Failure Mode of Rectangular Shallow Tunnel in Unsaturated Fine-Grained Soil Under Temperature Influence

Abstract

Failure analysis has always been among the key research focuses in underground tunneling, particularly in forecasting the collapse risk of tunnel crowns, which bears great engineering and practical significance for tunnel safety assessment. In practical engineering, the soil surrounding shallow tunnels and other underground chambers is typically unsaturated. With the advancement of tunneling technology, shallow tunnels affected by ground temperatures are increasingly common, making it essential to incorporate temperature effects into the stability analysis of unsaturated shallow tunnels. This paper proposes a novel framework for analyzing the stability of shallow rectangular tunnel crowns under temperature influence. By adopting a temperature-dependent effective stress model for unsaturated soils combined with the soil–water characteristic curve, temperature influence is integrated into the calculation of apparent cohesion in unsaturated soils. The upper bound theorem and a multi-rigid-block failure mechanism are adopted to assess crown stability, with the geometry of the failure mechanism determined through a compatible velocity field. New analytical expressions are derived. Through calculating the internal energy dissipation rate, considering temperature effects and external work rate, the critical support pressure at the tunnel crown is obtained using the Sequential Quadratic Programming (SQP). Discussions of temperature and other unsaturated soil parameters are carried out to explore their effects on the stability of shallow tunnels. Results demonstrate that temperature significantly influences the tunnel’s critical support pressure, with the extent of this impact primarily dependent on the unsaturated soil type and seepage conditions. Furthermore, the theoretical framework developed in this study provides a more accurate description for unsaturated fine-grained soils. This study introduces a novel integration of thermal influences into the upper bound theorem, applying this enhanced methodology to the stability assessment of shallow rectangular tunnel crowns. The resulting failure model and analytical framework establish a rigorous upper bound solution for crown stability, thereby furnishing a more accurate theoretical foundation for subsequent tunnel face support strategies.

1. Introduction

The stability of tunnels has consistently remained a focal concern for scholars in engineering. Over recent decades, the rapid development of urban subway tunnels and utility tunnels for power and communication has heightened attention toward shallow-tunnel stability, leading to a series of achievements across various research directions [1,2,3,4,5]. These accomplishments primarily fall into two categories, which are model testing and numerical simulation. Beyond these approaches, many scholars have also explored theoretical extensions and refined models to achieve more accurate and reliable stability assessments for tunnels [6,7,8].

Current theoretical research on tunnel stability mainly employs the limit equilibrium method, the slip-line method, and the limit analysis method. The slip-line method involves deriving fundamental differential equations and solving the problem by determining a “slip-line network.” The limit equilibrium method establishes a simplified failure mechanism that enables limit state analysis through force equilibrium equations. In contrast, the limit analysis method bypasses considerations of stress–strain states and complex loading conditions, directly solving for the ultimate failure load via energy balance equations. Consequently, it yields more precise solutions than alternative approaches. The limit analysis theorem is divided into the upper bound theorem and the lower bound theorem. The lower bound theorem requires constructing a statically admissible stress field, which imposes extremely stringent conditions that are rarely satisfied in general engineering practice, thus making it seldom employed. Conversely, the upper bound method defines an admissible velocity field and calculates solutions through the principle of energy balance. This approach avoids solving complex partial differential equations, thus gaining widespread application [9].

With the development of underground engineering, tunnels in unsaturated soil, including those in high-temperature regions and municipal energy tunnels, are increasingly prevalent. Temperature is a salient and non-negligible factor affecting their stability. Many scholars have studied tunnel problems under temperature influence. The effects of temperature and seasonal thawing-layer depth on rock mass stability during permafrost tunnel construction were investigated by Shen et al. [10], with their analysis encompassing temperature field distributions at varying depths, temporal variations in thaw depth, and consequent impacts on crown settlement and rock stability. Lei et al. [11] employed the finite element method (FEM) to analyze the structural stability of an underwater shield tunnel under varying temperature conditions, with a focus on the characteristics of temperature field variations, maximum principal stress, and tunnel settlement. Yao et al. [12] experimentally examined how temperature differentials between the tunnel lining’s inner and outer walls influenced thermally generated stresses, subsequently validating their empirical findings through numerical simulations. While these studies collectively emphasize the critical role of temperature in tunnel stability, research on collapse mechanisms in unsaturated tunnel crowns under thermal effects remains notably limited.

Crown collapse, also termed “roof fall” incidents, refers to the failure of surrounding rock at the tunnel crown that occurs after drill-and-blast excavation disturbs the rock mass, combined with delayed or insufficient support. In deep tunnels, such instability-induced collapses are typically confined to a limited area of the crown. This localized behavior arises because the undisturbed rock above forms a natural arching structure that redistributes overburden loads. Conversely, for shallow tunnels, such as municipal or metro tunnels, the proximity to the ground surface means instability can propagate upward, resulting in both overall subsidence of the overlying strata and localized surface collapse. To determine the collapse extent of unsupported faces in shallow unsaturated tunnels, the limit analysis theory has been applied by several researchers to assess the stability of surrounding rock at tunnel crowns. For shallow tunnels in cohesionless soil, model tests and limit analysis methods (upper bound and lower bound methods) were employed by Atkinson and Potts [13]. Davis et al. [14] proposed four distinct failure modes for shallow tunnels in cohesive soils under undrained conditions. They conducted a stability analysis of surrounding rock using the upper bound approach while simultaneously investigating face instability and localized failure phenomena. These researchers universally assumed uniformly distributed support pressure, primarily focusing on cohesionless soils ($c=0$) and undrained cohesive soils ($\phi =0$). Consequently, their applicability remains limited.

Through experimental observation of collapse modes and velocity distributions in tunnel crown rock at ultimate limit states, researchers have developed diverse two-dimensional failure models grounded in the upper bound theorem. Typical two-dimensional failure mechanisms include the multi-rigid-block mechanism [15] and the rotational failure mechanism [16]. Compared with three-dimensional failure mechanisms, two-dimensional mechanisms simplify the failure pattern by neglecting the three-dimensional soil arching effect. However, due to their straightforward construction, they remain widely adopted by many researchers for stability analysis.

