A Simplified CPT-Based Method for Estimating the Friction Angle of Partially Saturated Compacted Soils

Abstract

Embankments and other compacted earth structures are widely used in civil engineering, and their mechanical behaviour depends on the properties of compacted soils, which are often affected by unsaturated conditions and matric suction. This study presents an empirical methodology aimed at the preliminary estimation of the peak friction angle of compacted unsaturated soils from cone penetration test (CPT) data, intended to support engineering assessments when laboratory testing and site investigation are limited. The approach was developed using CPT measurements collected at three levee test sites in northeastern Italy (Grizzaga, Panaro, and Tagliamento Rivers), combined with laboratory shear test results on undisturbed samples and supplemented by selected data from the literature. Three empirical equations were proposed and compared using different normalization schemes of cone tip resistance with respect to in situ stress conditions, while considering the influence of matric suction. For the soils examined in this study, the estimated peak friction angles were generally consistent with laboratory results, suggesting that the proposed relationships may provide a reasonable first approximation under similar soil conditions. The proposed procedure is intended as a preliminary engineering tool to complement, rather than replace, detailed laboratory testing in the geotechnical characterization of levees, embankments, and other compacted earth structures.

1. Introduction

Earth structures such as embankments, engineered fills and levees are widely used in civil infrastructure, and their failure can result in significant risks to human safety and substantial economic losses. Although these are man-made systems, their composition and material properties are often uncertain, particularly when they were constructed decades ago or subsequently modified without adequate documentation. This lack of information makes safety assessments particularly challenging. Consequently, in situ geotechnical investigations play a crucial role in defining the stratigraphy and mechanical properties of the soils involved, providing the basis for reliable stability and performance evaluations. Cone penetration test (CPT) is a widely used in situ testing technique for site characterization thanks to its cost-effectiveness and versatility. It provides continuous measurements of cone tip resistance (q_c), and sleeve friction (f_s) with depth; in the piezocone (CPTU) pore water pressure (u) is measured too. These measurements can be used to recognize the soil type and to estimate various soil properties such as friction angle, undrained shear strength, and stiffness [1,2].

In flood risk assessment studies, CPT is very useful to obtain statistical distribution of material parameters thanks to the number of measurements available along depth and the possibility to perform tests at a close distance [3,4,5]. However, the interpretation of CPT in river levees may be inaccurate because these are compacted mixtures of sand, silt and clay in partially saturated conditions.

Unsaturated soils are characterized by the presence of negative pore water pressures (suction), which can significantly affect their mechanical behavior [6], leading to higher shear strength and stiffness compared to saturated conditions [7,8]. As a result, matric suction can significantly increase the cone tip resistance in unsaturated soils, leading to material misinterpretation and overestimation of soil friction angle when using correlations developed for saturated soils [9,10].

Several researchers have investigated the effects of unsaturated conditions on CPT measurements through experimental and numerical studies. Several studies reported seasonal variation in tip resistance due to partial saturation effects in the field [11,12,13,14], which could lead to misinterpretation of the material type and mechanical properties. Giacheti et al. [14] proposed procedures to correct the soil behavior type index accounting the effect of suction, while Lo Presti et al. [15] exploited the misclassification in soil behavior type to determine the in situ stress state. Other researchers [16,17,18], used calibration chamber tests to propose correlations between suction, tip resistance and relative density or state parameters. Miller et al. [19] proposed a procedure, based on bearing capacity theory, for predicting changes in tip resistance due to changes in water content and matric suction, using laboratory and field data. Numerical analyses, based on cavity expansion, finite elements (FEM), or finite difference (FDM) methods, have also been employed to investigate the effect of soil parameters and drainage conditions on tip resistance in un-saturated soils [20,21,22]. These studies highlighted that the behaviour of unsaturated soil is complex and, in addition to the stress state, many other material parameters affect the determination of q_c.

One of the most relevant parameters for levee stability assessment is soil friction angle, highlighting the need for a simple and reliable CPT-based estimation method. Empirical correlations are extremely useful thanks to their simplicity of use. Several equations have been presented in the literature to estimate soil friction angle from CPT in saturated or dry sand [10,23,24,25]. One of the most popular is Equation (1) [24], where q_t is the corrected tip resistance, σ_v0 represents the in situ vertical effective stress and P_atm is the atmospheric pressure.

$${\mathsf{\phi }}^{\prime }=17.6+11.0 \mathit{log} \left({\displaystyle \frac{\frac{q_c}{P_{\mathit{atm}}}}{\sqrt{\frac{{\sigma }_v^{\prime }}{P_{\mathit{atm}}}}}}\right)$$

(1)

Another famous equation was proposed by Chen and Juang [25] by fitting data sets from [23,26] (Equation (2)):

$$\mathit{tan} \left({\mathsf{\phi }}^{\prime }\right) = {\displaystyle \frac{1}{6.82}} \mathit{ln}\left({\displaystyle \frac{q_c}{0.266 {\sigma }_v^{\prime }}}\right)$$

(2)

Tang et al. [10] modified Equation (1) by including suction (s), and the degree of saturation (S_r) elevated at a material constant (κ) that depends on the liquid limit, plasticity index, and fines content of the soil, to account for partial saturation:

$${\mathsf{\phi }}^{\prime }=17.6+\left[1-0.22\left(s·{\displaystyle \frac{S_r^{\kappa }}{{\sigma }_v}}\right)\right]11.0 \mathit{log}\left({\displaystyle \frac{\frac{q_c}{P_{\mathit{atm}}}}{\sqrt{\frac{{\sigma }_v^{\prime }}{P_{\mathit{atm}}}}}}\right)$$

(3)

The applicability of these equations to compacted partially saturated soils, such as those encountered in river levees, may be limited [9]. Existing approaches for interpreting CPT data under unsaturated conditions include computationally intensive mechanistic formulations, such as cavity expansion and elastoplastic finite-element models, as well as empirical correlations developed for specific soil classes, including residual sands and silts.

The present study aims to develop a new empirical correlation for the estimation of the effective friction angle of partially saturated soils based on Cone Penetration Test (CPT) data. The proposed approach combines continuous CPT tip resistance (q_c) with Bishop’s extended effective stress framework, using degree of saturation and matric suction information derived from the soil water retention curve (SWRC). It is conceived as a preliminary and expedient methodology for geotechnical characterization, aimed at obtaining first-order estimates of the friction angle in contexts where laboratory testing data are scarce or unavailable, or where obtaining and testing undisturbed samples is particularly challenging. Within this framework, CPT-based interpretation may represent a practical and relatively rapid means of obtaining preliminary information on soil mechanical behaviour under partially saturated conditions, while explicitly acknowledging the uncertainties associated with the limited availability of experimental data.

The analysis is based on a dataset derived from three Italian levee case studies, further enriched by additional data available in the literature on the behavior of partially saturated soils. By combining new measurements with published datasets, the proposed correlation seeks to improve the accuracy and reliability of friction angle estimation. Furthermore, the paper proposes a simplified procedure to derive the friction angle in cases where no information is available on soil suction or water retention behavior, which represents the majority of practical engineering applications. This makes the proposed approach particularly suitable for routine design conditions, where advanced unsaturated soil parameters are rarely measured.

