Next Article in Journal
A Space Discretization Method for Smooth Trajectory Planning of a 5PUS-RPUR Parallel Robot
Previous Article in Journal
The Gradual Cyclical Process in Adaptive Gamified Learning: Generative Mechanisms for Motivational Transformation, Cognitive Advancement, and Knowledge Construction Strategy
Previous Article in Special Issue
Dynamic Inversion Method for Concrete Gravity Dam on Soft Rock Foundation
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Research on Cracking Mechanism and Crack Extension of Diversion Tunnel Lining Structure

1
Sichuan Energy Internet Research Institute, Tsinghua University, Chengdu 610039, China
2
State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
3
School of Emergency Managment, Xihua University, Chengdu 610039, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(16), 9210; https://doi.org/10.3390/app15169210 (registering DOI)
Submission received: 31 March 2025 / Revised: 28 June 2025 / Accepted: 23 July 2025 / Published: 21 August 2025

Abstract

Tunnel systems are often confronted with issues such as cracks, water seepage, and exposed tendons, all of which compromise their structural integrity. This study utilizes an advanced robotic system equipped with a 3D laser scanner to capture data on visible lining defects. By analyzing the distribution of defects across various tunnel segments, we explore the mechanisms underlying structural cracks. Finite element software is employed to assess stress, deformation, and crack progression within the tunnel linings. The result found that the diversion tunnel’s segments exhibit notable variations: 66.0% of the defects are concentrated in the upper flat section, while 34.0% are found in the inclined shaft segment. Cracks, primarily located in the vault area, characterize these defects. Under water pressure, stress deformation in the intact lining follows a linear escalation pattern. Specifically, after the formation of cracks measuring 0.1 m, 0.2 m, and 0.3 m, circumferential stresses increase by approximately 4.50%, 9.10%, and 15.10%, respectively. Numerical simulations reveal significant stress concentration near the cave entrance at the upper flat break. Crack propagation at the arch crown is found to pose a greater risk than at the sides of the arch waist. These findings offer valuable scientific insights and practical implications for improving safety and enabling intelligent monitoring of power station tunnels.

1. Introduction

With the continuous advancement of water conservancy and hydropower technologies, a large number of hydraulic tunnels have been constructed to support efficient water management, energy generation, and power supply. However, prolonged operation often leads to structural degradation. If such damage is not detected and addressed in a timely manner, it can pose serious threats to the safety and reliability of the infrastructure [1,2,3,4,5]. Manual inspection methods face several limitations, including high operational risk, extended inspection times, and potential inaccuracies in data acquisition. With the rapid development of advanced equipment and information technologies, the integration of artificial intelligence (AI) tools for tunnel scanning and defect detection has become increasingly important [6,7,8,9,10]. By analyzing the spatial distribution and formation mechanisms of tunnel defects, this study identifies key patterns that are critical for maintaining long-term structural integrity. These findings contribute to a better understanding of potential vulnerabilities and provide valuable guidance for proactive maintenance strategies, thereby enhancing the safety and resilience of hydraulic tunnel infrastructure.
Currently, investigations into defects within diversion tunnels predominantly concentrate on model testing, theoretical examinations, and computational simulations. Lin, C.J. et al. [11] studied the influence of crack location, Enhanced Safety Factors of Lining Structures: A Study Based on Crack Model Analysis and the Influence of Depth and Host Rock Resistance; according to Liu, C. et al. [12], the analysis of the lining cracking, conducted through rigorous model testing, revealed that the primary cause was the excessive tensile stress leading to cracks at the arch top and arch footing. Additionally, the side walls experienced significant localized pressure, contributing to their failure; Lu, A. et al.’s [13] study meticulously examined the cracking phenomena in the lining using field-measured pressure data from model tests. It explored the underlying causes, patterns, and distinctive characteristics of observed lining fractures; in a study by Li, G.D. et al. [14], the photoelastic patch method was employed to examine the cracking patterns, stability, and destabilizing damage in a concrete four-point shear beam test. This investigation covered various sizes and locations of cracks, revealing their expansion rules and characteristics comprehensively; Du, Jiaji’s [15] study employed cloud interferometry to examine the propagation dynamics of fractures in concrete during a three-point shear beam test; Li, Z.H. et al.’s [16] research introduced an advanced computational framework to assess fractures in lining masonry, drawing upon the principles of fracture mechanics. This innovative approach was subsequently juxtaposed with existing methodologies for analyzing cracks in similar structures, highlighting its unique contributions and enhanced accuracy; Khan, G. et al. [17] proposed a calculation model for the cracks of lining masonry based on the theory of fracture mechanics, analyzing the analysis of the cracks of lining masonry. Based on the fracture mechanics theory, Chen, S.’s [18] study introduced a novel model for calculating lining cracks, and conducted a comparative analysis on how various depths and angles of cracks influence the stability of lining structures; Karami, M. et al.’s [19] study delved into the influence of internal water pressure, tunnel radius, and lining thickness, among other variables, on the fracture patterns of the lining, employing advanced fracture calculation methodologies; Wang, G. et al. [20] proposed a structural methodology grounded in the longitudinal bending stiffness of a cracked lining, and conducted an exhaustive analysis on the impact of both isolated and multiple cracks on the mechanical attributes of tunnel linings. Chen, J.X. et al. [21], based on fracture mechanics theory, developed a sophisticated model for evaluating cracked linings. This model enables us to meticulously analyze the intricate stress and deformation characteristics inherent in mixed cracked lining systems; Peng, Z.T. et al.’s [22] study delved into the influence of reinforcement and surrounding rock conditions on the quantity and width of cracks by developing a numerical model focused on lining crack formation; Du, J.M. et al.’s [23] research focused on analyzing the maximal principal stresses, as well as the stresses at crack tips across varying depths of cracks within different components of lining structures, employing advanced numerical simulation techniques; Xu, Z.L.’s [24] study delved into the stress and deformation characteristics of cracks situated at critical junctions such as the arch summit, midsection, and base by constructing a sophisticated numerical model. This model incorporated both non-destructive and cracked linings, leveraging advanced finite element analysis techniques to simulate real-world conditions with enhanced precision. Lu, S. et al.’s [25] study delved into the characteristics, distribution patterns, and propagation mechanisms of fractures under varying conditions of surrounding rock loads, structural imperfections, and additional influencing factors; Lei, Y.M. et al.’s [26] study employed an integrated finite element-discrete element approach to meticulously simulate and analyze the distribution patterns and propagation dynamics of cracks within linings, considering variations in their locations, extents of influence, and magnitudes of applied loads. This investigation aimed at unraveling the intricate mechanisms governing crack evolution under diverse loading scenarios and geometrical configurations, thereby enhancing our understanding of structural integrity and failure modes in complex engineered systems.
Previous research has primarily focused on the stress characteristics and propagation behavior of cracks in lining structures. However, studies involving actual defect detection data from diversion tunnels remain limited, and the differences in lining defects and their underlying formation mechanisms are not yet well understood. This study utilizes inspection data collected from the diversion tunnel of a hydropower station in the Dadu River Basin to investigate the mechanisms of crack formation and patterns of crack propagation in the lining structure. The results aim to enhance the operational safety and efficiency of hydropower stations while contributing to the development of intelligent defect detection technologies.

