1 Introduction

For the purpose of designing engineering structures, shear strength is the most fundamental characteristic of soils that should be properly evaluated and understood. The ability of shallow foundations, retaining walls, excavations, pile foundations, slope stability, erosion risk prediction, as well as other geotechnical applications involving soil-structure interaction, are a few examples of geotechnical applications using soil shear strength (Yao and Yang 2017; Fattah et al. 2020; Nouzari et al. 2021; Zhang et al. 2021; Pham 2022a; Pham et al. 2022; Pham and Sutman 2023a). It is important to determine the shear strength of unsaturated soils and to be able to quantify changes in shear strength that might occur as a result of environmental conditions (Xu 2004; Dong et al. 2015; Mun et al. 2018; Yang et al. 2019; Pham and Sutman 2022a; Alassal et al. 2023; Pham et al. 2023a). Furthermore, because desaturation is a potential countermeasure to liquefaction, the unsaturated soil shear strength has been recognised as a significant element in liquefaction investigations (Mele and Flora 2019; Mele et al. 2022). However, the unsaturated shear strength measurement is challenging, expensive, and time-consuming. Therefore, theoretical predictions become a crucial approach, particularly in the preliminary design phase as well as for numerical modelling (Honda et al. 2011; Sheng et al. 2011; Pham 2020a; Pham et al. 2021, 2023b; Chali and Maleki 2021).

Several empirical models for describing the shear strength of unsaturated soils were presented, and they were all based on the theoretical framework of effective stresses. The soil property function, which establishes the connection between shear strength and matric suction, is the primary distinction between the available estimating models. Some different types of soil property functions were proposed such as using the saturation degree (Öberg and Sällfors 1997; Zhou et al. 2018; Zhai et al. 2019), volumetric water content (Lamborn 1986; Aubeny and Lytton 2003), volumetric air content (Graecen 1960), normalized water content (Vanapalli et al. 1996; Fredlund et al. 1996; Tarantino and Tombolato 2005; Oh and Vanapalli 2014), and air-entry value (Khalili and Khabbaz 1998; Lee et al. 2005; Kayadelen et al. 2007; Satyanaga and Rahardjo 2019). The literature review also demonstrated that many existing shear strength models were linear in form. However, the relationship between suction and shear strength can be nonlinear depending on the soil type or range of suction values (Karube and Kawai 2001; Sheng et al. 2011; Jiang et al. 2020; Pham and Sutman 2022b).

It should be also noted that numerous soil and environmental characteristics, including porosity, bulk density, stress state, water content, chemical composition, and soil fabric, have an impact on the unsaturated shear strength. Due to the complicated nature of the soil and the variation of environmental conditions, many different scenarios are made in practice. As a result, the demand to find a reliable model for the estimation of unsaturated shear strength is increasing. Unfortunately, almost all empirical models were established and verified using a limited number of data sets. Additionally, their applicability is frequently validated for a low suction range, but they did not prove how to apply those equations to forecast the shear strength of unsaturated soils at high suction (such as in the residual zone). As a result, it is not possible to state a particular model that is capable of accurately predicting the unsaturated shear strength over a wide suction range of every soil encountered. It is hence always desirable to verify the shear strength models with independent experimental data sets over a wide range of soil conditions (Marinho and do Amaral Vargas Jr 2020).

In other ways, geotechnical engineers are starting to move towards a more rational reliability-based design as they become more aware of the uncertainty resulting from empirical models for unsaturated soils (Sillers and Fredlund 2001; Bozorgzadeh et al. 2019; Ching et al. 2021; Guo et al. 2022; Guo et al. 2023; Pham et al. 2023c). It is important to assess how accurately the analytical models can be asserted using the statistical analysis method in order to persuade geotechnical engineers to apply unsaturated soil mechanics theory in routine practice. The uncertainty found in this paper will serve as a benchmark to decide which models are appropriate for the various types of soils.

This paper has four main objectives as follows. (1) To execute a comprehensive verification investigation employing thirty different test data sets for existing shear strength models. The advantages and disadvantages of each shear strength model are presented in relation to a statistical evaluation. (2) To assess the relative suitability of different analytical models and their applicability to unsaturated soil. (3) To provide a performance classification program that will be fundamental for selecting models to estimate reliably the shear strength of unsaturated soils under various soil conditions. (4) To discuss the validity associated with various shear-strength equations at high suctions or over a wide suction range.

2 Review of the Existing Shear Strength Models for Unsaturated Soils

2.1 Selected Criteria of Unsaturated Shear Strength Models

Numerous models were proposed in the literature to predict the shear strength of unsaturated soils. However, it is difficult to verify all the available shear strength models. Therefore, to select shear strength models for comparison, the following criteria are proposed: (i) selected models should not contain any fitting parameters, (ii) shear strength models should be formulated in the form of the Bishop’s stress approach or Fredlund’s independent stress approach, (iii) selected models should be recommended in a report, design guideline, standard, or should be well-known for prediction in practice, and (iv) selected shear strength models should include different soil property functions. Based on the four criteria mentioned above, seven well-known shear strength models are selected for comparison in this study. The following section briefly describes the characteristics of the selected shear strength models.

2.2 Description of the Selected Shear Strength Models

2.2.1 Global Material Equations-the Micromechanical Model

The normal and shear components of the stress tensors are connected mathematically by the shear strength constitutive relationship. Extending the effective stress theory and Mohr–Coulomb failure criterion to describe the shear strength of unsaturated soils is a common feature of most existing equations (Pham 2022b).

