Calculation of Surrounding Rock Pressure Design Value and the Stability of Support Structure for High-Stress Soft Rock Tunnel

Abstract

With the comprehensive implementation of the “Belt and Road” initiative and the Western Development Strategy, the scale of tunnel construction has been continuously expanding, with many tunnels being built in high ground stress and fractured soft rock strata. The design, construction, and operation of tunnels all rely on the surrounding rock pressure as a fundamental basis. Therefore, determining the surrounding rock pressure is essential for ensuring the safe construction of tunnels. However, due to the complexity of geological conditions, differences in construction methods, variations in support parameters, and time–space effects, it is challenging to accurately determine the surrounding rock pressure. This paper proposes a design approach using the surrounding rock pressure design value as the “support force” for the tunnel, starting with the reserved deformation of soft rock tunnels. Based on the calculation principle of the surrounding rock pressure design value, a relationship curve between the support force and the maximum deformation of surrounding rock in high ground stress soft rock tunnels is developed. By combining the surrounding rock deformation grade with the tunnel’s reserved deformation index, a calculation method for the surrounding rock pressure design value for high ground stress soft rock tunnels is proposed. The method is verified by the measured surrounding rock pressure data from the Mao County Tunnel of the Chengdu–Lanzhou Railway. Furthermore, the study integrates the creep characteristics and strain softening properties of soft rock to implement a secondary development of the viscoelastic–plastic strain softening mechanical model. Based on a custom-developed creep model and the calculation method for the surrounding rock pressure design value, the relationship among time, support force, and surrounding rock deformation is comprehensively considered. A calculation method for the surrounding rock pressure design value, accounting for time effects, is proposed. Based on this method, a time-history curve of the surrounding rock pressure design value is obtained and used as the input load. The safety factor time evolution of the rock-anchor bearing arch, spray layer, and secondary lining is derived using the load-structure method, and the overall safety factor time evolution of the tunnel support structure is evaluated. The overall stability of the support structure is assessed, and numerical simulations are compared with field measurements based on the mechanical behavior evolution law of the secondary lining of the Chengdu–Lanzhou Railway Mao County Tunnel. The results indicate that the monitoring data of the internal forces of the field support structure is in good agreement with the numerical calculation results, validating the rationality of the proposed calculation method.

1. Introduction

Tunnel engineering plays a pivotal role in modern infrastructure development. As underground space utilization continues to expand, the geological challenges encountered in tunnel construction have become increasingly complex [1]. In particular, under high in situ stress conditions in soft rock, the difficulty of tunnel design and construction significantly increases. Accurately assessing surrounding rock pressure and ensuring the stability of the support structure have therefore become critical issues that require urgent attention [2]. Due to the insufficient consideration of surrounding rock pressure during the tunnel design stage, especially in high in situ stress soft rock environments, the contact pressure between the surrounding rock and the support structure may become excessively large, potentially leading to structural instability. Tunnel design, construction, and operation all fundamentally depend on a reasonable calculation of surrounding rock pressure, making this a central theme in tunnel engineering research. Consequently, accurate calculations of surrounding rock pressure and stability analysis of support structures in high in situ stress soft rock tunnels are essential for improving the safety and efficiency of tunnel construction [3].

The determination of surrounding rock pressure is influenced by a range of factors, including the surrounding rock’s stress environment, rock mass grade, tunnel depth, tunnel cross-sectional dimensions, excavation methods, and construction techniques. Over the past century, extensive research has been conducted on surrounding rock pressure calculations under high in situ stress conditions, employing major methods such as theoretical calculations, empirical formulas, and numerical simulation techniques [4,5,6,7,8,9,10]. In terms of theoretical calculations, surrounding rock pressure typically manifests as both loose pressure and deformation pressure. Under high in situ stress conditions, surrounding rock pressure predominantly takes the form of deformation pressure. Key theoretical calculation methods include the Finna formula, Kaka formula, Castelnau formula, and Blei formula, which are primarily suited for deep-buried circular tunnel models [11,12,13,14]. Empirical formulas remain the most commonly applied method in tunnel design for calculating surrounding rock pressure. The Q-system classification method, proposed by Norwegian scholar Barton in 1994 [15,16,17,18,19,20], provides a widely adopted approach for surrounding rock pressure calculation. This method evaluates factors such as rock mass quality indicators, joint characteristics, groundwater presence, tunnel depth, and cross-sectional dimensions to form a quantitative assessment system. Similarly, the RMR classification method, proposed by South African scholar Bieniawski [21,22], considers six key rock mass parameters, including uniaxial compressive strength, rock mass quality index (RQD), joint spacing, joint conditions, groundwater, and joint orientation, and estimates surrounding rock pressure based on statistical analysis of numerous engineering cases. As advancements in geotechnical engineering, geological engineering, and computational mechanics have emerged, traditional elastoplastic theory, although valuable in certain scenarios, has proven insufficient in accurately modeling the complex behavior of non-elastic materials such as rock. Consequently, researchers have increasingly turned to more advanced numerical simulation methods, such as finite element methods and finite difference methods, to better simulate the intricate deformation and stress distribution of surrounding rocks [23,24,25,26,27,28,29].

Currently, the methods used for surrounding rock pressure calculations in tunnel design, construction, and operation largely rely on theoretical algorithms and empirical formulas under idealized conditions, which fail to adequately address complex geological environments. Moreover, these methods often neglect the time-dependent characteristics of the rock mass, leading to an underestimation of surrounding rock pressure and posing risks to tunnel safety. To overcome this challenge, this paper proposes a novel method for calculating the design value of surrounding rock pressure. The proposed method is based on the relationship between the support force and the maximum deformation of the surrounding rock, taking into account the creep and strain softening behavior of soft rock under high in situ stress conditions. This approach incorporates time-dependent effects, surrounding rock deformation, and support force in a comprehensive manner. By developing a viscoelastic–plastic strain softening creep mechanical model for soft rock and utilizing geological data from the Maoxian tunnel on the Chenglan Railway, we calculate the time-history curve of the surrounding rock pressure design value. Subsequently, the axial force and bending moment of the tunnel support structure at various time intervals are computed using the load-structure method, and the overall stability of the support structure is evaluated through a safety factor analysis. Finally, by comparing the field monitoring data with numerical calculation results, we demonstrate that this method significantly enhances the accuracy of surrounding rock pressure calculations and ensures the safe operation of the tunnel.

2. Calculation of Surrounding Rock Pressure Design Value for Soft Rock Tunnels Under High Geostress

2.1. Applicability Analysis of Surrounding Rock Pressure Calculation Methods

The calculation of surrounding rock pressure mainly focuses on two aspects: loose pressure and plastic deformation pressure. The applicability of various surrounding rock pressure calculation methods is summarized in

Table 1. Comparison of surrounding rock pressure calculation methods.

NO.	Calculation Method	Influencing Factors	Applicable Types
1	Taisha Base Theory	$\gamma ,B,H_t,{\phi }_0,H,\lambda$	Loose Pressure
2	Prouty Theory	$\gamma ,B,H_t,{\phi }_0$
3	Karko Formula	$c,\phi ,r_a,R_p,r_0$
4	Railway Tunnel Design Code	$h_q,S,\gamma ,B$
5	Fenner Formula	$\mathrm{r},c,\phi ,P_0,R_p$	Plastic Deformation Pressure
6	Railway Compression-type Surrounding Rock Tunnel Technical Specification	B, S, H

where $\gamma$ is the unit weight of the rock mass, B is the tunnel span, H_t is the tunnel height, H is the tunnel burial depth, and $\lambda$ is the lateral pressure coefficient. $r_a$ is the tunnel chamber radius, $\phi$ is the internal friction angle of the surrounding rock in the plastic zone, c is the cohesion of the surrounding rock in the plastic zone, P₀ is the initial geostress, R_p is the radius of the plastic zone, h_q is the equivalent height of the surrounding rock load, S is the surrounding rock grade, and B is the tunnel span.

