Upper-Bound Solutions for Active Face Failure in Shallow Rectangular Tunnels in Anisotropic and Non-homogeneous Undrained Clays

Urbanization is transforming the face of metropolitan areas around the world. As cities develop and populations grow, there is an ever-increasing demand for efficient, safe and sustainable underground infrastructures to accommodate them. In urban areas where there is soft ground and the overburden is relatively shallow, rectangular shield machines are employed to create various underground structures, including utility tunnels, pedestrian walkaways, and parking facilities (Khotbehsara et al. 2014; Mehdizadeh et al. 2022; He et al. 2024; Yu Huat et al. 2025). A rectangular tunnel improves space utilization ratio of the tunnel section by about 20% compared to a circular tunnel which can significantly reduce the amount of excavation required (Chen et al. 2021). Furthermore, circular tunnels must be backfilled to provide a flat base for certain infrastructure projects such as underground expressways and subways. Therefore, rectangular tunnels can lower costs and enhance construction efficiency by saving both time and resources. Vinod and Khabbaz (2019) conducted a numerical study comparing circular and rectangular tunnels to assess surface settlements induced by each type. Their findings indicate that rectangular tunnels cause less settlement than circular ones when constructed at shallow depths in soft ground conditions, making rectangular designs particularly advantageous for such environments. Recent years have seen a notable expansion and advancement in the use of rectangular tunnel boring machines (RTBMs), reflecting their growing prominence and technological evolution. The technology was developed in the 1970s in Japan. Since its development, this technology has become extensively utilized, with particularly significant adoption in China and Singapore in recent years (Mok and Ng 2021). Due to its advantages such as improving safety and time efficiency, as well as reducing traffic congestion and its minimal impact on surface structures, this technology is set to be a key player in the future development of underground spaces.

While excavating tunnels with RTBMs, it is crucial to ensure the stability of the tunnel face. If the applied face pressure is inadequate, the tunnel face may collapse, compromising the structural integrity of the excavation. It is therefore important to determine exactly what the working face pressure is when using shield machines in order to achieve optimal excavation efficiency. There have been several researchers who have conducted research on the stability of tunnel faces in the past several decades, using various approaches including analytical methods (e.g., Anagnostou and Kovári 1994; Davis et al. 1980; Leca and Dormieux 1990; Mollon et al. 2013, 2011; Seghateh Mojtahedi et al. 2021; Zhang et al. 2018a, 2018b), numerical simulations (e.g., Huang et al. 2018a; Mohammadifar et al. 2024b; Ukritchon et al. 2017; Vermeer et al. 2002; Zhang et al. 2020), and experimental investigations (e.g., Chambon and Corte 1994; Chen et al. 2013; Idinger et al. 2011; Liu et al. 2018; Mehmandari et al. 2024).

The limit equilibrium method is a commonly employed analytical technique for assessing the stability of tunnel faces. Horn (1961) developed an early theoretical model to study tunnel face stability using the limit equilibrium method, incorporating a wedge-shaped region in front of the tunnel face subjected to vertical loading from a prism. Afterwards, Anagnostou and Kovári (1994) examined the influences and relationships between various parameters related to slurry shields and tunnel face failure, utilizing an advanced wedge-prism model. The upper-bound theorem of limit analysis is a rigorous mathematical method that has been adopted by many researchers due to its reliability and efficiency in analyzing tunnel face stability. Leca and Dormieux (1990) introduced three-dimensional failure mechanisms for face collapse and blow-out in frictional soils, employing the translational movement of rigid conical blocks to more accurately model these phenomena. As an extension of Leca's model, Soubra et al. (2008) and Mollon et al. (2009) introduced a multi-block mechanism that offered greater flexibility and realism, providing a more adaptable and accurate representation of failure mechanisms compared to Leca’s approach. The results obtained from the multi-block mechanism was found to significantly improve Leca's upper-bound solutions and were relatively consistent with the numerical simulations and centrifuge tests results.