Consider the contact of two grains of unsaturated soil as illustrated in Fig. 1. The load transfer between the two grains is obtained partly through the inter-grain contact pressure (Pc), partly through the pore-water pressure (Pw = Pw1 + Pw2), and pore-air pressure (Pa = Pa1 + Pa2). For unsaturated soils, the equilibrium of the forces results in the following equation:

Fig. 1
figure 1

Model for micromechanical stress analysis

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$$P={P}_{c}+{P}_{w1}+{P}_{w2}+{P}_{a1}+{P}_{a2}$$
(1)

Equation (1) can also be re-written under the equilibrium of stresses as follows:

$$\sigma \cdot A={\sigma }_{c}.{A}_{c}+{u}_{w}\cdot {A}_{w}+{u}_{a}.{(A-{A}_{c}-A}_{w})$$
(2)

Division of Eq. (2) throughout by total cross-section area \(A\) results in

$$\sigma ={\sigma }^{^{\prime}}+{u}_{w}\cdot {A}_{w}/A+{u}_{a}\cdot {(1-{A}_{c}/A-A}_{w}/A)$$
(3)

or,

$${\sigma }^{^{\prime}}=\sigma -{u}_{a}+\left({u}_{a}-{u}_{w}\right)\cdot \frac{{A}_{w}}{A}+{u}_{a}\frac{{A}_{c}}{A}$$
(4)

where \(\sigma\) = total normal stress, \({\sigma }_{c}\) = contact stress, \({\sigma }^{^{\prime}}\)= effective stress or equivalent stress, \({u}_{a}\) = pore-air pressure, \({u}_{w}\) = pore-water pressure, \({A}_{w}\)= water area, \({A}_{c}\) = particle contact area, A = total cross-section area.

The grain contact proportion (\({A}_{c}/A)\) is commonly considered negligible and can be omitted. The general equation of shear strength for unsaturated soils can be transformed into the following form by using the Mohr–Coulomb failure criterion:

$$\tau ={c}^{^{\prime}}+\left(\sigma -{u}_{a}\right){\mathit{tan}\,\varphi }^{^{\prime}}+\left({u}_{a}-{u}_{w}\right)\cdot \left(\frac{{A}_{w}}{A}\right)\cdot {\mathit{tan}\,\varphi }^{^{\prime}}$$
(5)

where τ = unsaturated shear strength, (σ – \({u}_{a}\)) = net normal stress, (\({u}_{a}\) – \({u}_{w}\)) = ψ = matric suction, \({\varphi }^{^{\prime}}\)= effective friction angle, c' = effective cohesion.

If \({A}_{w}/A\) = χ, Eq. (5) is expressed as:

$$\tau ={c}^{^{\prime}}+\left(\sigma -{u}_{a}\right){\mathit{tan}\,\varphi }^{^{\prime}}+\left({u}_{a}-{u}_{w}\right)\cdot \upchi \cdot {\mathit{tan}\,\varphi }^{^{\prime}}$$
(6)

\(\upchi\) is sometimes referred to as Bishop's parameter. A fully saturated soil has a value of \(\upchi =1\) while dry soil has a value of \(\upchi =0\). It was suggested that the value \(\upchi\) would mainly depend on the level of saturation as well as the soil structure and stress state. It should be emphasized that the primary difference between existing shear strength models is the definition of normalized water function (\({A}_{w}/A)\) or the Bishop's parameter.

According to the capillary model, the term of matric suction \(\left({u}_{a}-{u}_{w}\right)\) in the formulas above denotes the difference between pore-air pressure and pore-water pressure as a result of surface tension. However, a change in the value of the matric suction does not correspond directly to a change in neutral stress since it represents a pressure difference due to the tension surface acting generally over only a part of the surface area of the soil particles. A significant change in the value of \(\upchi\) usually follows a change in the matric suction. Furthermore, the changes in soil structure among samples are caused by the presence of strong surface tension forces within the soils, which have proven to be particularly important in regard to the volume change.

2.2.2 Shear Strength Model of Greacen (1960)

Greacen (1960) conducted experiments on plastic clay with high porosity (void ratio of 1.2) by using a ring shear apparatus to investigate the effect of suction on the shear resistance of soils. It could be observed that shear strength and applied load have a linear relationship at low normal pressure (OA in Fig. 2). However, this relationship becomes highly nonlinear when the load is strong enough to compress the soil to saturation, as is the case with curve AD for plastic clay. Based on this observation, it was suggested that the suction acts as an added load over the internal contact area that increases the soil shear strength. The influence of suction on the unsaturated soil shear strength increase is considered through fractional air-filled voids that are given:

Fig. 2
figure 2

Maximum shear strength against applied normal pressure for plastic clay (data after Greacen 1960)

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$$\frac{{A}_{w}}{A}=1-\frac{{A}_{a}}{A}=(1-{\theta }_{a})$$
(7)
$${\theta }_{a}=n-{\theta }_{w}$$
(8)

And the expression of the shear strength equation by Greacen (1960) is as follows:

$$\tau = c^{\prime} + \left( {\sigma - u_{a} } \right)\tan \varphi ^{\prime} + \left( {u_{a} - u_{w} } \right)\left( {1 - \theta_{a} } \right)\tan \varphi ^{\prime}$$
(9)

where \(n\) = porosity, \({\theta }_{a}\) = volumetric air content, \({\theta }_{w}\) = volumetric water content.

2.2.3 Shear Strength Model of Lamborn (1986)

Lamborn (1986) proposed an equation for predicting the shear strength of unsaturated soils based on a micromechanical model. According to this model, the normalized water function is assumed to be equal to volumetric water content \(\left({A}_{w}/A={\theta }_{w}\right)\). As the soil compresses to saturation the suction becomes zero, \({\theta }_{w}\) approaches porosity, and the shear strength is described by the shear strength equation for the saturated soil. This approach was re-used in the design guideline of Aubeny and Lytton (2003). The simplified expression for the model of Lamborn (1986) is:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+({u}_{a}-{u}_{w})({\theta }_{w})\mathit{tan}\,\varphi {^{\prime}}$$
(10)

2.2.4 Shear Strength Model of Vanapalli et al. (1996)

One of the well-known shear strength models is the model proposed by Vanapalli et al. (1996), where the volumetric water contents at the saturated and residual conditions must be estimated from the soil–water characteristic curve (SWCC). The form of this model was used or modified by several other studies associated with numerical modeling of unsaturated soils (Tarantino 2007; Alonso et al. 2010; Lu et al. 2010; Zhou et al. 2012; Zhou and Sheng 2015; Lashkari and Kadivar 2016; Kim et al. 2016).

