Determination of HSS Model Parameters for Soft Clays in Hangzhou: Statistical Analysis and Engineering Validation

Abstract

The hardening soil model with small-strain stiffness (HSS model), capturing nonlinear stiffness of soils at small strains, offers advantages for deformation analysis of tunnels or deep excavations in soft clay areas such as Hangzhou City. However, its complex parameters are rarely determinable via conventional tests, and regional geological differences render parameter determination methods of other areas inapplicable to Hangzhou. To address this issue, this paper summarizes the geological genesis, spatial distribution, and physical–mechanical properties of Hangzhou soft clays, and clarifies significance and acquisition of HSS model parameters. Via statistical analysis of existing literature data, the relationships between key HSS model parameters and physical indices (e.g., void ratio) were established. A 3D finite element (FE) simulation of a Hangzhou excavation validated the proposed parameter determination method: simulated lateral retaining structure displacement and surface settlement closely matched field measurements. The simulation results employing the model parameters proposed herein are closer to the measurements than those based on the method of Shanghai, providing guidance for excavation design and geotechnical parameter selection in Hangzhou soft soil region.

1. Introduction

Hangzhou, a megacity in China, situated at the southern edge of the Yangtze River Delta and the western end of Hangzhou Bay, is a key transportation hub in southeastern China. With rapid economic growth, urban land has become increasingly scarce, spurring intensive underground-space development. Consequently, excavations and tunnels are now widely constructed across the city [1,2]. Many urban excavation projects are adjacent to existing structures and in environmentally sensitive settings, where small disturbances can trigger large deformations or even collapse of existing buildings, with substantial economic and societal consequences [3]. Accordingly, such projects must not only meet strength requirements but also strictly control their deformation. Traditional limit state bearing capacity analyses are insufficient for assessing influences of the excavations on surrounding structures, whereas finite-element (FE) modeling can effectively model soil-structure interaction, account for complex boundary conditions, and predict deformation of soils and structures. FE modeling has therefore become essential for deformation analysis and safety evaluations of excavations [4,5,6,7].

Mastering the physical and mechanical properties of geomaterials holds the key to engineering design and numerical simulations. As shown in Figure 1, Atkinson et al. [8] categorized soil behavior into three strain ranges: very small strain (smaller than 0.0001%), small strain (0.0001–0.1%), and large strain (greater than 0.1%). Extensive empirical data and analyses of excavations and tunnels [5,6,8,9,10,11,12] indicate that under typical working loads, the majority of soils exhibit small-strain behavior with highly nonlinear stiffness. Consequently, accurately describing the small-strain stiffness characteristics of soils is critical for predicting deformations induced by urban tunnel and excavations [5,6,10,11,12,13,14,15,16,17,18].

The soundness of the constitutive model and its parameters is the key determinant of the reliability of numerical simulation results. Accordingly, extensive research has been conducted on soil constitutive modeling, especially concerning the small-strain characteristics of soils, yielding substantial findings [16,17,18]. Among these studies, the hardening soil model with small-strain stiffness (HSS model), has gained increasing attention in underground engineering simulations, because it accurately captures both the nonlinear stiffness behavior of soils under small strains and their stress-path dependence [5,6,12,13,14,15,18].

The HSS model requires a relatively large number of parameters, many of which are typically obtained through advanced laboratory tests such as unloading–reloading triaxial, bender element, and resonant column tests. Because such procedures are rarely included in routine site investigations, direct parameter determination is often im-practical. To address this limitation, three principal strategies have been widely adopted in both domestic and international studies: back-analysis, advanced laboratory testing, and empirical or semi-empirical correlations. Back-analysis calibrates parameters by reproducing field monitoring results. This method has been applied in China (e.g., Mu et al. [19]; Zheng et al. [3]) and internationally in Canadian glacial tills [20], and Chicago excavations [5]. Laboratory testing provides direct measurements of stiffness and strength. Representative contributions include studies on Chinese soils by Liang et al. [21], Cao et al. [14], and Hu et al. [15], as well as investigations on Bangkok clays by Surarak et al. [22] and Wu et al. [23]’s experiment on soil from a highway in a certain region of the United States. Empirical or semi-empirical correlations estimate HSS parameters mainly through multiplier relationships among stiffness moduli. Surarak et al. [22] summarized such modulus ratios for Bangkok clays, while the PLAXIS manual [24] provides widely adopted recommendations for practical modeling. Huynh et al. [13] conducted a validation study on the simulation of deep foundation pits in Ho Chi Minh City using this method. In Shanghai, Gu et al. [12] developed a correlation-based framework linking HSS parameters to soil indices and modulus ratios, validated by engineering practice. These studies demonstrate the practicality and efficiency of ratio-based correlations, though their reliability remains strongly dependent on regional calibration and soil-specific conditions.

Although these strategies have advanced the practical application of the HSS model, their transferability across regions remains limited. The geological conditions in Hangzhou differ fundamentally from those in Shanghai, Southeast Asia, and North America, owing to variations in soil genesis, depositional environment, and microstructure. Consequently, the direct adoption of parameter determination schemes developed elsewhere is scientifically questionable. Several studies have attempted to characterize Hangzhou clays through laboratory investigations (e.g., Meng [25]; Deng [26]; Xia [27]; Chen [28]), yet a simple, robust, and region-specific framework for parameter determination has not been established to date.