Previous studies on the stability of tunnels in unsaturated soils have predominantly employed the Mohr–Coulomb failure criterion. While this criterion accurately calculates the shear strength of saturated soils, its application to unsaturated soils yields limited accuracy, exhibiting marked deviations from actual engineering outcomes [17,18,19]. Significant advances have been achieved in research on shear strength theories for unsaturated soils. Bishop et al. [20] pioneered the integration of unsaturated effective stress principles into the Mohr-Coulomb failure criterion, establishing a shear strength formulation for unsaturated soils. Their work posited a critical dependence of unsaturated soil shear strength on unsaturated state parameters, $\chi$. However, this approach suffers from theoretical deficiencies: when the unsaturated state parameter $\chi$ equals zero (i.e., in dry soils), it produces erroneous predictions of stress at zero suction, while accurate quantification of $\chi$ becomes highly problematic under high-suction conditions. Accordingly, Fredlund et al. [21] proposed a shear strength criterion incorporating two stress state variables. They asserted that matric suction and external loading constitute distinct stress state variables, necessitating separate characterization of matric suction and net normal stress to define the stress state in unsaturated soils. When evaluating particle size effects on the soil–water characteristic curve (SWCC), gradation parameters were adopted by Chiu et al. [22] to determine the $n$ and $\alpha$ coefficients in the van Genuchten model. Although these studies established expressions for unsaturated shear strength, neither accounted for thermal effects, leaving research on temperature-dependent shear strength in unsaturated soils notably underdeveloped.

This study investigates the relationship between temperature, matric suction, and shear strength within the framework of unsaturated soil mechanics, thereby analyzing temperature effects on the stability of shallow rectangular tunnels in unsaturated strata. Employing upper bound limit analysis, we established a collapse mechanism for the surrounding rock above the tunnel crown and formulated an equilibrium equation balancing external work rates with internal energy dissipation rate influenced by temperature. This approach yielded an upper bound solution for the support pressure of tunnel-surrounding rock. Furthermore, the influence of other key unsaturated soil parameters on crown stability was evaluated, offering valuable references for subsequent research on unsaturated shallow-tunnel stability.

2. Temperature Influence on Shear Strength in Unsaturated Soils

The influence of temperature on the stability of unsaturated tunnels primarily manifests through alterations in soil water content, which modify matric suction and thereby induce thermal variations in apparent cohesion. Consequently, this process alters the shear strength of the soil [23]. An analytical model for determining matric suction in unsaturated soils under undrained heating conditions was proposed by Thoat and Vahedifard [24], who observed a monotonic decrease in matric suction with rising temperature during undrained heating. Alsherif and McCartney [25] investigated the thermal behavior of unsaturated silt under high-temperature and high-suction conditions, revealing significant temperature-dependent effects on both shear strength and volumetric deformation. For constant water content, Wan et al. [26] demonstrated that rising temperatures induce a downward shift in the soil–water retention curve (SWRC), consequently reducing matric suction. This reduction in matric suction diminishes effective stress and lowers the factor of safety (FS). Under drained conditions, however, temperature increases enhance evaporation and reduce water content, thereby elevating matric suction and effective stress while augmenting soil shear strength [27]. A temperature-dependent matric suction equation and corresponding SWCC model incorporating thermal effects were formulated by Xiao et al. [28]. Their findings demonstrate that with constant matric suction, an increase in temperature leads to a decrease in saturation. Collectively, these studies confirm the significant influence of temperature on unsaturated soil strength, underscoring the necessity of incorporating thermal effects as a critical factor in stability analyses of unsaturated tunnel crowns.

Lu and Likos [29] extended and refined the classical Terzaghi effective stress principle based on Bishop’s framework, proposing a unified formulation for effective stress in both saturated and unsaturated soils:

$${\sigma }^{\prime }=\sigma -u_{\mathrm{a}}-{\sigma }^{\mathrm{s}}$$

(1)

where $\sigma$ denotes total stress, $u_{\mathrm{a}}$ represents pore air pressure, ${\sigma }^{\prime }$ signifies effective stress, $\left(\sigma -u_{\mathrm{a}}\right)$ is net normal stress, and ${\sigma }^{\mathrm{s}}$ corresponds to the soil suction stress characteristic curve (SSCC) with the general functional form:

$${\sigma }^{\mathrm{s}}=-\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)\hspace{1em}\hspace{1em}{u}_{\mathrm{a}}-u_{\mathrm{w}}\le 0$$

(2a)

$${\sigma }^{\mathrm{s}}=f\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)\hspace{1em}\hspace{1em}{u}_{\mathrm{a}}-u_{\mathrm{w}}>0$$

(2b)

where $u_{\mathrm{w}}$ denotes pore water pressure, and $f$ represents a scaling function relating suction stress to matric suction. Lu and Likos [29,30] demonstrated that the suction stress characteristic curve (SSCC) could be determined through either shear strength tests, tensile strength experiments, or theoretical formulations.

Lu and Godt [31] proposed an approximate expression for suction stress:

$${\sigma }^{\mathrm{s}}=-\left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)S_{\mathrm{e}}$$

(3)

where $S_{\mathrm{e}}$ denotes the effective degree of saturation. In the SWCC model proposed by van Genuchten [32], the effective degree of saturation $S_{\mathrm{e}}$ is expressed as

$$S_{\mathrm{e}}={\left\{{\displaystyle \frac{1}{1+{\left[\alpha \left(u_{\mathrm{a}}-u_{\mathrm{w}}\right)\right]}^{\mathrm{n}}}}\right\}}^{1-{\displaystyle \frac{1}{\mathrm{n}}}}$$

(4)

where $n$ is the pore size distribution parameter and $\alpha$ represents the reciprocal of the soil entry pressure. Both $n$ and $\alpha$ serve as empirical fitting parameters. Let the matric suction be denoted by the parameter $\psi$, which can be expressed as

$$\psi =u_{\mathrm{a}}-u_{\mathrm{w}}$$

(5)

Substituting Equations (4) and (5) into Equation (3) to eliminate the effective degree of saturation, $S_{\mathrm{e}}$ yields a closed-form expression for suction stress across the entire matric suction range, incorporating thermal dependence:

$${\sigma }^{\mathrm{s}}=-{\displaystyle \frac{\psi }{{\left[1+{\left(\alpha \psi \right)}^{\mathrm{n}}\right]}^{\mathrm{n}-1/\mathrm{n}}}}$$

(6)

Combining suction stress-based effective stress with the classical Mohr–Coulomb failure criterion yields the shear strength for unsaturated soils:

$$\tau =c^{\prime }+\left(\sigma -u_{\mathrm{a}}-{\sigma }^s\right)\tan {\phi }^{\prime }$$

(7)

where ${\phi }^{\prime }$ denotes the effective friction angle, $c^{\prime }$ represents the effective cohesion, and $\tau$ signifies the shear strength. In prior research, the contribution of matric suction to shear strength has been termed apparent cohesion [31,32,33], which can be expressed as

$$c_{\mathrm{app}}=-{\sigma }^{\mathrm{s}}\tan {\phi }^{\prime }$$

(8)

where $c_{\mathrm{app}}$ represents the apparent cohesion.