The starting point is the assumption that at lower values of matric suction (or higher degrees of saturation), the pore-water pressure acts directly to increase the effective stress in contributing to the shear strength. This condition applies until the soils begin to desaturate under an applied matric suction. The rate at which suction contributes towards shear strength can be related to the normalized area of water. By applying the Greens theorem, the normalized area of water is written as follows:

$$\frac{{A}_{w}}{A}={\left({\theta }_{w}\right)}^{\kappa }={\left(\frac{{\theta }_{w}-{\theta }_{r}}{{\theta }_{s}-{\theta }_{r}}\right)}^{\kappa }$$
(11)

where \({\theta }_{r}\) = residual volumetric water content, \({\theta }_{s}\) = saturated volumetric water content, \(\kappa\) =matching factor. For the sake of simplification, Vanapalli et al. (1996) assumed \(\kappa =1\) in their extended model. The unsaturated shear strength is therefore expressed as follows:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+({u}_{a}-{u}_{w})\left(\frac{\theta -{\theta }_{r}}{{\theta }_{s}-{\theta }_{r}}\right)\mathit{tan}\,\varphi {^{\prime}}$$
(12)

2.2.5 Adapted Shear Strength Model of Fredlund et al. (1996)

Fredlund et al. (1996) assumed that the matric suction contribution to the unsaturated soil shear strength is proportional to the normalized water area at a particular stress state. Garven and Vanapalli (2006) proposed linking the normalized water area with the plastic index of soils which is presented as follows:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+({u}_{a}-{u}_{w}){\left(\theta /{\theta }_{s}\right)}^{\kappa }\mathit{tan}\,\varphi {^{\prime}}$$
(13)
$$\kappa =-0.0016(PI{)}^{2}+0.0975(PI)+1$$
(14)

where \(PI\) = plastic index.

2.2.6 Shear Strength Model of Öberg and Sällfors (1997)

The matric suction is attributed to the shear strength via the effective stress parameter χ in the equation of Bishop and Blight (1963). However, the parameter χ is affected by many different factors such as the drying and wetting history, loading path, soil type, and internal soil structure of the specimen. Öberg and Sällfors (1997) proposed to assume that the parameter χ equals the degree of saturation. A similar form was also used by several researchers in modeling unsaturated soils (Jommi 2000; Sheng et al. 2004; Gens et al. 2006; Sun et al. 2007; Gallipoli et al. 2008; François and Laloui 2008; Abed and Vermeer 2009; Burton et al. 2020; Shahrokhabadi et al. 2020). The expression for this shear strength model is:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+({u}_{a}-{u}_{w})(S)\mathit{tan}\,\varphi {^{\prime}}$$
(15)

where \(S\) = degree of saturation.

2.2.7 Shear Strength Model of Khalili and Khabbaz (1998)

Khallili and Khabbaz (1998) have extended Bishop's equation by imposing an empirical constant for predicting the shear strength of unsaturated soils. Based on the data sets of 14 collected cases, the matric suction and air-entry value are suggested to decrease in response to an exponent power of − 0.55 for all soil types. Several researchers also reviewed this form by studying the constitutive behavior of unsaturated soils (Khalili et al. 2004; Russell and Khalili 2006; Hamidi and Tourchi 2018). The shear strength model is written as follows:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+({u}_{a}-{u}_{w}){\left(\frac{{u}_{a}-{u}_{w}}{AEV}\right)}^{-0.55}\mathit{tan}\,\varphi {^{\prime}}$$
(16)

where AEV = air-entry value.

2.2.8 Shear Strength Model of Tekinsoy et al. (2004)

Tekinsoy et al. (2004) presented a logarithmic model for linking matric suction and shear strength. Several other authors have also proposed shear strength equations with a similar form (Miao et al. 2002; Kayadelen et al. 2007; Satyanaga and Rahardjo 2019). This approach is relatively different from previous studies when the soil property function is described by the relation between the air-entry value, matric suction, and atmospheric pressure. The shear strength equation is expressed as follows:

$$\tau =c{^{\prime}}+(\sigma -{u}_{a})\mathit{tan}\,\varphi {^{\prime}}+(AEV+{p}_{at})\cdot \mathit{ln}\left[\frac{({u}_{a}-{u}_{w})+{p}_{at}}{{p}_{at}}\right]\mathit{tan}\,\varphi {^{\prime}}$$
(17)

where \({p}_{at}\) = atmospheric pressure (101.325 Pa).

3 Comparison of the Shear Strength Models with Measurements

3.1 Description of Published Experimental Cases

In order to have effective and reliable verification of shear strength models, 30 published data sets were selected for comparative purposes. For an experimental data set, some soil specimens with different physical properties were tested. In this study, the term ‘case’ is stipulated to define samples with their different physical properties such as different soil types, initial density, plasticity, or saturation degree. An overview of the basic characteristics of experimental data sets is presented in Table 1. It can be classified that almost all published data sets were conducted on four main soil types, which are sandy, clayey, silty, and sand-kaolin soils. The types of materials also include natural soils, expansive soils, tailing, and residual soils. While the clay percentage ranges from 0 to 69.9%, the soil density ranges from loose to medium-dense and dense levels. A wide range of matric suction between 0 and 1500 kPa and a wide range of saturation degrees between 21.2% and 95% were observed among all test cases. Moreover, most of the soils have plasticity index, PI, varying from 0 (sands) to 45 (clays). Therefore, the selected data is complete and can be considered sufficient and reliable to assess the prediction performance of the shear strength models. Furthermore, Table 2 shows a summary of the input parameters required for the shear strength calculation using theoretical models. Special attention was paid to the realistic determination of the parameters for the analytical calculations. It should be noted that the parameters of the air-entry value (AEV) and the residual volumetric water content were determined from soil–water characteristic curves, as illustrated in Fig. 3. The sensitivity of SWCC variables with unsaturated shear strength can be also found in several previous studies (Zhai et al. 2019; Pham 2022b; Pham and Sutman 2023b; Pham et al. 2023a, 2023b). It is indeed important to note that the SWCC and shearing tests are carried out on distinct soil specimens with varying confining pressure. As a result, specimens in those two types of experiments may have different initial void ratios. In order to have a better evaluation, calibrated SWCCs based on the real void ratio of soil samples in shear tests are employed to calculate the shear strength of unsaturated soils. The calibrated procedure for SWCCs can be found in the paper of Pham and Sutman (2022b).

Table 1 Summary of published data sets on shear strength measurement of unsaturated soils
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Table 2 Summary of input parameter determination for shear strength prediction
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Fig. 3
figure 3

Determination of variables from the soil–water characteristic curve

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The efficacy of current shear strength models is evaluated and categorized in this work based on the basic physical characteristics of soil samples, where water content, void ratio, and suction are frequently controlled to be constant. Engineers will find the evaluation based on the initial physical properties more useful and practical than taking into account the current state of soil samples, where water content and void ratio change with suction. This is because the initial physical properties are frequent and simple to measure. Finally, it is essential to emphasize that the current works did not allow us to evaluate the performance of the prediction models considering soil volume change. This is mainly because of the limited number of experimental studies and existing models on the variation of shear strength with soil volume change. Details of the comparison between measured and predicted results case by case are presented in the Appendix.