.

In high geostress soft rock tunnels, surrounding rock pressure mainly exists in the form of plastic deformation pressure. When solving using the Fenner formula, assumptions are required for the radius of the surrounding rock plastic zone. The results vary based on the geological conditions of the project. On the other hand, the railway compression-type surrounding rock tunnel technical specification is comprehensive and widely used for calculating plastic deformation pressure in soft rock tunnels. However, when the burial depth exceeds a certain value, the surrounding rock pressure tends to stabilize, which does not align with actual conditions. Therefore, the calculation methods for surrounding rock pressure in high geostress soft rock tunnels still have shortcomings and need further improvement.

2.2. Calculation Method for Surrounding Rock Pressure Design Value in High Geostress Soft Rock Tunnels

2.2.1. Proposal of Surrounding Rock Pressure Design Value

As the engineering geological environment faced by tunnels becomes increasingly harsh, with high-speed rail construction moving toward deeper, longer, and larger tunnels, high geostress and soft rock large deformations are frequently encountered during tunnel construction. Under long-term structural stress, the surrounding rock pressure mainly exists in the form of deformation pressure. Currently, due to the complexity of geological conditions, the surrounding rock pressure in high geostress soft rock tunnels is difficult to determine. Tunnel support structure design safety factor method can be used for load-structure design by determining the most unfavorable working condition near collapse or reaching the failure stage, introducing the surrounding rock pressure design value as the load design value for the tunnel support structure. This concept helps to solve the problem of uncertain surrounding rock pressure. It should be emphasized that the surrounding rock pressure design value is only used as a nominal load for structural calculation, not the actual pressure on the support structure.

2.2.2. Calculation Method for Surrounding Rock Pressure Design Value

Calculation Method

Based on the strength reduction method, a numerical simulation model is built using the stratigraphic structure method. The support force is adjusted until it equals the resistance required to balance the weight of the surrounding rock in the plastic zone of the tunnel arch. The minimum support force (P_imin) is then multiplied by a factor of 1.4 to determine the surrounding rock pressure design value (q). (Figure 1) The specific calculation steps are as follows:

(1)

Based on the actual tunnel survey data, a numerical simulation model is created, reducing the initial surrounding rock mechanical parameters with a reduction factor of 1.15. The excavation design profile is set, and support force is applied until convergence is achieved.

(2)

Adjust the support force until it equals the resistance required to balance the weight of the surrounding rock in the plastic zone of the tunnel arch. This is the minimum support force, P_imin. It is then equivalently applied as a uniform load, q_min, along the span. The horizontal load is calculated using the vertical load on the arch and the side pressure coefficient.

(3)

Multiply the minimum support force by a safety factor k (typically greater than 1.4) to obtain the surrounding rock pressure design value (q). The safety factor k can be adjusted depending on the specific situation.

Figure 1. Schematic diagram of the calculation principle of surrounding rock pressure.

Figure 1. Schematic diagram of the calculation principle of surrounding rock pressure.

Issues with the Calculation Method

The calculation method for the surrounding rock pressure design value proposed in the general safety factor method for tunnel support structure design determines the minimum support force (P_imin) required to maintain the ultimate equilibrium state of the surrounding rock after the tunnel excavation. When the support force provided by the support structure (P_i) is less than the minimum support force (P_imin), the tunnel will collapse. To ensure the long-term stability of the tunnel structure, and considering the practical importance of tunnel engineering, the surrounding rock pressure design value is calculated by multiplying the minimum support force (P_imin) by an appropriate safety factor. This calculation method is applicable to the surrounding rock pressure calculation for tunnels under any complex geological conditions and has general applicability.

However, this method is currently mainly applied to soft rock tunnels with a lateral pressure coefficient less than 1. In high-stress soft rock tunnels, the lateral pressure coefficient is generally greater than 1, and the total safety factor method does not specify whether this method is suitable for calculating the surrounding rock pressure design value for high-stress tunnels. Additionally, in high-stress soft rock tunnels, surrounding rock deformation is an important indicator for determining whether the tunnel will undergo significant deformation. However, the surrounding rock pressure calculation method does not take this into account, and therefore further improvements are needed.

2.2.3. Failure Modes of Embedded Foundations

This section proposes a calculation method for the surrounding rock pressure design value in high-stress soft rock tunnels. Based on the calculation method in Section Calculation Method, it introduces deformation level variables and considers the relationship between support force and surrounding rock deformation. The method uses the reserved deformation value under different deformation levels as a reference, taking 1.4 times the support force corresponding to this deformation as the surrounding rock pressure design value. The specific steps are as follows:

(1)

Based on actual tunnel survey data, establish a numerical simulation model, reduce the initial surrounding rock physical and mechanical parameters with a reduction factor (typically 1.15), choose an appropriate constitutive model, excavate the design profile, apply support force, and compute until convergence.

(2)

Adjust the support force and plot the relationship between the support force (P_i) and maximum surrounding rock deformation (u), as shown in the principle diagram in Figure 2.

(3)

Based on the reserved deformation amount at different deformation levels, determine the minimum support force (P_r) at the displacement control reference value, and further equivalently apply the arch vertical load P_r along the span direction as a uniformly distributed load (q_r). The tunnel’s reserved deformation refers to the displacement loss from excavation to measurement and the measured displacement. The measured displacement follows the control reference values for surrounding rock clearance displacement at different deformation levels, as outlined in the “Technical Code for Railway Squeezable Surrounding Rock Tunnels”. The displacement loss before measurement is approximately 15% of the measured displacement.

(4)

Take 1.4 times the minimum support force P_r as the surrounding rock pressure design value q. The horizontal load is selected as the product of the vertical load q at the arch and the lateral pressure coefficient $\lambda$. The lateral pressure coefficient $\lambda$ is taken from the parameter values in the “Technical Specification for Railway Squeezing Surrounding Rock Tunnels,” as shown in

Table 2. Clearance displacement control reference values for different deformation levels of surrounding rock (mm).

Deformation Level	I	II	III
Small and Medium Span (B ≤ 12 m)	300	500	700
Large Span (B > 12 m)	400	600	800

.

Figure 2. Calculation principle of surrounding rock pressure design value.

Figure 2. Calculation principle of surrounding rock pressure design value.

Taking Grade IV and Grade V surrounding rock, a lateral pressure coefficient of 1.5, and a double-track high-speed railway tunnel with a speed of 350 km/h as an example, the surrounding rock mechanical parameters are selected from the lower 1/3 percentile values in the Railway Tunnel Design Code (TB 1003—2016). The relationship curves between support force and maximum surrounding rock deformation are plotted (Figure 3). The calculated surrounding rock pressure design values at the tunnel crown are shown in

Table 3. Tunnel arch surrounding rock pressure design value (kPa).

Burial Depth Conditions	IV	V
600 m	279	1624
800 m	588	2968

.

2.2.4. Verification of the Calculation Method for Surrounding Rock Pressure Design Value in High Ground Stress Soft Rock Tunnels

References [30,31] measured the surrounding rock pressure in the high ground stress soft rock large deformation section at a depth of 450 m in the No. 1 inclined shaft of the Mao County Tunnel on the Chengdu–Lanzhou Railway, as shown in Figure 4. Using the area equivalence method, the measured surrounding rock pressure was equivalent to a uniformly distributed load, with the arch surrounding rock pressure being 1.33 MPa and the surrounding rock pressure on both sides being 0.76 MPa, as shown in Figure 5.

Based on the design data provided in the literature, the surrounding rock pressure design value for the Mao County Tunnel on the Chengdu–Lanzhou Railway was calculated using the method for calculating surrounding rock pressure design values in high ground stress soft rock tunnels proposed in Section 2.2.3. The surrounding rock mechanical parameters are shown in

Table 4. Surrounding rock calculation parameters for soft rock tunnel of Chengdu–Lanzhou Railway.