Several CPTs were conducted along three Italian river levees, each paired with a corresponding borehole performed in close proximity. Undisturbed soil samples were collected during boreholes and subjected to laboratory investigations, including classification and shear tests, to characterize the soils and assess their shear strength in terms of the friction angle. The dataset was subsequently supplemented with the results of calibration chamber tests reported in the literature in partially saturated soils [16,17,18].

The dataset is employed to develop new empirical correlations based on various normalized tip resistances that account for partial saturation effects. The accuracy of the proposed correlation is evaluated and compared with selected existing empirical relationships, in order to outline its applicability, as well as its main limitations. Although the behaviour of partially saturated soils is inherently complex and not all influencing parameters can be explicitly accounted for in the estimation of the peak friction angle, the comparison with the available laboratory data for the considered soil types indicates a generally satisfactory agreement, suggesting the potential usefulness of the proposed correlation within the investigated range of conditions.

2. Materials and Methods

The following section presents the process of material collection and data analysis. Section 2.1 details the field investigations conducted on three river levees in northern Italy, where undisturbed soil samples were retrieved from boreholes and CPT were carried out nearby. Section 2.2 then describes the laboratory procedures used to classify the soil samples and determine their friction angle, which is taken as the reference value. Section 2.3 is devoted to the description of the CPT data elaboration to develop the new correlation evaluating the friction angle in partially saturated soils, by taking into account the saturation degree and the consequent suction effects.

To enlarge the analyzed dataset, in addition to the values acquired from the three Italian test sites, further datasets including CPT data and corresponding friction angle estimations from laboratory tests were incorporated from the literature and processed in the same way. The additional data were drawn from previous studies in calibration chamber, including investigations on tailings [16] and unsaturated quartz sands under drained conditions [11,18], and CPTs performed at a test site located in Greensboro, Guilford County, North Carolina [10]. This extended dataset enabled a more robust analysis aimed at identifying potential correlations between the peak friction angle (φ′) and the cone tip resistance (q_c) obtained from CPTs.

2.1. Test Sites and CPT Investigation Database

Field investigations were conducted at three levee segments in the Emilia-Romagna and Veneto regions, in northern Italy. As shown in Figure 1, the three tested segments were located along the Grizzaga River, the Panaro River (both in the province of Modena) and the Tagliamento River (in the province of Venice). The geographic locations of the three test sites are shown in Figure 1: site (a) corresponds to the Grizzaga River, site (b) to the Panaro River, and site (c) to the Tagliamento River.

Figure 1. Map showing the rivers involved in the analysis and the location of the three test sites. Site (a): Grizzaga River; site (b) Panaro River;site (c) Tagliamento River (Copyright: Data SIO, NOAA, U.S. Navy, NGA, GEBCO Image Landsat/Copernicus).

At the first site (Grizzaga River), CPTs were performed during two repeated campaigns: the first during a dry period in 2022, and the second during the wet season in 2023, soon after a major flood event. Between 2022 and 2023 the levee crest was raised. The test considered in this analysis is GCPT2, that reached a depth of 9.50 m from the levee crest. At the same location, the borehole named GS1 was drilled, and four soil samples were collected from different layers (the samples are called GS1-C1, GS1-C2, GS1-C3, and GS1-C4). In the wet-season campaign, CPTs were repeated at the same location. Data from both campaigns were included in the present study.

At the second site, situated along the Panaro River, two CPTs (PCPT4 and PCPT5) and two boreholes (PS1 and PS2) were carried out in 2021. PCPT4 was located adjacent to borehole PS2, while PCPT5 was next to borehole PS1. Four soil samples were extracted from PS1 (PS1-C1 to PS1-C4), and two from PS2 (PS2-C3 and PS2-C4). All CPTs reached an approximate depth of 25.00 m from the levee crest.

At the third site, located along the Tagliamento River, the investigation campaign was carried out in 2019. The site characterization included n.8 CPTs, identified as TCPT2, TCPT3, TCPT4, TCPT5, TCPT7, TCPT11, TCPT12 and TCPT18, along with n.8 boreholes: TS2, TS3, TS5, TS6, TS9, TS7, TS10, and TPS7. Undisturbed soil samples were carried out from each borehole. Specifically, two samples were obtained from boreholes TS3, TS6, TS7, and TS10, labelled as TS3-C1 and TS3-C2, TS6-C1 and TS6-C2, TS7-C1 and TS7-C2, and TS10-C1 and TS10-C2, respectively. One undisturbed sample was collected from each of the remaining boreholes: TS2-C1, TS5-C1, TS9-C1 and TPS7-C1. Here the CPTs were performed down to a depth of approximately 20.00 m below the levee crest. Dry drilling is performed above water table, and soil samples were obtained using a piston sampler (Osterberg). Figure 2 shows the location of CPTs and boreholes along the three levee sections previously described. Table 1 lists all the samples with the most important characteristics. A total of 22 undisturbed samples were tested: 4 from Grizzaga, 6 from Panaro and 12 from Tagliamento. For Grizzaga, CPT data from the 2022 and 2023 campaigns are available for each sample, yielding two observations with different CPT-derived parameters but the same laboratory friction angle (φ′_LAB). The dataset therefore comprises 26 observations from 22 distinct samples.

Figure 2. Spatial distribution of CPTs and boreholes in the (a) Grizzaga site test (Image © 2025 Airbus), in the (b) Panaro site test (Image © 2025 Airbus) and in the (c) Tagliamento site test (Image © 2025 Airbus).

Table 1. Shear stress tests characteristics performed on soil samples collected from the three test sites.

River	Sample	Depth [m]	Shear Test	σ′3 [kPa]	σ′n [kPa]
Grizzaga	GS1-C1	1.60–2.30	RSS	-	50.00
	GS1-C2	3.10–3.80	RSS	-	50.00
	GS1-C3	4.60–5.30	RSS	-	100.00
	GS1-C4	6.10–6.80	RSS	-	100.00
Panaro	PS1-C1	1.10–1.80	Tx CIU	100.00
	PS1-C2	2.10–3.00	DS	-	100.00
	PS1-C3	3.80–4.50	Tx CID	100.00	-
	PS1-C4	6.20–7.10	DS	-	200.00
	PS2-C3	4.00–4.80	Tx CID	100.00	-
	PS2-C4	6.50–7.20	Tx CID	100.00	-
Tagliamento	TS2-C1	4.90–5.60	Tx CIU	100.00	-
	TS3-C1	2.80–3.60	Tx CIU	50.00	-
	TS3-C2	3.60–4.30	Tx CIU	100.00	-
	TS5-C1	5.70–6.60	Tx CIU	120.00	-
	TS6-C1	3.40–4.20	Tx CIU	70.00	-
	TS6-C2	5.90–6.70	Tx CIU	120.00	-
	TS7-C1	5.20–6.10	Tx CIU	110.00	-
	TS7-C2	6.70–7.60	Tx CIU	200.00	-
	TS9-C1	4.10–5.20	Tx CIU	100.00	-
	TS10-C1	2.70–3.70	Tx CIU	60.00	-
	TS10-C2	5.20–6.20	Tx CIU	110.00	-
	TPS7-C1	8.20–9.10	Tx CIU	170.00	-

Although the dataset may not seem very extensive, it is very rare to find groups of data including laboratory test results conducted on undisturbed soil samples, associated with field results obtained via CPT on nearby vertical sections in partially saturated soil conditions. For this reason, even when compared with datasets supporting previous studies, we believe this comparison is interesting and could prove fruitful.