3.2 Validation of Shear Strength Models

Figure 4 shows a comparison between predicted and measured shear strength-suction curves for a typical case utilizing the data sets of Escario and Saez (1986). The studies employed the soil–water characteristic curves recorded with a pressure of 120 kPa (Fig. 5a). The summary of soil properties and the saturated shear strength parameters of three different soil types is presented in Table 3. The measured shear strength values are indicated by the symbols in the figure, while the continuous lines represent the predicted shear strength values. It is interesting to note that the model of Vanapalli et al. (1996) produced the best agreement with experimental data for the case of clayey sand while the model of Fredlund et al. (1996) produced the best agreement between the predicted and measured values of the shear strength for the cases of silty clay and grey clay. It is also discovered that the Lamborn (1986) model produces a reasonable prediction for a low suction range. The ratio \({A}_{w}/A\) is involved as a function of water content which is a common feature among these models. However, the remaining models that were based on the correlation with the air-entry value showed a substantial overestimation.

Fig. 4
figure 4

Comparison of predicted and measured shear strength for a representative case (test data after Escario and Juca 1989): a soil–water characteristic curve, b silt clay, c clay sand, d gray clay

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Fig. 5
figure 5

Predicted against measured shear strength for sandy soils

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Table 3 Summary of soil properties tested by Escario and Saez (1986)
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Figure 5 compares the measured and predicted shear strengths for sandy soils. For the low matric suction range, it is noticeable that the results predicted by the shear strength models are quite similar to and fit well with measured data. But as the matric suction is increased, the prediction outcomes of the models begin to diverge dramatically. It should be noted that the fundamental difference among the shear strength models is considering the contribution of suction to the total shear strength. Matric suction makes a smaller contribution to the overall shear strength for low matric suctions compared to the component of net normal stress. This explains why the differences across the prediction models are so minimal with small matric suctions. When the matric suction is increased from low to high, all shear strength models give inconsistent results. The unsaturated shear strength is greatly overpredicted by Tekinsoy et al. (2004) whereas it is severely underestimated by Khalili and Khabbaz (1998). Two of these models define the soil property function based on the air-entry value without taking the soil degree of saturation into account, in contrast to the other remaining models. The soil property function defined by the model of Graecen (1960) also exhibits a notable underestimating of unsaturated shear strength due to the use of volumetric air content.

Five data sets of clayey soils were analysed, and Fig. 6 compares predicted shear strength to measured ones. The findings show that none of the chosen models produces outcomes that are consistent with the measured data across all comparison cases. According to the results, none of the selected models yields consistent results with the measured data for all different cases. Almost all selected models overpredict or underpredict the shear strength of unsaturated soils. It is evident that the range of suction and soil density have a substantial impact on the degree of agreement across various models. The results also show that, in contrast to other models, Greacen (1960) and Lamborn (1986) models create the most significant overprediction while Öberg and Sällfors (1997) and Tekinsoy et al. (2004) models produce the most significant underprediction in all cases. In contrast, of all the models chosen, the Fredlund et al. (1996) model typically produces the best agreement with measured data. Additionally, it is discovered that when matric suction increases, the discrepancy between predicted and measured results also increases.

Fig. 6
figure 6

Predicted against measured shear strength for clayey soils

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Figure 7 displays the predicted versus measured shear strengths of silty soils. It is discovered that none of the chosen models agrees well with the measured data. The models developed by Fredlund et al. (1996) and Vanapalli et al. (1996) provide a good degree of agreement with the results of measurements and are, in general, superior to other shear strength models. Even in the range of low matric suction, the models of Graecen (1960), Lamborn (1986), and Khalili and Khabbaz (1998) greatly underestimate the unsaturated shear strength. As a result, it may be said that these models are insufficient for estimating the shear strength of silty soils. In comparison to measure results, the models of Tekinsoy et al. (2004) overpredict greatly whereas the models of Öberg and Sällfors (1997) slightly overpredict.

Fig. 7
figure 7

Predicted against measured shear strength for silty soils

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Figure 8 presents the predicted against measured shear strength for sand-kaolin mixtures. It can be observed that the prediction potential of the models depends significantly on the suction range and the physical properties of soils. In general, for the shear strength range higher than 100 kPa, none of the chosen models yields a satisfactory match with the measured data. In general, the models of Tekinsoy et al. (2004) and Khalili and Khabbaz (1998) produce an overprediction, while the remaining models typically produce an underestimation. It should be noted that Öberg and Sällfors (1997) model frequently predicts a result that is lower than the measured shear strength. It is possible to significantly underestimate the shear strength of the sand-kaolin mixture by assuming that the component in the soil property function is equal to the degree of saturation.

Fig. 8
figure 8

Predicted against measured shear strength for sand-kaolin mixture

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4 Performance Classification of Shear Strength Models

4.1 Evaluation Criteria

It is well-known that the relationship between shear strength and matric suction is often followed by a nonlinear curve. The criterion on the degree of curve matches referring to the degree of convergence between measured and predicted results is used to assess the analytical models' performance. A better description of the data by the equation is implied by a lower difference between the predicted and the measured curves. The degree of curve match is represented by using an index, namely, average relative error (ARE). As indicated by the following equation, the average relative error is the proportion of a difference between the measured and predicted value:

$$ARE=\frac{1}{N}{\sum }_{1}^{n}\left|\frac{{\tau }_{predicted}-{\tau }_{measured}}{{\tau }_{measured}}\right|\cdot 100\%$$
(18)

where τmeasured is a measured shear strength value of ith data, τpredicted is the predicted shear strength of ith data, and N is the total number of data points available.