To address the regional gap in constitutive modeling, this study establishes a novel and practical parameter selection framework for the HSS model, specifically adapted to the geotechnical characteristics of Hangzhou soft clay. Distinct from existing approaches that rely heavily on advanced laboratory testing or adopt empirical relations developed for other regions, the proposed method integrates localized experimental findings and correlations with basic soil index properties. Moreover, the framework’s applicability is validated through numerical simulations of deep excavations, which show good agreement with field monitoring data. This research represents the first systematic effort to formulate a regionally customized, empirically informed, and practically applicable method for HSS parameter determination in Hangzhou, offering valuable guidance for the use of HSS models in soft clay engineering.

2. Basic Characteristics of Hangzhou Clays

2.1. Geologic Origin and Distribution

Hangzhou borders Hangzhou Bay to the east and the Qiantang River to the south, and lies in the southwestern part of the Hangzhou-Jiaxing-Huzhou Plain, where the topography and geomorphology are notably complex [29]. Since the Quaternary, recurrent tectonic activity has led to continuous compression of the city’s sedimentary soils. During the Early Holocene, rising sea levels facilitated the progressive evolution of sediments into estuarine, lacustrine, and marine soft soil layers. By the Mid-Holocene, gray muddy soft soils had gradually formed, with deposit thickness reaching 12–30 m. By the Late Holocene, lacustrine environments emerged, accompanied by the deposition of black muddy and soft soils. This sedimentary evolution ultimately resulted in the development of marine-lagoon soft soil deposits characteristic of the Hangzhou region [30]. The main 12 soil layers formed are shown in

Table 1. Geological layer of Hangzhou [31].

No	Soil Layer Name	Thickness	Note
1	Artificial fill	1–7 m
2	Silt, silty clay layer	6–10 m	First soft soil layer
3	Mucky clay layer	6–10 m
4	Silt, silty clay layer	5–8 m
5	Mucky clay layer	2–10 m	Second soft soil layer
6	Clay with silt interbed layer	3–4 m
7	Clay and silty clay layer	3–8 m
8	Qiantang River alluvial layer	4–6 m
9	Clay layer	2–5 m
10	Pre-hill flood, slope and ancient river valley silvial deposits
11	Reticulated clay, gravelly clay, expanded clay layer
12	Bedrock and bedrock weathering layer

[31]. Among these, the first and second soft clay layers with greater thickness exert significant engineering impacts.

2.2. Basic Physical and Mechanical Properties

According to relevant studies [31,32], the basic physical and mechanical properties of Hangzhou soft clays are summarized in

Table 2. Statistics of physical and mechanical properties of Hangzhou soft clays.

No.	Water Content$\mathit{w}$/%	Void Ratio $\mathit{e}$	Plasticity Index ${\mathit{I}}_{\mathit{p}}$	Liquidity Index ${\mathit{I}}_{\mathit{L}}$	Compression Coefficient ${\mathit{a}}_{1-2}/{\mathbf{M}\mathbf{P}\mathbf{a}}^{-1}$
First soft soil layer	$48\pm 10$	$1.3\pm 0.3$	$185\pm 10$	$1.3\pm 0.2$	$1.1\pm 0.3$
Second soft soil layer	$39\pm 7$	$1.1\pm 0.2$	$144\pm 10$	$1.2\pm 0.3$	$1.0\pm 0.4$

. The following characteristics are observed: (1) High water content, some strata reaching 50% and above; (2) Large void ratio, with the void ratio of most soft soils reaching 1.0 and above; (3) High compressibility and generally poor engineering performance. Consequently, in excavation stability analysis, the accuracy of soil parameters significantly influences deformation predictions.

3. HSS Model Parameters and Parameter Determination

In the deformation analysis of underground structures, selecting an appropriate constitutive model is critical. Kim et al. [5] observed that elastic models inadequately capture the plastic deformation of soil. Ideal plastic models, such as the Mohr–Coulomb and Drucker–Prager models, fail to provide reasonable foundation deformation predictions due to their reliance on a single stiffness value. In contrast, strain-hardening models like the Modified Cam-Clay (MCC) model and Hardening Soil (HS) model better satisfy engineering requirements by accounting for soil hardening behavior. The HS model, introduced by Schanz et al. [33] and based on the Mohr–Coulomb failure criterion, incorporates stress path-dependent stiffness variations. Based on previous research, Benz [18] comprehensively considered the small strain characteristics of soil and proposed HSS model The HSS model not only considers soil compressibility but also accurately represents stiffness nonlinearity at small strains. Consequently, it is widely adopted in underground engineering analyses, with existing studies demonstrating its effectiveness in predicting excavation deformations in Hangzhou [34,35].

3.1. HSS Model Parameters and Their Definitions

The HSS model has 13 parameters in total (shown in

Table 3. HSS model parameters and their definitions.