2.2. Laboratory Testing on Samples

The collected undisturbed samples were classified according to the Unified Soil Classification System (USCS) standards. Grain size distributions were performed using mechanical sieving and hydrometer tests. Additionally, shear tests were carried out to determine the friction angle (φ′_LAB) of the collected samples, to be used as reference for the comparison with the resistance parameters derived from CPT data (see Table 1):

• on the samples collected at Grizzaga River levee, Ring Shear Tests (RSS) [27] were performed;
• on the undisturbed samples collected along the Panaro river levee, triaxial test and direct shear tests were conducted: isotropically Consolidated Undrained Triaxial test (Tx CIU) [28] on PS1-C1, Direct Shear test (DS) [29] on samples PS1-C2 and PS1-C4, and isotropically consolidated Drained Triaxial test Tx CID [29] on samples PS1-C3, PS2-C3, and PS2-C4;
• on the undisturbed samples collected on the Tagliamento River site, isotropically consolidated Tx CIU tests were performed.

Tests were conducted at three levels of confining stress. For the triaxial and direct shear tests, the peak friction angle was determined from the tangent to the stress–strain curve corresponding to the stress level closest to the in situ conditions of the undisturbed samples, in recognition of the known dependence of peak friction angle on confining stress, while zero cohesion was adopted in all cases. For the ring shear tests, the friction angle was determined from the corresponding shear-strength response. Table 1 lists all the samples, the sampling depths, the performed type of test, the applied confining stress in the triaxial cell (σ₃) or the vertical stress (σ_n) applied in the direct shear box test.

Literature studies [30,31,32] have shown that friction angles obtained from different shear-strength testing methods may exhibit discrepancies because the tests impose different stress paths, drainage conditions, failure-plane constraints, and strain levels. The magnitude, and in some cases the sign, of these discrepancies may depend on soil type, including grain-size distribution and clay fraction, as well as density and stress level. However, the available literature does not support the application of a unique correction factor to harmonize DS and triaxial results. Considering the relatively narrow range of differences reported in the literature for silty soils and clean sands and the absence of a consistent correction trend across testing methods, no correction or harmonisation factor was applied to the laboratory measurements. Although these differences are relatively small compared with the overall variability of the dataset, applying a single correction factor would require assumptions that are not sufficiently constrained for the investigated materials and could introduce additional uncertainty rather than reduce it. The experimentally determined values of the laboratory friction angle were therefore retained as obtained from the corresponding laboratory tests. Differences among the laboratory shear-testing methods (RSS, Tx-CIU, Tx-CID and DS) may introduce additional scatter in the reference peak friction angles, φ′_LAB. This variability may contribute to the RMSE and MAE values reported in this study, although its specific contribution cannot be isolated or quantified using the available data.

2.3. CPT Dataset Elaborations

This section explains how CPT results are processed to develop new empirical correlations for estimating the peak friction angle (φ′_CPT) in partially saturated soils.

CPT measurements were recorded at 0.02 m depth intervals, then the continuous q_c profiles were processed by averaging the measured values over 0.25 m using the arithmetic mean. This preliminary averaging reduces the influence of small-scale local fluctuations in the measured q_c values. For each sample, the representative q_c value was calculated as the arithmetic mean of all the processed q_c values falling within the corresponding sample depth interval. The resulting representative q_c values were subsequently used as the CPT reference parameter in the correlation with the laboratory-derived friction angles. Raw data and elaborated data can be found in the Supplementary Material.

In the analyses, to determine the stress state along the depth of each CPT sounding, the unit weight of the soils collected from the embankments of the three rivers was assumed to be 19 kN/m³, corresponding to the average value obtained from both CPT measurements and laboratory tests on undisturbed samples.

It is assumed that the peak friction angle depends on a normalized tip resistance which incorporates the contribution of partial saturation. Several normalization approaches are evaluated, specifically three of the most widely used in the literature:

$$q_{c,\mathit{norm}1}=\left[{\displaystyle \frac{\left(q_c-p_0\right)}{p_0^{\prime }}}\right]+1$$

(4)

$$q_{c,\mathit{norm}2}={\displaystyle \frac{\left(\frac{q_c}{P_a}\right)}{{\left(\frac{{\sigma }_v^{\prime }}{P_a}\right)}^{0.5}}}$$

(5)

$$q_{c,\mathit{norm}3}={\displaystyle \frac{q_c}{{\sigma }_v^{\prime }}}$$

(6)

where q_c is the cone tip resistance, p₀ the mean total stress, p₀′ the mean effective stress, σ_v′ the vertical effective stress at the reference depth and Pa the atmospheric pressure (equal to 101.30 kPa). The first approach is applied by Russell et al. (2024) [16] to estimate the state parameter (ψ) from the cone tip resistance in saturated and unsaturated silty tailings, for estimating the peak friction angle under drained loading as well as their liquefaction potential during undrained loading. The second approach is applied by Kulhawy and Mayne [24] that observed how cone resistance increases with the square root of σ′_v, implying a less steep increase in resistance with depth compared to models assuming a linear dependency for normally consolidated and over consolidated sands. The third approach is proposed by Chen and Juang [25], based on cavity-expansion theory, for determining the drained friction angle of highly compressible saturated sands from CPT data.

In partially saturated soils, the mechanical behaviour cannot be described solely in terms of pore-water pressure, as in fully saturated soils, because matric suction contributes to the stress state of the soil skeleton. To account for this contribution, Bishop’s effective stress concept is adopted, which extends Terzaghi’s formulation by introducing a weighting parameter, χ, for the contribution of matric suction (Equation (7)) [33]:

σ′ = σ + χsI

(7)

where σ′ is the extended effective stress, σ is the total stress, s is suction, I is the identity matrix and χ is the Bishop’s effective stress parameter, ranging from 0 to 1 and representing the contribution of matric suction to the effective stress. In this study, χ is explicitly assumed to be equal to the degree of saturation, S_r, i.e., χ = S_r. The assumption χ = S_r has been adopted in several constitutive and shear-strength formulations for unsaturated soils [33,34,35,36,37,38,39,40,41,42]. Constant water content triaxial tests on sands and non-clayey mixtures have demonstrated that Bishop’s relationship adequately captures the effect of suction on the ultimate shear strength, supporting the use of this formulation in partially saturated granular soils as well [43]. Investigations on partially saturated pyroclastic deposits have experimentally confirmed that the coefficient χ derived from laboratory data agrees with theoretical predictions and that Bishop’s formulation can be considered valid for non-clayey soils in relation to the ultimate shear strength [44].

