The soil specimens used in the study were undisturbed soil samples taken from the strata at the construction site where the shield tunnel of Line eight in Zhengzhou crossed over the existing Line 2, as well as from the surrounding excavation sites. The sampling depths were 3.6–3.8 m, 8.6–8.8 m, 12.4–12.6 m, and 15.4–15.6 m, corresponding to four typical strata (silty clay①, silt②, silty clay③, and sand④). The distribution of these typical strata at the site is shown in Supplementary Figure S2. The basic physical characteristics of each stratum are presented in Table 2.

The thin-walled sampler drilling sampling method was used in this study to obtain undisturbed soil specimens. Using specialized soil sampling equipment, drilling and sampling were conducted in typical strata in accordance with the “Technical Code for Geological Exploration and Sampling in Building Engineering” (JGJT 87-2012) (China, 2011). The undisturbed soil specimens were classified as Grade II. To maintain the physical and chemical properties of the undisturbed soil specimens, each sample was immediately placed into a protective sleeve after being drilled out. Next, the sample was sealed to prevent moisture evaporation from affecting the water content of the sample. The preservation method used for the undisturbed soil specimens was wax sealing. During transportation, the specimens were wrapped in anti-vibration bubble wrap to minimize any potential impact on their structural integrity. The undisturbed soil specimens were stored in a constant-temperature and humidity curing chamber.

Conducting standard consolidation tests on soil specimens from different strata can provide the reference tangent modulus for the HSS model E o e d r e f .

The standard consolidation test was conducted using the WG series high-pressure consolidation apparatus. Cylindrical soil specimens with a diameter of 61.8 mm and a height of 20 mm were used. The loading sequence was 12.5 kPa, 25 kPa, 50 kPa, 100 kPa, 200 kPa, 300 kPa, 400 kPa, 800 kPa, and 1,600 kPa, with each loading stage lasting for 24 h.

After collecting the one-dimensional consolidation test data, further processing is carried out to plot the relationship between the axial strain (horizontal axis) and the consolidation pressure (vertical axis). A quadratic polynomial is used to fit this data, resulting in a fitted curve. By differentiating this curve, the derivative value corresponding to a consolidation pressure of 100 kPa is determined, which represents the reference tangent modulus of the soil E o e d r e f under standard consolidation test conditions. The results are shown in Figure 1.

Triaxial consolidated drained tests can be conducted on soil specimens from different strata to obtain the effective cohesion c ′ , effective internal friction angle φ ′ , failure ratio R f , and reference secant modulus E 50 r e f for the HSS model.

Triaxial consolidated drained tests were performed using a GDS stress-path triaxial testing system. Cylindrical soil specimens with a diameter of 38 mm and height of 76 mm were prepared. Before installation, the undisturbed soil specimens underwent vacuum saturation using a vacuum pump and were held in a vacuum chamber for 12 h. After specimen installation, back-pressure saturation was achieved by applying a confining pressure of 110 kPa and a back pressure of 110 kPa, followed by B-value verification to ensure saturation completeness. Consolidation was conducted under confining pressures of 100 kPa, 200 kPa, and 300 kPa, with a stabilization duration of 36 h for each stage. During the drained shearing phase, an advanced loading test module was employed with a shear rate of 0.0084 mm/min. To ensure sufficient deformation of the soil sample during the testing process, the test was terminated when the specimen reached 16% axial strain ε 1 .

By processing the test data, Mohr’s circles were plotted for the silty clay (Layer ③) and sand (Layer ④) under different confining pressures, as shown in Figure 2. The analysis of these Mohr’s circles can be used to determine the effective cohesion c ′ and the effective internal friction angle φ ′ .

Since the secant modulus under a confining pressure equal to the reference stress is defined as the reference secant modulus, the reference confining pressure is chosen to be equal to the reference stress p r e f , which is 100 kPa. The stress–strain relationship curves for typical strata of Zhengzhou at a reference confining pressure of 100 kPa are shown in Figure 3. The peak deviatoric stress at soil specimen failure is considered as the failure deviatoric stress q f . The slope of the straight line connecting the origin to the 1/2 q f point on the curve, as shown in the figure, is determined to be the reference secant modulus E 50 r e f .

In the HS model (Pramthawee et al., 2011), the relationship between the deviatoric stress q and the axial strain ε 1 in triaxial tests satisfies Equation 1. − ε 1 = 1 2 E 50 · q 1 − q / q a q < q f (1)

Equation 2 can be derived by transforming Equation 1. ε 1 q = ε 1 q a − 1 2 E 50 q < q f (2)

Equations 1, 2 represent the assumed relationships under ideal conditions. In practice, when transforming the relationship q − ε into relationship ε 1 / q − ε 1 , deviations from the theoretical model are inevitable, particularly at the onset and near the point of failure, where these deviations are more significant. To minimize the influence of human error, data points q / q f at 70% and 95% are selected for fitting, resulting in the ε 1 / q − ε 1 relationship curve illustrated in Figure 4. The slope of this straight line is q a , and q f / q a represents the failure ratio.