Equations (1)–(8) were originally established for ambient conditions. Building upon this foundation, Grant and Salehzadeh [34] proposed an expression for matric suction incorporating thermal effects:

$$\psi ={\psi }_{{\mathrm{T}}_{\mathrm{r}}}\left({\displaystyle \frac{\beta +T}{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}}\right)$$

(9)

where $T$ denotes temperature (in Kelvin). For intuitive representation, while Kelvin is used in calculations, Celsius is adopted for subsequent data and textual analyses. $T_{\mathrm{r}}$ signifies the reference temperature, ${\psi }_{{\mathrm{T}}_{\mathrm{r}}}$ represents matric suction at the reference temperature (0 °C), and ${\beta }_{{\mathrm{T}}_{\mathrm{r}}}$ signifies the regression parameter at the reference temperature, which depends on contact angle, surface tension, and immersion enthalpy per unit area. ${\beta }_{{\mathrm{T}}_{\mathrm{r}}}$ and $\beta$ can be calculated using the following expressions:

$${\beta }_{{\mathrm{T}}_{\mathrm{r}}}=-{\displaystyle \frac{\Delta h_{{\mathrm{T}}_{\mathrm{r}}}}{C_1}}$$

(10)

$$\beta ={\displaystyle \frac{-\Delta h}{-\Delta h+\left(\alpha +b_{\mathrm{n}}{T}_{\mathrm{r}}\right)\cos {\alpha }^{\prime }}}$$

(11)

where ${\alpha }^{\prime }$ represents the temperature-affected soil–water contact angle; $a$ and $b_{\mathrm{n}}$ are fitting parameters, which can be approximated as $a=0.11766 \mathrm{N}/\mathrm{m}$, $b_{\mathrm{n}}=-0.0001535 \mathrm{N}/\left(\mathrm{m}\cdot \mathrm{K}\right)$ [35]; and $\Delta h$ denotes the immersion enthalpy per unit area, whose temperature dependence is expressed by the following equation:

$$\Delta h=\Delta h_{{\mathrm{T}}_{\mathrm{r}}}{\left({\displaystyle \frac{1-T_{\mathrm{r}}}{1-T}}\right)}^{0.38}$$

(12)

where $\Delta h_{{\mathrm{T}}_{\mathrm{r}}}$ denotes the immersion enthalpy per unit area at reference temperature and the cosine of the soil–water contact angle can be expressed by the following equation [34]:

$$\cos {\alpha }^{\prime }={\displaystyle \frac{-\Delta h+T{C}_1}{a+b_{\mathrm{n}}T}}$$

(13)

where $C_1$ is a constant defined by the expression:

$$C_1={\displaystyle \frac{\Delta h_{{\mathrm{T}}_{\mathrm{r}}}+\left(a+b_{\mathrm{n}}{T}_{\mathrm{r}}\right)\cos {\alpha }_0}{T_{\mathrm{r}}}}$$

(14)

where $\cos {\alpha }_0$ represents the cosine of the soil–water contact angle at reference temperature $T_{\mathrm{r}}$, typically assigned an empirical value of 0.8.

Lu and Griffiths [36] described the vertical distribution of matric suction in unsaturated soils under steady-state conditions. Building upon their work, this study incorporates thermal effects, integrates Darcy’s law into the Gardner model, and accounts for the boundary condition at the water table ($z=0$ and $\psi =0$). Through this synthesis, we derived the depth-dependent distribution of matric suction in unsaturated soils:

$$\psi ={\displaystyle \frac{1}{\alpha }}\ln \left[\left(1+{\displaystyle \frac{q^{\prime }}{k_{\mathrm{s}}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-{\displaystyle \frac{q^{\prime }}{k_{\mathrm{s}}}}\right]\left({\displaystyle \frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}}\right)$$

(15)

where $k_{\mathrm{s}}$ denotes the saturated hydraulic conductivity, ${\gamma }_{\mathrm{w}}$ represents the unit weight of water, $q^{\prime }$ signifies the steady-state vertical fluid flux (negative for infiltration, positive for evaporation), and $z$ is the vertical distance from the water table to a specific point on the failure mechanism surface.

Substituting Equations (14) and (15) into Equations (6) and (8), the expressions for suction stress ${\sigma }^{\mathrm{s}}$ and apparent cohesion $c_{\mathrm{app}}$ can be obtained.

$${\sigma }^{\mathrm{s}}=-{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)}{{\left(1+{\left\{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)\right\}}^{\mathrm{n}}\right)}^{\mathrm{n}-1/\mathrm{n}}}}$$

(16)

$$c_{\mathrm{app}}={\displaystyle \frac{1}{\alpha }}\tan {\phi }^{\prime }{\displaystyle \frac{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)}{{\left(1+{\left\{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)\right\}}^{\mathrm{n}}\right)}^{\mathrm{n}-1/\mathrm{n}}}}$$

(17)

Substituting Equations (8) and (17) into Equation (7) gives the temperature-dependent expression for shear strength $\tau$:

$$\tau =c^{\prime }+\left(\sigma -u_{\mathrm{a}}\right)\tan {\phi }^{\prime }+{\displaystyle \frac{1}{\alpha }}\tan {\phi }^{\prime }{\displaystyle \frac{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)}{{\left(1+{\left\{\ln \left[\left(1+\frac{q^{\prime }}{k_{\mathrm{s}}}\right)e^{-\alpha z{\gamma }_{\mathrm{w}}}-\frac{q^{\prime }}{k_{\mathrm{s}}}\right]\left(\frac{{\beta }_{{\mathrm{T}}_{\mathrm{r}}}+T_{\mathrm{r}}}{\beta +T}\right)\right\}}^{\mathrm{n}}\right)}^{\mathrm{n}-1/\mathrm{n}}}}$$

(18)

3. Upper Bound Analysis of Crown Stability for Shallow Tunnel

3.1. Upper Bound Theorem

The stability analysis of tunnel surrounding rock essentially involves solving for an optimal solution under specific constraints, integrating stress and displacement boundary conditions. This optimal solution must satisfy three fundamental requirements: equilibrium conditions, geometric compatibility, and constitutive relationships. However, the complex mechanical behavior of soils often renders strict adherence to these conditions challenging. Consequently, limit analysis methods grounded in soil plasticity theory have emerged as powerful alternatives. Limit analysis idealizes soil stress–strain relationships through the normality rule and associated flow rule. Compared with traditional approaches like the slip-line theory or limit equilibrium methods, this framework circumvents the need to trace loading history and path-dependency, directly determining the ultimate limit state of soil while yielding results that are both rigorous and precise. Limit analysis comprises two theorems: the upper bound and lower bound theorems. In practical engineering scenarios involving complex geometries, constructing a statically admissible stress field—one that satisfies stress boundaries, equilibrium, and yield conditions—proves exceptionally difficult. Comparatively, establishing an admissible velocity field compatible with flow rules, velocity constraints, and boundary conditions is more tractable. Therefore, this study adopts the upper bound theorem for stability analysis of tunnel-surrounding rock.