Deformation Level	Unit Weight (kN/m3)	Elastic Modulus (GPa)	Poisson’s Ratio	Internal Friction Angle (°)	Cohesion (MPa)
Slight	23	1.2	0.32	32	2.1
Moderate	23	0.9	0.25	28	1.2
Severe	23	0.5	0.21	25	0.8

, and the reserved deformation values for the Chengdu–Lanzhou Railway soft rock tunnel are shown in

Table 5. Reserved deformation of soft rock tunnel in Chengdu–Lanzhou Railway (Unit: cm).

Deformation Level	Slight	Moderate	Severe	Extremely Severe
Single-track Tunnel	10~20	25~30	35~45	>45
Double-track Tunnel	20~30	25~40	40~60	>60

. The minimum support force is determined based on the reserved deformation values, and 1.4 times the minimum support force is taken as the surrounding rock pressure design value. The relationship curve between support force and maximum surrounding rock deformation is shown in Figure 6, resulting in a surrounding rock pressure of 1.62 MPa at the tunnel crown. It was found that the surrounding rock pressure design value obtained by the proposed method is approximately 21.8% higher than the field-measured equivalent (95% confidence interval: 15–28%). This deviation represents a conservative and code-compliant design margin. Therefore, the methodology presented herein provides a reliable basis for determining surrounding rock pressure in high ground stress soft rock tunnels.

2.3. Parameter Sensitivity Analysis

To identify the main influencing factors and their sensitivity on the design value of surrounding rock pressure in high ground stress soft rock tunnels, a systematic analysis was conducted using the orthogonal experimental design method. Four key factors were selected: surrounding rock mechanical parameters (A), tunnel burial depth (B), lateral pressure coefficient (C), and tunnel cross-section type (D). Each factor was set at four levels (

Table 6. Factors and levels for the orthogonal experiment.

Level	Surrounding Rock Parameters (A)	Tunnel Burial Depth (B)	Lateral Pressure Coefficient (C)	Tunnel Cross-Section Type (D)
1	Grade IVa (Lower 1/3 Fractile per Code)	400 m	1.2	160 km/h Single-Track
2	Grade IVb (Lower 2/3 Fractile per Code)	600 m	1.5	200 km/h Single-Track
3	Grade Va (Lower 1/3 Fractile per Code)	800 m	1.7	200 km/h Double-Track
4	Grade Vb (Lower 2/3 Fractile per Code)	1000 m	2.0	350 km/h Double-Track

), constructing an L₁₆(4⁴) orthogonal array, resulting in a total of 16 numerical simulation schemes.

2.3.1. Numerical Model Establishment

In the numerical model, to eliminate the boundary effects on tunnel excavation, the model dimensions were set to be no less than five times the tunnel diameter. The specific dimensions of the model are 200 m × 1 m × 140 m. A uniformly distributed load was applied on the top surface of the model to simulate the self-weight of the overlying rock mass. A vertical displacement constraint was applied at the bottom boundary, while stress boundaries were applied on the remaining surfaces. The Mohr–Coulomb constitutive model was adopted for the surrounding rock. The numerical model of the tunnel is illustrated in Figure 7.

2.3.2. Analysis of Orthogonal Experiment Results

Numerical simulations were conducted using the FLAC3D software (Version 6.0). Different rock mass conditions were simulated by varying the physical and mechanical parameters of the surrounding rock. Different tunnel burial depths and lateral pressure coefficients were simulated by altering the initial in situ stress conditions. Various tunnel cross-section types were modeled by changing the tunnel profile dimensions. The resulting design values of the surrounding rock pressure for different working conditions are presented in

Table 7. Calculation results of the design value of surrounding rock pressure.

Test No.	Surrounding Rock Grade	Burial Depth (m)	Lateral Pressure Coefficient	Cross-Section Type	Design Value of Surrounding Rock Pressure (MPa)
1	IVa	400	1.2	160 km/h, Single-track	0.110
2	IVa	600	1.5	200 km/h, Single-track	0.217
3	IVa	800	1.7	200 km/h, Double-track	0.496
4	IVa	1000	2.0	350 km/h, Double-track	0.542
5	IVb	400	1.5	200 km/h, Double-track	0.389
6	IVb	600	1.2	350 km/h, Double-track	0.480
7	IVb	800	2.0	160 km/h, Single-track	0.649
8	IVb	1000	1.7	200 km/h, Single-track	1.036
9	Va	400	1.7	350 km/h, Double-track	0.994
10	Va	600	2.0	200 km/h, Double-track	1.689
11	Va	800	1.2	200 km/h, Single-track	1.159
12	Va	1000	1.5	160 km/h, Single-track	1.771
13	Vb	400	2.0	200 km/h, Single-track	1.358
14	Vb	600	1.7	160 km/h, Single-track	1.401
15	Vb	800	1.5	350 km/h, Double-track	2.918
16	Vb	1000	1.2	200 km/h, Double-track	3.031

. Based on the surrounding rock deformation contour plots (Figure 8), the relationship curves between the support pressure and the maximum surrounding rock deformation for each test scheme were obtained, as shown in Figure 9. Due to space limitations, only selected deformation contour plots are provided.

The significance of each factor’s influence on the design value of surrounding rock pressure was quantitatively evaluated through Range Analysis and Analysis of Variance (ANOVA) performed on the results of the 16 experimental schemes, as shown in

Table 8. Sensitivity analysis results.

Factor	Range (R)	F-Statistic	Sensitivity Ranking
Surrounding Rock Parameters (A)	7.104	13.84	1
Tunnel Burial Depth (B)	3.180	5.66	2
Tunnel Cross-Section Type (D)	1.596	3.39	3
Lateral Pressure Coefficient (C)	1.258	1.86	4

.

The results indicate that the order of influence of the various factors on the design value of surrounding rock pressure is A (Surrounding Rock Parameters) > B (Tunnel Burial Depth) > D (Tunnel Cross-Section Type) > C (Lateral Pressure Coefficient). Among these, the surrounding rock mechanical parameters are the decisive factor, whose variation has the most significant impact on the design value; the tunnel burial depth has the next greatest influence; while the lateral pressure coefficient and tunnel cross-section type have a relatively smaller influence.

3. Calculation of Design Values for Surrounding Rock Pressure in High Ground Stress Soft Rock Tunnels Considering Time Effects

3.1. Mechanical Properties of Soft Rock

3.1.1. Strain Softening Characteristics of Soft Rock

Soft rock, as a unique heterogeneous material, exhibits degradation of rock mass parameters during the process of surrounding rock deformation. When the rock mass stress reaches its peak strength, the strength of the rock mass gradually decreases to a lower level as deformation increases, demonstrating significant strain softening characteristics. Additionally, after the excavation of soft rock tunnels, the surrounding rock stress undergoes multiple stress adjustments. The mechanical evolution of the surrounding rock gradually forms four zones from the deeper part to the tunnel wall: the elastic zone, the plastic hardening zone, the plastic softening zone, and the plastic flow zone. These correspond to the OA, AB, BC, and CD curves in the rock mass stress–strain curve, respectively. Taking a circular tunnel as an example, the schematic of the surrounding rock zoning is shown in Figure 10.

In studies of high ground stress soft rock tunnels, it is typically assumed that the surrounding rock is in a hydrostatic pressure state. After the tunnel excavation, the deformation of the surrounding rock can be classified into the following situations:

(1)

When the surrounding rock is only in the elastic zone, its stress–strain curve is represented by the OA segment in Figure 10, showing a linear relationship. At this stage, the rock mass can achieve self-stability through its own bearing capacity, and no support force is required.

(2)

When the surrounding rock enters the plastic hardening zone, the surrounding rock stress increases, but it remains in a stable state.

(3)

When the surrounding rock enters the plastic softening zone, the surrounding rock stress exceeds its ultimate bearing capacity. The surrounding rock can no longer remain stable and requires the application of support forces. This is the optimal time to install support, and at this stage, the surrounding rock deformation is not influenced by time variation.