2. Engineering Background

2.1. Tunnel Inspections

This study focuses on a power station located in the Dadu River Basin that includes a barrage, flood relief structures on either side of the river, and a diversion power generation system. The station features four parallel pressure pipelines designed to optimize water flow for energy generation. The pressure pipes are situated within the thin to medium-thick, and further thickening to massive laminated dolomitic greywacke and metamorphic greywacke formations, ranging from the 9th to 12th layer of the Lower Devonian System. This positioning covers various segments such as upper flat, upper curved, inclined, lower curved, and lower flat areas, all embedded within the microsilver rock strata, and the peripheral rocks are preliminarily classified as III1 to III2 types. Given the intricate environmental conditions of the diversion tunnel, including total darkness, standing water, jet streams, and sediment accumulation, we employ our custom-designed robot. This advanced robot is outfitted with a 3D laser scanner and high-definition microlight camera equipment. The inspection apparatus is carefully lowered into the tunnel using a specialized aerial basket, allowing for comprehensive scanning and examination of the tunnel’s lining surface. Here are the detailed steps involved in the robotic inspection process of the diversion tunnel:
Step 1: Gather comprehensive data on hydrological information for both flow and project areas from the commissioning of the diversion tunnel to the present. Step 2: The robot was deployed outside the tunnel for an inspection mission, navigating its way into the tunnel’s interior. It was equipped with high-definition cameras, 3D laser scanners, and lighting to capture detailed images and 3D point cloud coordinates of the tunnel. After completing the inspection, the robot returned by the same route it entered. Due to the challenging conditions—namely, darkness, lack of adequate lighting, and a slippery cave environment—the lower flat section of the water pump ceased functioning, failing to pump water up to 1.5 m or more. As a result, the inspection could only be conducted on the upper flat section and the inclined section of the tunnel (see Figure 1 and Figure 2).
Step 3: Following a thorough inspection, we generate a comprehensive tunnel and a LIDAR. This allows us to pinpoint and measure any visible defects on the laser image. Subsequently, the LIDAR orthophoto is meticulously edited to incorporate this defect information (see Figure 3).
Step 4: Result interpretation. Utilizing data gathered from the robotic assessment, defects undergo meticulous quantitative analysis. Subsequently, both mechanical and simulation models of the tunnel are devised to elucidate the underlying mechanical principles governing the spatial and geometric distribution patterns of the observed cracks.
From Figure 4, The inspection data reveals a total of 101 imperfections, comprising 77 fissures ranging from 0.38 m to 11.88 m; additionally, there are 21 instances of concrete seepage and three residual protruding rebar ends, the longest measuring up to 0.12 m. The diversion tunnel’s structural segments exhibit considerable variation in defect types. The upper flat section alone harbors 67 deficiencies, constituting 66.0% of the total defects tally, predominantly manifested as cracks accompanied by water seepage, with calcium deposits encircling most fissures, and minor water discharge observed. The primary cause is water seepage, with most cracks accompanied by calcium precipitation and fine water outflow. There are 34 defects in the inclined shaft section, accounting for 34.0% of the total number of defects. The distribution characteristics along the axial direction of the tunnel show that cracks are more concentrated in the upper-level section, particularly within the 0–40 m interval, indicating a higher risk area for the diversion tunnel.

2.2. Characterization of Defects Based on the Gray Correlation Method

The gray correlation approach evaluates the impact of defects on various structural components, facilitating a quantitative assessment and comparison of system evolution and trends. It determines correlation levels based on geometric similarity of parameter curves; higher resemblance indicates stronger correlation. The computation details follow [27].
The defects in different structural sections of the diversion tunnel are selected as the reference sequence X0 = {x0(k), k = 1, 2 … m}, and the cracks and leaks in different locations are selected as the comparison sequence Xi = {xi(k), k = 1, 2 … m}, and the cracks and leaks at different locations are compared with the sequence Xi = {xi(k), k = 1, 2 … m, i = 1, 2 … n}.
The compositional data series homogenization process is as follows:
Y 0 = x 0 ( k ) x 0 , k = 1 , 2 , 3 n ,   where :   x 0 = 1 n k = 1 n x 0 ( k )
Y i = x i ( k ) x i , k = 1 , 2 , 3 n ,   where :   x i = 1 n k = 1 n x i ( k )
where Y0 denotes the reference sequence priming process; Yi denotes the comparison sequence priming process.
Calculate the correlation coefficient of each point of the comparison sequence to each point of the reference sequence:
ξ i ( k ) = m + ρ   M Δ i ( k ) + ρ   M m = min i min k   Δ i ( k )       i = 1 , 2 , , n ,   k = 1 , 2 , , m M = max i max k   Δ i ( k ) Δ i ( k ) = Y 0 ( k ) Y i ( k )
In the formula, ξi(k) indicates the correlation coefficient; ρ indicates the resolution coefficient, ρ ∈ (0, +∞), and the larger value of the resolution will be smaller, usually taking the value of 0.5; ∆i(k) indicates the degree of proximity; m indicates the absolute value of the smallest difference in the smallest difference in the difference in distance between points I ∈ (1, 2, … n) and X0(k) on the curve Xi(k); M indicates the absolute value of the largest difference in the largest difference in the difference in distance between points I ∈ (1, 2, … n) and X0(k) on the curve Xi(k).
The degree of correlation between the reference sequence and the comparison sequence is as follows:
r i = 1 n k = 1 n ζ i ( k )
where ri represents the degree of association, when 0.5 ≤ ri ≤ 1, the reference sequence is associated with the comparison sequence, and the larger the value, the higher the degree of association; when 0 ≤ ri ≤ 0.5, the reference sequence is not associated with the comparison sequence.
The main defect correlation coefficients of the diversion tunnels were calculated by Equation (4), and the results are shown in Table 1.
From the table, it is evident that the correlation of major defects in both the upper-level section and the inclined shaft section exceeds 0.5. Particularly, the correlation among roof cracks, waist cracks, and waist leakage is notably high, while the correlation for roof leakage remains average. This suggests that a high incidence of cracks within the tunnel predominates as the primary defect type. In contrast, the number of leaks is relatively low and does not pose severe damage. The emergence of these cracks correlates with the extent of constraints on lining deformation [28,29]. Cracks will form in lining structures under stress during deformation, regardless of internal or external constraints. Thus, the impact of roof cracks on tunnel safety must be prioritized in practical engineering applications.

