Analytical Method for Tunnel Support Parameter Design Based on Surrounding Rock Failure Mode Identification

Abstract

Accurately identifying surrounding rock failure modes and designing matching support systems are critical to the safety of deep-earth and underground space engineering. We develop a graded classification scheme based on the rock strength-to-stress ratio and the Stress Reduction Factor (SRF) to quantify failure types and guide support design. Within the convergence–confinement method (CCM) framework, we establish analytical models for shotcrete, rock bolts, steel arches, and composite support systems, enabling parameterized calculations of stiffness, load-bearing capacity, and equilibrium conditions. We conduct single-factor sensitivity analyses to reveal how the Geological Strength Index (GSI), burial depth (H), and equivalent tunnel radius (R₀) govern the evolution of surrounding rock pressure and deformation. We propose targeted reinforcement strategies that address large-deformation and high-stress instabilities in practice by linking observed or predicted failure modes to specific support schemes. A large-deformation case study verifies that the proposed parameterized design method accurately predicts the equilibrium support pressure and radial deformation, and the designed support scheme markedly reduces convergence. Accordingly, this study provides a practical tool for tunnel support parameter design and an analytical platform for safe, reliable, and efficient decision making for initial support.

1. Introduction

As a key enabler of deep-earth and underground space development, tunnel construction relies on structural stability during construction, which directly affects construction safety and cost-effectiveness. Especially under conditions of high in situ stress, weak surrounding rock, and complex geology, the mechanisms of rock deformation and failure vary greatly, which makes support design particularly challenging. Support design must ensure safety while maintaining economic rationality. Consequently, accurately determining the failure modes of surrounding rock and efficiently designing support parameters have become core issues that urgently need to be addressed in tunneling and underground engineering [1,2].

Thanks to their high computational efficiency and transparent physical rationale, analytical methods have attracted wide attention for analyzing the interaction between surrounding rock and support systems in tunnels [3,4,5,6]. Compared with numerical simulations, analytical approaches can rapidly provide key parameters such as support demand, rock deformation, and stress distribution at the preliminary design stage, which makes them especially suitable for parameterized analysis and design scheme comparison [7,8,9]. Under deep burial and complex conditions, analytical methods can serve both as baseline solutions for numerical simulations and as rapid screening tools: on the one hand, they provide boundary conditions and initial parameter estimates for subsequent numerical analyses; on the other hand, they substantially reduce the time required for early design scheme comparison and thereby accelerate design convergence. Among these methods, the convergence–confinement method (CCM) stands out as the most widely used theoretical framework. Its fundamental principle is to plot the ground reaction curve (GRC) and the support characteristic curve (SCC) in the same coordinate system. Their intersection represents the equilibrium state between surrounding rock deformation and support reaction, thereby enabling the determination of support parameters as well as visualization of the design process [10,11,12].

In recent years, researchers have conducted extensive studies on the theory and application of the CCM. On the one hand, researchers have incorporated the Hoek–Brown criterion [13] and the unified strength theory [14,15] into analytical solutions to better capture the nonlinear mechanical behavior of the surrounding rock, thereby providing a more realistic representation of the stress–displacement relationship in the plastic zone. On the other hand, to reflect complex construction conditions, researchers have considered factors such as anisotropic stress fields [16,17], overburden effects [18], and construction delays [19,20]. These efforts have significantly expanded the applicability of the CCM. In addition, researchers have developed parameterized analytical solutions for special support systems, including circumferential yielding linings [21], steel–concrete composite supports [22], and staged supports [23]. These solutions provide practical tools for directly calculating the support stiffness and load-bearing capacity of engineering applications.

Accurate identification of failure modes in surrounding rock is a prerequisite for support design. Traditional approaches primarily rely on empirical classification systems (e.g., RMR and the Q-system) or a single stress-ratio criterion, but they often have limited applicability under high in situ stress or in weak rock masses. Within the CCM framework, researchers have proposed safety-factor criteria [24,25] and comprehensive failure-trend coefficients [26] and used analytical solutions to rapidly grade the stability of the surrounding rock under varying excavation and support conditions, thereby providing a quantitative basis for classification. In addition, researchers have incorporated analytical expressions for plastic-zone size [27] and for tectonic stress fields [28] to reveal how failure modes evolve under different conditions and to inform support-parameter matching.

Despite the solid foundation established by prior work, two critical gaps remain: (1) current practice often treats failure mode identification and support parameter design separately and lacks a systematic, executable procedure to integrate the identification results into the CCM analytical workflow, which impedes efficient design decision making under complex conditions, and (2) reusable mappings and criteria between key rock mass parameters and support design metrics remain underdeveloped (particularly under deep burial, high in situ stress, and large deformation), thereby constraining rapid screening of design schemes and convergence in preliminary design.

Accordingly, this study develops an integrated method for failure-mode identification in surrounding rock and support parameter design, using the CCM as the theoretical foundation. First, we employ criteria including the strength-to-stress ratio and the Stress Reduction Factor (SRF) to identify failure modes in surrounding rock quantitatively. Second, within the CCM framework, we derive analytical expressions for the support characteristic curves (SCCs) of shotcrete, rock bolts, steel arches, and composite support systems. Furthermore, we conduct a parameter sensitivity study to quantify how the Geological Strength Index (GSI), burial depth (H), and equivalent tunnel radius (R₀) affect surrounding rock pressure and deformation, thereby clarifying the parametric linkage from key rock mass parameters to support design metrics. We then analyze the correspondence between failure modes and support performance and propose optimized support strategies for complex conditions, including large-deformation and high in situ stress scenarios. Finally, case studies validate the applicability and engineering feasibility of the proposed method. This research establishes a reusable, parameterized design approach for tunnel support and provides an analytical basis for rapid screening and rational design of initial support at the preliminary design stage under complex geological conditions.

This study builds on well-established theoretical foundations, including the Hoek–Brown criterion, the convergence–confinement method (CCM), and the basic forms of the GRC and SCC. Unlike prior work that relies on single criteria or fragmented calculation approaches, the key innovation of this study is that we systematically integrate failure-mode identification into the CCM framework, thereby developing an integrated analytical workflow directly applicable to engineering practice. Within this framework, we further unify and systematize the analytical expressions of SCCs for various support elements. In addition, through sensitivity analysis, we elucidate the influence of key parameters on support performance and accordingly propose differentiated design strategies under large-deformation and high-stress conditions.