The upper bound theorem can be stated as follows: for a given admissible velocity field, the limit load calculated by equating the total energy dissipation rate to the external power should be not less than the actual collapse load. The specific expression is as follows:

$${\displaystyle \underset{\mathrm{V}}{\int }{\sigma }_{ij}}{\dot{\epsilon }}_{ij}\mathrm{d}\mathrm{V}\ge {\displaystyle \underset{\mathrm{S}}{\int }{T}_i}{\mathit{v}}_i\mathrm{d}\mathrm{S}+{\displaystyle \underset{\mathrm{V}}{\int }{F}_i}{\mathit{v}}_i\mathrm{d}\mathrm{V}$$

(19)

where $\mathrm{V}$ and $\mathrm{S}$ denote the volume and boundary of the failure mechanism, respectively, $T_i$ represents the surface force acting on boundary $\mathrm{S}$, $F_i$ is the body force acting within volume $\mathrm{V}$, ${\sigma }_{ij}$ and ${\dot{\epsilon }}_{ij}$ denote the stress tensor and strain rate tensor in the kinematically admissible velocity field, and ${\mathit{v}}_i$ signifies the velocity vector in the kinematically admissible velocity field.

3.2. Collapse Mechanism of Crown Surrounding Rock in Rectangular Shallow Tunnel

Within the framework of the upper bound theorem, the key to accurately assessing the stability of surrounding rock at tunnel crowns lies in establishing a kinematically reasonable and admissible failure mechanism. Mollon et al. [37] proposed a “point-to-point” discretization method based on the associated flow rule in limit analysis theory to generate rotational failure mechanisms for tunnel faces. Subsequently, Yang and Wang [38] adopted the limit analysis technique to generate block profiles and constructing a new failure mechanism for 2D tunnel-surrounding rock. Although this method achieves high accuracy, its computational complexity hinders practical implementation in engineering projects. Therefore, this study employs the translational failure mechanism for crown-surrounding rock in shallow rectangular tunnels to develop the failure model. This approach has been widely used in engineering by some researchers and published in several studies [39,40,41,42,43].

For the crown-surrounding rock, its shape is often hexagon. The failure mechanism of dividing the hexagon into several triangles is correct. The approach of keeping the entire hexagonal block intact is also correct. The references [39,40,41,42,43] adopt the hexagon intact block, not dividing the hexagon into several triangles. This paper also adopts the hexagon intact block, similar to the failure mechanism of the references [39,40,41,42,43].

The following discussions were adopted in developing the collapse mechanism for unsaturated shallow tunnels. (1) The groundwater table lies below the tunnel invert, ensuring the entire tunnel resides within unsaturated soil. (2) Collapse blocks are rigid bodies exhibiting zero volumetric strain during plastic flow. (3) The soil is idealized as a perfectly plastic material, obeying the associated flow rule, whereby the velocity vector at any point along a discontinuity surface forms an angle equal to the soil’s friction angle with the tangent to that surface. (4) Support pressure $q$ is uniformly distributed over the tunnel crown.

Based on the translational failure mechanism, the two-dimensional failure mechanism and velocity field of the unsaturated rectangular shallow tunnel are illustrated in Figure 1, respectively. The tunnel has a width of $2W$ and a height of $2H$, with $h$ representing the burial depth, ${\sigma }_{\mathrm{s}}$ denoting the surface load, and $q$ indicating the support pressure at the crown. The entire failure mechanism consists of a hexagonal translational block $AB{C}_{1}CD{D}_1$ and eight triangular translational blocks. Block $AB{C}_{1}CD{D}_1$ moves vertically downward at velocity $v_0$, while the eight triangular translational blocks move at velocities $v_1~v_4$, with relative velocities of $v_{0,1}~v_{3,4}$ compared with block $AB{C}_{1}CD{D}_1$. The geometry of the failure mechanism is defined by angles ${\alpha }_1~{\alpha }_4$ and ${\beta }_1~{\beta }_4$.

Figure 1. (a) Schematic diagram of the failure mechanism. (The arrows represent the directions of the velocity and force vectors.) (b) Velocity field.

According to the associated flow rule, the velocity vector along discontinuity lines between rigid blocks forms an angle $\phi$ with discontinuity lines and must satisfy vector closure conditions. It should be noted that since the tunnel is essentially in a water-free environment, pore water pressure in the soil can be neglected. Consequently, the friction angle $\phi$ and effective friction angle ${\phi }^{\prime }$ are practically equivalent ($\phi \approx {\phi }^{\prime }$). In subsequent calculations of internal and external work rates, while the friction angle $\phi$ is used to express variable relationships, the effective friction angle ${\phi }^{\prime }$ is employed in actual computations. Owing to the central symmetry of both the failure mechanism and velocity field in rectangular shallow tunnel, the work-rate balance analysis is confined to one symmetric half of the tunnel.