3. Research on Lining Cracking and Deformation Mechanism

Diversion tunnels show various structural flaws due to multiple factors like rock characteristics, material quality, design, construction, concrete mix, and maintenance. This paper focuses on the stress and deformation of intact and fractured linings under hydraulic pressure using a mechanical model (Figure 5) and thick-walled cylinder theory with several theoretical assumptions:
(1) The reinforced concrete lining and surrounding rock act as impermeable elastomers, adhering to Hooke’s law for displacement and deformation, ensuring structural integrity and stability; (2) The rock body’s exterior is devoid of any external load, experiencing solely the force of gravity; (3) The inner surface of the lining is subjected to water pressure, which manifests as a form of surface force; (4) The concrete lining and the surrounding rock structure are intimately linked, exhibiting no tendency for relative sliding.

3.1. Mechanical Characteristics of Non-Destructive Lining Structures

Figure 6 illustrates the force model of the lining under water pressure. Given that the lining is primarily subjected to uniform internal water pressure, the mechanical model can be simplified to an axial stress model. This simplification is based on the inverse solution method in elastic mechanics theory [30]; it is postulated that the stress function varies solely with the radial coordinate, r, and remains invariant to the polar angle. The specific form of this relationship is articulated as follows:
ϕ = ϕ ( r )
Then the stress expression is as follows:
σ r = 1 r d ϕ d r σ θ = d 2 ϕ d r 2 τ r θ = τ θ r = 0
where σr denotes radial stress; σθ denotes circumferential stress; and τrθ and τθr denote shear stress.
From Figure 7, The compatibility equation expressed in terms of the stress function can be simplified as follows:
( d 2 d r 2 + 1 r d d r ) 2 φ = 0
This is a fourth-order ordinary differential equation that provides the general solution of the stress function for the axisymmetric stress state as follows:
ϕ = A ln r + B r 2 ln r + C r 2 + D
where A, B, C, and D denote arbitrary constants.
Bringing Equation (8) into Equation (6) yields the stress component as follows:
σ r = A r 2 + B ( 1 + 2 ln r ) + 2 C σ θ = A r 2 + B ( 3 + 2 ln r ) + 2 C τ r θ = τ θ r = 0
However, the displacement component is expressed as follows:
u r = 1 E ( 1 + μ ) A r + 2 ( 1 μ ) B r ( ln r 1 ) +                   ( 1 3 μ ) B r + 2 ( 1 μ ) C r + I cos ϕ + K sin ϕ u θ = 4 B r ϕ E + H r I sin ϕ + K cos ϕ
where ur and uθ denote radial and circumferential displacements, respectively; A, B, C, H, I, and K denote constants; and E and μ denote the modulus of elasticity and Poisson’s ratio of the material, respectively.
In the thick-walled cylinder problem, since the closed cylinder is a multicontinuum, the displacement singular value condition needs to be considered. Also, because it is axisymmetric, the tangential displacement is zero, i.e., uθ = 0, so B = H = I = K = 0 in Equations (2)–(6).
Assuming that the inner diameter of the cylinder is, the outer diameter is, the internal pressure is, and the external pressure is, the boundary conditions and contact conditions are as follows:
σ r | r = r 1 = p 1 σ r | r = r 2 = p 2 τ r θ | r = r 1 = τ θ r | r = r 2 = 0
The boundary conditions are brought into Equations (9) and (10), and the coefficients A and C are obtained with the following expressions:
A = r 1 2 r 2 2 ( p 2 p 1 ) r 2 2 r 1 2 C = r 1 2 p 1 r 2 2 p 2 2 ( r 2 2 r 1 2 )
The lining stresses and displacements are obtained by bringing Equation (12) into Equations (9) and (10) with the following expression:
σ r = p 1 r 1 2 p 2 r 2 2 r 2 2 r 1 2 + r 1 2 r 2 2 ( p 2 p 1 ) r 2 2 r 1 2 1 r 2 σ θ = p 1 r 1 2 p 2 r 2 2 r 2 2 r 1 2 r 1 2 r 2 2 ( p 2 p 1 ) r 2 2 r 1 2 1 r 2       r 1 r r 2 u r = 1 + μ c E c ( r 2 2 r 1 2 ) ( 1 2 μ c ) r 2 + r 2 2 p 1 r 1 2                 ( 1 2 μ c ) r 2 + r 1 2 p 2 r 2 2
where σr denotes the radial stress of the lining; σθ denotes the tangential stress of the lining; ur denotes the displacement of the lining; r1 denotes the inner radius of the lining; r2 denotes the outer radius of the lining; Ec and μc denote the modulus of elasticity and Poisson’s ratio of the concrete lining, respectively; p1 denotes the pressure of water in the tunnel; and p2 denotes the additional pressure.
For the surrounding rock surface acting under uniform water pressure, the stress and displacement of the surrounding rock under water pressure can be calculated using the principles of elastic mechanics, with its formula being as follows:
σ r = r 2 2 r 2 p 2 σ θ = r 2 2 r 2 p 2       r 2 r u r = 1 + μ d E d r 2 2 r p 2
where σr denotes the radial stress of the surrounding rock; σθ denotes the tangential stress of the surrounding rock; ur denotes the displacement of the surrounding rock; r2 denotes the radius of the tunnel; Ed and μd denote the modulus of elasticity and Poisson’s ratio of the surrounding rock, respectively.
By applying the displacement condition at the contact interface between the surrounding rock and the lining, Equations (13) and (14) are utilized to determine the interaction force between the outer wall of the lining and the surrounding rock under internal water pressure. The resultant expression is as follows:
p 2 = p 1 2 ( 1 μ c 2 ) E d r 1 2 ( 1 + μ d ) E c ( r 2 2 r 1 2 ) + ( 1 + μ c ) E d r 1 2 + ( 1 μ c ) r 2 2
The Geological Investigation Report outlines the specific geometrical and material characteristics of the diversion tunnel, which are as follows: inner radius r1 = 4.25 m; outer radius r2 = 5.25 m; lining modulus of elasticity Ec = 30 GPa; lining Poisson’s ratio μc = 0.25; peripheral rock modulus of elasticity Ed = 10 GPa; peripheral Poisson’s ratio μd = 0.3; and water pressure = 0~1.35 MPa. The stress deformation diagrams of the non-destructive lining are shown in Figure 8, which shows a positive growth trend of cyclic stress and displacement. As shown in Figure 8, the stress and displacement of the undamaged lining are shown to be positively increasing, which indicates that the lining is currently in the stage of elastic deformation, i.e., the lining can effectively resist deformation through the annular stress when it is subjected to the water pressure.