2. Materials and Methods

2.1. Notation and In Situ Stress State

The initial in situ stress field in tunnels is commonly simplified to a triaxial principal-stress model, consisting of the maximum horizontal principal stress (σ_H), the minimum horizontal principal stress (σ_h), and the vertical principal stress (σ_v) [29,30]. Specifically, σ_H and σ_h are typically described by linear depth-dependent relations with nonzero intercepts, whereas σ_v arises primarily from the self-weight of the rock mass and is commonly approximated as the overburden (gravitational) stress, expressed as follows:

$$\begin{array}{l}{\sigma }_H={\eta }_{1}H+T_1\\ {\sigma }_h={\eta }_{2}H+T_2\\ {\sigma }_v=\gamma H\end{array}$$

(1)

where σ_H, σ_h, and σ_v denote the maximum and minimum horizontal principal stresses and the vertical principal stress, respectively; H is the burial depth; η and T are constants representing the in situ stress gradient and the ground-surface intercept, respectively.

Because the cross-sectional dimensions of underground excavations are limited, the initial in situ stress difference between the tunnel crown and invert is small and is generally neglected in design. Accordingly, the initial in situ stress in the surrounding rock is assumed to be essentially uniform across the tunnel cross-section. For a given burial depth, we express the principal stresses on the tunnel cross-section as follows:

$$\begin{array}{l}{\sigma }_1=K_1\gamma H\\ {\sigma }_2=K_2\gamma H\\ {\sigma }_3=K_3\gamma H\end{array}$$

(2)

where σ₁, σ₂, and σ₃ denote the maximum, intermediate, and minimum principal stresses, respectively; K is the stress ratio. This formulation provides the absolute stress levels and directly characterizes the principal stress ratios.

At greater burial depths, the maximum principal stress primarily governs surrounding rock failure;

Table 1. Classification of surrounding rock stress states based on K.

Stress Ratio (K)	Stress State Description	Deformation and Failure Characteristics
K < 1.50	Favorable in situ stress state	Stress differences have little effect; deformation is governed mainly by the tunnel cross-sectional geometry.
1.5 ≤ K < 2.0	Moderately unfavorable in situ stress state	Stress differences significantly affect surrounding rock deformation and stability. Deformation and failure tend to concentrate in tangential zones along the excavation contour, aligned with the direction of the maximum principal stress.
2.0 ≤ K < 3.0	Significantly unfavorable in situ stress state	Under similar rock mass conditions, deformation and failure tend to localize in specific regions.
K ≥ 3.0	Extremely unfavorable in situ stress state	Anomalous local geology usually causes highly localized deformation and failure in the surrounding rock.

shows how the actual failure and its severity relate to the in situ stress ratio of the surrounding rock.

2.2. Surrounding Rock Failure Modes and Identification Criteria

2.2.1. Potential Failure Modes of Surrounding Rock

Potential failure modes of the tunnel surrounding rock primarily include the following: (1) block failure controlled by structural planes; (2) squeezing deformation under the coupled effects of high in situ stress and low rock mass strength; and (3) failure induced by high stress under deep burial or other unfavorable stress conditions [31,32]. Block failure is generally negligible for small tunnels with limited jointing. In such cases, support design primarily focuses on large-deformation and high-stress failure.

Table 2. Identification criteria for problem types.

Problem Type	Determination Criterion
Deformation problem	σcm/σ1 < 0.45
High-stress problem	σcm/σ1 ≥ 0.45 and (3σ1 − σ3)/σci ≥ 0.6
Basic stability problem	σcm/σ1 ≥ 0.45 and (3σ1 − σ3)/σci < 0.6

Here, σ_cm denotes the uniaxial compressive strength of the rock mass, which can be derived from the Hoek–Brown criterion using the intact rock strength and the Geological Strength Index (GSI); and σ_ci denotes the uniaxial compressive strength of the intact rock, which can be obtained directly through laboratory uniaxial compression tests.

lists the identification criteria for potential failure in the surrounding rock.

These identification criteria mainly apply to tunnel projects with relatively small spans (generally not exceeding 15 m) and reflect practical experience from mining, transportation, and hydropower engineering. For spans greater than 20 m, project teams should undertake focused studies with numerical analysis to improve the accuracy of the stability assessment of the surrounding rock.

2.2.2. Method for Identifying Deformation Based on Convergence Strain (ε)

As shown in Table 2, whether the tunnel surrounding rock exhibits a deformation problem depends primarily on the relative magnitudes of σ_cm and σ₁. This criterion rests on a straightforward premise: support design focuses on plastic deformation, and the mismatch between the peak strength of the rock mass and the surrounding rock stress level governs its occurrence and severity. Therefore, conventional deformation indicators such as displacement or strain are not adopted. Unlike the high-stress failure criterion, the method considers only σ₁ and omits σ₃, because deformation problems typically occur in weak surrounding rock with relatively low strength, where high stress ratios are unlikely to develop.

Extensive engineering practice shows that surrounding rock deformation is jointly controlled by the rock mass strength, the in situ stress level, and the tunnel cross-sectional size. Define the convergence strain ε as the ratio of surrounding rock convergence to the tunnel radius R; this definition quantifies the deformation level [14,33]. Equation (3) describes the relationship between ε and σ_cm/σ₁. Using this relationship with site-specific rock mass parameters, one can assess the deformation level and propose corresponding support design recommendations (see

Table 3. Assessment method for tunnel deformation severity and recommended support schemes.

Grade	σcm/σ1	ε*100	Degree of Deformation	Analysis Method	Support Scheme
A	>0.45	<1	No squeezing deformation	Design support scheme based on rock mass classification	Rock bolts and shotcrete
B	0.28–0.45	1–2.5	Slight squeezing deformation	Use the CCM to predict the plastic zone and deformation; evaluate the effects of alternative support options	Conventional rock bolts and shotcrete; local reinforcement with light steel arches or steel lattice girders to enhance stability
C	0.2–0.28	2.5–5	Severe squeezing deformation	Apply 2D finite element analysis considering excavation sequence and support type	Use a steel arch support to ensure quality and rapid installation
D	0.14–0.2	5–10	Very severe squeezing deformation	Apply 2D finite element analysis, requiring pre-reinforcement support before excavation face deformation	Provide advanced support and steel arches with shotcrete
E	<0.14	>10	Extremely severe squeezing deformation	Perform 3D analysis; although design methods exist, most conclusions still rely on empirical experience	In addition to grade-D measures, adopt yielding (energy-absorbing) supports under extreme conditions

).

$$\epsilon =0.2/{({\sigma }_{cm}/{\sigma }_1)}^2$$

(3)

2.2.3. Method for Identifying High-Stress Failure Based on the Stress Reduction Factor (SRF)

The surrounding rock strength–stress ratio is a common criterion for identifying high-stress failure; it is defined as the ratio of the rock’s uniaxial compressive strength to the maximum initial in situ principal stress. However, this indicator ignores the minimum principal stress on the tunnel cross-section and thus the effect of the principal stress ratio. Field experience indicates that, in hard rock, the initial principal stress ratio on the cross-section strongly influences both the severity and the mode of surrounding rock failure, whereas in soft rock, where the principal stress ratio is close to 1.0, focusing on the maximum principal stress alone is sufficient. To address this limitation, this study adopts the Stress Reduction Factor (SRF) as the high-stress failure criterion [34,35], defined as follows:

$$\mathrm{SRF}={\displaystyle \frac{3{\sigma }_1-{\sigma }_3}{{\sigma }_{cm}}}$$

(4)

In the above equation, σ₁, σ₃, and σ_cm denote the initial maximum principal stress, the initial minimum principal stress on the tunnel cross-section, and the uniaxial compressive strength of the rock mass, respectively.