(4)

When the surrounding rock enters the plastic flow zone, the surrounding rock deformation becomes unstable, and the deformation of the surrounding rock in each zone changes over time.

3.1.2. Creep Characteristics of Soft Rock

The creep characteristics of soft rock refer to the gradual increase in rock strain over time under a constant load, rather than immediately reaching a fixed value, exhibiting a significant time accumulation effect. Even with relatively small stress, creep deformation may accumulate to a considerable extent over a long period. According to different external loads, the creep curve of the rock can be divided into steady-state creep and non-steady-state creep, as shown in Figure 11.

(a)

Steady-state creep: When the external load is small, the creep rate gradually decreases as time increases, entering a relatively stable stage with no significant acceleration or deceleration. Eventually, the creep rate tends to a constant value.

(b)

Non-steady-state creep: When the external load is large, the strain rate increases significantly as time passes. When the creep rate exceeds a certain value at a particular moment, the rock undergoes failure due to creep. This form of creep mainly manifests in four stages:

(1)

First stage: Instantaneous Elastic Stage (OA segment): After initial loading, the rock undergoes an instantaneous deformation.

(2)

Second stage: Deteriorating Creep Stage (AB segment): In this stage, the strain–time curve shows a gradual decrease in the creep rate as time increases.

(3)

Third stage: Steady Creep Stage (BC segment): After transient deformation, the rock enters a stable but gradually increasing creep rate stage. The characteristic of this stage is that the creep rate is proportional to time, meaning the creep rate remains relatively stable but continues to increase.

(4)

Fourth stage: Accelerating Creep Stage (CD segment): In this stage, when the external load exceeds the rock’s inherent bearing strength limit, the creep rate increases rapidly, causing rock failure and exhibiting significant nonlinear characteristics.

During the construction of high ground stress soft rock tunnels, the surrounding rock is subject to harsh environmental stress. Due to the strain softening and creep characteristics of soft rock itself, when the surrounding rock is under high surrounding rock pressure, if the support structure is unreasonable, the surrounding rock is prone to non-steady-state creep, which poses significant safety risks to tunnel construction. Therefore, it is necessary to select a creep constitutive model that matches the mechanical properties of soft rock to calculate the surrounding rock pressure design values for high ground stress soft rock tunnels, considering time effects.

3.2. Viscoelastic–Plastic Strain Softening Creep Mechanical Model for Soft Rock

Hu [32] comprehensively considered the strain softening and creep characteristics of high ground stress soft rock. He combined the Burgers model with the SS plastic strain softening element in series and proposed a viscoelastic–plastic strain softening creep model for soft rock. This model can effectively describe the instantaneous elasticity, decaying creep, steady-state creep, and accelerating creep deformation of soft rock. It compensates for the accelerating creep deformation caused by the decay of the rock’s inherent strength, better aligning with the mechanical properties of high ground stress soft rock. The mechanical model is shown in Figure 12.

3.2.1. Analysis of the Strain Softening Creep Mechanical Model for Soft Rock

The constitutive equation of the soft rock viscoelastic–plastic strain softening creep model proposed in reference [32] is as follows:

When $\sigma <{\sigma }_S$, the soft rock viscoelastic–plastic strain softening creep mechanical model can degrade into the typical Burgers model, and its constitutive equation is

$$\ddot{\sigma }+\left({\displaystyle \frac{G^K}{{\eta }^M}}+{\displaystyle \frac{G^K}{{\eta }^K}}+{\displaystyle \frac{G^M}{{\eta }^K}}\right)\stackrel{.}{\sigma }+{\displaystyle \frac{G^M{G}^K}{{\eta }^M{\eta }^K}}\sigma =G^M\ddot{\epsilon }+{\displaystyle \frac{G^M{G}^K}{{\eta }^K}}\stackrel{.}{\epsilon }$$

(1)

where $G^K,G^{\mathrm{M}}$ represent the Kelvin and Maxwell bulk shear modulus, respectively, while ${\eta }^K,{\eta }^M$ are the viscosity coefficients of the Kelvin and Maxwell bodies, respectively.

When $\sigma \ge {\sigma }_S$, the soft rock viscoelastic–plastic strain softening creep mechanical model is composed of the Burgers viscoelastic model and the SS viscoplastic model in series. The deviatoric behavior of model is expressed by the following relationship:

(1)

Total strain rate of the model:

$${\stackrel{.}{e}}_{ij}{=\stackrel{.}{e}}_{ij}^K+{ \stackrel{.}{e}}_{ij}^M+{ \stackrel{.}{e}}_{ij}^P$$

(2)

where ${ \stackrel{.}{e}}_{ij}^K$, ${ \stackrel{.}{e}}_{ij}^M$, ${ \stackrel{.}{e}}_{ij}^P$ represent the strain rates of the Kelvin body, Maxwell body, and SS body strain softening rate, respectively.

(2)

Kelvin body strain softening rate:

$$S_{ij}=2{\eta }^K{\stackrel{.}{e}}_{ij}^K+2G^K{e}_{ij}^K$$

(3)

(3)

Maxwell body strain softening rate:

$${ \stackrel{.}{e}}_{ij}^M={\displaystyle \frac{{\stackrel{.}{s}}_{ij}}{2G^M}}+{\displaystyle \frac{S_{ij}}{2{\eta }^M}}$$

(4)

(4)

SS body strain softening rate:

$${\stackrel{.}{e}}_{ij}^P=\lambda {\displaystyle \frac{\partial g}{\partial {\sigma }_{ij}}}-{\displaystyle \frac{1}{3}}{\stackrel{.}{e}}_{vol}^P{\delta }_{ij}$$

(5)

$${\stackrel{.}{e}}_{vol}^P=\lambda \left[{\displaystyle \frac{\partial g}{\partial {\sigma }_{11}}}+{\displaystyle \frac{\partial g}{\partial {\sigma }_{22}}}+{\displaystyle \frac{\partial g}{\partial {\sigma }_{33}}}\right]$$

(6)

The plastic flow law follows the Mohr–Coulomb subsequent yield function in the strain softening model, $f=0$, that is

$$f={\sigma }_1-{\sigma }_3\cdot {\displaystyle \frac{1+\sin \phi (\epsilon )}{1-\sin \phi (\epsilon )}}+2\cdot c(\epsilon )\cdot \sqrt{{\displaystyle \frac{1+\sin \phi (\epsilon )}{1-\sin \phi (\epsilon )}}}$$

(7)

The variation pattern of the shear strength of rock materials $c$, $\phi$ with respect to the parameters $\epsilon$ is shown in Equations (8) and (9), and the corresponding evolution curve is shown in Figure 13.

$$c(\epsilon )=\left\{\begin{array}{ll}{\mathrm{c}}_p& 0\le \epsilon <{\epsilon }_p\\ c(\epsilon ,{\epsilon }_p,{\epsilon }_r)& {\epsilon }_p\le \epsilon <{\epsilon }_r\\ {\mathrm{c}}_r& {\epsilon }_r\le \epsilon <\propto \end{array}\right.$$

(8)

$$\phi (\epsilon )=\left\{\begin{array}{ll}{\phi }_p& 0\le \epsilon <{\epsilon }_p\\ \phi (\epsilon ,{\epsilon }_p,{\epsilon }_r)& {\epsilon }_p\le \epsilon <{\epsilon }_r\\ {\phi }_r& {\epsilon }_r\le \epsilon <\propto \end{array}\right.$$

(9)

where ${\epsilon }_p$, ${\epsilon }_r$ represent the peak strain and residual strain of the rock mass, respectively; ${\mathrm{c}}_p$, ${\mathrm{c}}_r$ represent the cohesion corresponding to the peak strain and residual strain of the rock mass, respectively; ${\phi }_p$, ${\phi }_r$ represent the friction angle corresponding to the peak strain and residual strain of the rock mass, respectively.