Parameter	Definition
$c^{\prime }$	Effective cohesion of soil
$K_0$	Coefficient of static lateral earth pressure
${\phi }^{\prime }$	Effective friction angle
$m$	Power index associated with the modal stress level
$\Psi$	Stress dilatancy angle
$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	Tangential compression modulus at reference stress
$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	Secant modulus at reference stress
$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	Unloading–reloading modulus at reference stresses
$R_{\mathrm{f}}$	Failure ratio
${\nu }_{\mathrm{u}\mathrm{r}}$	Loading and unloading Poisson’s ratio
$G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}$	Maximum shear modulus at reference stress
${\gamma }_{0.7}$	Shear strain when the shear modulus decreases to 0.7 G0
$p^{\mathrm{r}\mathrm{e}\mathrm{f}}$	Reference stress, usually taken as 100 kPa

) [18].

The strength parameters include the effective cohesion (c$\prime$) and the effective friction angle (${\phi }^{\prime }$), which are commonly determined from triaxial tests. Notably, investigation reports typically provide cohesion and friction angle in terms of total stress, which should not be used directly. The stress dilatancy angle (Ψ) can be derived from drained consolidation tests; Ψ can be estimated as ${\phi }^{\prime }$−30° for sandy soils, while for cohesive soils, it is generally assumed to be 0 [36].

The coefficient of earth pressure at rest (K₀) represents the ratio of lateral stress to vertical stress in the soil under natural stress conditions, reflecting the soil’s horizontal stress state. It can be determined through a K₀ consolidation test or estimated using Equation (1):

$$K_0=1-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime },$$

(1)

The soil moduli vary with depth and are stress-dependent; thus, a reference stress (p^ref) of 100 kPa is defined.

The failure ratio R_f is defined as the ratio of the deviatoric stress at failure to the shear strength, which can be determined from triaxial tests [18]. It characterizes the hyperbolic relationship between deviatoric stress and axial strain, as assumed in the HSS model. It can be calculated using Equation (2):

$$-{\epsilon }_{\mathrm{a}}={\displaystyle \frac{1}{E_i}}\times {\displaystyle \frac{q}{1-\frac{q}{q_{\mathrm{a}}}}},R_{\mathrm{f}}={\displaystyle \frac{q_a}{q}},$$

(2)

where ε_a is the axial strain; q is the deviatoric stress; q_a the asymptotic value of deviatoric stress; q_f is the deviatoric stress at failure; E_f is the initial modulus.

The reference tangent modulus of the soil ($E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$) can be determined through standard consolidation tests. It represents the tangent compression modulus in the axial stress–strain curve when the consolidation stress equals the reference stress ($p^{\mathrm{r}\mathrm{e}\mathrm{f}}$). Additionally, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for different consolidation stresses by Equation (3):

$$E_{\mathrm{o}\mathrm{e}\mathrm{d}}=E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}{\left({\displaystyle \frac{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }-{\sigma }_1^{\prime }\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }+p^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}}\right)}^m,$$

(3)

where ${\sigma }_3^{\prime }$ is the initial horizontal stress in the field, with compressive stress as negative; ${\sigma }_1^{\prime }$ is the vertical stress, which can be calculated as ${\sigma }_3^{\prime }/K_0$; m is the power index related to the stress level.

The reference secant modulus ($E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$) can be determined from drained consolidation tests and corresponds to the slope of the line connecting the origin to the deviatoric stress at half of the maximum deviatoric stress in the stress–strain curve under a consolidation stress of $p^{\mathrm{r}\mathrm{e}\mathrm{f}}$. $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for different consolidation stresses can be converted by Equation (4):

$$E_{50}=E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}{\left({\displaystyle \frac{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }-{\sigma }_3^{\prime }\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }+p^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}}\right)}^m,$$

(4)

The reference unloading–reloading modulus $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ is obtained through the unloading–reloading drained consolidation tests. The stress–strain curve has a hysteresis loop during unloading–reloading. The slope of the straight line obtained by connecting the two endpoints is $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$. $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for different consolidation stresses can be converted by Equation (5):

$$E_{\mathrm{u}\mathrm{r}}=E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}{\left({\displaystyle \frac{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }-{\sigma }_3^{\prime }\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }+p^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}}\right)}^m,$$

(5)

The reference initial small strain shear modulus $G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}$ can be determined from field or laboratory tests. Field methods primarily involve wave velocity measurements, while common laboratory techniques include resonant column tests and bending element tests. Extensive experiment shows that $G_0$ depends on the average effective stress $p^{\prime }$ [18]. Consequently, $G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}$ at different stress levels should be calculated using Equation (6):

$$G_0=G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}{\left({\displaystyle \frac{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }-\frac{{\sigma }_1^{\prime }+2{\sigma }_3^{\prime }}{3}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}{c^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{\prime }+p^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }^{\prime }}}\right)}^m,$$

(6)

The shear strain ${\gamma }_{0.7}$ represents the shear strain at which the shear modulus decreases to $0.7G_0$. It characterizes the modulus degradation of soil and is typically determined from resonant column tests.

The unloading–reloading Poisson’s ratio ${\nu }_{\mathrm{u}\mathrm{r}}$ is the unloading–reloading Poisson’s ratio, which can be determined through unloading–reloading drained triaxial tests. It reflects the elastic characteristics of soil under small-strain conditions. In the absence of experimental data, the default value of 0.2 recommended in the PLAXIS manual [24] may be utilized as a reasonable approximation for this parameter.