The SRF quantifies the relative magnitude between rock mass compressive strength and the in situ stress state and thus provides an objective basis for assessing the risk of surrounding rock failure under high-stress conditions. Figure 1 shows the corresponding failure-severity assessment.

Accounting for the minimum principal stress on the tunnel cross-section, we use the Stress Reduction Factor (SRF) to quantify surrounding rock failure mechanisms under high-stress conditions.

Table 4. Failure types of surrounding rock under high-stress conditions.

Risk level	SRF	Failure Type (High Stress)
No risk	<0.45	Self-stable
Low risk	0.45–0.60	Damage
Moderate risk	0.6–0.9	Fracture
High risk	0.9–1.2	Spalling
Extremely high risk	>1.2	Rockburst

shows the SRF-based failure-type classification. Based on SRF, we classify high-stress surrounding rock failure into five grades: self-stable, damage, fracture, spalling, and rockburst.

Table 5. Observed phenomena and support strategies for different high-stress failure types of surrounding rock.

Failure Type (High-Stress)	Observed Phenomena	Support Strategy
Damage	Local occurrence of cavities, microcracks, or minor spalling.	Fill materials to repair damaged areas and enhance rock stability.
Fracture	Extensive rock mass fracturing characterized by crack propagation and the development of continuous fracture zones.	Install rock bolts, steel mesh, and shotcrete to improve rock mass integrity and load-bearing capacity.
Spalling	Large-area, thick-layer detachment forming slabs; buckling or toppling may occur.	Provide immediate face and perimeter support with rock bolts, steel mesh, steel-fiber-reinforced shotcrete, and prestressed anchors.
Rockburst	Violent failure and ejection of surrounding rock, e.g., large-scale rockburst or ejection of rock fragments.	Adopt energy-absorbing systems such as NPR bolts and prestressed grouted anchors.

summarizes the typical characteristics of each grade and the corresponding support strategies.

2.3. Fundamentals of the Convergence–Confinement Method

The convergence–confinement method (CCM) is an analytical framework based on a two-dimensional idealization that captures the three-dimensional interaction between the surrounding rock and the support in underground excavations [36,37]. It centers on three key curves: the ground reaction curve (GRC), the longitudinal displacement profile (LDP), and the support characteristic curve (SCC). The GRC relates the fictitious support pressure applied at the excavation boundary to the resulting radial wall displacement under unsupported conditions; as the fictitious pressure decreases to zero, the curve traces the deformation path and the transition of the surrounding rock from elastic to plastic behavior. The LDP describes the decay of radial displacement along the tunnel axis in an unsupported opening, revealing the face-advance (3D) effect on rock deformation. The SCC gives the relationship between the ground pressure on the support and the resulting radial deformation; it depends on installation timing and the mechanical properties of the support [38,39,40]. The CCM is used to determine the optimal installation time and the required support pressure, thereby mobilizing the self-supporting capacity of the surrounding rock and optimizing support design [41]. Figure 2 illustrates the fundamental principles of this method.

In the above figure, P_i denotes the support reaction; P₀ denotes the initial in situ stress; P_max denotes the ultimate capacity of the support structure; P_eq denotes the equilibrium support resistance between the surrounding rock and the support; u denotes the radial displacement at the tunnel boundary; u₀ denotes the initial radial displacement of the surrounding rock at the time of support installation; u_eq denotes the radial displacement at rock–support equilibrium; u_el denotes the maximum elastic deformation of the support structure; u_max denotes the maximum deformation of the support structure at failure.

2.3.1. Method for Determining the Equivalent Circular Radius of Noncircular Tunnel Cross-Sections

The GRC assumes a circular tunnel when deriving the deformation equation, whereas tunnel cross-sections are often noncircular [42]. For analytical convenience, irregular sections are commonly converted to an equivalent circle using equivalence-based formulations, a process known as normalization of irregular cross-sectional shapes. The two standard approaches are mechanical equivalence and area equivalence [43,44]. References [45,46] summarize several conventional methods for computing the equivalent circular radius of typical cross-sectional shapes, providing a solid theoretical basis for engineering analyses of noncircular sections, as illustrated in Figure 3.

For more complex tunnel cross-sections, the determination of the equivalent radius can follow two general principles. The first is the area equivalence method, in which the area of the equivalent circle is set equal to the excavation area of the actual tunnel. The second is the mechanical equivalence method, which ensures that the moment of inertia or other key mechanical properties of the equivalent circle match those of the original cross-section, thereby preserving similarity in deformation response [47,48]. For extremely complex or unconventional cross-sections, numerical inversion methods are recommended.

As shown in Figure 3a, for the circular arch with straight sidewalls, set the equivalent circular radius R₀ to the radius of the circumscribed circle, given by

$$R_0={\displaystyle \frac{\sqrt{h^2+{\left(l/2\right)}^2}}{2\cos \left[\arctan \frac{2h}{l}\right]}}$$

(5)

As shown in Figure 3b, for the horseshoe cross-section, set the equivalent circular radius R₀ to one-fourth of the sum of the span l and the height h, given by

$$R_0={\displaystyle \frac{l+h}{4}}$$

(6)

2.3.2. Elastic–Plastic Analytical Expressions for the Ground Reaction Curve (GRC)

The GRC depends only on the mechanical parameters of the surrounding rock and is independent of support pressure variations. It is commonly partitioned into elastic and plastic segments and can be treated as a piecewise function. Analytical solutions for a circular opening in an infinite medium and numerical simulations capture the surrounding rock response across the different deformation stages. Under hydrostatic far-field stress, for a deeply buried circular tunnel, the elastic convergence equation follows from the thick-walled cylinder theory, given by

$$u={\displaystyle \frac{1+\mu }{E}}{R}_0\left(P_0-P_i\right)$$

(7)

When the Mohr–Coulomb yield criterion is adopted, the plastic convergence equation reads

$$u={\displaystyle \frac{1+\mu }{E}}\left\{P_0-c\times \cot \phi \left[{\left({\displaystyle \frac{R_p}{R_0}}\right)}^{{\displaystyle \frac{2\sin \phi }{1-\sin \phi }}}-1\right]-P_i{\left({\displaystyle \frac{R_p}{R_0}}\right)}^{{\displaystyle \frac{2\sin \phi }{1-\sin \phi }}}\right\}{\displaystyle \frac{R_p^2}{R_0}}$$

(8)

where R_p is given by

$$R_p={\left[{\displaystyle \frac{\left(P_0+c\times \cot \phi \right)\left(1-\sin \phi \right)}{P_i+c\times \cot \phi }}\right]}^{{\displaystyle \frac{1-\sin \phi }{2\sin \phi }}}\times R_0$$

(9)

where R_p denotes the radius of the plastic zone.