In geotechnical engineering, there are two types of loading conditions: Drained and undrained. Drained condition refers to the situation where the soil mass is permitted to drain freely. This implies that any excess pore water pressure generated by the loading can dissipate through the soil. Drained loadings typically occur in situations where the soil is highly permeable, such as in sandy soils, or when there is sufficient time for the excess pore water pressure to dissipate (long-term condition). Conversely, in low-permeability soils and under short-term conditions, excess pore pressure cannot escape, resulting in an undrained condition. Undrained conditions are common in saturated clays where the water table is high (near the ground surface). In such conditions, the soil is analyzed in terms of total stresses, with a horizontal failure envelope (i.e., φ = 0), which means that the soil's shear strength is defined by its undrained shear strength (Budhu 2010; Das 2019). Given that tunnel face failure can happen abruptly during excavation, it is crucial to examine the stability of the tunnel face in low-permeability clays under undrained conditions. Extensive research has been carried out on tunnel face failure in undrained clays. Applying the kinematic approach of limit analysis, Davis et al. (1980) developed advanced two-dimensional and three-dimensional failure mechanisms, utilizing the translational movement of rigid blocks to thoroughly investigate the stability of tunnel faces in circular tunnels within purely cohesive soils. However, their model was overly simplistic, lacking the complexity needed to yield highly accurate results and thus requiring further refinement. Augarde et al. (2003) employed both finite element limit analysis and the upper-bound method to examine the stability of two-dimensional tunnel face in non-homogeneous clays under undrained conditions. Huang and Song (2013) investigated the same issue using a multi-rigid-block mechanism, achieving enhanced results compared to the solutions provided by Davis et al. (1980). Mollon et al. (2013) introduced two novel failure mechanisms that incorporate continuous deformation of the soil mass, providing a more detailed analysis of the face stability in circular tunnels constructed within cohesive clays. Ukritchon et al. (2017) performed a three-dimensional finite element analysis to investigate the stability of tunnel faces under undrained conditions, factoring in the effects of linearly increasing shear strength with depth. Li et al. (2019) built upon the work of Soubra et al. (2008) by introducing a multi-block failure mechanism incorporating a vertically distributed force applied to the final block, aiming to analyze the face stability of shallow tunnels within multilayered clays. The results from their model were found to align well with both existing solutions and numerical simulations conducted using FLAC^3D software.

A substantial portion of research on face stability assumes soil to be homogeneous and isotropic material. In reality, however, soils often exhibit anisotropic and non-homogeneous behavior in their strength characteristics. There are many factors contributing to soil anisotropy and heterogeneity, including soil deposition, stress states and cementation bonds (Al-Karni and Al-Shamrani 2000; Lee and Rowe 1989). Ignoring anisotropy and non-homogeneity can lead to inaccurate predictions of stability and potentially catastrophic failures. This makes it important to take these factors into consideration when investigating the stability of geotechnical infrastructures. In recent years, a number of researchers have investigated tunnel face stability by integrating the effects of soil anisotropy and non-homogeneity into their models. Yang et al. (2015) developed a two-dimensional failure mechanism within the upper-bound limit analysis framework, utilizing the spatial discretization technique initially developed by Mollon et al. (2010). They examined how anisotropy and non-homogeneity influence both the support pressure and the shape of the failure mechanism in frictional-cohesive soils. Pan and Dias (2016) devised an advanced three-dimensional failure mechanism for assessing the face stability of circular tunnels in frictional-cohesive soils, effectively accounting for the effects of soil anisotropy and non-homogeneity. According to them, both anisotropy and non-homogeneity greatly affect the support pressure, particularly in weak soil layers or soils with a high cohesion. Taking into account the anisotropy and heterogeneity of undrained shear strength in clays, Huang et al. (2018b) proposed a failure mechanism featuring a shear zone coupled with a cylindrical rigid block to analyze the stability of circular tunnel faces under undrained conditions. Using a 3-D rotational failure mechanism, Li and Yang (2020) examined the tunnel face stability in two-layer soils considering anisotropy and non-homogeneity of different soil parameters. Wang et al. (2023b) investigated the face stability of circular tunnels using a double discrete method, focusing on the anisotropy of soil cohesion as well as the non-homogeneity of unit weight, cohesion, and internal friction angle.