The power index m, which reflects the stress-dependency of the modulus, indicates the sensitivity of the modulus to changes in the stress state [37]. Although different moduli correspond to distinct m values, the relevant software permits only a single input value. Existing sensitivity analyses suggest that the small-strain shear modulus is more sensitive than other parameters [14,38]. Therefore, the m value corresponding to the small-strain shear modulus is selected as the input value.

3.2. Parameter Determination

In engineering practice, obtaining all parameters of the HSS model through site investigations and laboratory tests is often challenging. A common approach is to estimate the elastic moduli ($E_{\mathrm{s}1-2}$, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$) using data from site investigation reports. This method provides a practical solution for engineering applications [18,21,34].

The relationship between $E_{\mathrm{s}1-2}$, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ is illustrated in Figure 2 [25,26,27,34,39,40]. It can be seen that all moduli exhibit a strong linear correlation. Specifically, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=0.84E_{\mathrm{s}1-2}$, where the coefficient (0.84) is slightly higher than that of Shanghai soil (0.81). Similarly, $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}=1.29E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, with a coefficient (1.29) closely matching that of Shanghai soil (1.26) [12]. The relationship between $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ is presented in Figure 2c. As shown in the figure, the fitted line representing the relationship between the two variables does not intersect the origin, indicating that for softer soils (lower stiffness) the gap between $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ becomes greater. Moreover, $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ demonstrate a stronger correlation with each other than with $E_{\mathrm{s}1-2}$, because both are derived from drained consolidation tests.

The relationship between the moduli $E_{\mathrm{s}1-2}$, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, $E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and the void ratio $e$ is presented in Figure 3 [25,26,27,34,39,40]. As shown in the figure, each modulus exhibits a strong logarithmic correlation with the void ratio, with most data points falling within $\pm 30\%$ of the predicted values. The overall fitting results differ slightly from the values obtained for Shanghai soil [12]. The relationship between $E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and $e$ shows greater variability, which reflects the distinct unloading–reloading characteristics of different soft clays.

Figure 4 illustrates the relationship between effective friction angle (${\phi }^{\prime }$) and void ratio ($e$), revealing a strong logarithmic correlation [25,26,27,34,39]. While conventional site investigation reports typically provide shear strength in terms of total stress from consolidation tests, this parameter cannot theoretically serve as the strength parameters for the HSS model. Therefore, establishing a relationship between the void ratio and the effective friction angle is crucial for determining ${\phi }^{\prime }$ when the effective strength data are unavailable. Unlike the power function relationship observed in Shanghai clay [12], the logarithmic form more accurately describes the ${\phi }^{\prime }-e$ correlation in Hangzhou clay. Additionally, Figure 5 demonstrates a clear linear relationship between the effective friction angle and the effective cohesion ($c^{\prime }$) [25,26,27,34,39]. Thus, ${\phi }^{\prime }$ can first be estimated from $e$, after which $c^{\prime }$ can be derived from ${\phi }^{\prime }$.

Figure 6 illustrates the relationship between the failure ratio $R_{\mathrm{f}}$ and the void ratio $e$ [25,26,27,28,34,39] In theory, when the soil $e$ is small, based on the hyperbolic stress–strain relationship, it can be inferred that the $R_{\mathrm{f}}$ of the soil is large. Compared to Shanghai clay, the overall trend is similar. However, at the same void ratio, the failure ratio is comparatively smaller.

Figure 7 illustrates the relationship between $K_0$ and the void ratio $e$ [41,42,43]. Overall, the $K_0$ values for Hangzhou clay range from 0.4 to 0.6 and exhibit an increasing trend with increasing void ratios. Compared with Shanghai clay, Hangzhou soft clay demonstrates a more pronounced increase in $K_0$ with increasing $e$.

In this study, empirical correlations between void ratio and HSS stiffness parameters (Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7) are established via regression analysis. While the functional forms are data-driven, their trends are supported by microstructural theory. According to Zheng et al. [44], the compressibility of clays relates to fractal pore-size distributions, indicating that higher void ratios correspond to looser, less connected pore structures and lower stiffness. The observed relationships thus reflect not only statistical trends but also underlying structural mechanisms, suggesting the influence of pore-scale fractal characteristics.

Figure 8 presents the $G_0$ derived from in situ wave velocity tests conducted at the Hangzhou Juzhou road station excavation. Previous studies have demonstrated that $G_0$ values obtained from laboratory tests are typically lower than those from field wave velocity tests due to soil disturbance [45,46]. Consequently, this study focuses on $G_0$ statistics based on in situ shear wave velocity measurements. Hardin’s formula, widely employed for $G_0$ prediction, is expressed in Equation (7).

$$G_0=AF\left(e\right){\left({\displaystyle \frac{p^{\prime }}{p_{\mathrm{a}}}}\right)}^m,$$

(7)

where $A$ and $m$ are the fit parameters, $p^{\prime }$ is the mean effective stress, and $p_{\mathrm{a}}$ is the reference stress (taken as 100 kPa in this study), $F\left(e\right)$ is a void ratio function. The two most common forms of $F\left(e\right)$ are given in the following equations and this study adopts the form of the first equation [46]:

$$\begin{array}{ll}& F\left(e\right)=e^{-a}\\ & F\left(e\right)={\displaystyle \frac{{\left(a-e\right)}^2}{1+e}}\end{array},$$

(8)

where $a$ is the fit parameters.