3.2. Mechanical Characteristics of Crack Lining Structures

Given that the surrounding rock remains in an elastic stress state without any damage following the emergence of cracks in the lining, according to the principles of elasticity, the plane differential equation for the axisymmetric plane problem in polar coordinates [30] can be expressed as follows:
σ r r + σ r σ θ r = 0
The lining structure produces cracks with tangential stresses σθ = 0, and the stress balance equation is then expressed as follows:
σ r r + σ r r = 0
Integrating Equation (18) yields from the boundary conditions σ r | r = r 1 = p 1 :
σ r = r 1 r p 1       r 1 r r 2
Therefore, the stress in the cracked lining can be expressed as follows:
σ r k = r 1 r k p 1       r 1 r k r 2
where σrk denotes the radial stress of the cracked lining; rk denotes the radius of the cracked lining; r1 denotes the inner radius of the lining; and p1 denotes the water pressure in the tunnel.
It is presumed that the lining at the location where no crack has formed remains in an elastic condition, with the water pressure on its surface being pk = −rk. The structural stresses and displacements of the undamaged lining are derived using Equation (6) along with the provided expression:
σ r = p k r k 2 p 2 r 2 2 r 2 2 r k 2 + r k 2 r 2 2 ( p 2 p k ) r 2 2 r k 2 1 r 2 σ θ = p 1 r k 2 p 2 r 2 2 r 2 2 r k 2 r k 2 r 2 2 ( p 2 p k ) r 2 2 r k 2 1 r 2       r k r r 2 u r = 1 + μ c E c ( r 2 2 r k 2 ) ( 1 2 μ c ) r 2 + r 2 2 p 1 r k 2                 ( 1 2 μ c ) r 2 + r k 2 p 2 r 2 2
where σr denotes the radial stress at the lining without cracks; σθ denotes the tangential stress at the lining without cracks; ur denotes the displacement at the lining without cracks; Ec and μc denote the modulus of elasticity and Poisson’s ratio of the concrete lining, respectively; p1 denotes the water load in the tunnel; and p2 denotes the additional pressure.
The stresses and displacements of the surrounding rock subjected to water pressure can be determined through elastodynamic analysis, utilizing the provided expression:
σ r = r 2 2 r 2 p 2 σ θ = r 2 2 r 2 p 2       r 2 r u r = 1 + μ d E d r 2 2 r p 2
where σr denotes the radial stress of the surrounding rock; σθ denotes the tangential stress of the surrounding rock; ur denotes the displacement of the surrounding rock; r2 denotes the radius of the tunnel; and Ed and μd denote the modulus of elasticity and Poisson’s ratio of the surrounding rock, respectively.
In the same vein, Equations (21) and (22) are employed to determine the interaction force between the outer wall of the lining and the surrounding rock under the influence of internal water pressure. This force is represented as follows:
p 2 = p k 2 ( 1 μ c 2 ) E d r k 2 ( 1 + μ d ) E c ( r 2 2 r k 2 ) + ( 1 + μ c ) E d r k 2 + ( 1 μ c ) r 2 2
Based on the Geological Investigation Report for the diversion tunnel, we computed stresses and deformations of the crack lining structure under hydraulic pressure. As shown in Figure 9, there is a linear progression in both annular stress and displacement of the crack lining. With increasing crack depth, stress and displacement magnitudes escalate. This is due to stress transmission and deformation effects from water pressure, with cracks compromising the lining structure. Compared to an unblemished lining, the circumferential stress of the lining increased by approximately 4.50% with 0.1 m cracks, 9.10% with 0.2 m cracks, and 15.10% with 0.3 m cracks. A cracked lining fails to distribute stress evenly, reducing its resistance to water pressure interference and leading to heightened circumferential stress near the cracks. Over time, this stress concentration further degrades the lining structure.