The velocity vectors satisfy the following kinematic relationships:

$$v_1={\displaystyle \frac{\sin \left({\displaystyle \sum _{i=1}^4{\beta }_i}+\phi \right)}{\sin \left({\alpha }_1+{\beta }_1+2\phi \right)}}{v}_0$$

(20)

$$v_2={\displaystyle \frac{\sin \left({\alpha }_1+2\phi \right)}{\sin \left({\alpha }_2+{\beta }_2+2\phi \right)}}{v}_1$$

(21)

$$v_3={\displaystyle \frac{\sin \left({\alpha }_2+2\phi \right)}{\sin \left({\alpha }_3+{\beta }_3+2\phi \right)}}{v}_2$$

(22)

$$v_4={\displaystyle \frac{\sin \left({\alpha }_3+2\phi \right)}{\sin \left({\alpha }_4+{\beta }_4+2\phi \right)}}{v}_3$$

(23)

$$v_{0,1}={\displaystyle \frac{\sin \left({\alpha }_1-{\beta }_2-{\beta }_3-{\beta }_4+\phi \right)}{\sin \left({\alpha }_1+{\beta }_1+2\phi \right)}}{v}_0$$

(24)

$$v_{1,2}={\displaystyle \frac{\sin \left({\alpha }_2-{\alpha }_1+{\beta }_2\right)}{\sin \left({\alpha }_2+{\beta }_2+2\phi \right)}}{v}_1$$

(25)

$$v_{2,3}={\displaystyle \frac{\sin \left({\alpha }_3-{\alpha }_2+{\beta }_3\right)}{\sin \left({\alpha }_3+{\beta }_3+2\phi \right)}}{v}_2$$

(26)

$$v_{3,4}={\displaystyle \frac{\sin \left({\alpha }_4-{\alpha }_3+{\beta }_4\right)}{\sin \left({\alpha }_4+{\beta }_4+2\phi \right)}}{v}_3$$

(27)

The lengths of the velocity discontinuity lines are

$$C{C}_4=2H{\displaystyle \frac{\sin {\alpha }_4}{\sin \left({\alpha }_4+{\beta }_4\right)}}$$

(28)

$$C{C}_3={\displaystyle \frac{\sin {\alpha }_3}{\sin \left({\alpha }_3+{\beta }_3\right)}}C{C}_4$$

(29)

$$C{C}_2={\displaystyle \frac{\sin {\alpha }_2}{\sin \left({\alpha }_2+{\beta }_2\right)}}C{C}_3$$

(30)

$$C{C}_1={\displaystyle \frac{\sin {\alpha }_1}{\sin \left({\alpha }_1+{\beta }_1\right)}}C{C}_2$$

(31)

$$C_1{C}_2={\displaystyle \frac{\sin {\beta }_1}{\sin \left({\alpha }_1+{\beta }_1\right)}}C{C}_2$$

(32)

$$C_2{C}_3={\displaystyle \frac{\sin {\beta }_2}{\sin \left({\alpha }_2+{\beta }_2\right)}}C{C}_3$$

(33)

$$C_3{C}_4={\displaystyle \frac{\sin {\beta }_3}{\sin \left({\alpha }_3+{\beta }_3\right)}}C{C}_4$$

(34)

$$C_{4}E={\displaystyle \frac{\sin {\beta }_4}{\sin \left({\alpha }_4+{\beta }_4\right)}}CE$$

(35)

$$B{C}_1={\displaystyle \frac{h+C{C}_1\cos \left({\displaystyle \sum _{i=1}^4{\beta }_i}\right)}{\cos \phi }}$$

(36)

The areas of the rigid blocks are

$$S_{\Delta C{C}_1{C}_2}={\displaystyle \frac{1}{2}}C{C}_1\cdot C{C}_2\cdot \sin {\beta }_1$$

(37)

$$S_{\Delta C{C}_2{C}_3}={\displaystyle \frac{1}{2}}C{C}_2\cdot C{C}_3\cdot \sin {\beta }_2$$

(38)

$$S_{\Delta C{C}_3{C}_4}={\displaystyle \frac{1}{2}}C{C}_3\cdot C{C}_4\cdot \sin {\beta }_3$$

(39)

$$S_{\Delta C{C}_{4}E}={\displaystyle \frac{1}{2}}C{C}_4\cdot CE\cdot \sin {\beta }_4$$

(40)

$${\displaystyle \frac{1}{2}}{S}_{AB{C}_{1}CD{D}_1}={\displaystyle \frac{1}{2}}B{C}_1\cdot C{C}_1\cdot \sin \left({\displaystyle \sum _{i=1}^4{\beta }_i-\phi }\right)+{\displaystyle \frac{h}{2}}\left[2W-C{C}_1\cdot \sin \left({\displaystyle \sum _{i=1}^4{\beta }_i}\right)+B{C}_1\cdot \sin \phi \right]$$

(41)

The gravity power of the failure mechanism is

$$\begin{array}{ll}{P}_{\gamma }=& \frac{1}{2}{S}_{AB{C}_{1}CD{D}_1}\cdot \gamma \cdot v_0+S_{\Delta C{C}_1{C}_2}\cdot \gamma \cdot v_1\cdot \cos \left[{\alpha }_1-{\beta }_2-{\beta }_3-{\beta }_4+\phi \right]\\ & +S_{\Delta C{C}_2{C}_3}\cdot \gamma \cdot v_2\cdot \cos \left({\alpha }_2-{\beta }_3-{\beta }_4+\phi \right)+S_{\Delta C{C}_3{C}_4}\cdot \gamma \cdot v_3\cdot \cos \left({\alpha }_3-{\beta }_4+\phi \right)\\ & +S_{\Delta C{C}_{4}E}\cdot \gamma \cdot v_4\cdot \cos \left({\alpha }_4+\phi \right)\end{array}$$

(42)

The power generated by the surface load is

$$P_S={\sigma }_{\mathrm{s}}\cdot v_0\cdot \left[C{C}_1\cdot \cos \left({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4-{\displaystyle \frac{\pi }{2}}\right)+W-B{C}_1\cdot \sin \phi \right]$$

(43)

The power provided by the tunnel support structure is

$$P_T=-W\cdot q\cdot v_0-2H\cdot q\cdot v_4\cdot \sin \left({\alpha }_4+\phi \right)$$

(44)

The internal energy dissipation rate under thermal effects is

$$\begin{array}{ll}{P}_C=& \left(c^{\prime }+c_{\mathrm{app}}\right)\cdot \cos \phi \cdot (B{C}_1\cdot v_0+C_1{C}_2\cdot v_1+C_2{C}_3\cdot v_2+C_3{C}_4\cdot v_3+C_{4}E\cdot v_4\\ & +C{C}_1\cdot v_{0,1}+C{C}_2\cdot v_{1,2}+C{C}_3\cdot v_{2,3}+C{C}_4\cdot v_{3,4})\end{array}$$

(45)

Substituting Equations (20)–(45) into the upper bound theorem, the critical support pressure $q_{\mathrm{cr}}$ of the tunnel at the limit state can be determined as

$$q_{\mathrm{cr}}={\displaystyle \frac{P_{\gamma }+P_S-P_C}{W\cdot v_0+2H\cdot v_4\cdot \sin ({\alpha }_4+\phi )}}$$