Conducting triaxial loading–unloading tests on different strata can help obtain the reference loading–unloading modulus E u r r e f for the HSS model.

The equipment, specimen installation, saturation, B-value testing, and consolidation process in this test are the same as those used in the triaxial consolidated drained test. For the loading–unloading stage, an advanced loading module is used. The modulus obtained from the tests varies with the applied confining pressure. To obtain the reference loading–unloading modulus E u r r e f , the applied confining pressure is set equal to the reference stress p r e f , which is 100 kPa. Based on the results of the triaxial consolidated drained test on samples extracted from the same stratum, the expected failure deviatoric stress q f of the specimen is predicted. To simulate the initial stress state that soil may experience in practical engineering applications and to enable the specimen to achieve a stable stress-strain state at a relatively low stress level The specimen is first loaded to 40% of the predicted failure deviatoric stress q f and then axially unloaded. Because reloading the specimen to 60% of the predicted failure deviatoric stress q f is sufficient to form a hysteresis loop with a shape and data quality that meet the requirements for modulus calculation, the specimen is reloaded to 60% of the predicted failure deviatoric stress q f (Wang et al., 2023) after unloading is complete to ensure the efficiency of the test. This marks the end of the loading and unloading process.

The test data were obtained, and a relationship curve between deviatoric stress and axial deformation was plotted, as shown in Figure 5. During the initial loading stage, the curve steadily rises, with stress increasing as strain increases. Once the stress reaches 40% of the predicted failure value, unloading begins, at which point the axial strain continues to increase slowly. Subsequently, both stress and strain begin to decrease. Upon reloading, the stress–strain curve becomes very steep. Once it reaches the predicted level, the loading curve rises slowly and follows the same trend as the previous unloading until the end of the test. Throughout this process, hysteresis loops can be observed in the curves of all strata. These hysteresis loops indicate that the soil exhibits significant energy dissipation and non-linear behavior under cyclic loading. Specifically, the area enclosed by the hysteresis loops represents the energy dissipated during the unloading and loading cycles. a larger area of the hysteresis loop suggests higher energy dissipation and more pronounced plastic deformation, while a smaller area indicates more elastic behavior. By connecting the upper and lower endpoints of the hysteresis loop with a straight line, the slope of this line represents the reference loading–unloading modulus.

Conducting K 0 consolidation tests on different strata can help obtain the initial lateral pressure coefficient K 0 for the HSS model.

The equipment, sample installation, back-pressure saturation, and B-value testing process in the K 0 consolidation test are the same as those used in the triaxial consolidated drained test. In the consolidation stage, the K 0 ​ consolidation module in the CDS Lab Operating System is selected. To accurately simulate the loading conditions encountered in practical engineering, ensure the uniform distribution of pore pressure throughout the testing process, and mitigate the potential interference from the soil’s instantaneous response on the test outcomes, the pressure loading rate set at 0.1 kPa/min. Additionally, the shear strain rate corresponding to 0.1 kPa/min is within the range of 0.1%/min-0.5%/min as stipulated by the “Standard for Geotechnical Testing Method” (GB/T 50,123-2019) (Institute, 2019). To ensure that the soil specimen has sufficient time to fully consolidate under the applied load and thereby guarantee the accuracy of the test results, the test duration was set to 16 h.

The results of the triaxial K 0 ​ consolidation tests on the typical strata of silty clay①, silt②, silty clay③, and sand④ in the Zhengzhou area are shown in Figure 6. It is evident that, except for sand, the difference in the K 0 ​ coefficient between the other typical strata specimens is not significant. The K 0 ​ coefficient of sand is the highest at 0.58, whereas those of silty clay①, silt②, and silty clay③ are 0.51, 0.53, and 0.51, respectively. For sand, the interaction between particles is predominantly governed by frictional forces, with minimal cohesion, and it typically exhibits a relatively low K 0 ​ value. However, the experimental results in this study reveal a relatively higher K 0 ​ value for sand. This deviation is attributed to the fact that the sand used in the experiment was sampled from a deeper sedimentary layer, where it experienced a higher vertical effective stress. This stress caused the sand particles to be rearranged during the long - term sedimentation process, resulting in a more compact structure. Consequently, the measured K 0 ​ value in the experiment was higher than expected.

Conducting resonant column tests on different strata can help obtain the small-strain parameters of the HSS model, including the initial dynamic shear modulus G 0 r e f and the shear strain corresponding to the initial dynamic shear modulus at 70% strain γ 0.7 .

The resonant column test was conducted using a GDS resonant column apparatus with cylindrical soil samples that had a diameter of 50 mm and a height of 100 mm. Consolidation was performed at a reference confining pressure of 100 kPa. The test voltages selected were 0.01 V, 0.02 V, 0.04 V, 0.08 V, 0.016 V, 0.032 V, 0.064 V, 0.128 V, and 0.256 V. Torsional excitation was applied in the order of these voltages.