3.3. Mechanical Characteristics of Different Structural Segments

The text describes the features of lining cracks in axial coordinates. The diversion tunnel includes an upper flat section, an inclined shaft section, and a lower flat section. Data show a notable difference in defect numbers between the upper flat and inclined shaft sections. This difference stems from the unique force characteristics of each structural segment, affecting defect formation and distribution.
Figure 10 presents a detailed schematic diagram illustrating the comprehensive force analysis of the diversion tunnel. This analysis is meticulously computed in accordance with the guidelines stipulated by the Road Tunnel Design Code and incorporates fundamental principles of rock mechanics theory to ensure accuracy and reliability [31]. In this context, the distinction between shallow and deeply buried tunnels is delineated by employing a formula that calculates the load equivalent height:
H p = 2.5 h q h q = q γ
where Hp denotes the dividing depth between deep and shallow buried tunnels; hq denotes the load equivalent height; γ denotes the weight of the surrounding rock; and q denotes the vertical pressure of the surrounding rock.
Given that the deepest burial point of the diversion tunnel in this project exceeds the critical depth Hp as stipulated by Equation (23), it is categorized as a deeply embedded tunnel. Consequently, the lining structure of the diversion tunnel experiences vertical stress corresponding to the pressure exerted by the surrounding rock mass of the tunnel:
σ z = γ h = γ × 0.45 × 2 s 1 w
where σz denotes the vertical pressure; γ denotes the perimeter rock capacity; s denotes the perimeter rock level; w denotes the width influence coefficient, w = 1 + i (B − 5); B denotes the width of the tunnel; and i is taken as 0.1 when B > 5 m.
The lining structure is subjected to horizontal pressure, which is expressed as follows:
e 1 = q tan 2 ( 45 ° φ c 2 ) e 2 = ( q + γ h ) tan 2 ( 45 ° φ c 2 ) σ x = ( e 1 + e 2 ) 2 = K 0 σ Z K 0 = μ 1 μ
In the formula, e1, e2 indicates the horizontal pressure of surrounding rock from the top to the bottom of the tunnel arch; e indicates the horizontal mean pressure; h indicates the height of the tunnel; φc indicates the friction angle of surrounding rock, and the value of 65° is taken in this project; K0 indicates the lateral pressure coefficient of surrounding rock, and the value of 0.43 is taken in the calculation of this project; and μ indicates the Poisson’s ratio of surrounding rock.
The architecture of the upper flat and inclined shaft segments mainly addresses vertical pressure from the surrounding rock at the diversion tunnel’s apex. Lateral pressures are evenly distributed, neutralizing each other. Thus, the focus is on vertical pressure impact. Due to lining inclination, the pressure on the vertical face of the inclined shaft segment is articulated as follows [32]:
σ x i e = σ z cos θ < σ z
where σxie denotes the component of pressure in the vertical direction, and θ denotes the angle between the inclined shaft section and the horizontal line.
The shaft’s angle relative to the horizontal creates a vertical stress component that reduces perpendicular pressure on the surrounding rock. This leads to fewer defects in the inclined shaft compared to the upper level, as confirmed by defect inspection data from both sections of the diversion tunnel.

4. Numerical Simulation and Analysis of Diversion Tunnels

Numerical analysis is less costly and faster. To further verify the results of the field tests, the continuous medium software ABAQUS (Version ABAQUS-2023) was used to continue the numerical model analysis.

4.1. Three-Dimensional Modeling

Based on actual test conditions, this paper investigates the stress and strain response of the diversion tunnel lining structure using a 3D numerical model. Based on a comprehensive review of the existing literature and a detailed analysis of the engineering geological structure, a sophisticated three-dimensional simulation model of the diversion tunnel has been developed. The model is configured with comprehensive mechanical and seepage parameters: elastic modulus E = 50 GPa, Poisson’s ratio ν = 0.24, uniaxial compressive strength UCS = 110 MPa, tensile strength σt = 7 MPa, internal friction angle ϕ = 32°, permeability coefficient k = 1 × 10−7 m/s, and porosity n = 0.05. Figure 11 shows that the model’s geometric dimensions are precisely defined: the upper flat section spans 87.4 m, the inclined shaft section measures 58.3 m, and the lower flat section extends 55.3 m. The tunnel has a diameter of 10.5 m and a lining thickness of 1.0 m, supported by rebar at intervals of every 2 m. Longitudinal cracks are prioritized for investigation due to the specific structural configuration and loading conditions of the lining, while circumferential cracks are excluded from this study [33]. When formulating the model with cracks, crack components are set with an average length of 4 m and varying depths of 0.1 m, 0.2 m, and 0.3 m. The tunnel lining utilizes an elastic–plastic model based on actual engineering values. Material parameter values are detailed in Table 2. A vertical load is applied to the top of the lining (see Equation (24)), a horizontal load to the left and right boundaries (see Equation (25)), and a water load from 0 to 1.35 MPa inside the lining.

4.2. Simulation Results

4.2.1. Deformation Characteristics of Non-Destructive Lining Structures

Figure 12 shows the simulated circumferential stress deformation of the lining. A negative sign indicates compression, and a positive sign indicates tension. In Figure 12a, the lining displays varying stress levels, with the vault in tension. The maximum tensile stress exceeds 1.43 MPa, raising the risk of tensile cracking. The pressure on both sides of the cave waist is significant, but the peak compressive stress is below the concrete’s compressive strength, indicating a low risk of compression fractures. Figure 12b shows that the lining near the entrance deforms more, with greater displacement in the upper flat section than in the inclined shaft section. This suggests internal water pressure negatively impacts the lining’s bearing capacity. However, the overall displacement cloud diagram shows symmetrical and minimal deformation, ensuring the lining’s safety and stability remain intact.
To gain deeper insights into the distribution patterns of lining stress and displacement, we have selected specific points from both the upper flat section and the inclined shaft section. These numerical results are depicted in Figure 13, which illustrates their evolution over time through a series of curves. Figure 14 illustrates that the patterns of stress and deformation in both the lining arch and arch waist are fundamentally consistent. The displacement values across points A to E exhibit minor variations, with the maximum displacement observed at point A. This suggests a uniform deformation response of the tunnel structure as a whole, albeit with notably larger deformations nearer the entrance. Since the magnitude of the displacement value does not directly dictate the deformation characteristics of the lining, it necessitates analysis from the perspective of internal force distribution. Figure 14c illustrates that the variation in circumferential stresses at points A to C along the lining is minimal, whereas the disparity at points D to F is substantial, suggesting a relatively uniform stress distribution across the upper flat section. When compared to the section points depicted in Figure 14d, the stress alteration at the apex of the arch is more pronounced. This phenomenon may be attributed to the heightened tensile stress induced by the vertical pressure exerted by the surrounding rock at the arch’s crown.
To verify the non-destructive lining model, analytical solutions are compared with simulated stress values at the annular region. Discrepancies, detailed in Table 3, show errors within a 15% margin. Despite numerical deviations, both methods concur on the overall stress trends of the lining structure, validating the theoretical computations.