All of the studies discussed thus far have predominantly focused on the face stability of circular tunnels. However, in practice, rectangular and square tunnels are more commonly excavated due to their distinct advantages over circular designs (Abbo et al. 2013). Despite their prevalence in the field, research on the face stability of rectangular tunnels remains relatively scarce (Song et al. 2024). The analytical models proposed by previous researchers are not directly applicable to rectangular tunnels, as the geometric differences prevent accurate estimations of limit support pressure for wide rectangular tunnels. Studies by Lai et al. (2023), Shiau et al. (2024) and Song et al. (2024) have shown that the failure mechanisms governing rectangular tunnels differ significantly from those of circular tunnels. This underscores the need for failure models tailored specifically to rectangular tunnels. Wilson et al. (2017) employed upper-bound rigid block method as well as finite element limit analysis to explore the stability of rectangular tunnels in purely cohesive soils under plane strain condition. However, since tunnel stability is inherently a three-dimensional problem, two-dimensional models are often inadequate due to oversimplification. Using spatial discretization and Terzaghi earth pressure theory, Chen et al. (2019) utilized spatial discretization in combination with Terzaghi’s earth pressure theory to introduce an enhanced failure mechanism rooted in the kinematic approach of limit analysis. Their model was designed to assess the face stability of shallow square tunnels in frictional-cohesive soils, explicitly accounting for the heterogeneity in the friction angle. However, Chen's failure model cannot be applied to purely cohesive soils under undrained conditions. Liu et al. (2021) studied face stability in rectangular tunnels within frictional-cohesive soils by proposing a two-dimensional rotational mechanism. However, a 2-D mechanism for rectangular tunnels is limited as it cannot adequately incorporate the width-to-height ratio effects. Jafari and Fahimifar (2022) conducted a comprehensive investigation into the face stability of rectangular shield-driven tunnels in undrained clays. They introduced two novel failure mechanisms specifically designed to evaluate the face pressure required to prevent both collapse and blowout scenarios. Nonetheless, their results did not achieve high accuracy when compared to numerical simulations, and the combined effects of anisotropy and non-homogeneity were not considered. (Shiau et al. 2024) investigated rectangular tunnel stability using finite element limit analysis (FELA) in cohesive soils but did not propose an analytical model or consider soil anisotropy and non-homogeneity. Wang et al. (2023a, b) examined rectangular tunnel stability in cohesive-frictional soils by introducing a 3-D rotational failure mechanism based on the discretization method; however, this model is also unsuitable for undrained clays and does not account for anisotropy or non-homogeneity. Using numerical simulations, Song et al. (2024) investigated rectangular tunnel face stability with the limit equilibrium method, exploring the effect of varying height-to-width ratios on critical support pressure.

In purely cohesive soils, the combined influence of anisotropy and non-homogeneity on undrained shear strength have yet to be examined when assessing the face stability of rectangular tunnels. This study aims to bridge this gap by presenting a rigorous upper-bound solution for evaluating the face stability of shallow rectangular tunnels in clays exhibiting both anisotropic and non-homogeneous properties. In this study, we focus exclusively on the active failure of the tunnel face, a scenario that is more commonly encountered in practical applications. Employing the upper-bound theorem of limit analysis, an advanced three-dimensional multi-block failure mechanism is proposed. This new model integrates soil arching effects and incorporates the complexities of anisotropy and non-homogeneity in undrained shear strength, offering an enhanced framework for a more accurate and nuanced assessment of face stability in shallow rectangular tunnels. For practical implementation, we offer a closed-form expression to calculate the critical support pressure. To ensure the robustness and precision of the proposed failure mechanism, we rigorously compare the upper-bound solution results with finite element simulations and other established methods. Our model's performance is further benchmarked against existing approaches for circular tunnels, assessing its applicability to shallow circular tunnel face stability. Finally, an in-depth analysis is conducted to evaluate how various geometrical and geotechnical factors influence the stability of rectangular tunnel faces.