The normalised sum of squared error (SSE), which measures how well the shear strength models can predict outcomes, is used as the second criterion. If the normalised sum of squared error is lower, the results that the analytical model is more reliable to be used. The following is a definition of the sum of square errors:

$$SSE=\sum _{1}^{n}{\left(\frac{{\tau }_{predicted}-{\tau }_{measured}}{{\tau }_{measured}}\right)}^{2}$$
(19)

4.2 Overall Reliability Evaluation with the Statistical Analysis Model

Two crucial factors, the quality, and consistency of the predicted results are described in the statistical analysis method, which gives a broad view of the theoretical models' ability to predict results. For the chosen shear strength models over all 30 examined cases, Fig. 9 shows the normalised sum of squared errors and the average relative error. The vast range of the ARE and SSE distributions highlights the fact that none of the analytical models offers a consistent agreement with the measured data in all conceivable scenarios. The complexity behind the mechanisms governing the unsaturated soil behavior may be the reason for this inconsistency. According to the results, the overall average relative error is equal to 12.1% for the model of Fredlund et al. (1996), 13.75% for the model of Vanapalli et al. (1996), 15.3% for the model of Öberg and Sällfors (1997), 17.1% for the model of Lamborn (1986), 16.8% for the model of Khalili and Khabbaz (1998), 25.7% for the model of Greacen (1960), and 29.2% for models of Tekinsoy et al. (2004). It is evident that the three shear strength models developed by Fredlund et al. (1996), Vanapalli et al. (1996), Öberg and Sällfors (1997) perform better in terms of prediction than the other models. The outcomes also highlight how much one model's ability to predict depends on the physical characteristics and suction range of soils.

Fig. 9
figure 9figure 9figure 9figure 9

Statistical analysis of seven shear strength models: a model of Graecen (1960); b model of Lamborn (1986), c model of Fredlund et al. (1996); d model of Vanapalli et al. (1996); e model of Öberg and Sällfors (1997), f model of Khalili and Khabbaz (1998); g model of Tekinsoy et al. (2004)

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4.3 Performance Classification Based on Soil Types

Four major soil types (sandy, clayey, silty, and sand-kaolin soils) were tested among the published data sets, as was covered in the section previously. It should be emphasised that each type of soil appears to have a unique complex interaction between the solid and liquid phases. Thus, it is crucial to categorise the shear strength model performance for various soil types and determine the best model for each soil type.

Figure 10 displays the average relative error (ARE) for the four different soil groups for the chosen shear strength models. As can be observed, the value of ARE for sands is substantially lower than that of the other soil types, indicating that the analytical models perform better at predicting sandy soil shear strength than clayey ones. It should be mentioned that clayey soils frequently exhibit more complex behaviour than granular soils due to their characteristics in terms of shape, size, structural arrangement, and water distribution phenomenon. Additionally, the suction range of sands is far lower than clays, making models possible to anticipate shear strength more accurately. The models by Fredlund et al. (1996) and Öberg and Sällfors (1997) are shown to be the two most effective options for forecasting the unsaturated shear strength of sandy soils. It should be noted that the plastic index of sandy soils is often small, and parameter κ (Eq. 14) approaches a value of 1.0. As a result, the Öberg and Sällfors (1997) model and the Fredlund et al. (1996) model become similar, which explains why the average relative error of both models for sandy soils is so close. It can be concluded that the assumption of the Bishop parameter χ being equal to the saturation degree is more suitable for sands but not reasonable for clays. The model developed by Vanapalli et al. (1996) is generally superior to all other models for clayey soils and can be regarded as the best choice for forecasting unsaturated shear strength. However, the one proposed by Vanapalli et al. (1996) requires the residual volumetric water content which is sensitive to the soil structure change and volumetric strain. It has been also found that the models predict the unsaturated shear strength for silty soils more accurately than that for sand-kaolin mixtures. Sand-kaolin mixtures are compacted during sampling, resulting in a high density and a much greater matric suction range. Three models from Öberg and Sällfors (1997), Vanapalli et al. (1996), and Fredlund et al. (1996) have also been discovered to be reliable candidates for estimating the shear strength of silty soils and sand-kaolin mixtures. Additionally, it is noteworthy that the models (Greacen, Khalili and Khabbaz, Tekinsoy et al.) that relate suction and shear strength using the air-entry value or air volumetric water content have poor prediction ability in comparison to the models using the water content.

Fig. 10
figure 10

Performance of shear strength models based on soil types

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Figure 11 shows the differences in microstructure between coarse-grained (sands) and fine-grained (clays) soils to demonstrate how different soil types affect the performance of the prediction models. It should be noted that while clay particles often have a platy shape, sands typically have a nearly spherical shape. Numerous earlier studies noted that clay formations are complicated and that more particles are reoriented parallel to one another rather than rolling as it occurs in sand structures based on findings from photomicrographs (Dhadse et al. 2023). Additionally, the microstructure of compacted clays exhibit a double porosity network with two levels of interaction: a microstructural level that corresponds to the active clay minerals and their vicinity (intra-aggregate voids), which is predominated by physicochemical interaction phenomena, and a macrostructural level that accounts for the larger scale structure of the soil (inter-aggregate voids) (Wan et al. 1995; Romero et al. 1999; Rojas 2008; Airò Farulla et al. 2010; Eyo et al. 2019). Due to attraction of the highly charged particles for water and capillary action, the smaller intra-aggregate voids would reach water saturation before the inter-aggregate voids. This indicates that in compacted clays, there is a qualitative relationship between matric suction, water content, and intra-aggregate soil structure. As a result, in addition to the global water content, local water content in intra-aggregate voids also affects the contribution level of matric suction to shear strength. Therefore, the contribution of matric suction to shear strength not only depends on the global water content of the soil but also on the local water content in intra-aggregate voids. However, most of the shear strength equations are frequently developed using simple regression techniques for the experimental data and overlook the underlying physical principles. For this, prediction model performance for clays is much lower than that of sands.

Fig. 11
figure 11

Schematic layout of microstructure and capillary forces: a sand cluster; b clay cluster, c Meniscus between sand particles, d meniscus between clay particles

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4.4 Performance Classification Based on the Matric suction

The level of matric suction has a strong influence on the difference in results predicted by shear strength models. Therefore, it is necessary to evaluate the shear strength model performance based on the matric suction range. In this section, the term “low level” is defined as matric suction being smaller than or equal to 50 kPa, “medium level” is defined as matric suction being between 50 and 100 kPa, and “high level” is defined as matric suction being between 100 and 200 kPa, and “very high” is defined as matric suction being larger than 200 kPa.