From a physical standpoint, relating χ to S_r provides a simple way of weighting the contribution of matric suction according to the water content of the pore space, without introducing an additional empirical fitting parameter. For the present dataset, which consists predominantly of soils classified as ML, CL and SM, χ = S_r is considered a reasonable simplifying approximation. However, χ is not generally equal to S_r, and their relationship may depend on soil type and hydraulic conditions. The applicability of this assumption should therefore be reassessed for high-plasticity clays and for conditions outside the range investigated in this study.

Accordingly, in the subsequent analyses, the value of χ assigned to each soil sample was taken as the corresponding degree of saturation. For each sample, the degree of saturation was determined from the SWRC characterizing that sample, at the corresponding matric suction value. The resulting sample-specific value of χ was then used to calculate the suction-corrected effective stress for the corresponding CPT data.

Determining the variation in the degree of saturation (S_r) and matric suction (s) with depth in earth structures is a challenging task due to the complex soil–water–atmosphere–vegetation interaction processes governing near-surface conditions. This complexity is particularly evident in river embankments and levees, where hydraulic boundary conditions, atmospheric actions, and vegetation effects interact simultaneously, leading to highly variable suction and saturation profiles along the vertical direction. In particular, when a sufficient amount of experimental data is not available to define the suction profile with depth and to determine the SWRC for each partially saturated soil encountered along the stratigraphic profile, simplifying assumptions become unavoidable.

Under such conditions, a linear variation in matric suction with depth is commonly adopted as a practical approximation. However, in the near-surface zone, often referred to as the active or evapotranspiration layer, typically extending from a few tens of centimeters to about one meter, the suction profile may deviate from this simplified trend. In this zone, the combined effects of rainfall infiltration, evaporation, and root water uptake induce seasonal and short-term fluctuations that locally alter the suction distribution. Consequently, while the linear assumption provides a reasonable representation of the overall suction profile, it should be regarded as a simplified approximation in the uppermost soil layer.

Several studies in the literature have investigated the influence of partial saturation on levee stability and have commonly adopted an approximately linear suction profile with depth, except within the near-surface active zone. This assumption has been supported by in situ measurements obtained from tensiometers installed in earthen levees along the Secchia River, which showed a nearly linear decrease in suction with depth below the evapotranspiration layer [45,46,47]. Additionally, long-term volumetric water content (VWC) measurements on the Grizzaga River levee revealed that rainfall infiltration alters the humidity of the first 0.90 m, while water table governs deeper soil layers [48]. Similarly, a recent study on the long-term hydrostatic stability of a river embankment [49] employed a linear distribution of suction with depth to simulate pore-water pressure conditions in partially saturated soils.

Based on the experimental and numerical evidence from the aforementioned studies, the simplifying assumption of a linear suction profile with depth was adopted in the present case study, as expressed in Equation (8). This approach represents a reasonable compromise between physical realism and the limited availability of site-specific hydraulic and water retention data. With reference to the sampling depths listed in Table 1, the shallowest samples (GS1-C1, 1.60–2.30 m and PS1-C1, 1.10–1.80 m) are the most likely to be affected by deviations from the assumed linear suction profile associated with near-surface evapotranspiration. All other samples are located deeper than 2 m. For these shallow samples, if the actual suction is lower than the linear estimate (e.g., following rainfall), the contribution of the χ∙s term to Bishop’s effective stress would be overestimated, potentially leading to an underestimation of the friction angle inferred from the CPT; the opposite tendency may occur during dry periods. The magnitude of this effect depends on the actual suction conditions and cannot be established directly from the available data. Since the same suction-profile assumption was adopted in the calibration, its influence may also be partly reflected in the fitted regression coefficients.

s = (z_w − z)γ_w

(8)

where z_w − z is the distance between the considered point (at depth z) and the water table (at depth z_w), and γ_w = 9.81 kN/m³ is the water unit weight. The groundwater table depth is measured from boreholes adjacent to the CPT locations. The degree of saturation is calculated as a function of suction, following the SWRC.

The SWRC is a material property that provides the relationship between soil water content and matric suction (or water potential). For the soil samples collected from the Grizzaga and Panaro river sites, experimental SWRCs on undisturbed samples were available for each soil type. Experimental determination of SWRCs was carried out through two types of laboratory tests. The first was performed using the Ku-pF apparatus (Umwelt-Geräte-Technik GmbH, Müncheberg, Germany), which allowed measurement of the SWRC from saturation (0 kPa) down to the lower limit of tensiometer readings (approximately –70 kPa) [50]. For higher suction values, measurements were performed using the WP4C dewpoint potentiometer (METER Group, Inc., Pullman, WA, USA). WP4C provides the total water potential as the sum of the osmotic and matric potentials. The osmotic potential was estimated from the electrical conductivity (ECe) of the saturated paste extract, and the matric potential was obtained by subtracting the osmotic component from the total water potential [51].

In contrast, along the Tagliamento River, experimental soil water retention curves were not available. Therefore, the SWRCs are estimated using two of the most widely adopted pedotransfer functions reported in the literature: the formulation proposed by Aubertin et al. [52] for silty sands and the formulation proposed by Fredlund and Xing [53] for non-plastic soils. These formulations derive hydraulic retention properties from fundamental geotechnical parameters, namely grain-size distribution, porosity, and plasticity. Further details are provided in Appendix A.

The approach was validated by comparing the estimated SWRCs with experimental curves obtained from soil samples for which laboratory retention data are available, specifically those collected from the Grizzaga River and the Panaro River. The comparison between the modeled and experimental saturation curves was performed by calculating the root mean squared error (RMSE). Within the range of suction values relevant to this study, the agreement between modeled and experimental curves is satisfactory, as illustrated in Figure A1.

After validation, the method was applied to estimate the degree of saturation for each sample collected from the Tagliamento River and to derive empirical relationships linking the peak friction angle to the cone tip resistance. Further details on the construction and comparison of the SWRCs are provided in Appendix A.

For the first normalization approach (Equation (4)), the mean total stress is calculated starting from the mean effective stress as Equation (9):

$${{\mathit{p}}^{\prime }}_0={\displaystyle \frac{{{\mathsf{\sigma }}^{\prime }}_v+\left(2·{{\mathsf{\sigma }}^{\prime }}_h\right)}{3}}={\displaystyle \frac{{{\mathsf{\sigma }}^{\prime }}_v+\left(2·K_0{{\mathsf{\sigma }}^{\prime }}_v\right)}{3}}$$

(9)

the coefficient of earth pressure at rest (K₀) is assumed to be equal to 0.8, as suggested by DMT measurements in Panaro and Tagliamento river levees.

To establish how accurately the different empirical relationships estimate the peak friction angle of the soils, a quantitative performance metrics were evaluated. The Root Mean Squared Error (RMSE) was chosen for this purpose, as it provides a measure of the deviation between the predicted and laboratory-measured values. The RMSE was calculated according to Equation (10):

$$\mathit{RMSE}= \sqrt{{\displaystyle \frac{1}{N}}·\sum _{i=1}^n{\left({{\mathsf{\phi }}^{\prime }}_{\mathit{LAB},i}-{{\mathsf{\phi }}^{\prime }}_{\mathit{CPT},i}\right)}^2}$$

(10)

where φ′_LAB,i is the peak friction angle of the i-th sample measured in the laboratory, φ′_CPT,i is the estimated corresponding value, and N is the number of soil samples included in the analysis.