As shown in Figure 8, the in situ wave velocity test results for determining $G_0$ exhibit some scatter, yet overall demonstrate strong regularity. This observation that $G_0$ is highly sensitive to the disturbance. Consequently, special attention must be paid to potential disturbances from the surrounding environment when conducting field wave velocity tests.

Since the degradation characteristics of the shear modulus cannot be directly obtained from in situ measurements, this study compiles relevant laboratory resonant column test results. The ${\gamma }_{0.7}$ values derived from these tests are summarized in Figure 9 [47,48]. However, due to the limited number of resonant column test studies on Hangzhou clay, the available data are rather limited. The existing data indicate that ${\gamma }_{0.7}$ for Hangzhou clay ranges from $1.1\times {10}^{-4}$ to $3.7\times {10}^{-4}$, with a mean of $2.4\times {10}^{-4}$. Additional resonant column tests on Hangzhou clay are recommended in future work to further improve the dataset.

In this study, the parameters for the HSS model were determined using empirical regression relationships based on physical indices, such as void ratio. As highlighted by Zheng et al. [49], the pore structure of clays plays a crucial role in determining their mechanical behavior. A decrease in void ratio leads to a more compact soil structure, resulting in greater pore connectivity and consequently higher stiffness [50,51]. This trend is reflected in our parameter determination method, where a reduction in void ratio corresponds to an increase in stiffness parameters. Additionally, Zhang et al. [52] demonstrated that, as the void ratio decreases, particle interactions, including van der Waals and electrical double-layer forces, become more significant. These interactions contribute to an increase in shear modulus and small-strain stiffness.

By incorporating these microstructural insights, parameter determination method aligns with the underlying physical mechanisms that govern the mechanical behavior of Hangzhou soft clays [53]. This approach enhances the reliability and applicability of the derived HSS model parameters.

The proposed parameter determination method is distinct from previous studies primarily due to its regional focus and practical approach. Unlike methods based on general empirical correlations from other regions, this method simplifies the process by relying on physical indices (e.g., void ratio) and regional correlations, making it more applicable to routine engineering practice, without the need for advanced laboratory tests.

3.3. Summary of Model Parameters

Table 4. Comparison of HSS model parameters for clays in Hangzhou and Shanghai.

Parameter	Hangzhou Clays (This Paper)	Shanghai Clays (Gu et al., 2021) [12]
$c^{\prime }$	$c^{\prime }=41.03-1.33{\phi }^{\prime }$	$c^{\prime }=43.56-1.32{\phi }^{\prime }$
$K_0$	${\phi }^{\prime }=-9.32e+27.62$	${\phi }^{\prime }=30.4e-0.65$
${\psi }^{\prime }$	$K_0=0.26e+0.25$	$K_0=0.049e+0.02f+0.139$
$m$	$m=0.52$ (from wave velocity test)	$m=0.65$ (from wave velocity test)
$\Psi$	$0$	$0$
$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=0.84E_{\mathrm{s}1-2}$,$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-6.58\mathrm{l}\mathrm{n}\left(e\right)+3.02$	$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=0.81E_{\mathrm{s}1-2}$, $E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-4.32\mathrm{l}\mathrm{n}\left(e\right)+3.51$
$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}=1.29E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$,$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-5.04\mathrm{l}\mathrm{n}\left(e\right)+4.73$	$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}=1.26E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$,$E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-5.33\mathrm{l}\mathrm{n}\left(e\right)+3.95$
$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$	$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=3.07E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}+14.63$,$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-30.2\mathrm{l}\mathrm{n}\left(e\right)+28.3$	$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=4.12E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}+7.24$,$E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}=-28.8\mathrm{l}\mathrm{n}\left(e\right)+24.5$
$R_{\mathrm{f}}$	$R_{\mathrm{f}}=0.60$$\left(e>1.21\right)$$, R_{\mathrm{f}}=0.83$$\left(e<0.97\right)$$, R_{\mathrm{f}}=-0.95e+1.75$, $(0.97\le e\le 1.21$)	$R_{\mathrm{f}}=0.50$$\left(e>1.5\right)$$, R_{\mathrm{f}}=0.95$$\left(e<1.0\right)$$, R_{\mathrm{f}}=-0.9e+1.85$$(1.0\le e\le 1.5$)
${\nu }_{\mathrm{u}\mathrm{r}}$	0.2 (from PLAXIS manual)	0.2 (from PLAXIS manual)
$G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}$	$G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}=49.22e^{-0.90}$	$G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}=67.5e^{-1.57}$
${\gamma }_{0.7}$	$2.0\sim 2.8\times {10}^{-4}$	$2.4\sim 4.4\times {10}^{-4}$
$p^{\mathrm{r}\mathrm{e}\mathrm{f}}$	100 kPa	100 kPa

presents the parameter selection for the HSS model of soft clay in Hangzhou, determined from statistical analysis of extensive test data from the Hangzhou area. Most parameters can be derived from the soil’s physical properties such as the void ratio, while others are based on empirical or recommended values.