A comparison of the selected shear strength models among the different ranges of matric suction is shown in Fig. 12. It should be noted that at low-value ranges, the contribution of the matric suction to the total shear strength is small when compared to the net normal stress component. The shear strength models hence show a relatively good performance in predicting unsaturated shear strength for low matric suction (average relative error lower than 10%). The prediction performance of models changes and generally decreases with increasing the matric suction. It is also found that the models of Fredlund et al. (1996), Öberg and Sällfors (1997) show the best performance in predicting the shear strength of unsaturated soils for a suction range smaller than 100 kPa. However, the model of Vanapalli et al. (1996) gives the lowest average relative error, which yields the best performance in predicting the shear strength for a range of higher matric suction (> 100 kPa). The results proved that the assumption of the Bishop parameter χ being equal to the saturation degree is more suitable for low matric suctions but not reasonable for high matric suctions. The interaction between air and water phases becomes more complicated with increasing the matric suction, and the effective saturation degree may be more sufficient to describe the dependence of shear strength with the soil suction. Furthermore, low prediction performance is found for all other remaining models (Greacen, Lamborn, Khalili and Khabbaz, Tekinsoy et al.), particularly for high matric suctions.

Fig. 12
figure 12

Performance of shear strength models based on ranges of matric suction

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The boundary effect state, the transition state, and the residual state are identified as the three distinguishable stages of desaturation. Figure 13 shows how the distribution of water varies with various matric suction states. When there is a boundary effect or low suction, water fills almost all the soil pores, forming water menisci between the aggregates or soil particles. The water content significantly decreases when the suction is increased in the transition state. Because of this, the water menisci area in contact with the soil or aggregates is not continuous and begins to decrease at this stage. Large increases in suction eventually result in relatively minor changes in water content, and at the residual stage where the water menisci are rarely noticeable. It should be noted that the greatest nonlinear increase in shear strength occurs in the transition zone. Beyond the residual suction condition, the shear strength of unsaturated soils can increase, decrease, or remain approximately constant during further desaturation. It is possible that the shear strength decreases in some circumstances, especially in soils that desaturate quickly (such sands and silts). In actuality, the water content in sands and silts under residual suction conditions can be quite low, which may make it difficult to properly transmit suction to the soil particles or aggregate contact points. As a result, shear strength is unlikely to rise significantly even with a large increase in suction. Clay, on the other hand, might not have a clearly defined residual state because of its intra-aggregate voids that are filled with many layers of adsorbed water. A large portion of the inter-aggregate voids and the entirety of the intra-aggregate voids stay saturated when the applied suction is low. When suction increases, only the microstructure remains saturated, which contributes toward increases in shear strength. As a result, if the change in water content with different phases of desaturation is taken into account during modelling, the prediction will be more accurate. In comparison to other models, the Vanapalli et al. (1996) model exhibits a better prediction performance for high suction range by employing effective degree of saturation, which is directly related to residual water content. Other models ignoring the mechanism of the residual stage demonstrated an unsuccessful prediction for a high suction range.

Fig. 13
figure 13

Variation of water area with suction in different stages

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4.5 Performance Classification Based on the Saturation Degree

It is important to recognize that the shear strength of unsaturated soils is controlled by two separate factors, matric suction, and saturation degree. The saturation degree is frequently included in the soil property function used to characterise the contribution of matric suction to shear strength. The influence of the saturation degree varies among the various shear strength models as well. As a result, it is critical to evaluate how well existing shear strength models perform over various saturation degree ranges. The degree of saturation is ranged for three corresponding levels: Low saturation degree (S \(\le\) 50%), medium saturation degree (50 \(<\) S < 75%), and high saturation degree (75 \(\le\) S < 100%).

Figure 14 displays the effectiveness of theoretical models for the various saturation degree ranges. The findings show that when the saturation degree increases, the performance of all models declines. However, compared to the matric suction one, the sensitivity of models with saturation degree is smaller. It should be highlighted that the prediction performances of the models by Khalili and Khabbaz (1998) and Tekinsoy et al. (2004) are poor for a variety of saturation degree ranges. The soil property function is defined using the models of Khalili and Khabbaz (1998) and Tekinsoy et al. (2004) based on air-entry value without taking the degree of saturation into account. This is an explanation of why these models produce a high disagreement for any range of saturation degree. Meanwhile, other remaining models produce a good performance in predicting the shear strength of unsaturated soils for saturation degrees higher than 50%. Regarding the model of Graecen (1960), the volumetric air content is used to define the soil property function. With a saturation degree smaller than 50%, the volumetric air content is approximate to the volumetric water content. This explains why the model of Greacen (1960) shows a good prediction for low saturation degrees. When the saturation degree is larger than 50%, the models of Vanapalli et al. (1996), Öberg and Sällfors (1997) produce prediction performance better than other models. Among all selected candidates, three models of Vanapalli et al. (1996), Fredlund et al. (1996), Öberg and Sällfors (1997) give the best results to predict the shear strength of unsaturated soils over a different range of saturation degrees.

Fig. 14
figure 14

Performance of shear strength models based on ranges of saturation degree

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For the low-value range of saturation degree, it can be seen that almost all models currently perform poorly in terms of prediction. The contribution of pore water to the mechanical behaviour of unsaturated soils, therefore, must be taken into account in order to properly comprehend the limitation of these models. The equilibrium of the stress components at the surface and interior of the soil element is shown in Fig. 15 in the cross-section of a typical unsaturated soil. It has been observed that the entire cross-sectional area of unsaturated soil is obtained by combining the saturated part area and the unsaturated part area. Three types of pore water can be identified in unsaturated soil, which are bulk water, capillary water, and adsorbed water. The expression of pore water could be written as follows:

$$S={{S}_{b}+S}_{c}+{S}_{a}$$
(20)

where S = overall degree of saturation, \({S}_{b}\), \({S}_{c}, {S}_{a}\) = components of bulk water, capillary water, and adsorbed water.