4.2.2. Deformation Characteristics of Cracked Lining Structures

Based on the mechanical characterization of the non-destructive lining, the upper flat section with superior stress deformation properties is chosen for investigating crack propagation.
Throughout the crack expansion progression, STATUS indicates the extent of the crack. A STATUSXFEM value of one signifies complete obliteration of the lining. Figure 15 shows a fully cracked region from 0 to 90 degrees at the top of the arch, with cloud map values at the tip between zero and one, indicating imminent cracking that will spread as steps increase. Water pressure keeps the crack open, and the incremental step method simulates crack expansion by gradually decomposing the overall loading. In the initial phase (steps 0–20), cracks of 0.1 m, 0.2 m, and 0.3 m expand along depth; over time (steps 20–40), cracks penetrate through the lining, potentially causing significant fractures; beyond step 40, crack expansion shifts towards the lining wall until halted.
Extracting a point at the crack and recording its development with a curve, it can be seen in Figure 16b,d,f,h that the displacement value at 90° of the vault > 0° of the vault > 60° of the vault > 30° of the vault. It shows that the water pressure has a greater influence on the top of the arch at 60~90°, and the deeper the crack, the higher the displacement value. This phenomenon likely arises due to the arch waist experiencing not only heightened circumferential rock compression but also uneven hydrostatic pressure. The presence of cracks exacerbates displacement magnitudes, suggesting that the extent of displacement in a fissured lining alone does not encapsulate its deformation behavior comprehensively. Instead, an in-depth analysis of stress distribution patterns is imperative to fully grasp the nuances of lining deformation characteristics. From the stress distribution depicted in Figure 16a,c,e,g, it is evident that the circumferential stress at angles 0–30° off the vault exceeds the circumferential stress found at angles 60–90° off the vault. The data reveals a significant tensile stress concentration at the apex of the lining, with stress magnitude escalating as the crack deepens. This phenomenon can likely be attributed to heightened surrounding rock tension at the crown, which disrupts the normal stress distribution within the lining structure. Consequently, an augmented tensile force manifests at the crack periphery, leading to a more pronounced localized stress concentration as the crack progresses deeper into the structure [34].
Furthermore, to ensure the validity of the crack lining model presented above, a portion of the analytical solutions was chosen for comparison against the simulated values for the maximum annular stress. As demonstrated in Table 4, the error margins between the theoretical predictions and the simulated outcomes for crack linings at 0.1 m, 0.2 m, and 0.3 m are all within a 15% range. Although there exists some discrepancy between the two sets of calculations, the overall trend in the stress characteristics of the lining structure derived from both methods is consistent, thereby validating the reasonableness of the model assumptions.

5. Conclusions

This study investigates the inspection results of a diversion tunnel in a hydropower station located in the Dadu River Basin, with a focus on the formation mechanisms and simulation analysis of lining cracks. The main conclusions are summarized as follows:
(1) Tunnel lining inspections reveal significant spatial variation in defect distribution across different structural sections. The upper horizontal section exhibits a markedly higher frequency of defects compared to the inclined shaft, with most defects concentrated near the tunnel crown. Cracks at the crown are identified as having a more pronounced impact on structural safety.
(2) A two-dimensional axial force model was developed based on elastic mechanics for both intact and cracked linings. The analysis indicates a rising trend in both hoop stress and displacement. The presence of cracks at varying depths results in localized stress concentrations, potentially accelerating long-term structural degradation. The inclined shaft demonstrates a lower tendency for defect formation under stress, which aligns with field inspection findings.
(3) A three-dimensional finite element model was employed to simulate stress distribution. The results show that the undamaged lining in the upper horizontal section is subjected to greater circumferential stress than that in the inclined shaft. For cracked linings, circumferential stress is higher in the arch range of 0–30° than in 60–90°, indicating that cracks in the upper arch pose a greater threat to structural integrity. The simulation results are consistent with theoretical predictions, with deviations maintained within an acceptable error margin of 15%.

Author Contributions

Conceptualization, H.X., X.Z. and K.L.; Methodology, H.X. and H.W.; Formal analysis, L.Y.; Investigation, H.X., Z.L. and K.L.; Writing—original draft, H.X., H.W., Y.C. and Z.L.; Visualization, Y.C.; Supervision, X.Z.; Funding acquisition, H.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Key R&D Program of China (No. 2022YFB4703404), the National Natural Science Foundation of China (No. U21A20157), and the Sichuan Science and Technology Program (No. 2023YFS0410).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