(46)

The upper bound optimal solution of $q_{\mathrm{cr}}$ can be searched using the Sequential Quadratic Programming (SQP) method subject to the constraints. As indicated by the velocity field vector relationships, the angles between all velocity vectors must be acute, with the constraints expressed as

$$\left\{\begin{array}{cc}\begin{array}{c}{P}_{\gamma }+P_S+P_T=P_C\\ {\alpha }_1-{\beta }_2-{\beta }_3-{\beta }_4+\phi >0\\ \pi -{\alpha }_i-{\beta }_i-2\phi >0\\ {\alpha }_2-{\alpha }_1+{\beta }_2>0\hspace{1em} \\ {\alpha }_3-{\alpha }_2+{\beta }_3>0\hspace{1em} \\ {\alpha }_4-{\alpha }_3+{\beta }_4>0\hspace{1em} \\ 0<{\alpha }_i<{\displaystyle \frac{\pi }{2}}\\ 0<{\beta }_i<{\displaystyle \frac{\pi }{2}}\end{array}& \left(i=1,2,3,4\right)\end{array}\right.$$

(47)

4. Parametric Discussions

To investigate the effects of temperature and main unsaturated soil parameters on the critical support pressure, based on previous researchers [44,45,46,47], this study conducts a parametric analysis using established parameter values, with some data in Table 1.

Table 1. Main parameters of different kinds of soil types.

Soil Type	c′ (kPa)	$\mathit{n}$	α (kPa−1)	$∆{\mathit{h}}_{{\mathit{T}}_{\mathit{r}}}$ (J/m2)	${\mathit{k}}_{\mathit{s}}$ (m/s)	${\mathit{\phi }}^{\mathbf{\prime }}$ (°)	γ (kN/m3)	${\mathsf{\theta }}_{\mathbf{s}}$	${\mathsf{\theta }}_{\mathbf{r}}$
Clay	$10$	$2$	$0.005$	$-0.516$	$5\times {10}^{-8}$	$25$	$20.5$	0.58	0.05
Silt	$2$	$3$	$0.01$	$-0.516$	$5\times {10}^{-7}$	$30$	$21.5$	0.45	0.10
Sand	$0$	$4$	$0.1$	$-0.285$	$3\times {10}^{-5}$	$35$	$22.0$	0.41	0.05

In this table, ${\theta }_{\mathrm{s}}$ and ${\theta }_{\mathrm{r}}$ represent the saturated volumetric water content and residual volumetric water content, respectively. Their relational formula with the volumetric water content $\theta$ is given by:

$$\theta =S_{\mathrm{e}}\left({\theta }_{\mathrm{s}}-{\theta }_{\mathrm{r}}\right)+{\theta }_{\mathrm{r}}$$

(48)

Volumetric water content $\theta$ refers to the proportion of the volume of water in a unit volume of soil to the total volume. It quantifies the moisture content in the soil and reflects the humidity state of the soil. Temperature fundamentally changes the matric suction by altering the soil water content, thereby modifying the strength parameters of the soil. Therefore, defining the effective humidity range in the unsaturated shallow-tunnel model in this paper can better align with reality and meet the needs of engineering practice.

Generally, the burial depth of shallow tunnels is typically between 10 and 20 m. Within this depth range, the temperature fluctuation in shallow tunnels in typical regions around the world is shown in Table 2.

Table 2. Temperature fluctuation ranges and engineering information at depths of 10~20 m in representative regions worldwide.

Region	Temperature Range (°C)	Main Unsaturated Soil Type	Actual Engineering Case
North China Plain, China	13~15	sand, silt, clay	Luochuan Tunnel
Sydney, Australia	17–25	clay, sand	Sydney Metro West
Delhi, India	18~33	clay	Delhi Metro Phase IV
Vienna, Austria	9~12	clay, loess	Vienna U-Bahn extensions
Yakutsk, Russia	−10~−2	ice-rich silty soil	Underground Laboratory of the Melnikov Permafrost Institute

From the aforementioned table, it is evident that the temperature variations in unsaturated soil shallow tunnels with burial depths ranging from 10 to 20 m worldwide are predominantly concentrated within the interval of −10 to 35 °C. Consequently, the temperature parameter range examined in this study is set to −10 to 35 °C.

The variation range of volumetric water content $\theta$ under the temperature conditions from −10 to 35 °C can be calculated using Equation (48), with the data summarized in Table 3.

Table 3. The variation ranges of volumetric water content under temperature variation.

Soil Type	$\mathit{\theta }$
Clay	0.48~0.56
Silt	0.26~0.40
Sand	0.05~0.10

The above table delineates the applicable moisture ranges for different soil types in the unsaturated soil shallow tunnel model presented in this paper. Based thereon, the computational results of the parameter analysis have been derived.

4.1. Influence of Temperature

With the tunnel half-height $H=5 \mathrm{m}$, half-width $W=6 \mathrm{m}$ and ${\gamma }_w=10 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, ${\sigma }_{\mathrm{s}}=50 \mathrm{kPa}$, $h=10 \mathrm{m}$, $q^{\prime }=0$. Figure 2 depicts the variation in the critical support pressure $q_{\mathrm{cr}}$ for clay, silt, and sand under different vertical distances $z_0$ (0 m, 10 m, 20 m) from the tunnel invert to the horizontal plane, across the temperature range of −10~35 °C.

Figure 2. Versus temperature for different soil types and $z_0$ conditions, (a) clay; (b) silt; and (c) sand.

As shown in Figure 2, the critical support pressure $q_{\mathrm{cr}}$ of both clay and silt increases linearly with rising temperature under different $z_0$ conditions, while that of sand decreases linearly at a very slow rate. This indicates that with increasing temperature, the stability of the tunnel surrounding rock in clay and silt gradually decreases, while that in sand increases very slowly. Furthermore, $q_{\mathrm{cr}}$ decreases with the increase of $z_0$ for clay and silt but increases for sand. Given this inverse response of sand to $z_0$ and temperature variations compared with clay and silt, we hypothesize a suction stress dominance and thermal-induced suction enhancement behavior in sand.