Fig. 15
figure 15

Stress equilibrium of a cross-section in unsaturated soil

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It should be emphasized that the contact stress component that is transferred from one boundary surface to another through the soil skeleton is directly responsible for balancing a portion of the applied load, the remaining of which is carried by the pore-water pressure. The bulk water contains some contact points between soil particles, and it can support an external load alongside the soil skeleton, as well (Karube and Kawai 2001). Additionally, there is capillary water between soil particles, pressure of which influences the contact stress between soil particles, affecting the shear strength. Adsorbed water does, however, contribute very small to the soil's shear strength (Konrad and Lebeau 2015). In principle, this is because the surface of each soil particle is completely covered with adsorbed water, which almost has no impact on the contact stress between soil particles. The correct effective stress, in this case, should be rewritten as a function of bulk and capillary waters:

$${\sigma }^{^{\prime}}=\sigma -{u}_{a}+\left({u}_{a}-{u}_{w}\right)\cdot \frac{{A}_{w}^{b}+{A}_{w}^{m}}{A}+{u}_{a}\frac{{A}_{c}}{A}$$
(21)

However, none of the shear strength models that were considered in this study were able to distinguish between the contributions of bulk water and capillary water with adsorbed water to the shear strength of unsaturated soils. As a result, almost all models typically overpredict results, especially at low levels of saturation where the impact of absorbed water is more pronounced.

4.6 Performance Classification Based on the Density of the Soil

The shear strength of unsaturated soils does not only depend on the water amount but also on the soil density. This is because the pore size has an important impact on how water, air, and solid phases interact (Bencheikh and Messast 2023). An element of soil is frequently viewed as a simple three-phase system made up of pore air, pore water, and solid particles according to recent unsaturated soil mechanics theories. The curvature of the air–water meniscus is related to the capillary actions that cause suction, which is attributed to the interactions between the air–water menisci by:

$$\psi ={u}_{a}-{u}_{w}=\frac{2{\sigma }_{s}\cdot cos\alpha }{{r}_{i}}=\frac{2{\sigma }_{s}}{{R}_{m}}$$
(22)

where \({r}_{i}\) = pore radius, \({\sigma }_{s}\) = surface tension, \(\alpha\) = air–water contact angle, \({R}_{m}\) = curvature length of air–water meniscus.

The total differential of suction \(d\psi\) with respect to the void ratio can be written as follows:

$$\frac{d\psi }{de}=\frac{\psi }{{\sigma }_{s}}\cdot \left(\frac{\partial {\sigma }_{s}}{\partial e}\right)+\frac{\psi }{cos\alpha }.\left(\frac{\partial cos\alpha }{\partial e}\right)-\frac{\psi }{{r}_{i}}\cdot \left(\frac{\partial {r}_{i}}{\partial e}\right)$$
(23)

Soil density has been observed to resemble to the arrangement of solid particles. Therefore, any variation in soil density results in changes in the degree of saturation and volumetric moisture content. A change in density would also affect the degree that the soil particles are packed, which would have an impact on the contact angle, curvatures of the air–water menisci, and therefore, the matric suction in the soil (Fig. 16).

Fig. 16
figure 16

Effect of soil density on air–water meniscus curvature

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In order to understand how shear strength models, react to changes in pore size, it is essential to classify performance according to soil density. The average relative error of the analytical models with the initial soil density variation is shown in Fig. 17. With increasing soil density, the prediction accuracy of models declines. None of the selected shear strength models yields results that are in good agreement with the measured results for dense soils. None of the shear strength models are used to produce outcomes that are in agreement with the measured data for dense soils. All shear strength models typically have an average relative error of dense soils that is more than 15%. The complex interaction nature accompanied by the pore size and water distribution phenomenon in dense soils can be considered as an explanation of why the value ARE for dense soils is significantly higher than for loose soils. Moreover, it should be noted that soils with a higher density will have smaller pore sizes, and their matric suction will, therefore, become very high. As a result, the shear strength models' prediction accuracy declines as soil density increases. Additionally, it is discovered that the influence of soil density on the theoretical models' prediction ability is more significant than that of matric suction and degree of saturation. Unfortunately, the soil density influence is ignored in all the existing shear strength models. Unfortunately, none of the shear strength models that are currently in use take soil density into account. It can be shown that while the model of Vanapalli et al. (1996) generates the highest performance in predicting the shear strength for dense soils, the model of Fredlund et al. (1996) exhibits the best prediction performance for loose soils. Additionally, throughout a range of soil densities, three models by Vanapalli et al. (1996), Fredlund et al. (1996), Öberg and Sällfors (1997) often outperform the others in terms of prediction performance.

Fig. 17
figure 17

Performance of shear strength models based on the soil density

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4.7 Performance Classification Based on the Plasticity Index of the Soil

The plasticity index (PI), which is typically larger the finer the soil, measures the range of water content within which soil is plastic. However, it is well known that variations in soil water content frequently lead to changes in soil shear strength. The plasticity index allows to consider how altering the amount of water in the soil may affect its strength and compressibility. A change in water content has a limited effect on these characteristics in granular soil, whereas cohesive soil tends to become significantly stronger and less compressible as the water content decreases. As a result, the plasticity index may be a significant feature to consider when assessing sensitivity and classifying theoretical models' performance. The plasticity of soils is divided into three categories based on the plasticity index: low plasticity (\(PI\le\) 10), medium plasticity (\(10<PI<\) 20), and high plasticity (\(PI\ge\) 20).

Capillary cohesion, a sophisticated physical system, is the outcome of the attraction between soil particles and water molecules. The moisture content is thus by far the most obvious factor influencing the cohesive properties of the soil. When the soil moisture content falls below the plastic limit (low plasticity index), the soil becomes dry and loose, and an annular water film forms where the soil particles touch one another, but they are insufficient to form a network. As a result, the impact of matric suction on shear strength is minimal for soils with low plasticity. When the matric suction is increased during the residual stage, the shear strength curve in this instance has a tendency to decrease (Fig. 18a).

Fig. 18
figure 18

Soil plasticity and shear strength: a low plasticity, b high plasticity

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For soils with high plasticity, the water films between the soil voids are sufficient enough to link and form a network, increasing the cohesion of soils. This is due to the significant specific surface area of clay particles as well as the presence of intra-aggregate voids. As a result, the impact of matric suction on shear strength is more significant for soils with high plasticity. The shear strength curve in this case tends to increase or roughly remain constant when the matric suction is raised during the residual stage (Fig. 18b).

The effectiveness of shear strength models at three different soil plasticity levels is shown in Fig. 19. The findings show that all analytical models perform poorly for medium plasticity soils, with Vanapalli et al. (1996) having the greatest prediction performance with an ARE of about 15%. On the other hand, the model developed by Fredlund et al. (1996) exhibits the best outcomes for high-plasticity soils. Except for the models of Graecen (1960) and Tekinsoy et al. (2004), the performance of analytical models for low-plasticity soils is remarkably similar. Furthermore, the models of Graecen (1960), Lamborn (1986), Khalili and Khabbaz (1998) are noticeably more sensitive to soil plasticity than models that employ normalised water content, like those of Fredlund et al. (1996), Vanapalli et al. (1996), Öberg and Sällfors (1997). It should be also emphasized that higher plasticity soils yield higher matric suction, therefore theoretical model prediction performance is typically reduced.