  1. Yu, J.; Zhong, D.H.; Ren, B.Y.; Tong, D.W.; Hong, K. Probabilistic Risk Analysis of Diversion Tunnel Construction Simulation. Comput. Aided Civ. Infrastruct. Eng. 2017, 32, 748–771. [Google Scholar] [CrossRef]
  2. McAdams, M.; Wang, J. Gunnison Tunnel: Engineering History of an Early American Reclamation Project. J. Perform. Constr. Facil. 2013, 27, 826–835. [Google Scholar] [CrossRef]
  3. Liu, K.; Chen, Y.C.; Wang, H.R.; Xie, H.; Liu, Z.W. Reducing inconsistencies of FAHP in structural safety assessment of diversion tunnels. Appl. Soft Comput. 2023, 146, 14. [Google Scholar] [CrossRef]
  4. He, Y.P.; Sun, X.J.; Zhang, M.X. Investigation on the Deformation of Segment Linings in Cross-Fault Tunnel Considering the Creep Behavior of Surrounding Rock during Construction-Operation Period. Buildings 2022, 12, 1648. [Google Scholar] [CrossRef]
  5. Deng, J.; Xiao, M. Dynamic response analysis of concrete lining structure in high pressure diversion tunnel under seismic load. J. Vibroeng. 2016, 18, 1016–1030. [Google Scholar] [CrossRef]
  6. Yao, J.; Wang, Y.; Feng, D. Application of Signal Imaging Analysis Technology in Prediction and Treatment of Water Inrush in Diversion Tunnel. Trait. Du Signal 2022, 39, 1729–1736. [Google Scholar] [CrossRef]
  7. Xia, M.P.; Li, H.B.; Jiang, N.; Chen, J.L.; Zhou, J.W. Risk Assessment and Mitigation Evaluation for Rockfall Hazards at the Diversion Tunnel Inlet Slope of Jinchuan Hydropower Station by Using Three-dimensional Terrestrial Scanning Technology. KSCE J. Civ. Eng. 2023, 27, 181–197. [Google Scholar] [CrossRef]
  8. Wu, Y.; Li, Y.Z.; Qiao, W.G.; Fan, Z.W.; Zhang, S.; Chen, K.; Zhang, L. Water Seepage in Rocks at Micro-Scale. Water 2022, 14, 2827. [Google Scholar] [CrossRef]
  9. Hao, X.J.; Feng, X.T.; Yang, C.X.; Jiang, Q.; Li, S.J. Analysis of EDZ Development of Columnar Jointed Rock Mass in the Baihetan Diversion Tunnel. Rock Mech. Rock Eng. 2016, 49, 1289–1312. [Google Scholar] [CrossRef]
  10. Dong, L.H.; Chen, J.D.; Song, D.Q.; Wang, C.W.; Liu, X.L.; Liu, M.X.; Wang, E.Z. Application of Long-Range Cross-Hole Acoustic Wave Detection Technology in Geotechnical Engineering Detection: Case Studies of Tunnel-Surrounding Rock, Foundation and Subgrade. Sustainability 2022, 14, 16947. [Google Scholar] [CrossRef]
  11. Lin, C.J.; Wang, X.T.; Li, Y.; Zhang, F.K.; Xu, Z.H.; Du, Y.H. Forward Modelling and GPR Imaging in Leakage Detection and Grouting Evaluation in Tunnel Lining. KSCE J. Civ. Eng. 2020, 24, 278–294. [Google Scholar] [CrossRef]
  12. Liu, C.; Zhang, D.L.; Zhang, S.L. Characteristics and treatment measures of lining damage: A case study on a mountain tunnel. Eng. Fail. Anal. 2021, 128, 13. [Google Scholar] [CrossRef]
  13. Lu, A.; Yan, P.; Lu, W.B.; Li, X.F.; Liu, X.; Luo, S.; Huang, S.L.; Grasselli, G. Crack propagation mechanism of smooth blasting holes for tunnel excavation under high in-situ stress. Eng. Fract. Mech. 2024, 304, 19. [Google Scholar] [CrossRef]
  14. Li, G.D.; Ren, Z.Y.; Yu, J.J. Mixed-Mode I-II Fracture Process Zone Characteristic of the Four-Point Shearing Concrete Beam. Materials 2020, 13, 3203. [Google Scholar] [CrossRef]
  15. Du, J. A Fracture Mechanics Study of Concrete Failure. Ph.D. Thesis, University of Washington, Seattle, WA, USA, 1988. [Google Scholar]
  16. Li, Z.H.; Gong, Y.J.; Chen, F.J. Simulation of mixed mode I-II crack propagation in concrete using toughness-based crack initiation-propagation criterion with modified fracture energy. Theor. Appl. Fract. Mech. 2023, 123, 16. [Google Scholar] [CrossRef]
  17. Khan, G.; Ahmed, A.; Liu, Y.; Tafsirojjaman, T.; Ahmad, A.; Iqbal, M. Phase field model for mixed mode fracture in concrete. Eng. Fract. Mech. 2023, 289, 15. [Google Scholar] [CrossRef]
  18. Chen, S.; Yang, Z.; Liu, S.; Li, L.F.; Zheng, Y.B.; Yuan, Y. Numerical simulation and analysis of crack disease in tunnel lining structure. Front. Mater. 2022, 9, 17. [Google Scholar] [CrossRef]
  19. Karami, M.; Kabiri-Samani, A.; Nazari-Sharabian, M.; Karakouzian, M. Investigating the effects of transient flow in concrete-lined pressure tunnels, and developing a new analytical formula for pressure wave velocity. Tunn. Undergr. Space Technol. 2019, 91, 13. [Google Scholar] [CrossRef]
  20. Wang, G.; Fang, Q.; Du, J.M.; Wang, J. Semi-analytical solution for internal forces of tunnel lining with multiple longitudinal cracks. J. Rock Mech. Geotech. Eng. 2023, 15, 2013–2024. [Google Scholar] [CrossRef]
  21. Chen, J.X.; Hu, T.T.; Hu, X.; Jia, K. Study on the influence of crack depth on the safety of tunnel lining structure. Tunn. Undergr. Space Technol. 2024, 143, 16. [Google Scholar] [CrossRef]
  22. Peng, Z.T.; Fang, Q.; Ai, Q.; Jiang, X.M.; Wang, H.; Huang, X.C.; Yuan, Y. Identifying the dominant influencing factors of secondary lining cracking risk in an operating mountain tunnel. Int. J. Struct. Integr. 2024, 15, 731–756. [Google Scholar] [CrossRef]
  23. Du, J.M.; Shu, Y.H.; Xu, G.W.; He, C.; Yao, C.F. Study on the influence of geo-stress field on the fracture pattern of secondary tunnel lining. Eng. Fail. Anal. 2023, 152, 28. [Google Scholar] [CrossRef]
  24. Xu, Z.L.; Chen, J.X.; Luo, Y.B.; Zhu, H.Y.; Liu, W.W.; Shi, Z.; Song, Z.G. Geomechanical model test for mechanical properties and cracking features of Large-section tunnel lining under periodic temperature. Tunn. Undergr. Space Technol. 2022, 123, 15. [Google Scholar] [CrossRef]
  25. Lu, S.; Sun, Z.Y.; Zhang, D.L.; Liu, C.; Wang, J.C.; Huangfu, N.Q. Numerical modelling and field observations on the failure mechanisms of deep tunnels in layered surrounding rock. Eng. Fail. Anal. 2023, 153, 27. [Google Scholar] [CrossRef]
  26. Lei, Y.M.; Yang, X.H.; Liu, Q.S.; Liu, H.; Chu, Z.F.; Wen, J.T.; Huang, Y.H. An enhanced polar-based GPGPU-parallelized contact detection algorithm for 3D FDEM and its application to cracking analysis of shield tunnel segmental linings. Tunn. Undergr. Space Technol. 2024, 148, 22. [Google Scholar] [CrossRef]
  27. Li, L.Y.; Liu, Z.; Du, X.L. Improvement of Analytic Hierarchy Process Based on Grey Correlation Model and Its Engineering Application. Asce-Asme J. Risk Uncertain. Eng. Syst. Part A Civ. Eng. 2021, 7, 12. [Google Scholar] [CrossRef]