The Mean Average Error (MAE) was additionally calculated to quantify the average magnitude of the prediction errors, irrespective of their direction. It is defined as the mean of the absolute differences between the laboratory-measured and estimated friction angles as shown in Equation (11):

$$\mathit{MAE}= {\displaystyle \frac{1}{N}}·\sum _{i=1}^n\left|{{\mathsf{\phi }}^{\prime }}_{\mathit{LAB},i}-{{\phi }^{\prime }}_{\mathit{CPT},i}\right|$$

(11)

The MAE provides a direct measure of the average absolute discrepancy between the estimated and measured values and is less sensitive to large individual errors than the RMSE. Since the peak friction angle is expressed in degrees, both RMSE and MAE are reported in degrees.

3. Results

By using the analytical methodologies previously described, new correlations have been developed among CPT data (normalized cone resistance) and friction angle, extended to the entire dataset composed of measurements relative to river levees collected in the framework of this study and literature data.

The remaining partial results are described as follows:

• Dataset of the laboratory test results including soil classification and determination of the friction angle through shear tests;
• In situ test data: derived from CPTs;
• SWRCs for the Tagliamento river soils;
• Determination of empirical correlations through different approaches of q_c normalized quantities.

3.1. Laboratory Tests Results

Table 2 reports the results obtained by the laboratory tests conducted on soil samples in terms of USCS classification, fine fraction and friction angle. The subsurface soils of the investigated levee segments are predominantly classified as ML (low-plasticity silt) and CL (low-plasticity clay) according to the USCS standard. Only the PS1-C2 and PS1-C4 of the Panaro River and TS10-C2 of the Tagliamento River were classified as silty sand (SM), indicating more granular composition material in these specific locations. The last column reports the friction angle derived from the laboratory tests described in Section 2.2. No correction is applied, in this table, to homogenize results of different shear tests.

Table 2. Geotechnical classification and characterization of soil samples from the Grizzaga, Panaro, and Tagliamento river sites.

Sample	Depth [m]	Soil Type	Fine [%]	φ′LAB [°]
GS1-C1	1.60–2.30	ML	65.7	34.3
GS1-C2	3.10–3.80	CL	85.4	31.3
GS1-C3	4.60–5.30	CL	90.2	31.6
GS1-C4	6.10–6.80	CL	89.1	29.8
PS1-C1	1.10–1.80	ML	72.3	35.0
PS1-C2	2.10–3.00	SM	35.8	33.5
PS1-C3	3.80–4.50	ML	65.4	31.0
PS1-C4	6.20–7.10	SM	48.6	31.0
PS2-C3	4.00–4.80	ML	77.2	32.0
PS2-C4	6.50–7.20	CL	79.9	32.0
TS2-C1	4.90–5.60	CL	99.0	32.5
TS3-C1	2.80–3.60	ML	89.9	35.0
TS3-C2	3.60–4.30	ML	95.1	29.7
TS5-C1	5.70–6.60	ML	95.5	34.4
TS6-C1	3.40–4.20	ML	68.0	36.5
TS6-C2	5.90–6.70	CL	95.0	34.5
TS7-C1	5.20–6.10	ML	95.0	34.0
TS7-C2	6.70–7.60	ML	99.0	29.0
TS9-C1	4.10–5.20	ML	59.6	33.2
TS10-C1	2.70–3.70	ML	65.0	36.5
TS10-C2	5.20–6.20	SM	45.0	37.5
TPS7-C1	8.20–9.10	CL	100.0	30.0

In addition to the USCS classification, Figure 3 presents a summary of the grain-size distribution curves for each soil sample collected from the three investigated Italian sites. These curves provide useful information for the identification and classification of the different soil types analyzed in this study. The grain-size distribution curve of each individual sample is also reported separately in the data repository available at the link provided at the end of the article.

Figure 3. Grain-size distribution curves of soil samples collected from the Grizzaga River, Panaro River, and Tagliamento River.

3.2. In Situ Tests Results

Table 3 summarizes the reference tip resistance, total stress, suction and degree of saturation corresponding to each sample. In particular, the table reports the cone tip resistance (q_c) and the total vertical stress (σ_v). In the last two columns, the table reports the degree of saturation (S_r) and suction (s), derived from the SWRCs, either experimentally determined or predicted for each soil sample, as previously described in Section 2.3. Additionally, Table 3 reports groundwater level measurements obtained during the investigation campaigns. For the Grizzaga River, groundwater levels measured during both the dry and wet periods were considered in the analysis to account for the observed seasonal variability and to estimate the peak friction angle under the corresponding suction conditions for each period. The groundwater level measured during the wet season was approximately 1.80 m higher than that measured during the dry period. Note that the Grizzaga River levee was raised by approximately 60 cm between 2022 and 2023, and this generates and increment of total vertical stress that is considered for the interpretation of CPT. The sample depth is referred to the 2023 campaign; therefore, to obtain the stress state corresponding to the 2022 campaign, 0.60 m should be subtracted from the depth used in the calculation.

Table 3. Results obtained from CPTs for each site test.

CPT	Water Table Depth [m]	Reference Sample	Reference Depth [m]	qc [MPa]	σv [kPa]	Sr [-]	s [kPa]
GCPT2-2022	6.50	GS1-C1	1.60–2.30	4.27 2.05	25.65 37.05	0.345 0.562	46.60 26.98
GCPT2-2023	4.30
GCPT2-2022	6.50	GS1-C2	3.10–3.80	2.52 1.84	52.25 63.65	0.902 0.959	34.34 12.26
GCPT2-2023	4.30
GCPT2-2022	6.50	GS1-C3	4.60–5.30	1.91 1.23	82.65 94.05	0.905 1.000	17.17 0.00
GCPT2-2023	4.30
GCPT2-2022	6.50	GS1-C4	6.10–6.80	1.06 0.69	111.15 122.25	0.988 1.000	2.45 0.00
GCPT2-2023	4.30
PCPT5	10.00	PS1-C1	1.10–1.80	6.11	27.55	0.701	82.40
		PS1-C2	2.10–3.00	2.17	48.45	0.732	71.81
		PS1-C3	3.80–4.50	2.89	78.85	0.767	56.51
		PS1-C4	6.20–7.10	2.57	126.35	0.461	31.78
PCPT4	9.00	PS2-C3	4.00–4.80	1.63	83.60	0.900	43.56
		PS2-C4	6.50–7.20	1.57	130.15	0.850	22.37
TCPT3	7.10	TS2-C1	4.90–5.60	2.04	93.10	1.000	18.39
TCPT2	7.80	TS3-C1	2.80–3.60	1.67	60.80	0.355	44.15
		TS3-C2	3.60–4.30	1.73	75.05	0.774	36.79
TCPT4	12.40	TS5-C1	5.70–6.60	2.64	116.85	0.800	60.09
TCPT11	10.00	TS6-C1	3.40–4.20	2.51	72.20	0.235	60.09
		TS6-C2	5.90–6.70	1.77	115.90	0.750	33.56
TCPT5	10.40	TS7-C1	5.20–6.10	2.49	107.35	0.824	45.37
		TS7-C2	6.70–7.60	2.53	135.85	0.509	30.66
TCPT7	10.15	TS9-C1	4.10–5.20	8.77	88.35	0.628	52.73
TCPT18	6.80	TS10-C1	2.70–3.70	1.55	60.80	0.143	35.56
		TS10-C2	5.20–6.20	3.01	108.30	0.462	11.04
TCPT12	10.20	TPS7-C1	8.20–9.10	2.06	164.35	1.000	13.49