Compared with soft clays in Shanghai, the engineering properties of Hangzhou soft clays are significantly poorer. Therefore, when conducting engineering analyses in Hangzhou, region-specific empirical relationships should be applied. In subsequent research, in situ shear-wave velocity measurements and resonant column tests will be conducted to further improve the parameter determination method.

To facilitate readers’ intuitive understanding of the HSS model parameter determination method for Hangzhou soft clays and the logic of its engineering application,

Table 5. Step-by-step procedure of parameter determination.

Step	Procedure	Input Data	Output Parameter
1	Determine basic physical index of Hangzhou clay	Site investigation report	$e$$, E_{s1-2}$
2	Calculate strength parameters	$e$	$c^{\prime }$$, {\phi }^{\prime }$
3	Calculate stress state parameter	$e$	$K_0, R_{\mathrm{f}}$
4	Calculate stiffness moduli	$e$$\mathrm{or} E_{s1-2}$	$E_{\mathrm{o}\mathrm{e}\mathrm{d}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$$, E_{50}^{\mathrm{r}\mathrm{e}\mathrm{f}}$$, E_{\mathrm{u}\mathrm{r}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$
5	Calculate small-strain parameters	e	$G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}$$, {\gamma }_{0.7}$
6	Assign fixed/recommended parameters	PLAXIS manual	$p^{\mathrm{r}\mathrm{e}\mathrm{f}}$$, {\nu }_{\mathrm{u}\mathrm{r}}$$, \Psi$$, m$
7	Parameters Application	3D FE simulation	Parameter rationality check

presents a step-by-step procedure table. This table systematically organizes the entire process from the acquisition of basic physical indices to the final application of parameters, and clearly lists the input data, calculation basis, and corresponding output parameters for each link. Following the logic of this table, readers can efficiently grasp the core steps of the method and directly apply it to practical scenarios such as parameter selection in geotechnical numerical simulations.

4. Case Study: Excavation of Juzhou Road Station, Hangzhou Metro Line 5

4.1. Project Overview

The Juzhou Road Station on Hangzhou Metro Line 5 is located at the intersection of Juzhou Road and Second Road in southeastern urban area of Hangzhou. The project location is shown in Figure 10. The underground station structure adopts a two-story, two-column three-span reinforced concrete frame configuration with critical dimensions as follows: a 12.6 m wide platform section, 21.3 m wide standard sections, and 25.4 m wide shield shaft sections. The excavation depth ranges from 16.6 m to 18.6 m below ground level, with a total station length of 190.35 m. The total constructed area measures 12,073 m².

The excavation support system uses an 800 mm-thick diaphragm wall (D-wall) as the primary retaining structure, supplemented by a hybrid internal bracing system. In the standard sections, five levels of support were installed: the first level comprises 800 mm × 1000 mm cast-in-place reinforced concrete struts, whereas the remaining four levels adopt steel pipe struts (nominal diameter 609 mm, wall thickness 16 mm). The horizontal spacing ranges from 6 to 9 m for concrete struts and from 2 to 4 m for steel struts, with the longitudinal spacing between adjacent supports of approximately 3 m. The cross-section, geological conditions and excavation sequence are shown in Figure 11.

4.2. Three-Dimensional Finite Element Model Analysis

Based on the parameter determination method for the HSS model of soft clay in Hangzhou summarized herein, the HSS model parameters determined for each soil layer at Juzhou Road Station are detailed in

Table 6. Input parameters of soil layers for the HSS model.

Soil Layer	$\mathit{\gamma }/\mathbf{k}\mathbf{N}/\mathbf{m}$	${\mathit{c}}^{\prime }/\mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\phi }}^{\prime }/°$	${\mathit{K}}_0$	${\mathit{E}}_{\mathbf{o}\mathbf{e}\mathbf{d}}^{\mathbf{r}\mathbf{e}\mathbf{f}}/\mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{E}}_{50}^{\mathbf{r}\mathbf{e}\mathbf{f}}/\mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{E}}_{\mathbf{u}\mathbf{r}}^{\mathbf{r}\mathbf{e}\mathbf{f}}/\mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{G}}_0^{\mathbf{r}\mathbf{e}\mathbf{f}}/\mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\gamma }}_{0.7}$/10−4	$\mathit{m}$
Filling soil	18.73	1.6	29.7	0.47	5.94	4.60	32.90	66.6	2.4	0.52
Very soft clay	16.86	4.0	25.3	0.55	2.33	1.81	21.85	40.1	2.4	0.52
Silty clay	19.30	25.3	28.7	0.39	7.42	5.75	37.44	72.8	2.4	0.52
Clay	18.54	10.0	27.7	0.49	5.45	4.23	31.41	60.7	2.4	0.52
Very soft silty clay	17.93	4.0	25.9	0.54	3.94	3.05	26.76	52.8	2.4	0.52
Soft silty clay with silty sand	17.95	3.0	27.3	0.54	3.66	2.84	25.91	52.6	2.4	0.52
Silty clay	19.47	26.0	28.2	0.40	6.99	5.43	36.16	75.1	2.4	0.52
Silty clay	19.43	21.3	27.9	0.41	6.90	5.35	35.84	74.1	2.4	0.52
Clay	17.95	4.5	27.5	0.45	4.83	3.74	29.49	53.3	2.4	0.52

. Noted that Table 1 presents the general geological stratigraphy of Hangzhou soft clays, while Table 5 summarizes the soil layers and parameters obtained from the site-specific geotechnical investigation of the case study project.