2.3.3. Expression of the Longitudinal Displacement Profile (LDP)

Hoek proposed the following empirical relationship to describe how tunnel radial displacement evolves with distance ahead of the excavation face [49]:

$${\displaystyle \frac{u_r}{u_r^M}}={\left[1+\exp \left({\displaystyle \frac{-x/R}{1.10}}\right)\right]}^{-1.7}$$

(10)

where u_r denotes the tunnel radial displacement under face-advance (three-dimensional) effects, x is the distance from the tunnel face, $u_r^M$ represents the maximum radial displacement of the cross-section after excavation is completed (i.e., the ultimate displacement after full release), and R is the tunnel radius. This empirical curve characterizes the evolution of convergence deformation ahead of the face and can be regarded as a standardized pattern of longitudinal displacement evolution. The basic premise is that the rock mass ahead of the excavation face undergoes pre-deformation due to stress release, and as the distance from the face increases, the deformation progressively accumulates and approaches the ultimate value $u_r^M$.

2.4. Support Design Basis for Large-Deformation Conditions: Convergence Control and Safety Margin

2.4.1. Principles for Determining Surrounding Rock Equilibrium Displacement in the CCM Framework

Within the CCM framework described above, large-deformation analysis focuses on the coordinated behavior of the GRC, LDP, and SCC (Figure 4). By jointly analyzing these three curves, we accurately characterize the deformation evolution of the surrounding rock, thereby enabling optimization of support parameters and ensuring overall tunnel stability.

The GRC relates tunnel-perimeter radial displacement to support pressure under the assumptions of negligible face effects and support installation sufficiently far behind the tunnel face. The support system must provide support pressure commensurate with that target to keep the final tunnel convergence within a prescribed range. Accordingly, configuring the support types and parameters to deliver the target support pressure is the core task of support design [50].

The LDP is a key basis for assessing surrounding rock stability because it profiles tunnel-perimeter radial displacement as a function of distance from the tunnel face [15]. Specifically, most deformation concentrates within approximately 25 m behind the face, where radial displacement varies markedly among cross-sections. Beyond this distance, deformation gradually stabilizes and differences between sections become minimal, indicating that the surrounding rock is approaching a stable state.

The SCC describes the interaction between the support pressure provided by a given support design and the tunnel-perimeter radial deformation of the surrounding rock. Here, P_max denotes the ultimate bearing capacity of the support system, corresponding to the ordinate of the horizontal segment of the SCC. P_eq represents the equilibrium support reaction, defined as the ordinate at the intersection between the inclined segment of the SCC and the GRC. The abscissa value of this intersection, u_eq, indicates the actual radial convergence of the tunnel in its stable state.

2.4.2. Criteria for Evaluating the Safety Factor of Support Structures

The support structure must remain strictly elastic throughout its service life. Once it yields plastically, its mechanical performance degrades significantly, leading to a loss of control over surrounding rock deformation and potentially triggering overall instability. Accordingly, we introduce the structural safety factor F_s as a key criterion [17]:

$$F_s={\displaystyle \frac{P_{max}}{P_{eq}}}$$

(11)

where P_max denotes the ultimate capacity of the support system, and P_eq represents the actual support pressure. If F_s > 1, the support system has an adequate safety margin and can maintain surrounding rock stability; if F_s ≤ 1, the structure is at risk of instability, and immediate reinforcement is required.

2.5. Design Basis for Support Under High-Stress Conditions: Failure Depth, Support Pressure, and Rock Bolt Ductility

Under high-stress conditions, support design must address three aspects simultaneously: the load capacity of the support system (support pressure), deformation capacity, and energy absorption capacity. The load capacity reflects the maximum reaction the system can provide at the ultimate limit state; energy absorption is critical under rockburst risk to resist dynamic impacts. By contrast, deformation capacity is central to the design of supports for deeply buried tunnels because, under high stress, hard-rock failure is time-dependent, and the extent of damage in the surrounding rock increases over time. Although the timely installation of shotcrete and rock bolts can suppress the damage process, time-dependent stress buildup in the bolts may jeopardize the long-term stability of the system [51]. Flexible components such as shotcrete layers and steel mesh increase the deformation capacity of the support system, thereby reconciling stress release with structural safety.

Therefore, tunnel support design under high-stress conditions should focus on two key aspects: (1) support pressure, namely the maximum design pressure (P_design) that the support system can deliver at the ultimate state, and (2) the bolt elongation (ε_e), representing the cumulative deformation capacity of bolts in the plastic stage. Both design indicators are derived from the mechanical response of the surrounding rock during high-stress failure, and their evaluation relies primarily on the potential collapse zone and the dilatant deformation that occurs during yielding. In other words, failure depth (D_f) and potential dilatancy are the critical parameters for quantifying the safety reserve and deformation adaptability of the support system. The specific design process is as follows:

(1)

Failure Depth of Surrounding Rock (D_f)

For surrounding rock with good integrity, rock strength, and the in situ stress state around the tunnel cross-section govern the failure depth. Compute D_f as follows [35,52]:

$$D_f=R_0(1.25\cdot SRF-D_0)$$

(12)

$$D_0=0.53-0.00167(mi-9)$$

(13)

where D_f is the failure depth of the surrounding rock, and mi is the intact-rock material constant of the rock in the Hoek–Brown failure criterion.

(2)

Support Pressure (P_design)

Under high-stress conditions, support pressure design must ensure stability within the failure zone to prevent collapse and limit the dilatant deformation during yielding. To this end, the calculation of P_design accounts for the combined effects of these two mechanisms [52]:

$$P_{design}=K_p\left(W_1+W_2\right)$$

(14)

$$W_1={\displaystyle \frac{9.8\gamma D_f}{{10}^6}}(w_1+w_2)$$

(15)

$$w_1+w_2=100$$

(16)

$$W_2=\left\{\begin{array}{c}0,SRF<0.6\\ (SRF-0.6)/10SRF,SRF\ge 0.6\end{array}\right\}$$

(17)

where K_p is the safety factor, typically ranging from 1.0 to 1.5; W₁ and W₂ represent the support loads generated by collapse and dilatancy mechanisms, respectively, during the yielding failure of the surrounding rock. The sum of the two constitutes the pressure the surrounding rock exerts.