3.3. Determinations of Empirical Correlations Between Friction Angle and Tip Resistance

Building on the quantities introduced earlier and the considerations discussed in Section 2.3, a series of plots was developed to explore empirical relationships for the estimation of the peak friction angle of partially saturated soils, with particular reference to levee embankment materials. The resulting correlations are intended as a practical and expedient tool for the preliminary estimation of the peak friction angle from CPT data, particularly in situations where direct laboratory measurements are limited or unavailable. Owing to their empirical nature, these relationships should be regarded as first-order approximations applicable within the range of soil and hydro-mechanical conditions investigated in this study. Figure 4 illustrates the relationship obtained using the first normalization approach for the tip resistance (Equation (12)), plotted along the x-axis on a logarithmic scale. The results of the shear tests display a tendency to align along a linear trend, the corresponding regression equation of which is reported in the figure. The coefficient of determination (R²) is 0.64, indicating a moderate correlation between the normalized tip resistance and the measured peak friction angle.

Figure 4. Equation obtained through the [(q_c − p₀)/p₀′] + 1 normalization approach. The solid lines represent the 95% confidence intervals for the mean value of φ’_LAB at a given level of normalised cone resistance [10,11,16,18].

The peak friction angles of soils obtained from the three levee test sites are predominantly situated in the lower portion of the diagram. This distribution is primarily attributed to their lithological composition, which consists mainly of silts and clays. These fine-grained materials exhibit comparatively lower shear strength than coarser-grained soils, thereby explaining their position in the plots.

Figure 5 illustrates the second normalization approach proposed in this study, in which the cone tip resistance is divided by the square root of the vertical effective stress (Equation (13)). As shown in the graph, the experimental data points for the peak friction angle exhibit a closer alignment with the regression line compared to the previous case. The coefficient of determination (R²) is 0.60, which is slightly lower than that obtained for the first normalization approach (R² = 0.64), indicating that the second normalization does not result in an improvement in the overall explained variance. The corresponding 95% confidence band is narrower, indicating a reduction in the uncertainty associated with the estimated mean regression relationship and, consequently, an improvement in the precision of the fitted model. It should be noted that the uncertainty bounds shown in the figure refer to the confidence interval of the mean predicted response rather than to the variability of individual observations. Accordingly, the 95% interval should be interpreted as a confidence interval for the mean value of φ′_LAB at a given level of normalized cone resistance, rather than as a prediction interval for single measurements.

Figure 5. Equation obtained through the (q_c/P_a)/(σ_v′/P_a)^0.5 normalization approach. The solid lines represent the 95% confidence intervals for the mean value of φ’_LAB at a given level of normalised cone resistance [10,11,16,18].

Finally, Figure 6 presents the logarithmic relationship derived using the normalization of cone resistance with vertical effective stress (Equation (14)). As in the previous case, the copper tailings were excluded from the analysis.

Figure 6. Equation obtained through the q_c/σ_v′ normalization approach. The solid lines represent the confidence intervals for the mean value of φ′_LAB at a given level of normalised cone resistance (95% confidence level) [10,11,16,18].

The 95% confidence band around the regression line is narrower than in the previous normalization. This result indicates a lower dispersion of the estimates in the relationship between peak friction angle and the natural logarithm of the q_c/σ′_v ratio. The coefficient of determination (R²) is 0.69, representing the highest value among the three normalization approaches considered in this study and indicating the strongest overall correspondence between the measured and estimated peak friction angles.

The formulations derived above can be expressed as shown in Equations (12)–(14):

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.6·\mathit{ln}\left({\displaystyle \frac{q_c-p_0}{{{\mathit{p}}^{\prime }}_0}}+1\right)+22.3$$

(12)

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.7·\mathit{ln}\left[{\displaystyle \frac{\frac{q_c}{P_a}}{{\left(\frac{{{\mathsf{\sigma }}^{\prime }}_v}{P_a}\right)}^{0.5}}}\right]+22.5$$

(13)

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.6·\mathit{ln}\left({\displaystyle \frac{q_c}{{{\mathsf{\sigma }}^{\prime }}_v}}\right)+22.6$$

(14)

By substituting Equation (7) into Equations (12)–(14), the expressions can be reformulated to explicitly highlight the contribution of suction, as follows in Equations (15)–(17), where χ is equal to the degree of saturation of the soil (S_r):

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.6·\mathit{ln}\left[{\displaystyle \frac{q_c-p_0}{{{\mathit{p}}^{\prime }}_0+\left(\chi ·s\right)}}+1\right]+22.3$$

(15)

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.7·\mathit{ln}\left[{\displaystyle \frac{\frac{q_c}{P_a}}{{\left(\frac{\left({\sigma }_v+\left(\chi ·s\right)\right)}{P_a}\right)}^{0.5}}}\right]+22.5$$

(16)

$${{\mathsf{\phi }}^{\prime }}_{\mathit{CPT}}=3.6·\mathit{ln}\left[{\displaystyle \frac{q_c}{{\sigma }_v+\left(\chi ·s\right)}}\right]+22.6$$

(17)

4. Discussion

The following section presents and discusses the results obtained from the application of the proposed empirical relationships, with a comparison to those available in the literature, in light of the assumptions outlined in the previous sections and the related considerations. Furthermore, practical implications of the study and limitations of the methodology as applied to the present case study are discussed.

4.1. Performance of Proposed Correlations

New empirical correlations are developed to relate cone tip resistance to the friction angle under partially saturated conditions, leading to Equations (15)–(17). The accuracy of these relationships is assessed through the RMSE metric. In addition, the effect of neglecting suction is analyzed by computing the friction angles assuming the vertical effective stress equal to the total vertical stress (σ′_v = σ_v), as typically done when interpreting CPT in partially saturated soils. Figure 7 compares the results obtained considering (on the left) or neglecting (on the right column) the suction effects.

Figure 7. Comparison between predicted and laboratory-measured peak friction angle using (a,b) Equation (15), (c,d) Equation (16), (e,f) Equation (17). Left panels (a,c,e) account for partial saturation, whereas right panels (b,d,f) neglect it. The dashed line denotes the 1:1 relationship, the solid lines indicate ±2° deviation, and RMSE values are reported in each panel [10,11,16,18].