Figure 1 provides a schematic overview of the problem addressed in this study. We examine a rectangular tunnel, characterized by a height \(D\) and width \(B\), excavated by an RTBM beneath a cover depth \(C\) within an anisotropic and non-homogeneous purely cohesive clay. In this idealized scenario, a rigid lining is positioned in immediate proximity to the tunnel face. A uniform support pressure \(\sigma_{t}\) is applied to the tunnel face, and a uniform vertical surcharge \(\sigma_{s}\) is exerted on the ground surface. In shield tunneling, it is reasonable to assume uniform support pressure if compressed air is used to support the face (Mollon et al. 2010). The Tresca failure criterion is utilized in the analysis, which means that the soil's strength is defined by its undrained shear strength \(c_{u}\). Due to vertical soil deposition, the undrained shear strength of clays is not constant with depth. To develop a comprehensive failure model that yields accurate results, it is essential to account for the variation of undrained shear strength with depth. This study adopts a linear model for undrained shear strength variation, as illustrated in Fig. 1, which is a reasonable assumption for normally consolidated clays, as discussed by Augarde et al. (2003). The non-homogeneous undrained shear strength, represented as increasing linearly with depth, is expressed mathematically by the following formula:

In this expression, \(c_{u0}\) denotes the horizontal undrained shear strength at the ground surface, \(y\) indicates the depth beneath the ground surface. The parameter \(\rho\) reflects the rate at which the undrained strength increases with depth. If \(\rho = 0\), the soil is homogeneous, exhibiting a constant shear strength throughout its depth.

Anisotropy in soil mechanics refers to the condition where a soil's mechanical properties vary depending on the direction. This characteristic is notably more common in clayey soils compared to granular soils (Das 2019). Numerous studies have delved into the anisotropy of undrained shear strength, examining how this directional variability impacts soil behavior (Casagrande and Carillo 1944; Hansen and Gibson 1949; Lee and Rowe 1989; Lo 1965). The curves illustrated in Fig. 2 describe the directional variations in undrained shear strength. Considering that \(i\) denotes the angle between the direction of the major principal stress and the vertical direction, the anisotropic undrained shear strength \(c_{ui}\) along the major principal stress direction can be represented by the following expression (Casagrande and Carillo 1944):

In the above expression, \(c_{uh}\) and \(c_{uv}\) represent the horizontal and vertical undrained shear strengths whose maximum principal stress directions coinciding with horizontal and vertical directions, respectively. It is clear that \(c_{ui} = c_{uh} = c_{uv}\) in isotropic soils. \(c_{uh}\) and \(c_{uv}\) can be determined by geotechnical experiments. In the case of undrained clays, it is assumed that the failure surface forms an angle of \({\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-0pt} 4}\) with the direction of the major principal stress. This assumption is based on Tresca failure criterion and possible slip planes in purely cohesive soils as mentioned by Ding et al. (2019). Thus, the anisotropic angle \(i\) can be determined based on the geometry of the proposed failure mechanism. According to Lo (1965), the ratio of horizontal to vertical principal undrained strengths (\({{c_{uh} } \mathord{\left/ {\vphantom {{c_{uh} } {c_{uv} }}} \right. \kern-0pt} {c_{uv} }}\)) remains constant throughout the soil mass. As a convenient way to approach the problem of anisotropy, a coefficient, denoted by \(k = {{c_{uh} } \mathord{\left/ {\vphantom {{c_{uh} } {c_{uv} }}} \right. \kern-0pt} {c_{uv} }}\), is introduced to be as a representative of the degree of anisotropy in soil. Various studies have reported different ranges for the parameter k. Lo (1965) identified values between approximately 0.6 and 1.3, while Davis and Booker (1973) found a range from 0.75 to around 1.56. Additionally, Lee and Rowe (1989) determined that k varies from 0.77 to 1.27 across different soil types. In this study, the considered range for k is from 0.6 to 1.4. According to Lee and Rowe (1989), \(k < 1\) pertains to normally or lightly overconsolidated clays, \(k > 1\) represents highly overconsolidated clays, and a value of \(k = 1\) denotes isotropic soil conditions. Consequently, Eq. (2) can be reformulated as:

It is assumed that \(c_{u0}\) represents the horizontal principal undrained shear strength at the ground surface. Thus, by substituting Eq. (1) into Eq. (3) the expression for undrained shear strength accounting for both anisotropy and non-homogeneity can be derived as follows:

To thoroughly evaluate the face stability of rectangular tunnels in non-homogeneous and anisotropic undrained clays, nine essential variables \(\left\{ {\sigma_{t} ,\sigma_{s} ,c_{u0} ,\rho ,k,\gamma ,C,B,D} \right\}\) need to be considered. Dimensional analysis is a powerful tool for simplifying intricate problems by reducing them to fundamental dimensions. This technique is used to derive seven dimensionless parameters, which are as follows:

Under undrained conditions, Davis et al. (1980) suggested that the two parameters \({{\sigma_{t} } \mathord{\left/ {\vphantom {{\sigma_{t} } {c_{u} }}} \right. \kern-0pt} {c_{u} }}\) and \({{\sigma_{s} } \mathord{\left/ {\vphantom {{\sigma_{s} } {c_{u} }}} \right. \kern-0pt} {c_{u} }}\) could be combined into one parameter, denoted as \({{\left( {\sigma_{s} - \sigma_{t} } \right)} \mathord{\left/ {\vphantom {{\left( {\sigma_{s} - \sigma_{t} } \right)} {c_{u} }}} \right. \kern-0pt} {c_{u} }}\). As a result, the face stability analysis is governed by five dimensionless parameters: load parameter \({{\left( {\sigma_{s} - \sigma_{t} } \right)} \mathord{\left/ {\vphantom {{\left( {\sigma_{s} - \sigma_{t} } \right)} {c_{u0} }}} \right. \kern-0pt} {c_{u0} }}\), soil unit weight ratio \({{\gamma D} \mathord{\left/ {\vphantom {{\gamma D} {c_{u0} }}} \right. \kern-0pt} {c_{u0} }}\), cover depth ratio \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-0pt} D}\), width ratio \({B \mathord{\left/ {\vphantom {B D}} \right. \kern-0pt} D}\), non-homogeneity ratio \({{\rho D} \mathord{\left/ {\vphantom {{\rho D} {c_{u0} }}} \right. \kern-0pt} {c_{u0} }}\) and anisotropy factor \(k\). To streamline the evaluation of tunnel face stability in purely cohesive soils, Broms and Bennermark (1967) proposed the use of a stability number \(N\), which is defined as follows:

Analytical methods have become an indispensable tool in geotechnical engineering, particularly when it comes to assessing the stability of various structures. Owing to its precision and robust framework, the limit analysis method has gained more attention from researchers compared with the limit equilibrium and slip line methods. Chen (1975) utilized both the lower-bound and upper-bound theorems of limit analysis to systematically study and evaluate the stability of various geotechnical structures. This study employs the upper-bound technique of limit analysis to rigorously examine the face stability of rectangular tunnels. The upper-bound method of limit analysis states that by equating the rate of work done by external forces to the rate at which internal energy is dissipated within a kinematically feasible failure mechanism, one can establish an upper bound for the real failure load (Chen 1975). The pressure applied to the tunnel face counteracts the forces driving active failure, functioning as a negative load by resisting soil movement. Consequently, the upper bound on the failure load derived from this method will either be less than or equal to the real load needed to maintain stability. The key to effectively applying the upper-bound method is to identify a failure mechanism that closely aligns with both experimental data and numerical simulations. This approach ensures that the computed collapse pressure is the highest possible upper bound, providing a value that closely approximates the true failure load.