(3)

Bolt Length and Elongation

In rock bolt design, the bolt length must extend through the failure zone of the surrounding rock and anchor in competent rock; therefore, D_f directly governs the required length, and the design must also account for the overall deformation characteristics of the surrounding rock. The calculation of ε_e primarily relies on the dilatant deformation induced by yield failure of the surrounding rock. Its magnitude is constrained not only by the extent of failure in the surrounding rock but also by the confinement imposed by the support system. Consequently, the design must first quantify the deformation of the surrounding rock with support in place, denoted as U, using the following equation [52]:

$$U=10D_{f}BF$$

(18)

$$BF=\left\{\begin{array}{l}0, P_{design}=0\\ 3.1-2\ln (P_{design}),P_{design}\ne 0\end{array}\right\}$$

(19)

$$L_{bolt}=D_f\left(1+BF/100\right)+2$$

(20)

$$\epsilon =\left\{\begin{array}{l}0,L<2\\ (U/L),L\ge 2\end{array}\right\}$$

(21)

where U is the surrounding rock deformation, L_bolt is the calculated bolt length, and ε_e is the elongation rate of the rock bolt.

Under high-stress conditions, once the P_design is determined, we then compute the maximum support pressure of the configured system using the CCM. This method comprehensively accounts for the interaction between surrounding rock deformation and the support system, ensuring that the maximum support pressure meets design requirements and safety standards.

2.6. Analytical Design Workflow

Building on the preceding sections, this study integrates surrounding rock failure-mode identification with analytical support-parameter design, thereby establishing a systematic workflow for tunnel support parameter design based on surrounding rock failure modes (Figure 5).

The process begins with geological engineering and in situ stress parameters, and at its core, it adopts the two criteria σ_cm/σ₁ and SRF to quantitatively identify potential squeezing deformation and high-stress failure in the surrounding rock. These results then guide differentiated support design strategies. Subsequently, the workflow is incorporated into the CCM analytical framework, where the GRC and SCC are constructed to solve for the equilibrium state of the surrounding rock–support interaction system. F_s serves as the safety evaluation index, and the workflow verifies and optimizes the support scheme until the required safety margin is achieved. The proposed analytical method is suitable for rapid evaluation and parameter optimization at the preliminary design stage and can also provide initial boundary conditions and reasonable parameter ranges for subsequent numerical simulations. For complex geological conditions or projects requiring refined analysis, numerical simulations should be combined with this method for further verification and optimization.

3. Results and Discussion

3.1. Analytical Formulation and Parametric Sensitivity Analysis of the Support Characteristic Curve (SCC) for Typical Support Structures

As shown in Figure 6, the graphical method locates the intersection of the SCC and GRC; this point yields the equilibrium support pressure P_eq and the corresponding radial tunnel-boundary displacement u_eq at rock–support equilibrium. The support exhibits three response stages: for u₀ < u < u_el, the support behaves elastically; for u_el < u < u_max, it yields; and for u > u_max, it fails.

The support characteristic equation is given by [53]:

$$P=Ku$$

(22)

where K denotes the stiffness of the support structure.

3.1.1. Analytical SCC Formulation and Parametric Sensitivity Analysis for Shotcrete Lining

The expressions for the shotcrete support stiffness K_shot and the ultimate capacity P_shot,max are given by [53]:

$$K_{shot}={\displaystyle \frac{E_{con}}{\left(1+{\upsilon }_{con}\right)}}{\displaystyle \frac{\left[R_0^2-{\left(R_0-t_{shot}\right)}^2\right]}{\left(1-2{\upsilon }_{con}\right)R_0^2+{\left(R_0-t_{shot}\right)}^2}}{\displaystyle \frac{1}{R_0}}$$

(23)

$$P_{shot,\max }={\displaystyle \frac{1}{2}}{\sigma }_{con}\left[1-{\displaystyle \frac{{\left(R_0-t_{shot}\right)}^2}{R_0^2}}\right]$$

(24)

where E_con denotes the elastic modulus of shotcrete; υ_con denotes Poisson’s ratio of shotcrete; t_shot denotes the shotcrete thickness; and σ_con denotes the uniaxial compressive strength of the shotcrete material.

Accordingly, the deformation of the shotcrete structure is given by [53]:

$$u_{shot,el}={\displaystyle \frac{P_{shot,\max }}{K_{shot}}}$$

(25)

$$u_{shot,\max }=u_{shot,el}+{\epsilon }_{br,con}\left(R_0-t\right)-{\displaystyle \frac{2\left(1-{\upsilon }_{con}\right)}{{\left(R_0-t_{shot}\right)}^2+\left(1-2{\upsilon }_{con}\right)R_0^2}}\times {\displaystyle \frac{P_{shot,\max }}{K_{shot}}}$$

(26)

where u_shot,el denotes the elastic deformation of shotcrete; u_shot,max denotes the maximum deformation of shotcrete; and ε_br,con denotes the ultimate strain of shotcrete.

We consider a hydraulic tunnel with a horseshoe cross-section as a representative case. The tunnel has a span l = 10.22 m and a height h = 10.93 m. Using Equation (5), the equivalent circular radius is R₀ = 5.29 m.

Table 6. Parameters of shotcrete.

Concrete Strength Grade	tshot (cm)	Econ (GPa)	υcon	σcon (MPa)
C20	20	23	0.2	9.6

lists the shotcrete parameters used. To examine how the t_shot and the strength grade affect support performance and structural deformation, we perform calculations for five thicknesses (10, 15, 20, 25, and 30 cm) and five strength grades (C20, C25, C30, C35, and C40), and we plot the resulting shotcrete SCC in Figure 7.

Figure 7a shows that changes in shotcrete thickness markedly affect support performance. Specifically, when t_shot increases from 10 cm to 30 cm, P_max rises from 0.18 MPa to 0.53 MPa (about a 194% increase). Meanwhile, u_el decreases from 2.07 mm to 1.97 mm and u_max from 28.01 mm to 26.91 mm, reductions of roughly 4.83% and 3.92%, respectively. These results indicate that increasing shotcrete thickness can effectively suppress surrounding rock deformation, although the magnitude of its influence remains relatively limited.

Figure 7b shows that increasing the concrete strength grade affects support pressure. When the strength increases from C20 to C40, P_max increases from 0.36 to 0.71 MPa (about a 97% increase). Meanwhile, u_el increases from 2.02 to 2.94 mm and u_max from 27.46 to 28.37 mm, increases of roughly 46% and 3.31%, respectively. Thus, high-strength shotcrete provides higher support pressure but leads to greater elastic displacement and a slight increase in maximum deformation.

Overall, t_shot and concrete strength grade affect support-system performance through different mechanisms. The former primarily enhances load-bearing capacity and reduces deformation by increasing support stiffness; the latter increases load-bearing capacity and leads to greater elastic deformation. Therefore, support design should balance overall structural stability and deformation-control requirements.