The figures show that including suction in the calculation of vertical effective stress leads to a better estimation of the friction angle. In all three normalization approaches, lower RMSE values are observed when partial saturation is taken into account. The method remains preliminary, but it provides more consistent results when the influence of suction on soil behavior is considered. To provide a clearer assessment of the effect of partial saturation on the predictive performance of the three models, the RMSE values obtained by accounting for suction were compared with those obtained when suction was neglected. The effect is not uniform across the three equations and depends on the specific formulation considered. For Equation (15), the RMSE decreases from 2.32° when suction is neglected to 1.96° when it is accounted for, corresponding to a reduction of −15.5%. For Equation (16), a smaller reduction is observed, from 2.18° to 2.06° (−5.5%), whereas Equation (17) shows the largest reduction, from 2.18° to 1.81° (−16.9%). A similar trend is observed for the MAE, which decreases from 1.79° to 1.56° (−12.8%) for Equation (15), from 1.76° to 1.63° (−7.4%) for Equation (16), and from 1.74° to 1.42° (−18.4%) for Equation (17). Overall, the inclusion of suction is associated with lower RMSE and MAE values for all three formulations, although the magnitude of the reduction varies among the equations. The largest reduction is observed for Equation (17), which also provides the lowest RMSE (1.81°) and MAE (1.42°) among the three formulations when suction is considered.

The performance of previous formulations available in the literature for the estimation of friction angle is evaluated. Figure 8 presents the results obtained using the empirical correlations proposed by Equation (1) [24], Equation (2) [25] and Equation (3) [10]. For Equation (1) and Equation (2), the peak friction angle is determined both by accounting for the effect of suction within the effective stress state (Bishop’s hypothesis) and by neglecting this contribution. In Equation (3), the exponent (κ) depends on the liquid limit, plasticity index, and fines content of the soil; however, these data were not available in the literature for the soils under consideration, thus it is assumed equal to 1.0. With this equation, when suction is not included, the expression reduces to the formulation of Kulhawy and Mayne [24].

Figure 8. Comparison between laboratory-measured peak friction angles (φ′_LAB) and those estimated from CPT data (φ′_CPT) using the empirical formulation proposed by (a,b) Kulhawy and Mayne (1990) [24], by (c,d) Chen and Juang (1996) [25] by (e,f) Tang et al. (2017) [10] with κ set to 1.00. In the left column, data obtained accounting partial saturation and, in the right column, neglecting partial saturation. Dashed black line is the perfect match and the solid line indicate a shift of ±2° [10,11,16,18].

When partial saturation is considered, the RMSE values obtained are 2.37° for Equation (1) [24], 2.38° for Equation (2) [25], and 3.07° for Equation (3) [10], while the corresponding MAE values are 1.94°, 1.99°, and 2.48°, respectively. In all cases, both RMSE and MAE are higher than those obtained with the equations proposed in this study (Equations (15)–(17)), indicating lower prediction errors for the proposed formulations for the dataset considered. When partial saturation is neglected, the RMSE increases to 2.40° for Equation (1) [24], 3.03° for Equation (2) [25], but decreases to 2.40° for Equation (3) [10], as Equation (3) is equal to Equation (1) under saturated conditions. The corresponding MAE values are 1.94°, 2.48°, and 1.94°, respectively. These results indicate a slight increase in RMSE for Equation (1), whereas a more pronounced increase is observed for Equation (2). Although the magnitude of these variations differs between the two formulations, the results suggest that accounting for suction may contribute to reducing the prediction error, particularly for Equation (2).

Figure 9 displays the vertical profiles of cone tip resistance and estimated friction angle along the GCPT2-2022 from the Grizzaga River, PCPT5 from the Panaro River, and TCPT2 from the Tagliamento River, also reporting the stratigraphic sequence. Friction angle is estimated with empirical equations from the literature, i.e., Equation (1) [24], Equation (2) [25] and Equation (3) [10] and with the newly proposed Equations (15)–(17). In all cases, the suction effect is included. The red dots represent the values of friction angle measured on samples in the laboratory, at the sampling depth.

Figure 9. Interpretation along the depth of (a) GCPT2-2022, (b) PCPT5, and (c) TCPT2 [10,24,25].

All the relationships considered exhibit similar trends with depth, reflecting the observed variation in cone resistance (q_c). The largest differences among the estimated friction angles occur at shallow depths, corresponding to low stress levels, where discrepancies may exceed 20° within the first 2.0 m, whereas at greater depths they generally reduce to about 2°. The expressions proposed by [24,25] tend to provide higher friction angle estimates in GCPT2-2022 (Figure 9a) and TCPT2 (Figure 9c), while lower estimates are obtained in PCPT5 (Figure 9b), where suction effects are more pronounced. Conversely, the empirical relationship proposed by [10] tends to yield lower friction angle estimates under low stress conditions. The relationships proposed in Equations (15)–(17) provide estimates that are generally comparable to those obtained using the existing correlations, with differences that are more evident at shallow depths and progressively decrease with increasing confinement.

Overall, the estimated friction angles show reasonable agreement with the laboratory measurements, supporting the potential applicability of the proposed approach as a practical tool for preliminary peak friction angle assessment. The largest discrepancies, reaching approximately 4°, occur within the first 2 m, particularly for the GCPT2-2022 and PCPT5 profiles (GS1-C1 and PS1-C1), which are characterised by higher suction and lower degrees of saturation. With increasing depth, the discrepancies progressively decrease to about 1°, and the estimated friction angles become more consistent with the laboratory measurements.

This trend may reflect the combined influence of the simplified suction profile and the low confining stresses prevailing near the ground surface. At shallow depths, matric suction can make a substantial contribution to Bishop’s effective stress. For sample PS1-C1, for example, the χ·s term is approximately 57.8 kPa, compared with a total vertical stress of approximately 27.6 kPa (Table 3). Consequently, the estimated friction angle may be sensitive to the assumed suction profile in this depth range. The sensitivity analysis reported in Appendix B indicates that variations in matric suction can produce changes in the estimated friction angle of approximately 2–5° at a depth of 1 m over the investigated suction range. However, this sensitivity analysis does not establish the magnitude of the error associated with the actual deviation of the suction profile from the assumed linear distribution. Therefore, the simplified suction profile may contribute to the discrepancies observed in Figure 9, but its contribution cannot be quantified from the available data. Differences between laboratory testing conditions and in situ stress conditions may also contribute to these discrepancies. The decrease in discrepancies with depth is consistent with a reduced relative influence of matric suction and increasing confining stress, although the individual contributions of these factors cannot be established from the present results.

4.2. Practical Implications

The application of the proposed formulations to CPT data in partially saturated soils requires knowledge of the hydraulic properties of the investigated materials, particularly in terms of SWRC, and thus suction and degree of saturation, in order to account for and remove the additional resistance associated with partial saturation.

To this end, the SWRC must be defined for each soil type encountered along the investigated vertical profile. The assumption of a linear variation in suction with depth is necessary to estimate the distribution of this parameter within each stratigraphic layer. Through the SWRC, whether determined experimentally or estimated via pedotransfer functions, it is then possible to evaluate the degree of saturation along the depth profile down to the groundwater table.

Pedotransfer functions (such as those adopted in the present case study) allow prediction of the SWRC based on basic soil properties obtained from routine laboratory classification tests, including grain size distribution, specific gravity (G_s), unit weight (γ), Atterberg limits, void ratio (e), and porosity (n). Once these parameters are defined, the retention curve can be calibrated to obtain an estimate of the soil’s hydraulic behavior, acknowledging the associated uncertainty.