Based on Equation (1):

$${\displaystyle \frac{\partial {\sigma }^{\prime }}{\partial z}}=-\gamma -{\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial z}}$$

(49)

while

$${\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial z}}={\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial {\sigma }^{\prime }}}\cdot {\displaystyle \frac{\partial {\sigma }^{\prime }}{\partial z}}={\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial {\sigma }^{\prime }}}\cdot \left(-\gamma -{\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial z}}\right)$$

(50)

When the effective stress ${\sigma }^{\prime }$ increases, the shear strength increases, leading to a reduction in the critical support pressure, i.e., ${\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial {\sigma }^{\prime }}}<0$.

When $-\gamma -{\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial z}}<0$, ${\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial z}}>0$, which corresponds to the phenomenon of increasing critical support pressure $q_{\mathrm{cr}}$ with increasing $z_0$ in sand.

The anomalous $z_0$ behavior in sand is attributed to its higher $\alpha$ and $n$ values compared with clay or silt, resulting in higher ${\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial z}}$ in sand and making ${\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial z}}>-\gamma$.

Regarding the temperature response of sand, Equation (1) demonstrates that

$${\displaystyle \frac{\partial {\sigma }^{\prime }}{\partial T}}={\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial T}}$$

(51)

while

$${\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial T}}={\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial {\sigma }^{\prime }}}\cdot {\displaystyle \frac{\partial {\sigma }^{\prime }}{\partial T}}={\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial {\sigma }^{\prime }}}\cdot {\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial T}}$$

(52)

When ${\displaystyle \frac{\partial {\sigma }^{\mathrm{s}}}{\partial T}}>0$, ${\displaystyle \frac{\partial q_{\mathrm{cr}}}{\partial T}}<0$, which corresponds to the phenomenon of decreasing critical support pressure $q_{\mathrm{cr}}$ with increasing temperature $T$ in sand.

The above provides a mathematical interpretation of the content displayed in Figure 2. In reality, a decline in the groundwater level (i.e., an increase in $z_0$) leads to a reduction in the volumetric water content of the soil, thereby augmenting the matric suction within the soil mass. This enhancement elevates the shear strength, consequently diminishing the critical support pressure $q_{\mathrm{cr}}$ for the tunnel. However, this contribution is contingent upon the soil type’s water retention capacity and capillary rise height, wherein coarse-grained soils (such as sand) exhibit a modest capillary height (typically 0.1–1 m), whereas fine-grained soils (such as clay and silt) demonstrate substantial capillary height (potentially exceeding 10–100 m). When the groundwater level decline causes the soil position to surpass the capillary rise height, the soil forfeits the apparent cohesion contributed by matric suction, resulting in a diminution of shear strength. Given the diminutive capillary height of sand, its shear strength exhibits a pronounced reduction concomitant with the rapid decline in groundwater level. Conversely, clay and silt possess considerable capillary heights, such that the groundwater level decline does not exceed the capillary rise height at the soil position; instead, it induces a decrease in water content, thereby intensifying matric suction and augmenting their shear strength.

Regarding temperature, the trend of critical support pressure $q_{\mathrm{cr}}$ in clay and silt is diametrically opposite to that in sand. This divergence stems from the different primary components of shear strength in coarse-grained and fine-grained soils. For sand, which is mainly composed of coarse particles, its shear strength primarily originates from the internal friction angle between particles, i.e., inter-particle friction, while cohesion is usually very small or close to zero. When temperature rises, the viscosity of pore water between sand particles decreases, reducing the impeding effect of water films. This allows particles to come into closer contact, possibly slightly enhancing inter-particle friction and thus increasing its shear strength. In contrast, for clay and silt, their shear strength significantly depends on cohesion. The high $\left|\Delta h_{T_r}\right|$ in clay and silt makes them very sensitive to temperature changes. When temperature increases, their soil skeleton expands significantly, increasing the soil volume. This necessitates more water to fill these enlarged pore spaces to maintain saturation. Data from the model in this paper also confirm a slight increase in the volumetric water content of clay and silt with rising temperature. This situation leads to a reduction in their cohesion, which in turn decreases their shear strength.

4.2. Influence of Unsaturated Parameters α and n

To investigate the influence of unsaturated parameters $\alpha$ and $n$ on the critical support pressure $q_{\mathrm{cr}}$, with ${\gamma }_w=10 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, $\gamma =20 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, $c^{\prime }=10 \mathrm{kPa}$, ${\phi }^{\prime }=25°$, ${\sigma }_s=50 \mathrm{kPa}$, $h=10 \mathrm{m}$, $q^{\prime }=0$, $z_0=0$ and $T=20 °\mathrm{C}$. Figure 3 demonstrates the variation in $q_{\mathrm{cr}}$ with α-values under different n-values. Simultaneously, the figure reveals the effects induced by varying width-to-height ratios ($W/H=1.2,2.0,3.0$).

Figure 3. Versus $1/\alpha$ for different $n$, (a) $H=5 \mathrm{m},W=6 \mathrm{m}$; (b) $H=5 \mathrm{m},W=10 \mathrm{m}$; (c) $H=5 \mathrm{m}$, $W=15 \mathrm{m}$.

The results demonstrate that as $1/\alpha$ increases (i.e., $\alpha$ decreases), the critical support pressure $q_{\mathrm{cr}}$ initially declines rapidly, then exhibits rapid deceleration in its rate of decrease, and finally converges to a specific value. For a given width-to-height ratio, higher $n$ values accelerate convergence while reducing initial $q_{\mathrm{cr}}$. With increasing tunnel width-to-height ratio, initial $q_{\mathrm{cr}}$ increases at fixed $n$ and converged $q_{\mathrm{cr}}$ increases monotonically.

This study provides an explanation for the convergence phenomenon as $1/\alpha$ increase (i.e., $\alpha \to 0$), the suction stress ${\sigma }^{\mathrm{s}}$ calculation simplifies to

$${\sigma }^{\mathrm{s}}=-{\displaystyle \frac{\psi }{{\left[1+{\left(\alpha \psi \right)}^{\mathrm{n}}\right]}^{\mathrm{n}-1/\mathrm{n}}}}\to -\psi$$

(53)

Beyond this point, further reduction in $\alpha$ ceases to alter suction stress, resulting in stabilized apparent cohesion and shear strength, thereby causing $q_{\mathrm{cr}}$ to converge to a threshold value.

When the tunnel half-height $H$ remains constant while the half-width $W$ increases, the volume of rigid blocks expands. This leads to a faster growth in gravity power ($P_{\gamma }\propto W^2$) compared with the growth of surface load power ($P_S\propto W$) and internal energy dissipation rate ($P_C\propto W$). Consequently, wider tunnels must bear greater soil self-weight, resulting in increased initial $q_{\mathrm{cr}}$ and converged values.