3.1.2. Analytical SCC Formulation and Parametric Sensitivity Analysis for Rock Bolts

Rock bolt support stabilizes the surrounding rock primarily through suspension and composite action [54]. With bolts installed around the tunnel boundary, each bolt develops its anchorage, and their collective action enhances overall sliding resistance. The SCC relationship is given by [55]:

$$K_{bolt}={\displaystyle \frac{1}{S_c{S}_l\left[\frac{4L_{bolt}}{\pi d_{bolt}^2{E}_{bolt}}+Q\right]}}$$

(27)

$$P_{bolt,\max }={\displaystyle \frac{T_{\max }}{S_c{S}_l}}$$

(28)

$$u_{bolt,el}={\displaystyle \frac{P_{bolt,\max }}{K_{bolt}}}$$

(29)

$$u_{bolt,\max }=u_{bolt,el}+{\epsilon }_{br,bolt}{L}_{bolt}$$

(30)

where K_bolt denotes the stiffness of the rock bolt support; P_bolt,max denotes the ultimate capacity of the bolts; E_bolt denotes the elastic modulus of the bolt material; S_c denotes the circumferential bolt spacing; S_l denotes the longitudinal bolt spacing; L_bolt denotes the bolt length; d_bolt denotes the bolt diameter; T_max denotes the ultimate pullout strength measured in pullout tests; Q denotes a characteristic constant related to the bolt body, bearing plate, and bolt head; u_bolt,el denotes the elastic deformation of the rock bolts; u_bolt,max denotes the maximum deformation of the rock bolts; ε_br,bolt denotes the failure strain of the rock bolts.

To systematically investigate the influence of bolt support performance on structural safety, this study separately examined the effects of L_bolt, d_bolt, S_c × S_l, and steel grade on support pressure and structural deformation. The parameters of the shotcrete used are summarized in

Table 7. Parameters of rock bolts.

Steel Grade	dbolt (mm)	Lbolt (m)	Ebolt (GPa)	Sc (m)	Sl m
HPB300	25	6	210	1.5	1.5

. We set L_bolt to 5.0, 5.5, 6.0, 6.5, and 7.0 m; d_bolt to 17, 19, 21, 23, and 25 mm; and S_c × S_l to 1.3 × 1.3, 1.4 × 1.4, 1.5 × 1.5, 1.6 × 1.6, and 1.7 × 1.7 m. We also compare three steel grades (HPB300, HRB400, and HRB500). Figure 8 shows how these parameters affect the load-bearing capacity and deformation characteristics of the support system.

Figure 8a,b, respectively, show how L_bolt and d_bolt affect the SCC. In Figure 8a, increasing L_bolt from 5.0 to 7.0 m keeps P_max nearly constant at about 0.11 MPa, whereas u_el increases from 23.75 to 28.42 mm and u_max from 348.75 to 483.42 mm. This pattern arises because a longer bolt-free length permits greater elastic elongation before yielding; under the same support pressure, the system exhibits larger elastic deformation. In addition, total bolt elongation scales with length, so more plastic-stage deformation can accumulate, which raises u_max.

Figure 8b shows that increasing d_bolt from 17 to 25 mm raises P_max from 0.05 to 0.11 MPa (nearly doubling). Meanwhile, u_el increases from 19.53 to 26.09 mm, and u_max from 409.53 to 416.09 mm. This increase arises because a larger d_bolt increases the bolt cross-sectional area, allowing it to carry a higher load at the same strain and thus significantly raising the upper limit of the support pressure. However, since the bolt ductility and free length remain unchanged, the elastic deformation increases accordingly. Consequently, u_max grows slightly, indicating that a modest increase in plastic-stage deformation accompanies higher load capacity. Nevertheless, relative to changes in bolt length, the effect of d_bolt on u_max is relatively limited.

As shown in Figure 8c, increasing the spacing S_c×S_l from 1.3 m × 1.3 m to 1.7 m × 1.7 m reduces P_max from 0.14 to 0.08 MPa (nearly 43%) without increasing deformation. These results indicate that changing the spacing primarily affects the overall stiffness of the support system and its load-bearing capacity, while exerting only a limited influence on the deformation characteristics of individual bolts. In other words, reducing the spacing increases the number of bolts per unit length, which raises the overall support stiffness and therefore permits higher support pressure.

Figure 8d shows that upgrading the steel grade from HPB300 to HRB500 raises P_max from 0.11 to 0.15 MPa (about 36%). Meanwhile, u_el increases from 26.09 mm (HPB300) to 36.57 mm (HRB500). This change arises because higher-grade bolts sustain higher loads at the same strain and store more energy during the elastic stage. By contrast, u_max decreases from 416.09 mm to 336.57 mm, indicating that the higher-grade steel exhibits reduced plastic deformation capacity and lower ultimate elongation, thereby reducing the cumulative deformation in the plastic stage.

In summary, increasing rock bolt strength increases the maximum load-bearing capacity of the support system but also increases elastic deformation. The value of u_max is governed primarily by the ultimate elongation of the bolts. Once the bolts enter the plastic stage, their support capacity is already near its limit, and any further significant deformation would markedly compromise the overall safety. Therefore, support design should avoid excessive plastic deformation of rock bolts. The support system can achieve high load-bearing capacity by optimizing parameters bolt strength, length, diameter, and installation spacing.

3.1.3. Analytical SCC Formulation and Parametric Sensitivity Analysis for Steel Arches

Without wooden blocks, the support characteristic curve (SCC) for the steel arch is given by [55]:

$$K_{set}={\displaystyle \frac{E_{set}{A}_{set}}{S{\left(R_0-h_{set}/2\right)}^2}}$$

(31)

$$P_{set,\max }={\displaystyle \frac{{\sigma }_{set}{A}_{set}}{S\left(R_0-h_{set}/2\right)}}$$

(32)

$$u_{set,el}={\displaystyle \frac{P_{set,\max }}{K_{set}}}$$

(33)

$$u_{set,\max }=u_{set,el}+{\epsilon }_{br,set}\left(R_0-{\displaystyle \frac{h_{set}}{2}}\right)$$

(34)

where K_set denotes the stiffness of the steel arch support; E_set denotes the elastic modulus of the steel; A_set denotes the cross-sectional area of the steel arch; S denotes the arch spacing; h_set denotes the height of the steel arch cross-section; P_set,max denotes the ultimate capacity of the steel arch; σ_set denotes the yield strength of the steel; u_set,el denotes the elastic deformation of the steel arch; u_set,max denotes the maximum deformation of the steel arch; ε_br,set denotes the failure strain of the steel arch.

The selected parameters of the steel arches are given in

Table 8. Parameters of steel arches.

Steel Grade	Section Type	Aset (cm2)	hset (mm)	Eset (GPa)	S m
Q235	I20a	35.58	200	206	1.5

. To examine the effects of arch type, steel grade, and spacing on support pressure and structural deformation, we set the arch spacing S to 0.8, 1.0, 1.2, 1.4, and 1.6 m; selected steel grades Q195, Q215, Q235, Q255, and Q275; and analyzed five I-beam sections: I18, I20a, I20b, I22a, and I22b. Figure 9 shows the characteristic curves of the steel arch support, which illustrate how these parameters influence overall support performance.