A three-dimensional finite element analysis of the Juzhou Road Station excavation was performed using PLAXIS 3D. To minimize the influence of boundary effects on calculation results, the model height was set to 2–4 times the excavation depth, with one side of the model positioned at 3–4 times the excavation depth from the pit edge [54]. The standard section excavation depth of the main pit is 16.62 m, while the left and right shafts have excavation depths of 18.76 m and 18.05 m, respectively. Accordingly, the model height was determined to be 45 m, with the left edge positioned 55 m from the excavation, ultimately establishing a computational model measuring 300 m × 150 m × 45 m. The finite element model is shown in Figure 12.

The numerical model incorporated three primary components: (1) the soil matrix, discretized using 10-node tetrahedral solid elements; (2) the retaining system (D-wall), represented by 6-node triangular plate elements; (3) the internal bracing system, modeled as 2-node point-to-point anchor elements.

The support configuration precisely replicated the construction design specifications. Material properties were assigned as follows: both the D-wall and primary concrete strut had an elastic modulus of 30 GPa, while steel pipe struts had a modulus of 206 GPa. The final mesh configuration comprised 178,756 finite elements with 246,159 nodal points, ensuring adequate resolution for the analysis.

The construction steps are listed in

Table 7. Three-dimensional finite element excavation construction steps.

Construction Step	Working Condition
Phase 0	Calculation of initial stress field
Phase 1	Construction of the retaining structure (D-wall)
Phase 2	Excavation to 1.4 m below ground level (GL), followed by installation of the first-level concrete strut.
Phase 3	Excavation to 5.0 m below GL, with deployment of the second-level steel strut.
Phase 4	Excavation to 8.5 m below GL, coupled with the third-level steel strut installation.
Phase 5	Excavation to 11.5 m below GL, incorporating the fourth-level steel strut.
Phase 6	Excavation to 14.1 m below GL, concluding with the fifth-level steel strut.
Phase 7	Final excavation stage: Standard section: Reached 16.62 m below GL Left shaft: Excavated to 18.76 m below GL Right shaft: Terminated at 18.05 m below GL

. Due to the presence of grouting at the pit bottom and a cut-off wall along the sides, combined with the low permeability of the clay layer, the excavation’s anti-seepage measures are highly effective. As a result, the dewatering process inside the pit has a negligible impact on the external groundwater table.

The dewatering process was simulated using the similar—group groundwater head interpolation method. In each excavation step, the groundwater head of the soil mass within the pit was set to 0.5 m below the excavation surface, thereby simulating the actual in—pit water level condition. Conversely, the groundwater head outside the pit remained constant. As the excavation advanced incrementally, the groundwater head inside the pit decreased progressively.

4.3. Analysis of Three-Dimensional Finite Element Calculation Results

The finite element deformation results are illustrated in Figure 13. Both the field-measured deformation of the excavation’s D-wall and the finite element simulated results exhibit distinct three-dimensional spatial characteristics, with more significant deformation in the central region and relatively minor deformation at both end zones. This observation is consistent with in situ measurements reported by Tan et al. [55] for strip-type subway excavations in soft soil regions, showing that retaining-wall deformation at the pit extremities is less pronounced than at the mid-sections of the long sides. The layout of the monitoring points is shown in Figure 14.

Figure 15 compares the measured and simulated lateral displacements of the diaphragm wall (D-wall) at multiple monitoring points. Two parameter determination methods for HSS model are evaluated: the Hangzhou-specific parameters proposed in this study and the Shanghai regional parameters reported by Gu et al. [12]. A 20% error band around the measured values is also provided. Using the Hangzhou regional parameters, the simulated lateral displacement profiles agree well with the field observations and consistently exhibit the characteristic bulging shape; the peak lateral dis-placements occur at essentially the same depth, slightly below the mid-depth of the excavation.

In general, the simulation results based on the Hangzhou parameters summarized in this study demonstrate a closer correspondence with the measured values, confirming the regional specificity of HSS parameters. The use of a locality-adapted parameter determination method leads to predictions that are more consistent with observed excavation performance. Notably, the simulation results obtained using Hangzhou parameters slightly overestimate the measured displacements, while those using Shanghai parameters tend to underestimate them. From a design safety perspective, this conservative prediction pattern supports the use of Hangzhou parameters in local engineering practice. At a pit depth of −5 m, the simulation results using the Hangzhou parameters exhibit noticeable deviations from the field data at several monitoring points. This discrepancy is likely due to limitations in characterizing the shallow soil profile. First, the stiffness of the heterogeneous fill layers in the upper strata may have been underestimated during parameter selection, especially since these layers exhibit high variability and are difficult to characterize accurately. Second, the dewatering process inside the excavation was simplified in the model as a static reduction in hydrostatic pressure, without accounting for time-dependent consolidation effects that can stiffen shallow soils. Third, the stiffness of the diaphragm wall may have been conservatively estimated, further influencing the deformation pattern. These factors collectively suggest that while the proposed parameter framework offers reliable predictions overall, refinements are still needed for better characterization of near-surface behavior.