Finite element analysis (FEA) has emerged as a pivotal technique for solving complex stability problems in geotechnical engineering, providing detailed insights into intricate structural behaviors. In this research, we utilize PLAXIS 3D CONNECT Edition V21, a premier commercial software extensively adopted in the field, to evaluate the face stability of rectangular tunnels embedded in purely cohesive soils and to validate the proposed failure mechanism. Recognizing the symmetrical nature of rectangular tunnels, the numerical model is constructed for half of the tunnel's cross-section to enhance computational efficiency. The soil is modeled as an elastic-perfectly plastic material, following the Tresca failure criterion. To enhance computational efficiency, we select a Young’s modulus of 500c_u and a Poisson's ratio of 0.495. This high modulus is chosen to speed up calculations while minimizing its impact on the limit support pressure (Vermeer et al. 2002). In the simulations, we implement the following boundary conditions: the base of the model is fully constrained to prevent displacement; the ground surface is allowed unrestricted movement in all directions to represent natural surface behavior; and the vertical boundaries are constrained to prevent any horizontal displacement, ensuring accurate representation of tunnel interactions with the surrounding soil.

Figure 12 presents a detailed illustration of the meshed model for a square tunnel in PLAXIS software. To ensure high precision, the mesh is meticulously refined in front of the tunnel face, where failure is anticipated to occur. The dimensions of the numerical models are chosen to be sufficiently large to ensure that the failure pattern is not influenced by the boundaries of the model. Optimal dimensions for the models are determined through a systematic approach, with minimal changes in results observed even with larger dimensions. While typical tunnel excavation is represented through staged construction in PLAXIS software, this study focuses solely on analyzing face stability and displacement. As a result, the model assumes the tunnel is “pre-installed”, with excavation, lining, and face pressure application treated as a single-phase process. This approach simplifies the analysis and aligns with methods employed in previous studies (Ukritchon et al. 2017; Vermeer et al. 2002).

A uniform initial support pressure, similar to that applied when using compressed air shields, is imposed on the tunnel face. To determine the critical support pressure, a stress-controlled method is employed. In this procedure, tunnel face failure is simulated by incrementally lowering the applied support pressure until collapse is observed. With each reduction, the tunnel face’s deformation is carefully tracked. Following the methodology suggested by Vermeer et al. (2002), displacements at several points along the tunnel face are recorded, with the point experiencing the greatest displacement identified as the control point. Figure 13 illustrates a representative curve showing how face support pressure varies with the horizontal displacement at the control point. As the support pressure decreases, the control point's displacement grows progressively larger. This continues until a critical threshold is reached, where even a marginal reduction in pressure results in a sharp rise in displacement. At this critical juncture, the pressure–displacement curve flattens, nearing a horizontal line, signifying the onset of tunnel face failure. The corresponding support pressure at this point is identified as the critical collapse pressure. To accurately pinpoint this pressure, extensive numerical simulations are conducted, refining the results and ensuring precision in identifying the critical collapse pressure. Figure 14 presents shadings of incremental displacements at the moment of failure, derived from our numerical results. The incremental displacements at failure serve as valuable indicators for observing potential failure mechanisms. The figure demonstrates that failure initiates at the tunnel face and propagates upward toward the ground surface, developing a chimney-like failure pattern. This behavior is consistent with earlier findings from other numerical investigations (Huang et al. 2018a; Li et al. 2019; Ukritchon et al. 2017). The control point, identified as the location experiencing the greatest displacement, is positioned below the center of the tunnel face. Furthermore, it is noteworthy that the failure zone is precisely confined within the refined mesh area, ensuring the precision and reliability of the results.

