wall (under conditions and specifications corresponding to the properties of the soils studied in this research) has been solved, and the values of the lateral forces acting on the wall in the at-rest and active states of the backfill soil have been calculated. Accordingly, two different conditions, namely, unsaturated (for the proposed model) and ordinary conditions (for conventional geotechnical engineering), were considered for the backfill soil materials. Quantitative comparisons between the obtained results indicate that the proposed DSC model not only led to more precise values of the lateral forces acting on the retaining walls (due to the use of more realistic soil conditions) but also reduced the project costs since it has resulted in lower values for design loads.