Figure 16 illustrates the simulated results of surface settlements outside the pit at various monitoring points. The simulation outcomes align with the deformation pattern of conventional slab-supported excavation retaining structures, exhibiting a uniformly groove shaped distribution [55,56].

Figure 17 presents a comparison between the measured and simulated maximum surface settlements at different monitoring points. Except for Monitoring Point 1, the simulated maximum surface settlements at the remaining points are in close agreement with the measured values, with an absolute error that does not exceed 3.3 mm. The error bar is set to 20% of the measured value, and the calculated values at Observation Points 2 and 3 fall within the range of the measured value error bars, indicating that the simulated results are generally consistent with the measured values, and the fitting is satisfactory. At Monitoring Point 1, the measured settlement exceeds the simulated value, which may be attributed to the dewatering process within the excavation. This caused a decline in the groundwater level outside the pit at Monitoring Point 1, resulting in additional consolidation settlement of the soil.

5. Discussions

Hangzhou soft clay, a typical marine-lagoonal sediment, exhibits unique engineering characteristics, including high water content (≥50%) and a high void ratio (≥1.0), resulting in high compressibility and significant nonlinear deformation. These properties necessitate a region-specific approach to HSS model parameter determination. This study presents a method that integrates physical indices, such as void ratio (e), with regional empirical data and provides a tailored solution for Hangzhou soft clay.

However, the HSS model has notable limitations. It assumes isotropy, which may be invalid for naturally anisotropic clays and can bias predictions under deep excavations or under lateral loading. Moreover, while HSS captures small-strain stiffness and strain-dependent degradation, it does not explicitly model rate effects under cyclic or earthquake loading. Even so, its ability to represent hardening/softening and distinct loading–unloading stiffness makes it effective for many practical problems. These limitations—anisotropy and rate dependence—should be addressed in future extensions and validations.

Compared with other published methods, the proposed parameter determination method offers distinct advantages. Unlike traditional methods based on empirical relationships or generalized correlations, our approach integrates region-specific empirical data and physical indices (such as void ratio) to determine key parameters like G₀ and γ₀.₇. This method better reflects local soil behavior, especially in regions with unique characteristics, such as Hangzhou. In contrast, methods based on Shanghai or other regional soils may fail to capture local variations, as shown by our comparison, which highlighted discrepancies when Shanghai parameters were applied to Hangzhou soils.

Nevertheless, the robustness of the proposed method under varying stress states warrants further investigation. While it performed well with the excavation data from Juzhou Road Station, real-world stress conditions can be more complex. Future research should assess the method under dynamic loading and high confining pressures, and conduct sensitivity analyses to evaluate how different stress states affect parameter estimates.

In conclusion, this study provides a reliable method for determining HSS model parameters for Hangzhou soft clay. While the method effectively captures small-strain behavior and nonlinear characteristics, future work should address its limitations, including anisotropy and rate effects, and further test its robustness under complex stress conditions to ensure its broader applicability in engineering projects.

6. Conclusions

This paper commences with an analysis of the geological origin and basic physical properties of Hangzhou soft clay, and clarifies the definitions of all HSS model parameters and their acquisition methods. Subsequently, it performs statistical analysis on an extensive set of laboratory test data and in situ measurement data fron the Hangzhou area. Finally, a parameter-determination framework for the HSS model applicable to Hangzhou soft clay is established and validated using the Juzhou Road Station excavation of Hangzhou Metro Overall, this study presents the first comprehensive and validated framework for HSS parameter determination tailored to the unique properties of Hangzhou soft clay. The findings fill a regional gap in constitutive modeling and provide a practical and transferable method for improving numerical simulations in soft clay engineering.

• The soft clay in Hangzhou is classified as marine soft clay, characterized by two distinct soft clay layers, an average water content of 50%, an average void ratio of 1.0, high compressibility, and poor engineering properties. These unique characteristics must be accounted for in numerical simulations.
• A correlation-based method for HSS parameter determination was established by integrating laboratory tests and field shear wave velocity measurements, with particular attention to the influence of void ratio. Comparative analysis with Shanghai soft clay reveals significant regional differences, especially in small-strain stiffness parameters, underscoring the need for localized models.
• The parameter determination method obtained in this study was applied to the three-dimensional finite element analysis of excavation deformation at Juzhou Road Station on Hangzhou Metro. The simulated lateral displacement curve of the D-wall and the position where the maximum displacement occurs are almost consistent with the measured values, and the surface settlement outside the excavation is relatively close to the measured values, thereby verifying the reliability of the parameter determination method in this study. Comparisons with simulations using Shanghai parameters show that the proposed method yields results closer to the measurements.