3.2.2. Implementation and Verification of the Viscoelastic–Plastic Strain Softening Creep Model for Soft Rock

Model Secondary Development

According to the constitutive equations of the soft rock viscoelastic–plastic strain softening creep model in Section 3.2.1, custom secondary development is carried out using the source files (.cpp) and header files (.h) provided by the built-in Cvisc creep constitutive model in FLAC3D. Visual Studio 2019 software is used to perform the development, generating a dynamic link library file (.dll). Then, the dynamic link library file (.dll) is configured by using the input command Model configure plugins, enabling FLAC3D to recognize the custom constitutive model and allowing the configuration of the custom-developed program.

(1)

Visual Studio 2019 Environment Setup

① Based on the custom soft rock viscoelastic–plastic strain softening creep mechanical model, this paper selects Cvisc from the built-in constitutive model library of FLAC3D as the prototype model for the development of modelcvisc-soft. The reason for choosing Cvisc is mainly that it introduces the Mohr–Coulomb yield criterion based on the Burgers model. The custom model modelcvisc-soft modifies the shear strength parameters of the rock mass, making a plastic stress correction on the Mohr–Coulomb yield criterion. Therefore, it is reasonable to choose Cvisc as the prototype for custom development.

② Copy the modelcvisc folder and rename it as modelcvisc-soft. Run Visual Studio 2019, execute the command Create New Project—Itasca Constitutive Model—Configure New Project, and create the modelcvisc-soft dynamic library project.

③ In the Solution Explorer window, rename the header file (modelcvisc.h) and source file (modelcvisc.cpp) to modelcvisc-soft.h and modelcvisc-soft.cpp, respectively. Add them to the newly created project. Then, in the project properties dialog, modify the additional header file directory and additional library directory.

④ In the modified header file (modelcvisc-soft.h) and source file (modelcvisc-soft.cpp), replace all instances of modelcvisc with modelcvisc-soft.

(2)

Modify the modelcvisc-soft.h file

Redefine the private variables. The custom creep model variables in modelcvisc-soft introduced in this paper, compared to those in Cvisc, include additional parameters: cTable and fTable. cTable and fTable represent the values of cohesion and friction angle of the rock material as they change with the surrounding rock deformation.

(3)

Modification of modelcvisc-soft.cpp file

This section mainly involves the description of the base class and its member functions, including the Properties function for material parameters, the State function, the GetProperty function for parameter assignment, the Copy function for the base class copy, the Initialize function, and the Run function. Among these, the most critical development program is the Run function.

During the calculation process, the Run function needs to be called in each iteration and for each sub-unit calculation. It first calculates the initial stress and deviatoric stress of the element. Then, using the Mohr–Coulomb subsequent yield function in the strain softening model, it determines whether the element has entered the plastic stage. The time is recorded as t = t0. If the element has not entered the plastic stage, it returns to the previous step. When the element enters the plastic stage, the strain softening parameters are updated, and the new deviatoric stress of the element is calculated. The time ts = t − t0 is also recorded.

The main program calculates the unbalanced forces, nodal velocities, and nodal displacements until the entire creep process is completed. The program development flowchart is shown in Figure 14.

Model Validation

To validate the custom-developed soft rock viscoelastic–plastic strain softening constitutive model, a triaxial compression simulation test was performed using the model. The dimensions of the model were 1 m × 1 m × 1 m, with a confining pressure set to 6 MPa and a vertical load set to 17.79 MPa. The computational model is shown in Figure 15, and the model parameters are provided in

Table 9. Model creep parameters.

K	${\mathit{G}}^{\mathit{M}}$	${\mathit{G}}^{\mathit{K}}$	${\mathit{\eta }}^{\mathit{M}}$	${\mathit{\eta }}^{\mathit{K}}$	C	$\mathit{\phi }$	${\mathit{\sigma }}^{\mathit{t}}$
(GPa)	(GPa)	(GPa)	(GPa·h)	(GPa·h)	(MPa)	(°)	(MPa)
1.89	3.32	0.38	744.2	1.06	0.6	27	0.8

and

Table 10. Model strain softening parameters.

Plastic Strain $\mathit{\epsilon }$	0	0.005	0.01	0.15	0.2
Cohesionc (MPa)	0.6	0.54	0.46	0.3	0.15
Friction Angle
$\phi$ (°)	27	26.6	25.9	24.5	24

.

The axial strain variation with time at the model’s center point was plotted. The results were compared between the Cvisc model and the soft rock viscoelastic–plastic strain softening model, as shown in Figure 16. When the strain softening parameters are constant, the soft rock viscoelastic–plastic strain softening constitutive model can degenerate into the Cvisc model. Compared to the built-in Cvisc model in FLAC3D, the custom-developed model can effectively simulate the unstable creep behavior of soft rock under high stress, capturing the accelerated creep deformation induced by post-peak strain softening in the rock mass. This model provides a better representation of the creep deformation in high-stress soft rock. Therefore, the soft rock viscoelastic–plastic strain softening constitutive model is adopted for all subsequent creep simulations.

3.3. Calculation Method for Design Values of Tunnel Surrounding Rock Pressure Considering Time Effects in High-Stress Soft Rock

During the construction of tunnels in high-stress soft rock, the surrounding rock deformation exhibits significant time accumulation effects. During this period, the contact pressure between the tunnel surrounding rock and the support structure continuously increases. If the support force provided by the support structure is less than the support force required to stabilize the surrounding rock deformation, the tunnel may experience large deformation and failure. Therefore, based on a comprehensive consideration of the relationship between time, support force, and surrounding rock deformation, a calculation method for the design value of surrounding rock pressure in high-stress soft rock tunnels considering time effects is proposed. This method can serve as a reference for the design of support structures in high-stress soft rock tunnels.

3.3.1. Calculation Principle

The most direct evaluation criterion for determining whether a high-stress soft rock tunnel meets safety requirements during the operational phase is the ability of the support structure to fulfill its specific functional requirements. This includes ensuring that the structure provides adequate support throughout tunnel operation. Specifically, this means whether the support force provided by the support structure can bear the continuous increase in surrounding rock pressure due to surrounding rock deterioration and other factors during the operational period. The method proposed in Section 2, which determines the surrounding rock pressure design value through reserved deformation in the tunnel, does not consider the influence of creep time on surrounding rock deformation and surrounding rock pressure. Therefore, further improvement is necessary.

This section builds upon the calculation method for high-stress soft rock tunnel surrounding rock pressure design values presented in Section 2.2 and introduces the traditional surrounding rock characteristic curve based on elastoplasticity theory, incorporating creep time. The surrounding rock characteristic surface is used to represent the relationship between the three physical quantities: creep time, support force, and surrounding rock deformation, as shown in Figure 17. The functional relationship of this surface is expressed as

$$P=f(u,t)$$

(10)

As shown in Figure 17, when t = 0, the support force provided by the support structure is P_e, and at this time, the surrounding rock deformation is u_e. When t > 0, in order to maintain the surrounding rock deformation at u_e, the support structure must be strengthened until the support force provided by the structure exceeds P_c, ensuring that the surrounding rock remains in a stable state. If a larger support force cannot be provided, the surrounding rock deformation will continue to increase with time, generating new creep displacement u_c. At this point, the total surrounding rock deformation u is the sum of u_e and u_c. When the total surrounding rock deformation reaches u, the support structure may exceed its ultimate load-bearing capacity due to excessive surrounding rock pressure, leading to large deformation failure.

Therefore, during the design phase of high-stress soft rock tunnels, the relationship between surrounding rock deformation, time, and support force given in Figure 17 can be used to determine the required support force provided by the support structure under different times and surrounding rock deformations. This allows for the determination of appropriate support measures and timing for tunnel support structures.