Figure 9a shows that as the steel grade increases from Q195 to Q275, the yield strength rises accordingly, and P_max increases from 0.11 to 0.16 MPa, an increase of approximately 45%. Meanwhile, u_el increases from 4.91 to 6.93 mm, and u_max increases from 30.85 to 32.86 mm, with increases of about 41% and 6.52%, respectively. These results indicate that changes in steel grade primarily affect the support pressure, whereas the support stiffness remains unchanged.

Figure 9b shows that increasing the section from I18 to I22b raises P_max from 0.12 to 0.18 MPa, an increase of approximately 50%. By contrast, u_el decreases slightly from 5.93 to 5.91 mm and u_max from 31.92 to 31.79 mm, leaving both essentially unchanged. The increase arises because a larger section increases the flexural stiffness and load-bearing capacity of the steel arch support, thereby increasing the upper limit of the support pressure. However, changing the section size has little effect on the ductility and ultimate elongation of the steel; consequently, plastic-stage deformation changes only negligibly.

As shown in Figure 9c, when the spacing increases from 0.8 m to 1.6 m, P_max decreases from 0.20 to 0.10 MPa (a 50% reduction), while u_el and u_max remain essentially constant at about 5.92 mm and 31.86 mm, respectively. These results indicate that reducing the spacing significantly increases the overall stiffness of the steel arch support system, enabling it to sustain higher loads at lower deformation levels. However, as long as the section, material grade, and length of individual arches remain unchanged, their inherent elastic–plastic deformation capacity varies little. Consequently, the overall deformation level remains nearly unchanged.

3.1.4. Analytical Formulation of the SCC Stiffness Model for Combined Support

Combined support systems comprise rock bolts, cable bolts, steel arches, and shotcrete, all installed sequentially in the same excavation zone under a unified construction plan. In such a system, the overall support structure fails once any individual component reaches its ultimate deformation and fails [56]. In theory, the effective stiffness of the combined support equals the sum of the component stiffnesses [55]:

$$K_{com}={\displaystyle \sum _{i=1}^n{K}_i}$$

(35)

When the combined support consists of rock bolts, steel arches, and shotcrete, its overall support stiffness is given as follows:

$$K_{com}=K_{shot}+K_{bolt}+K_{set}$$

(36)

The SCC for the combined support is given by

$$P_{com}=K_{com}{u}_{com}$$

(37)

where P_com denotes the support reaction of the combined support, and u_com denotes its radial deformation.

According to the stability criterion for parallel systems, the smallest allowable deformation among the components governs the maximum allowable deformation of the support system:

$$u_{com,\max }=\min \left\{u_{shot,el},u_{bolt,el},u_{set,el}\right\}$$

(38)

The ultimate capacity of the combined support, P_com,max, is given by

$$P_{com,\max }=K_{com}{u}_{com,\max }$$

(39)

3.2. Single-Parameter Sensitivity Analysis of Rock Mass Pressure and Deformation

Equations (7)–(9) show that rock mass pressure is closely related to tunnel deformation and is also influenced by parameters such as the Geological Strength Index (GSI) and burial depth (H). To elucidate the roles of these parameters, we conducted a single-parameter sensitivity analysis using the following baseline parameters: GSI = 30, H = 250 m, γ = 0.024 MN/m³, UCS = 35 MPa, m_i = 8, and R₀ = 5.29 m. To ensure the representativeness and generalizability of the analysis results, the parameter ranges were determined based on the case conditions and then extended to cover the typical ranges encountered in tunnel engineering.

3.2.1. Effect of GSI on Surrounding Rock Pressure and Deformation

Figure 10 shows the coupled relationship between surrounding rock pressure and tunnel-perimeter deformation for four GSI levels (20, 30, 40, and 50). Figure 10b shows how the surrounding rock stress changes at the elastic–plastic boundary, further elucidating the role of GSI in the elastic–plastic transition.

As shown in Figure 10, when the GSI increases from 20 to 50, the surrounding rock pressure at the elastic–plastic transition drops from 2.70 MPa to 1.57 MPa (a decrease of about 42%), while the radial displacement at the tunnel boundary decreases from 21.37 mm to 4.93 mm (a decrease of about 77%). A higher GSI indicates better rock mass integrity and stronger self-supporting capacity. Under the same stress conditions, the plastic zone is smaller and radial deformation is lower. Accordingly, high-quality rock masses experience only limited plastic deformation under excavation-induced disturbances, require lower support pressure, and are easier to control with respect to deformation.

3.2.2. Effect of Burial Depth (H) on Surrounding Rock Pressure and Deformation

Figure 11 shows the coupled relationship between surrounding rock pressure and tunnel-perimeter deformation for burial depths (H = 200–350 m), and the stress variation at the elastic–plastic boundary, thereby revealing the role of burial depth in the elastic–plastic transition.

As shown in Figure 11, when burial depth increases from 200 m to 350 m, the surrounding rock pressure at the elastic–plastic boundary rises from 1.68 MPa to 3.75 MPa, an increase of 123%, and the tunnel-perimeter radial displacement increases from 11.26 mm to 16.77 mm, an increase of 49%. The increase arises because greater depth significantly raises the initial in situ stress, which expands the plastic zone in the surrounding rock while reducing the proportion of elastic deformation. The support must provide higher stiffness and confining pressure to keep the rock mass elastic. By contrast, allowing limited plastic deformation better mobilizes the self-supporting capacity of the rock mass and markedly lowers the required support pressure. In this process, flexible support is key: by permitting a moderate release of surrounding rock displacement, it both reduces the load demand on the support structure and relieves the squeezing effect of high in situ stress on the tunnel rock–support system, thereby promoting a more coordinated and stable stress–strain state.

3.2.3. Effect of R₀ on Surrounding Rock Pressure and Tunnel-Perimeter Deformation

To examine how R₀ affects the mechanical response of the surrounding rock, we consider four cases (R₀ = 3.29, 5.29, 7.29, and 9.29 m). For each case, we computed the coupled relationship between surrounding rock pressure and tunnel-perimeter deformation (Figure 12a) and analyzed the stress variation at the elastic–plastic boundary to clarify the role of R₀ in the elastic–plastic transition (Figure 12b).

As shown in Figure 12, as R₀ increases from 3.29 to 9.29 m, the tunnel-perimeter radial displacement at the elastic–plastic boundary rises from 8.23 to 23.25 mm (an increase of about 183%), while the surrounding rock pressure remains relatively constant. The increase arises because a larger radius significantly reduces the overall stiffness of the surrounding rock, flattens the elastic segment of the ground-reaction curve, and expands the plastic zone, which markedly increases displacement. In other words, larger-radius tunnels disturb a wider zone and produce stronger displacement responses, although the bearing capacity of the surrounding rock essentially remains unchanged. Accordingly, increasing R₀ markedly amplifies the risk of surrounding rock displacement, thereby necessitating targeted optimization of support parameters and construction methods to ensure tunnel stability.