4.3. Influence of Burial Depth $\mathit{h}$

Based on the research by Griffiths and Lu [48], this study categorizes seepage conditions into three types: high evaporation ($q^{\prime }=1.15\times {10}^{-8} \mathrm{m}/\mathrm{s}$), no-flow ($q^{\prime }=0$), and high infiltration ($q^{\prime }=-3.14\times {10}^{-8} \mathrm{m}/\mathrm{s}$). With the tunnel half-height $H=5 \mathrm{m}$ and half-width $W=6 \mathrm{m}$, ${\gamma }_w=10 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, ${\sigma }_s=50 \mathrm{kPa}$, $z_0=0$, $T=20 °\mathrm{C}$. Figure 4 illustrates the variation in critical support pressure $q_{\mathrm{cr}}$ with burial depth $h$ for three soil types under different seepage conditions.

Figure 4. Versus $h$ for different soil types and seepage conditions, (a) clay; (b) silt; and (c) sand.

Figure analysis reveals distinct depth-dependent characteristics: for all three soil types, $q_{\mathrm{cr}}$ increases with increasing burial depth. Moreover, clay and silt exhibit maximum $q_{\mathrm{cr}}$ values under high infiltration conditions and minimum values under high evaporation conditions, whereas sand shows the opposite trend. Among the three soil types, silt and sand exhibit very low sensitivity to seepage conditions, while clay shows the highest responsiveness.

4.4. Influence of Vertical Seepage Velocity ${\mathit{q}}^{\prime }$

Within the seepage velocity range of three seepage velocity, with the tunnel half-height $H=5 \mathrm{m}$, half-width $W=6 \mathrm{m}$, and ${\gamma }_w=10 \mathrm{k}\mathrm{N}/{\mathrm{m}}^3$, ${\sigma }_s=50 \mathrm{kPa}$, $h=10 \mathrm{m}$, $z_0=0$. Figure 5 depicts the variation in the critical support pressure $q_{\mathrm{cr}}$ with vertical seepage velocity $q^{\prime }$ for three soil types under different temperature conditions ($T=-10,10,35 °\mathrm{C}$).

Figure 5. Versus $q^{\prime }$ for different soil types and temperature, (a) clay; (b) silt; and (c) sand.

As depicted in the figure, increasing vertical seepage velocity $q^{\prime }$ reduces $q_{\mathrm{cr}}$ for both clay and silt, with clay exhibiting a more significant decline. This validates the inference that silt is less sensitive to vertical seepage velocity variations than clay. Conversely, sand show increasing $q_{\mathrm{cr}}$ with rising $q^{\prime }$, eventually stabilizing at higher seepage rates. Furthermore, while there was an elevated temperature increase $q_{\mathrm{cr}}$ in clay and silt, it reduced in sand.

For the opposite response of sand compared with the other two soil types regarding $q^{\prime }$, this study provides a possible explanation. Model-calculated results indicate that the volumetric water content of sand is lower under high infiltration conditions and higher under high evaporation conditions, whereas clay and silt exhibit lower volumetric water content under high evaporation and higher values under high infiltration conditions. This suggests that the matric suction of sand is greater during infiltration than during evaporation, leading to a larger contribution of apparent cohesion and thus higher shear strength, causing $q_{\mathrm{cr}}$ to increase with rising $q^{\prime }$. In contrast, the matric suction of clay and silt is smaller during infiltration than during evaporation, resulting in a smaller contribution of apparent cohesion and reduced shear strength, thereby causing $q_{\mathrm{cr}}$ to decrease with increasing $q^{\prime }$.

The fact that sand exhibits a lower volumetric water content under high infiltration conditions than under high evaporation conditions may appear counterintuitive, but it actually results from the combined effects of the physical properties of sand and its water migration mechanisms. As a coarse-grained soil, sand has large pores and strong drainage capacity. When water infiltrates from the ground surface, sand has almost no water retention ability, and the infiltrated water quickly percolates downward or drains away, resulting in little increase in water content. Under evaporation conditions, however, water loss from the surface forms capillary rise channels within the pores, allowing water from deeper layers to be continuously drawn upward to compensate for surface evaporation. Although the surface dries, during the early or intermediate stages of evaporation the volumetric water content in the capillary zone may actually increase, especially within shallow layers. Therefore, this phenomenon is indeed possible.

5. Conclusions

This study establishes an effective framework for unsaturated soils, incorporating thermal effects into the stability analysis of shallow rectangular tunnel crown. By extending the generalized effective stress principle, we developed strength expressions for unsaturated soils under temperature influence. The critical support pressure $q_{\mathrm{cr}}$ at the tunnel crown was calculated through balancing the internal energy dissipation rate and external work rate. The research analyzes the impact of temperature and other main unsaturated soil parameters on $q_{\mathrm{cr}}$, with the main conclusions as follows:

There are significant differences in the critical support force $q_{\mathrm{cr}}$ of tunnel crown under the influence of temperature changes in different soil types. According to the average percentage change in $q_{\mathrm{cr}}$ with temperature changes under different $z_0$ conditions, among the three soil types, sand is the least sensitive to temperature changes, while clay and silt exhibit similar sensitivity. The order of sensitivity to temperature variations is clay $\approx$ silt > sand. Furthermore, clay and silt exhibit a similar sensitivity to changes in $z_0$, whereas sand is the least sensitive. The order of sensitivity to $z_0$ variations is clay $\approx$ silt > sand.

The minimum $q_{\mathrm{cr}}$ for clay and silt occurs under high evaporation conditions, while the opposite is true for sand. Overall, clay is the most sensitive to vertical seepage rate $q^{\prime }$ and sand is the least. The order of sensitivity to $q^{\prime }$ variations is clay > silt > sand. The effects of temperature variations on all three soil types are relatively insignificant, being less pronounced than the influence of vertical seepage rate $q^{\prime }$.

This study innovatively incorporates temperature effects into the upper bound theorem and applies it to the stability analysis of the crown of shallow rectangular tunnels. The failure model and analysis framework thus developed provides a strict upper bound solution for crown stability, which can offer a more precise theoretical basis for subsequent tunnel face support. The failure model proposed in this study currently lacks validation from actual engineering practice and is still in the theoretical design stage. Future research can combine the upper bound theorem with numerical simulations (finite element and finite difference methods) to more accurately assess the impact of temperature on shallow unsaturated tunnels. This will provide a more scientific basis for evaluating the stability of shallow tunnel crowns.