3.3.2. Calculation Method

Based on the above computational principles, the calculation method for the design value of surrounding rock pressure in highland stress soft rock tunnels considering the time effect is as follows:

(1)

Numerical simulation model setup: Based on the actual tunnel survey data, a numerical simulation model is established. The initial surrounding rock’s physical and mechanical parameters are reduced using a strength reduction factor, typically set at 1.15. A suitable elastoplastic constitutive model is selected, the excavation design contour is determined, radial support forces are applied, and the calculation is carried out until convergence.

(2)

Without considering time effect: Plot the relationship curve between the support force (P) and the maximum deformation (u) of the surrounding rock. Based on the reserved deformation amount (u_e) of the tunnel, the support force (P_e) under the deformation control reference value is determined, as shown in Figure 18.

(3)

Considering time effect: The surrounding rock’s constitutive relation adopts a viscoelastic–plastic constitutive model. Numerical simulations are conducted based on creep parameters obtained from creep testing machines and triaxial tests. The relationship curve between the support force and the maximum deformation of the surrounding rock at different creep times is plotted, as shown in Figure 19.

(4)

Determining support forces at different creep times: Based on the relationship curves between support force and maximum deformation at different times, and following the principle that the reserved deformation of the tunnel remains unchanged, the support forces at different creep times (P₀, P₁, P₂, …, P_n) are determined. The support forces (P₀, P₁, P₂, …, P_n) are then plotted into a smooth curve according to the time sequence, representing the relationship between support force and time, as shown in Figure 20. Here, P₀ refers to the support force (P_e) corresponding to the reserved deformation (u_e) of the tunnel without considering the time effect.

(5)

Determining Minimum Support Force: From the obtained relationship curve of support force changing with time (Figure 21), the minimum support force (P_c) is determined as the point where the support force no longer changes over time. Further, the vertical load at the crown (P_c) is equivalently converted into a uniform load (q_c).

(6)

Design Value of Surrounding Rock Pressure: The design value of surrounding rock pressure (q) is taken as 1.4 times the minimum support force (P_c). The horizontal load is selected by multiplying the vertical load at the crown (q) by the side pressure coefficient, which is taken according to the parameter values specified in the technical specification for railway squeezing surrounding rock tunnel.

3.3.3. Calculation Example

Due to the differences in the mechanical properties of the rock mass, the deformation of the surrounding rock in highland stress soft rock tunnels will gradually stabilize with the passage of time after tunnel excavation. This process may take a month, a year, or even longer, and exhibits a significant time accumulation effect. Ma [33], using shale as the research subject, found from the in situ measured surrounding rock deformation data of the Chengdu–Lanzhou Railway that after the secondary lining was applied, the deformation of the surrounding rock continued to increase. Even after 300 days of excavation of the tunnel cross-section, the surrounding rock deformation had not stabilized and maintained a growth trend, as shown in Figure 21a. Cao [34], using laminated soft rock as the research subject, analyzed the surrounding rock deformation of the Xinchengzi Tunnel on the Lanzhou–Yunnan Railway. The surrounding rock deformation stabilized after 30 days of excavation of the tunnel cross-section, as shown in Figure 21b.

Based on the above two engineering cases, due to the differences in the mechanical properties of soft rock, the time required for surrounding rock deformation to stabilize shows significant variation. Research indicates that the difference in the mechanical properties of soft rock is mainly reflected in the creep parameters of soft rock. Therefore, this section calculates the minimum support force corresponding to different time effects under two creep parameters for a high-speed railway double-line tunnel with a burial depth of 600 m, a side pressure coefficient of 1.2, and a speed of 350 km/h. The model’s creep parameters are shown in

Table 11. Group 1 of creep parameters.

K	${\mathit{G}}^{\mathit{M}}$	${\mathit{G}}^{\mathit{K}}$	${\mathit{\eta }}^{\mathit{M}}$	${\mathit{\eta }}^{\mathit{K}}$	C	$\mathit{\phi }$
(GPa)	(GPa)	(GPa)	(GPa·d)	(GPa·d)	(MPa)	(°)
1.89	0.6	2.13	185185	3819	0.15	24.67

and

Table 12. Group 2 of creep parameters.

K	${\mathit{G}}^{\mathit{M}}$	${\mathit{G}}^{\mathit{K}}$	${\mathit{\eta }}^{\mathit{M}}$	${\mathit{\eta }}^{\mathit{K}}$	C	$\mathit{\phi }$
(GPa)	(GPa)	(GPa)	(GPa·d)	(GPa·d)	(MPa)	(°)
1.89	0.9	1.5	1000	7800	0.15	24.67

below.

The selection principle of creep parameters is based on the reference [35]. The literature points out that the larger $G^M$, the smaller the instantaneous creep deformation. ${\eta }^M$ mainly controls the steady-state creep rate, with a larger value resulting in a smaller steady-state creep rate. $G^M$ also controls the final deformation of the decay creep phase, with a larger value leading to a smaller final deformation in the decay phase. ${\eta }^K$ primarily controls the duration of the decay creep phase, with a larger value resulting in a longer decay creep duration. Therefore, based on group 1 of creep parameters, to reflect the influence of the mechanical properties of soft rock on the surrounding rock pressure design value, an expansion of the $G^M$, ${\eta }^K$ range and a reduction in the ${\eta }^M$, $G^K$ range were adopted. Figure 22 shows the relationship curves between support force and the maximum deformation of the surrounding rock considering the time effect for the group 1 and 2 of creep parameters, respectively.

Assuming the tunnel reserved deformation is 400 mm, the minimum support force corresponding to different times under group 1 and 2 of creep parameters is obtained, as shown in

Table 13. Minimum support force corresponding to different time periods (MPa).

Time	t = 0	t = 0.5 a	t = 1 a	t = 5 a	t = 10 a	t = 30 a	t = 60 a
Group 1	1.05	1.09	1.12	1.22	1.26	1.32	1.36
Group 2	1.05	1.12	1.28	1.82	2.16	2.68	2.96

. The minimum support force variation curve over time is plotted, as shown in Figure 23. it can be observed that when the creep time is 60 years, the minimum support force required to stabilize the surrounding rock under the group 2 of creep parameters is more than twice that of the group 2 of creep parameters, and the surrounding rock is still in an unstable state. This indicates that the mechanical properties of soft rock, to some extent, determine the magnitude of the tunnel surrounding rock pressure.

4. Stability Evaluation of Support Structure for Highland Stress Soft Rock Tunnels

4.1. Total Safety Factor Method for Support Structures in Highland Stress Soft Rock Tunnels

In highland stress soft rock tunnels, the composite bearing structure primarily consists of the rock-anchor bearing arch, sprayed layer, and secondary lining. Based on previous research, Xiao [36] proposed the total safety factor method for tunnel support structures, which organically combines the surrounding rock, load, and support structure. This method calculates the surrounding rock pressure design value using the geological structure method, then calculates the internal forces of the support structure using the load-structure method. The safety factor for each layer of the support structure is calculated using the damage stage method. Additionally, it fully considers the coordinated deformation between each layer of the composite lining and their respective contributions, allowing for the calculation of the overall safety factor of the composite lining structure. This method addresses the limitation in standards where stability is only evaluated based on a single component of the support structure.