Fig. 19
figure 19

Performance of shear strength models based on the index of plasticity

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4.8 Performance Classification Based on the Activity of the Clay

It is also important to note that unsaturated shear strength is significantly influenced by the capillary action surrounding soil particles. The tensile strength of unsaturated soils is affected by the interparticle interactions and water distribution over particle areas that are significantly altered by the presence of different clay minerals. Thus, it may be advantageous to establish a connection between clay activity and the effectiveness of shear strength models. The plasticity index to the percentage of clay particles in the soil is used to calculate the clay activity (\(A=\frac{PI}\%{\mathrm{clay}}\)), which identifies the swelling potential of clay soils. In this study, the soils are classified into three different levels of clay activity, which are low activity or inactive (\(A\le\) 0.5), medium activity or normal (\(0.5<A<\) 1.0), and high activity or active (\(A\ge\) 1.0).

It should be mentioned that the value of matric suction and the amount of pore water determine the contribution level of matric suction to the shear strength of unsaturated soils. If matric suction and water content are raised simultaneously, the shear strength will increase to its maximum value. However, there is an inversely proportionate function between matric suction and water content. As a result, there is always an optimal water content ratio that gives unsaturated soils the highest shear strength, and this ratio significantly depends on the presence of fine soil contents. Figure 20 depicts how the soil microstructure relates to the three levels of activity. It should be noted that low activity relates to scenarios in which there is low plasticity index and high clay content, whereas high activity corresponds to situations in which there is high plasticity index and low clay content. However, the higher clay content also indicates a greater capacity to hold water, and plastic index may be therefore larger. The effective contact between the sand particles is improved when the amount of fine-grained soil increases, as well as the adsorption force. As can be seen from the analysis above, adding a limited amount of clays to unsaturated soils can also result in a very complex behaviour.

Fig. 20
figure 20

Correlations between activity, water content, and clay content: a low activity, b medium activity, c high activity

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Figure 21 shows the prediction performance of the shear strength models with variation in clay activity. It is important to note that, particularly for medium and high-activity soils, none of the analytical models produce results that are in good agreement with the measured values. For every clay activity level, the average relative error of all shear strength models is often higher than 10%. The value of ARE for the Vanapalli et al. (1996) model is interesting to notice because it is roughly the same for all three levels of activity, leading to the conclusion that the activity has less of an impact on the model's performance. One of the important reasons for this tendency is that the model of Vanapalli et al. (1996) considers the residual degree of saturation in the prediction and thus the influence of clay activity is minimized. Among the models chosen, it was found that the Fredlund et al. (1996) model had the greatest prediction performance for soils with low and medium activity. Regarding the remaining models, it is far more intriguing to emphasise that the accuracy of prediction models typically declines with increasing clay activity. The difference in clay activity has a considerable impact on several models, most notably those of Graecen (1960) and Lamborn (1986). Using volumetric water content to link unsaturated shear strength with suction, while this parameter depends strongly on the clay percentage in soils, can be considered as the reason for this sensitivity.

Fig. 21
figure 21

Performance of shear strength models based on the activity of the clay

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4.9 Performance Summary of Analytical Models

A statistical summary is presented in order to provide a benchmark that design engineers can use to determine which models are appropriate for the various soil conditions. The performance of each model is assessed using the averaged information criterion (Pham and Sutman 2022a; Pham 2020b; Pham and Dias 2021). Table 4 provides an overview of how each model performed under various soil conditions. The best model to use for each associated parameter will be the one with the lowest ARE. The model chosen to predict unsaturated shear strength should, in general, have a value ARE less than 20%, to be within the acceptable range.

Table 4 Performance summary of each model with different conditions
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5 Conclusions

This paper presents a comprehensive verification study based on statistical analysis to investigate the validity of theoretical models. Several well-known shear strength models were selected and compared with measured data from thirty published case studies. Following are some conclusions that can be made from this study:

For the purposes of comparison, thirty test data sets representing a variety of soil types, matric suction, saturation degree, density, and plasticity of soils have been compiled. The selection of the soil characteristics for the computation models received special consideration. The input soil parameters and measured shear strength results extracted from the published data sets provide a database as well as a benchmark for verifying these models in the future.

Based on comparisons between the measured and calculated results, it is concluded that the predicted models showed an inconsistent agreement with the experimental results. The performance of the shear strength models depends significantly on the soil types, density, plasticity, clay activity, range of matric suction, and saturation degree. This study found that the prediction capability of the shear strength models is influenced by the following factors in descending order: (1) soil density, (2) matric suction, (3) soil type, (4) saturation degree, (5) plasticity index, and (6) activity of clay.

The findings show that the theoretical model can estimate the shear strength of sandy and silty soils better accurately than clayey soil. Additionally, the performance of shear strength models decreases with an increase in suction stress (due to increasing matric suction), an increase in initial soil density and plasticity, and with a decrease in saturation degree. None of the selected shear strength models yields results that are in good agreement with the measured results for dense soils. This finding suggests that the estimation models should be carefully selected based on the soil properties in calculating the unsaturated shear strength.

Three models of Vanapalli et al. (1996), Fredlund et al. (1996), Öberg and Sällfors (1997) generally show a prediction performance higher than other models over a wide range of density, suction, and saturation degrees. It has also been found that the prediction models linking matric suction with the shear strength based on the water content are more efficient and reliable than the ones based on the air-entry value (models of Tekinsoy et al., Khalili and Khabbaz) or volumetric air content (model of Greacen). Besides, the analysis results pointed out that the choice to assume the factor \(\chi\) in the equation of Bishop equals the degree of saturation is only suitable for medium-dense soils with low matric suction. This assumption is particularly not effective for clayey soils, or dense soils with high matric suction.

Finally, some special observations can be made (i) under the same net normal stress, higher matric suctions result in higher shear strengths; (ii) the influence of the matric suction on the shear strength depends significantly on the saturation degree and soil density; and (iii) the relationship between the matric suction and the shear strength is highly non-linear.