Overall, the proposed approach provides a practical framework for incorporating the effects of partial saturation into the interpretation of CPT data, while maintaining a level of simplicity compatible with routine geotechnical practice. The equations can also be applied to fully saturated soils by assuming a degree of saturation S_r = 1.0 and suction s = 0, yielding reliable results along the entire depth profile, as shown in Figure 9.

Despite the assumptions involved, particularly those related to suction distribution and the estimation of the SWRC, the results indicate that the proposed methodology is robust and provides consistent estimates of the peak friction angle. As shown in Appendix A (Table A2), peak friction angles (φ′_PEAK) derived using different pedotransfer functions are generally very similar, with only minor differences. For instance, for sample GS1-C1 (2022), φ_PEAK ranges from 39.7° to 37.2°, corresponding to a maximum difference (Δφ′_PEAK) of 2.5°, which represents the largest variation observed among all samples. For the remaining GS1 samples, as well as for the PS series, differences are significantly smaller, typically below 1.0°.

Comparable results are obtained when applying alternative empirical formulations (Equations (15) and (16)), confirming the overall consistency of the approach. The magnitude of these differences falls well within the typical variability associated with the experimental determination of soil friction angle, indicating that the influence of the selected SWRC on the computed φ′_PEAK is limited.

This reduced sensitivity further supports the applicability of simplified predictive methods when direct measurements are unavailable. Consequently, the proposed formulations could be effectively implemented in engineering practice to enhance the reliability of soil strength assessments under partially saturated conditions, while maintaining awareness of the inherent uncertainties and limitations.

4.3. Limitation of the Method

The adopted approach presents several limitations that should be considered, as they arise from the simplifications introduced.

The first concerns the assumption that matric suction varies linearly with depth, from the groundwater table to the ground surface or, in the case of embankments, to the crest of the levee. This assumption may not adequately reproduce the actual suction profile, particularly at shallow depths, and its influence on the estimated friction angle is expected to depend on soil type, with potentially larger deviations in finer-grained materials, which can develop higher suction levels. The sensitivity of the proposed correlations to the soil suction at shallow depth was investigated in Appendix B.

The availability of the SWRC represents a further source of uncertainty, as it may not be available in many site characterization studies, affecting the estimation of the χ·s term. In such cases, predictive approaches based on basic soil properties [52,53] may provide a practical alternative for estimating the SWRC. In the present analysis, their use resulted in a variation in the estimated friction angle of less than 2.5°, although this result should be considered specific to the available dataset. Further details are in Appendix A.

The reference friction angle φ′_LAB was obtained using different laboratory shear-testing techniques (RSS, Tx-CIU, Tx-CID and DS), as discussed in Section 2.2. All 26 observations, corresponding to 22 distinct physical samples, were retained in the analysis to preserve the available experimental information. However, the contribution of the laboratory testing procedure cannot be isolated from other sources of variability, because the different testing techniques are not evenly distributed among the investigated sites and are associated with differences in soil characteristics and hydraulic-state characterization. In particular, the distribution of the laboratory testing techniques varies among the investigated sites, with Tx-CIU tests being more prevalent in the Tagliamento dataset, while RSS, Tx-CID, and DS tests are mainly represented in the Grizzaga and Panaro datasets. Consequently, the present dataset does not allow the specific influence of the laboratory testing procedure on the proposed correlations to be quantified independently. A more comprehensive assessment would require a larger and more balanced dataset, in which the same soils are characterized using different laboratory testing procedures under comparable conditions.

The formulation of Equation (15) also involves the assumption of an at-rest earth pressure coefficient K₀ equal to 0.8 This value is consistent with average values reported for in situ analyses of river embankments; however, adopting a lower value (e.g., K₀ = 0.5) results in an increase in the estimated friction angle of approximately 1° for the present dataset.

Finally, the proposed correlations were calibrated using a combined dataset comprising the field levee data collected in this study and supplementary calibration-chamber data from the literature. As discussed in Appendix C, including the calibration-chamber data allows the calibration to cover a wider range of normalised cone resistance and peak friction angle. The R² values obtained by calibrating the correlations using only the field data are lower (0.18–0.23, Appendix C) than those obtained using the combined dataset (0.60–0.69). This difference is consistent with the narrower ranges of normalised cone resistance and peak friction angle covered by the field data, which limit the variability that can be captured by regression. For Case A, the RMSE calculated for the field-data calibration ranges from 2.15° to 2.22°, whereas for Case B, the RMSE calculated for the combined-dataset calibration ranges from 1.81° to 2.06° (Table A5). However, when both calibrations are evaluated against the field data only, the RMSE increases from 2.15–2.22° for Case A to 2.62–2.76° for Case B, corresponding to an increase of approximately 0.4–0.6°. These results highlight the importance of interpreting R² alongside the range and distribution of the calibration data and the corresponding prediction errors. The limitations associated with the restricted range of the field data are further discussed in Appendix C.

Despite these limitations, the proposed relationships may provide a practical and expedient tool for the preliminary estimation of the peak friction angle of partially saturated soils from CPT data. The correlations were developed and assessed using a database consisting mainly of low-plasticity silts (ML), low-plasticity clays (CL), and silty sands (SM), characterized by fines contents ranging from approximately 35 to 100%, matric suction values between 0 and 82 kPa, degrees of saturation between 0.14 and 1.00, and net vertical stresses between about 26 and 168 kPa. Within these ranges, the proposed expressions provide friction angle estimates that are generally consistent with the laboratory measurements. Given the empirical nature of the approach and the range of soil and hydro-mechanical conditions represented in the database, the relationships should primarily be considered for soils and hydro-mechanical conditions broadly comparable to those investigated in this study. Further validation using independent datasets covering a wider range of soil types, stress states, suction levels, and fines contents would be useful to assess their applicability beyond the present database.

5. Conclusions

This study explored the possibility of estimating the peak friction angle of partially saturated soils from CPT data through a set of empirical relationships calibrated using data collected at three Italian test sites and supplemented with information available in the literature. Three formulations were developed and assessed, providing comparable estimates of the peak friction angle within the range of soil and hydro-mechanical conditions considered in this study. The results indicate that accounting for suction through the SWRC may improve the consistency of the estimated friction angles with laboratory measurements, highlighting the potential relevance of partial saturation effects in the interpretation of CPT data. However, a simplified linear distribution of suction within the soil profile was assumed, and the inherent spatial variability of suction and pore-water pressure remains a significant source of uncertainty in practical applications.

The proposed relationships should therefore be regarded as a first level, practical and expedient tool for the preliminary estimation of the peak friction angle when direct laboratory information is limited or unavailable. Their applicability should be considered within the range of conditions investigated, which mainly include low-plasticity silts, low-plasticity clays, and silty sands subjected to moderate suction levels and relatively low stress conditions. Overall, the study provides additional insights into the interpretation of CPT results in partially saturated soils and may support preliminary geotechnical assessments of earthworks and embankments. Nevertheless, further studies using independent datasets will be needed to validate and refine the proposed relationships and assess their applicability to a broader range of soil types, fines contents, stress states, suction conditions, and environmental settings than those considered herein. Future research will extend their application to other soil conditions and datasets.