3.3. Case Study: Application and Analysis of a Support Strategy Based on the CCM

In southwestern China, a hydraulic tunnel with a horseshoe-shaped cross-section has a span l = 10.22 m, a height h = 10.93 m, and an equivalent circular radius R₀ = 5.29 m. The burial depth ranges from 45 to 330 m, reaching a maximum of 362 m. The surrounding rock mainly consists of argillaceous siltstone with a UCS of 20–40 MPa and is classified overall as class IV. At a depth of about 250 m, regional tectonic effects have produced well-developed joints, fissures, and small faults, some of which contain clay infilling. Consequently, the rock mass shows poor integrity and pronounced stability problems. Geological survey data and laboratory tests indicate a unit weight of γ = 0.024 MN/m³ for this tunnel segment. According to the GSI classification table and the engineering geological conditions, the GSI is approximately 30. Combined laboratory and field tests give UCS = 35 MPa and mi = 8.

According to the aforementioned strength criterion, the surrounding rock exhibits significant deformation potential. The calculated convergence strain is ε = 2.5%, which reaches the onset threshold of severe squeezing and indicates that the tunnel is under a severe squeezing condition governed by the coupled effects of high in situ stress and weak surrounding rock. To effectively control the severe squeezing deformation that may occur during construction, the initial support design adopted a combined support system consisting of shotcrete, mortar bolts, reinforcement mesh, and steel arches to enhance structural stiffness and improve surrounding rock stability. The specific parameters are listed in

Table 9. Main parameters of initial support system components.

Initial Support Components	Parameter Description
Shotcrete	Strength grade C20; thickness 20 cm
Bolts	Mortar bolts; diameter 25 mm; length 6 m; spacing 1.5 × 1.5 m; steel grade HPB300
Reinforcement mesh	Diameter 8 mm; mesh spacing 20 × 20 cm; steel grade HRB335
Steel arches	Type I20a; longitudinal spacing 1.2 m; section height 200 mm; flange width 100 mm; web thickness 7.0 mm; flange thickness 11.4 mm; steel grade Q235

. The secondary lining was made of 60 cm of C30 concrete.

Using Equation (8), the free-convergence displacement of the surrounding rock is $u_r^M$= 51.136 mm, which represents the ultimate convergence deformation that may develop in the absence of any support constraint. In practice, a certain support lag distance is inevitable during initial support installation, and for this tunnel section, the actual lag distance is about 1 m. Based on Equation (10), the convergence displacement at the time of support installation is u₀ = 18.10 mm, indicating the amount of convergence deformation that had already occurred before the support began to take effect. Using Equations (22)–(34), we systematically calculated the K, P_max, u_el, and u_max of the three support elements (shotcrete, bolts, and steel arches) and summarized the results in

Table 10. Results of SCC calculations.

Column Headers	K (MPa·m−1)	Pmax (MPa)	uel (mm)	umax (mm)
Shotcrete	176.33	0.36	2.02	27.46
Rock bolts	4.11	0.11	26.08	416.08
Steel arches	22.70	0.13	5.92	31.86
Secondary lining	681.63	1.53	2.25	27.68
Combined support scheme	884.76	1.79	2.02	27.46

. In combination with the GRC curve, we plotted a CCM analysis chart (Figure 13) to visually illustrate the mechanical equilibrium state of this tunnel section.

As shown in Figure 13, we apply the CCM to analyze the interaction between the surrounding rock and the support system, which yields an equilibrium state characterized by u_eq = 19.57 mm, P_eq = 1.24 MPa, and F_s = 1.44, confirming that the adopted support scheme provides sufficient load-bearing capacity to effectively control surrounding rock deformation and ensure structural stability. Further calculations show that the equilibrium strain is ε = 0.37%, which fully meets the deformation control and safety requirements. At the final equilibrium state, the values of u_eq and P_eq serve as important input parameters for secondary lining design, providing key boundary conditions for the structural system and thereby ensuring consistency and safety in tunnel design.

4. Conclusions

This study addresses engineering requirements for tunnel surrounding rock stability and support design. We developed a classification system for surrounding rock failure modes using analytical criteria based on the stress ratio and the Stress Reduction Factor (SRF). Within the CCM framework, we further established characteristic curves for shotcrete, rock bolts, and steel arches, thereby unifying the analysis of surrounding rock failure mechanisms with the determination of support design parameters. The main conclusions are as follows:

• We introduced the σ_cm/σ₁ criterion and SRF and systematically analyzed the relationships among surrounding rock instability modes, strength parameters, and initial in situ stress. We identified the applicability conditions and control indicators for each failure type, thereby providing quantitative criteria for surrounding rock stability assessment. Building on the CCM framework, we formulated analytical expressions for the SCCs of shotcrete, rock bolts, and steel arches, and established a rapid evaluation method that does not rely on numerical simulations. This approach directly evaluates the load-bearing capacity and deformation control effectiveness of different support schemes.
• Parametric analysis shows that increasing t_shot significantly enhances P_max, but has only a limited effect on suppressing surrounding rock deformation. Raising the strength grade also increases P_max, yet simultaneously increases u_el and u_max, so the design must balance load-bearing and deformation control. Lengthening rock bolts does not increase capacity but increases deformation; by contrast, reducing bolt spacing markedly increases P_max. Increasing bolt diameter or strength also improves capacity, but increases elastic deformation. Accordingly, design should prioritize the strategy of “smaller spacing combined with increased diameter/strength” to enhance capacity, strictly limit bolt lengthening, and ensure an adequate ductility reserve. For steel arches, enlarging the section or reducing spacing substantially increases capacity, while u_el and u_max remain unchanged.
• Sensitivity analysis indicates that increasing GSI significantly reduces both surrounding rock pressure and deformation. By contrast, increasing H raises the in situ stress level and thereby markedly increases surrounding rock pressure and deformation. Increasing R₀ leaves surrounding rock pressure essentially unchanged but substantially heightens the risk of excessive displacement.
• Using a hydraulic tunnel at a burial depth of 250 m as a case study, we applied the criteria and identified severe squeezing of the surrounding rock (ε = 2.5%). Accordingly, we adopted an initial support system consisting of shotcrete, grouted rock bolts, wire mesh, and I20a steel arches. Within the CCM framework, we obtained u_eq = 19.57 mm, P_eq = 1.24 MPa, and F_s = 1.44; the corresponding equilibrium equivalent convergence strain is ε = 0.37%. These results meet the requirements for load-bearing capacity and deformation control, thereby verifying the method’s applicability and engineering feasibility for underground space engineering under large deformation conditions.