After calculating the safety factor of each layer of the support structure at different times, the minimum total safety factor for each time is calculated using Formulas (11)–(13):

Construction stage (before secondary lining is applied):

$$K_{\mathrm{c}\left(\mathrm{t}\right)}={\eta }_{(\mathrm{t})}{K}_{1(t)}+K_{2(t)}$$

(11)

$${\eta }_{\left(\mathrm{t}\right)}={\displaystyle \frac{{\epsilon }_u{E}_0}{{\sigma }_{1(\mathrm{t})}}}$$

(12)

Operational period:

$$K_{\mathrm{op}\left(\mathrm{t}\right)}={\eta }_{(t)}{K}_{1(\mathrm{t})}+\xi K_{2(\mathrm{t})}+K_{3(\mathrm{t})}$$

(13)

where K_1(t), K_2(t), K_3(t) represent the safety factors of the rock-anchor bearing arch, sprayed layer, and secondary lining at different time periods, respectively. ${\eta }_{(\mathrm{t})}$ is the adjustment coefficient for the safety factor of the rock-anchor bearing arch at different time periods, calculated according to Formula (12). $\xi$ is the correction coefficient for the safety factor of the sprayed layer, and it is given as $\xi =1$. ${\epsilon }_u$ is the ultimate strain of concrete, and it is given as ${\epsilon }_u=2\%$. E₀ is the elastic modulus of the rock-anchor bearing arch. ${\sigma }_{1(\mathrm{t})}$ is the time-dependent function of the compressive strength of the rock-anchor bearing arch, calculated according to Formula (14). The limit strength of the surrounding rock within the bearing arch range only considers the strength increase in the surrounding rock itself after the support structure is applied. It is calculated using the following Formula (14). The lateral confining forces ${\sigma }_{3(\mathrm{t})}$ are provided by ${\sigma }_{31}$ (the anchor bolts), ${\sigma }_{32(\mathrm{t})}$ (the concrete sprayed layer), and ${\sigma }_{33(\mathrm{t})}$ (the secondary lining). Based on the Mohr–Coulomb strength criterion, the surrounding rock’s strength ${\sigma }_{1(\mathrm{t})}$ is calculated, as shown in Figure 24, and this value is taken as the limit strength of the bearing arch itself.

$$\left[{\sigma }_{\mathrm{c}\left(\mathrm{t}\right)}\right]={\sigma }_{1(\mathrm{t})}=2\mathrm{c}\tan (45°+{\displaystyle \frac{\phi }{2}})+{\sigma }_{3(\mathrm{t})}{\tan }^2({45}^{°}+{\displaystyle \frac{\phi }{2}})$$

(14)

$$\mathrm{c}={\displaystyle \frac{0.35A_{\mathrm{s}}{f}_y}{bs}}+c_p$$

(15)

$${\sigma }_{31}={\displaystyle \frac{0.5T_1}{bs}}(or{\displaystyle \frac{0.4T_2}{bs}})$$

(16)

$${\sigma }_{32}\left(\mathrm{t}\right)=0.5K_{2(\mathrm{t})}\cdot q_{\left(\mathrm{t}\right)}$$

(17)

$${\sigma }_{33}\left(\mathrm{t}\right)=0.5K_{3(\mathrm{t})}\cdot q_{\left(\mathrm{t}\right)}$$

(18)

where c is the cohesion of the rock-anchor bearing arch, calculated using Formula (15); $\phi$ is the internal friction angle of the rock-anchor bearing arch; A_s is the cross-sectional area of the anchor bolts; f_y is the yield strength of the anchor rod; b and s represent the circumferential and longitudinal spacing of the anchor rods, respectively; c_p is the residual cohesion of the rock-anchor bearing arch. In the absence of experimental results, c_p is taken as 50%, 70%, and 90% of the initial value for surrounding rock classes III, IV, and V, respectively; q_(t) is the surrounding rock pressure design value as a function of time; T₁ and T₂ are the strength and tensile strength of the anchor rod, calculated using Formulas (19) and (20); and K_2(t) and K_3(t) are the safety factors of the concrete sprayed layer and secondary lining at different times, respectively.

$$T_1={\mathrm{f}}_y\cdot \pi d^2/4$$

(19)

$$T_2={\mathrm{f}}_{\mathrm{rd}}\cdot \pi d_g\cdot l_g$$

(20)

where d is the diameter of the anchor rod; f_rb is the ultimate bond strength between the anchor grout and the surrounding rock; d_g is the outer diameter of the anchor grout; l_g is the anchorage length of the anchor tendon and grout.

The total safety factor considers the coordinated deformation between the initial support and the secondary lining, as well as each layer’s contribution. Combined with the structural characteristics of highland stress soft rock tunnel engineering under complex geological conditions, the load-structure combination model, construction quality, and other factors, the suggested values for the total safety factor are proposed:

During the operating phase, when the secondary lining is made of reinforced concrete, the total safety factor should be greater than 3.0; when the secondary lining is made of plain concrete, the total safety factor should be greater than 3.6.

Based on the field monitoring data of the secondary lining from the Mao County Tunnel on the Chengdu–Lanzhou Railway, the internal force evolution of the support structure is obtained through numerical simulation. Then, by comparing the numerical simulation results with the field monitoring data, the total safety factor of the support structure is used to evaluate the overall stability of the support structure.

4.2. Engineering Case

The No. 1 inclined shaft of the Mao County Tunnel on the Chengdu–Lanzhou Railway (near the F1 fault) passes through fractured shale and weak strata. The in situ stress field is mainly dominated by horizontal stress, with measured results of 12.7 MPa, 21.4 MPa, and 15.1 MPa. The uniaxial saturated compressive strength of the rock mass is 2.41 MPa, and the strength-stress ratio is only 0.11. During tunnel excavation, significant deformation occurred, including severe twisting of the steel frame and cracking of the concrete sprayed layer, as shown in Figure 25.

Therefore, when the tunnel’s main excavation crosses the F1 fault, the support structure’s stiffness is increased to control surrounding rock deformation. The initial support uses 28 cm thick C25 sprayed concrete, H175 type steel arches with a longitudinal spacing of 0.6 m, 4.5 m long cement–sand anchor rods, and the secondary lining uses 60 cm thick C35 sprayed concrete. Figure 26 shows the field-measured bending moment and axial force evolution of the secondary lining. Referring to the Railway Tunnel Design Code, which recommends a safety factor of not less than 2.0 for reinforced concrete linings without anti-cracking requirements, the time-history curve of the secondary lining safety factor is calculated based on the field data, as shown in Figure 26c [37].

4.3. Establishment of the Calculation Model

Based on the in situ stress data of the Mao County Tunnel on the Chengdu–Lanzhou Railway, a numerical calculation model is established using FLAC3D finite difference software, as shown in Figure 27. In the calculation model, a uniform distributed load is applied to the upper surface to simulate the self-weight of the overlying rock mass. A vertical displacement constraint is applied to the bottom boundary, and stress boundaries are applied to the remaining surfaces. The surrounding rock is modeled using a customized soft rock viscoelastic–plastic creep constitutive model. The model’s creep parameters are shown in

Table 14. Creep parameters of the model.

${\mathit{G}}^{\mathit{M}}$	${\mathit{G}}^{\mathit{K}}$	${\mathit{\eta }}^{\mathit{M}}$	${\mathit{\eta }}^{\mathit{K}}$	C	$\mathit{\phi }$	${\mathit{\sigma }}_{\mathbf{t}}$	$\mathit{\gamma }$
(GPa)	(GPa)	(GPa·d)	(GPa·d)	(MPa)	(°)	(MPa)	(kN/m3)
0.7187	1.4778	42,000	5,800,000	0.6	27	0.176	24.5

and

Table 15. Strain softening parameters of the model.

Plastic Strain $\mathit{\epsilon }$	0	0.005	0.01	0.15	0.2
Cohesionc (MPa)	0.6	0.54	0.46	0.3	0.15
Friction Angle $\phi$ (°)	27	26.5	25.9	24.5	24

.

4.4. Simulation Results Analysis

Based on the established model, the calculation method for the surrounding rock pressure design value of highland stress soft rock tunnels considering time effects, as proposed in Section 3.3, is applied. The relationship curve between support force and maximum surrounding rock deformation at different creep times is plotted (Figure 28). This allows for extracting the minimum support force corresponding to different times for the reserved deformation of the Mao County Tunnel on the Chengdu–Lanzhou Railway. Then, multiplying by a safety factor of 1.4 gives the time-history curve of the surrounding rock pressure design value, as shown in Figure 29. Since the secondary lining of the Mao County Tunnel on the Chengdu–Lanzhou Railway bears 67.4% of the load carried by the initial support, the surrounding rock pressure design value for the secondary lining is taken as 67.4% of the sprayed concrete surrounding rock pressure design value. Based on the time-history curve of the surrounding rock pressure design value for the Mao County Tunnel (Figure 29), the surrounding rock pressure design values corresponding to different creep times are shown in

Table 16. Design values of surrounding rock pressure at different creep times (MPa).