This section delves into the influence of various geometrical and geotechnical parameters on the stability of rectangular tunnel faces. We also investigate the effects of non-homogeneity and anisotropy on the feature of the proposed failure mechanism.

This study has examined the face stability of shallow rectangular tunnels excavated by RTBMs in purely cohesive soils. We developed a novel failure mechanism for evaluating active tunnel face failure, using the upper-bound theorem of limit analysis. This mechanism incorporates soil arching effects along with considerations of anisotropy and non-homogeneity in undrained shear strength. By optimizing the geometric parameters of this failure model, we obtained the optimum upper-bound solutions for collapse pressure. We also formulated design fornulas to facilitate efficient computation of limit support pressure, thereby enhancing practical usability. The analytical solutions derived from this new failure mechanism were validated through comparisons with finite element simulations and existing solutions across multiple cases. Furthermore, a comparison between existing solutions for circular tunnels and the specific case of square tunnels was conducted. An extensive analysis was conducted to assess how different geometrical and geotechnical factors affect the stability of rectangular tunnel faces. The principal findings are summarized as follows:

The proposed model offers significant potential for practical application in geotechnical engineering. After conducting geotechnical investigations and determining site-specific ground characteristics, the design formulas derived in this study can serve as a valuable resource for estimating the required support pressure prior to excavation. The flowchart provided in Fig. 11 further streamlines the practical implementation of the model, enabling engineers to integrate its use efficiently during the early stages of tunnel excavation planning. This model is particularly well-suited for rectangular tunnels with shallow cover depths in anisotropic and non-homogeneous clays. By addressing the anisotropy and non-homogeneity of undrained shear strength, the model allows for a more precise and tailored analysis of tunnel face stability. Such precision not only enhances safety but also optimizes resource allocation by reducing the risk of conservative overdesigns or underestimated support requirements, which can lead to costly errors in tunneling operations. Furthermore, the findings highlight the importance of considering both anisotropy and non-homogeneity in stability analyses. Ignoring anisotropy in lightly overconsolidated clays may lead to overly conservative support pressure estimates, increasing project costs unnecessarily. Conversely, failing to account for anisotropy in heavily overconsolidated clays may result in critical underestimations, posing safety risks. Similarly, neglecting non-homogeneity can lead to significant overestimations of support pressure, reducing the efficiency of tunnel designs. By addressing these factors, the proposed model provides engineers with a reliable framework for improving safety and optimizing resources in tunneling projects.

While the proposed model provides significant insights into the face stability of shallow rectangular tunnels in undrained clays, it is subject to several limitations. The assumption of uniform support pressure, while valid for compressed air shields, may not accurately represent the non-uniform pressure distributions observed with slurry shield machines. Future research should analyze the impact of depth-dependent support pressure to refine face stability predictions for rectangular tunnels. The model demonstrates reliable performance for square and wide rectangular tunnels (\({B \mathord{\left/ {\vphantom {B D}} \right. \kern-0pt} D} \ge 1\)), but its accuracy diminishes for tall rectangular tunnels (\({B \mathord{\left/ {\vphantom {B D}} \right. \kern-0pt} D} < 1\)). To address this, future studies should propose new failure mechanisms specifically designed for tall rectangular tunnels. Additionally, this study assumes a linear variation of undrained shear strength to represent non-homogeneity, which could be extended in future research to include alternative soil profiles, such as layered soils, making the model applicable to more complex geotechnical conditions. Finally, the proposed model proves effective for shallow tunnels (\({C \mathord{\left/ {\vphantom {C D}} \right. \kern-0pt} D} < 1.5\)) but shows limitations when applied to deeper tunnels. Developing advanced failure mechanisms for deeper rectangular tunnels remains a promising direction for future research. By addressing these limitations, future studies can provide more comprehensive solutions tailored to diverse rectangular tunneling conditions.