According to the analysis by Hardin et al. (Hardin and Drnevich, 1972), The relationship between shear strain and dynamic shear stress obtained from the resonant column test can be expressed as follows: τ = γ a + b γ (3)

The dynamic shear modulus G can be defined as G = τ γ (4)

From Equations 3, 4, 5 can be obtained. G = 1 a + b γ (5)

The initial dynamic shear modulus is the dynamic shear modulus when the shear strain γ approaches 0. At this point, the dynamic shear modulus reaches its maximum value. The initial dynamic shear modulus G 0 at a reference confining pressure of 100 kPa is the required small-strain parameter G 0 r e f . The other small-strain parameter γ 0.7 is the shear strain corresponding to 70% of the initial dynamic shear modulus. Figure 7 shows the attenuation curves of the dynamic shear modulus for silty clay and sand strata under a reference confining pressure of 100 kPa. By analyzing the shear modulus attenuation curves of different typical strata, the small-strain parameters can be determined. The test results of these parameters are shown Figure 7.

The soil layers included in the model consist of miscellaneous fill, ②31 clayey silt, ②21 silty clay, ②33 clayey silt, ②22 silty clay, ②41 silty sand, ②51 fine sand, ③37 clayey silt, and ③51 fine sand. Undisturbed soil specimens collected from the strata at the construction site, where the Line eight shield tunnel passes over the existing Line two in the Zhengzhou area, were tested using the methods introduced in Section 2. The parameters of the HSS model for these soil samples are shown in Table 3.

The lining segments of both the new tunnel and the existing tunnel are cast with C50 concrete. When performing calculations, the effect of segment joints and assembly on overall stiffness is considered, with a reduction factor of 90% being applied. The main body of the reinforced concrete slab, as well as the main and secondary beams, are cast with C30 concrete. The slab is modeled using solid elements, while the main and secondary beams are modeled using beam elements in PLAXIS 3D. The uplift piles are cast with C35 concrete and are modeled using embedded pile elements in PLAXIS 3D. The physical properties of the structural materials used in the model are listed in Table 4.

According to the meshing functions in PLAXIS 3D, the solid elements are 10-node tetrahedral elements, beam elements are 3-node line elements that are compatible with the 3-node edges of the solid elements, and plate elements are 6-node elements. To simulate the interaction between soil and structures, a 12-node element is used for the interface. The embedded pile element can be regarded as a combination of beam elements and embedded interface elements. The model consists of 321,965 mesh elements and 479,186 nodes.

In engineering, the grouting pressure is approximately 240–280 kPa. In the model, the grouting pressure at the tunnel axis depth is taken as 260 kPa. Considering the specific weight of the grout, an incremental pressure of 20 kPa/m is applied along the depth. The model simulates ground loss caused by construction activities, such as the contraction of the shield machine and the lining segments, with a ground loss rate set at 0.2%. The model simulates the jack pressure by applying a uniformly distributed load in the direction opposite to that of excavation on the lining segments behind the shield tail. For Line 8, the left line is set at 1844 kPa, and the right line is set at 1,266 kPa.

By selecting the plane in the y-direction at the intersection of Line 8’s left line axis and Line 2’s left line axis as the monitoring section, Figure 8A illustrates a comparison between the simulated and measured values of this section. The surface at the monitoring section exhibits a bimodal uplift, with the peaks located directly above the axes of the two newly constructed tunnels. The maximum uplift on the surface is observed directly above the axis of the right line of Line 8, with a value of 7.038 mm. Additionally, the uplift above the axis of the left line of Line eight is 5.794 mm. The trend presented by the simulated values and the measured values is generally consistent.

By selecting the plane in the y-direction at the intersection of Line 8’s left line axis and Line 2’s left line axis as the monitoring section, Figure 8B illustrates the relationship curve between the final vertical deformation and the depth within the strata from the surface to −4 m above Line 8’s axis at the monitoring section, as well as the measured values at the corresponding locations. The figure reveals that the maximum simulated uplift value is 44.11 mm, occurring at the strata at a depth of −4 m, while the minimum uplift value is 5.61 mm, observed at the surface. Above −1.5 m, the uplift among the strata is almost uniform, while below −1.5 m, the uplift increases with depth and follows a generally linear trend. The overall pattern of strata deformation is largely consistent with the measured values.

Based on the comparative analysis above, it can be seen that the simulated values obtained from the finite-element model calculations using the HSS model show a relatively small difference when compared to the actual measured values. The deformation patterns of surface settlement and stratified settlement of the strata obtained through the model are largely consistent with the measured values. These findings suggest that the application of the HSS model in the construction of close-distance overpassing shield tunnels demonstrates both rationality and accuracy.