Creep Time		t = 30 d	t = 90 d	t = 150 d	t = 210 d	t = 240 d	t = 300 d	t = 730 d
Anchor Rock Bearing Arch and Shotcrete Layer	Vertical Load	1.232	1.442	1.526	1.554	1.564	1.575	1.597
	Horizontal Load	1.478	1.73	1.831	1.865	1.877	1.89	1.917
Secondary Lining	Vertical Load	0.825	0.966	1.022	1.041	1.048	1.055	1.07
	Horizontal Load	0.991	1.159	1.227	1.249	1.257	1.266	1.284

. Based on the calculation principle of the total safety factor method and after obtaining the time-history curve of the surrounding rock pressure design value, the GTS Madis finite element software was employed to perform calculations using the load-structure model. The calculation parameters for the sprayed layer and secondary lining are shown in

Table 17. Calculation parameters of shotcrete and secondary lining.

Structure Type	Thickness (cm)	Concrete Grade	Elastic Modulus (GPa)	Poisson’s Ratio	Material Density (kg/m3)	Foundation Stiffness Coefficient (MPa/m)
Shotcrete	28	C25	30	0.2	2300	120
Secondary Lining	60	C35	32.5	0.2	2400	120

. The load-structure models used are the calculation models from Section 4.2, with the calculation model for the sprayed layer illustrated in Figure 30. The axial forces and bending moments of the anchored rock bearing arch, sprayed layer, and secondary lining were solved based on the load combinations in Table 16, and the total safety factor of the support structure was calculated. The axial force diagrams and bending moment diagrams of the anchored rock bearing arch, sprayed layer, and secondary lining at different creep times are shown in Figure 31, Figure 32 and Figure 33.

From Figure 31, Figure 32 and Figure 33, it can be observed that the maximum axial force values for each layer of the support structure are distributed at the crown position. The axial force values gradually decrease from the crown to the sidewall, with the minimum axial force at the sidewall. This phenomenon is consistent with the axial force distribution of the secondary lining of the Mao County Tunnel on the Chengdu–Lanzhou Railway (Figure 26a). Therefore, in this study, the minimum safety factor section of the secondary lining of the Mao County Tunnel is selected, which corresponds to the crown position, to analyze the total safety factor of the support structure.

The bending moment and axial force values at the crown position are extracted from the bending moment and axial force diagrams for different creep times, as shown in

Table 18. Axial force and bending moment of support structures at different creep times.

Creep Time	Rock-Anchor Bearing Arch		Shotcrete		Secondary Lining
	Axial Force (kN)	Bending Moment (kN·m)	Axial Force (kN)	Bending Moment (kN·m)	Axial Force (kN)	Bending Moment (kN·m)
t = 30 d	8008	−3688.54	9881	11.71	5565	−19.18
t = 90 d	9315	−4373.36	10,887	3.33	6509	−93.92
t = 150 d	9867	−4646.28	11,409	4.85	7307	−142.93
t = 210 d	9878	−4934.72	11,615	8.75	8123	−174.06
t = 240 d	9887	−4963.94	12,431	19.89	8178	−175.12
t = 300 d	9896	−4985.49	13,106	30.56	8225	−175.56
t = 730 d	10,153	−5072.15	13,843	38.62	8353	−177.25

. Based on the total safety factor calculation method, the total safety factor corresponding to different creep times is calculated, as shown in

Table 19. Overall safety factor of support structures at different creep times.

Creep Time	Rock-Anchor Bearing Arch	Shotcrete	Secondary Lining	Overall Safety Factor
t = 30 d	0.81	0.67	2.8	3.91
t = 90 d	0.67	0.54	2.4	3.32
t = 150 d	0.511	0.515	2.135	2.998
t = 210 d	0.485	0.506	1.92	2.77
t = 240 d	0.481	0.473	1.907	2.72
t = 300 d	0.478	0.458	1.896	2.69
t = 730 d	0.476	0.446	1.876	2.66

, and the total safety factor time-history evolution curve is plotted, as shown in Figure 34. From Figure 34, it can be seen that when the creep time is 30 days, the total safety factor of the support structure is 3.91, which is greater than the lower limit of 3.0 in the total safety factor method, indicating that the support structure is in a stable state. When the creep time is 90 days, the total safety factor of the support structure is 2.998, which is in an unstable state, and cracks in the secondary lining may occur. Based on the secondary lining safety factor calculated from the measured bending moments and axial forces on the Chengdu–Lanzhou Railway Mao County Tunnel (Figure 26c), when the creep time exceeds 120 days, the safety factor of the secondary lining falls below 2.0, leading to the formation of circumferential, oblique, and longitudinal cracks in the secondary lining, as shown in Figure 35.

In summary, based on the secondary lining safety factors calculated from the field monitoring data, alarm signals are triggered around 120 days. The total safety factor method calculates that the support structure may undergo instability or failure around 90 days after the secondary lining is applied. The calculation results envelop the most unfavorable situation for the secondary lining. Therefore, the total safety factor method can be effectively applied in the design of support structures for high geostress soft rock tunnels.

5. Conclusions

In this study, the concept of “support force” is used as the design approach for the calculation of the rock pressure design value in high geostress soft rock tunnels. By considering the relationship between rock deformation, support force, and creep time, two methods for calculating the rock pressure design value are proposed: one without considering the creep time and one that includes the creep time. Based on these, the rock pressure design value time-series curve is obtained and used as the input load. Through a comparison with the mechanical behavior evolution of the secondary lining in the Chengdu–Lanzhou Railway Mao County Tunnel, the main conclusions are as follows:

(1)

Based on the principle of rock pressure design value calculation, a method is proposed that is applicable to high geostress soft rock tunnels. This method determines the minimum support force using the relationship curve between support force and maximum rock deformation, combined with the deformation level of the rock and the tunnel’s reserved deformation amount. The minimum support force is multiplied by 1.4 to obtain the rock pressure design value. Comparing the measured rock pressure values of the Chengdu–Lanzhou Railway Mao County Tunnel with numerical calculation results, the rock pressure design value is 21.8% higher than the measured rock pressure, confirming the validity of this method.

(2)

Based on the mechanical characteristics of soft rock, a secondary development of the soft rock viscoelastic–plastic strain softening mechanical model is realized and validated for correctness. Furthermore, considering the relationship between time, support force, and rock deformation, a method for calculating rock pressure design values with time effects for high geostress soft rock tunnels is proposed. This method uses the relationship curve between support force and maximum rock deformation at different creep times and determines the minimum support force based on the principle that the reserved deformation of the tunnel does not change. The minimum support force is multiplied by 1.4 to determine the rock pressure design value.

(3)

Based on the total safety factor method and the rock pressure design value time-series curve calculated considering time effects, input load is obtained and the safety factors of the anchor bearing arch, sprayed layer, and secondary lining are determined through load-structure analysis. Comparing the safety factors calculated from the secondary lining axial force and bending moment field monitoring data with the total safety factors, it is found that the total safety factor method predicts the possible instability or failure of the support structure 30 days ahead of the field monitoring results. The results are relatively consistent, showing that the total safety factor method can be effectively applied to tunnel support structure design.

Notwithstanding its contributions, this study has limitations that point to future directions. The continuum mechanics-based model, while effective in this context, may not fully capture behaviors in localized fault zones or intensely jointed rock masses, and the analysis excludes long-term hydro–mechanical–chemical processes. Addressing these by integrating discontinuum methods and coupled H–M–C models will be the focus of subsequent research to enhance the prediction of Pc(t).