  28. Lang, L.; Zhu, Z.M.; Zhang, X.S.; Qiu, H.; Zhou, C.L. Investigation of crack dynamic parameters and crack arresting technique in concrete under impacts. Constr. Build. Mater. 2019, 199, 321–334. [Google Scholar] [CrossRef]
  29. Peng, J.X.; Hu, S.W.; Zhang, J.R.; Cai, C.S.; Li, L.Y. Influence of cracks on chloride diffusivity in concrete: A five-phase mesoscale model approach. Constr. Build. Mater. 2019, 197, 587–596. [Google Scholar] [CrossRef]
  30. Mittelstedt, C.; Becker, W. Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method. Int. J. Solids Struct. 2006, 43, 2868–2903. [Google Scholar] [CrossRef]
  31. Xie, H.P.; Lu, J.; Li, C.B.; Li, M.H.; Gao, M.Z. Experimental study on the mechanical and failure behaviors of deep rock subjected to true triaxial stress: A review. Int. J. Min. Sci. Technol. 2022, 32, 915–950. [Google Scholar] [CrossRef]
  32. Li, C.B.; Yang, D.C.; Xie, H.P.; Ren, L.; Wang, J. Size effect of fracture characteristics for anisotropic quasi-brittle geomaterials. Int. J. Min. Sci. Technol. 2023, 33, 201–213. [Google Scholar] [CrossRef]
  33. Zhang, N.; Zhu, X.J.; Ren, Y.F. Analysis and Study on Crack Characteristics of Highway Tunnel Lining. Civ. Eng. J. Tehran 2019, 5, 1119–1123. [Google Scholar] [CrossRef]
  34. Rodríguez, C.A.; Rodríguez-Pérez, Á.M.; López, R.; Hernández-Torres, J.A.; Caparrós-Mancera, J.J. A Finite Element Method Integrated with Terzaghi’s Principle to Estimate Settlement of a Building Due to Tunnel Construction. Buildings 2023, 13, 1343. [Google Scholar] [CrossRef]
Figure 1. Schematic diagram of the inspection path and implementation process.
Figure 1. Schematic diagram of the inspection path and implementation process.
Applsci 15 09210 g001
Figure 2. Map of site operations.
Figure 2. Map of site operations.
Applsci 15 09210 g002
Figure 3. Infographic of diversion tunnel defects.
Figure 3. Infographic of diversion tunnel defects.
Applsci 15 09210 g003
Figure 4. Characteristic distribution of the number of cracks along the axis of the tunnel.
Figure 4. Characteristic distribution of the number of cracks along the axis of the tunnel.
Applsci 15 09210 g004
Figure 5. Schematic force diagram of the diversion tunnel.
Figure 5. Schematic force diagram of the diversion tunnel.
Applsci 15 09210 g005
Figure 6. Lining force diagram under water pressure.
Figure 6. Lining force diagram under water pressure.
Applsci 15 09210 g006
Figure 7. Force diagram of surrounding rock under water pressure.
Figure 7. Force diagram of surrounding rock under water pressure.
Applsci 15 09210 g007
Figure 8. Stress-deflection diagram for non-destructive lining.
Figure 8. Stress-deflection diagram for non-destructive lining.
Applsci 15 09210 g008
Figure 9. Stress deformation diagram for defective lining.
Figure 9. Stress deformation diagram for defective lining.
Applsci 15 09210 g009
Figure 10. Schematic diagram of the overall force analysis of the lining.
Figure 10. Schematic diagram of the overall force analysis of the lining.
Applsci 15 09210 g010
Figure 11. Diversion tunnel component models.
Figure 11. Diversion tunnel component models.
Applsci 15 09210 g011
Figure 12. Circumferential stress deformation of diversion tunnel lining cloud map.
Figure 12. Circumferential stress deformation of diversion tunnel lining cloud map.
Applsci 15 09210 g012
Figure 13. Schematic diagram of cross-section points of diversion tunnels.
Figure 13. Schematic diagram of cross-section points of diversion tunnels.
Applsci 15 09210 g013
Figure 14. Circumferential stress deformation curve of diversion tunnel lining.
Figure 14. Circumferential stress deformation curve of diversion tunnel lining.
Applsci 15 09210 g014
Figure 15. Diagram of the expansion process of the cracked lining.
Figure 15. Diagram of the expansion process of the cracked lining.
Applsci 15 09210 g015aApplsci 15 09210 g015b
Figure 16. Circumferential stress deformation curve of cracked lining of diversion tunnel.
Figure 16. Circumferential stress deformation curve of cracked lining of diversion tunnel.
Applsci 15 09210 g016aApplsci 15 09210 g016b
Table 1. Correlation of major defects in diversion tunnels.
Table 1. Correlation of major defects in diversion tunnels.
PositionVault CrackWaist CrackSoffit LeakageArch Girdlet Leakage
Upper flat section10.880.620.91
Inclined section0.990.880.620.91
Table 2. Table of values for each parameter.
Table 2. Table of values for each parameter.
TypeUnit Weight
(kN·m−3)
Modulus of Elasticity
(GPa)
Poisson’s RatioAngle of Internal Friction
(°)
Cohesion (MPa)Tensile Strength (MPa)Breaking Energy (N/M)
Surrounding rock27.55.380.30451.20
Lining2530.00.25--1.43114.35
Shotcrete2531.50.20--1.57
Anchor bolt78200.00.20--
Table 3. Comparison table between theoretical and simulated results of non-destructive lining.
Table 3. Comparison table between theoretical and simulated results of non-destructive lining.
Water Load in the Cave (MPa)Analytical Solution for Cyclic Stress (MPa)Numerical Solution for the Maximum Cyclic Stress (MPa)Error Value
(%)
0.451.131.2914.15
0.82.002.2814.00
1.22.753.1514.55
1.353.373.8413.95
Table 4. Comparison table between theoretical and simulated results for crack lining.
Table 4. Comparison table between theoretical and simulated results for crack lining.
(a) Comparison of 0.1 m crack results
Water Load in the Cave (MPa)Analytical Solution for Cyclic Stress (MPa)Numerical Solution for the Maximum Cyclic Stress (MPa)Error Value
(%)
0.451.171.3111.97
0.82.082.3512.98
1.22.863.2814.68
1.353.514.0314.81
(b) Comparison of 0.2 m crack results
Water Load in the Cave (MPa)Analytical Solution for Cyclic Stress (MPa)Numerical Solution for the Maximum Cyclic Stress (MPa)Error Value
(%)
0.451.221.3913.93
0.82.172.4412.44
1.22.993.4214.38
1.353.664.1914.48
(c) Comparison of 0.3 m crack results
Water Load in the Cave (MPa)Analytical Solution for Cyclic Stress (MPa)Numerical Solution for the Maximum Cyclic Stress (MPa)Error Value
(%)
0.451.291.4512.40
0.82.292.5712.27
1.23.143.5713.69
1.353.864.3813.47
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Xie, H.; Wang, H.; Zou, X.; Chen, Y.; Liu, Z.; Yang, L.; Liu, K. Research on Cracking Mechanism and Crack Extension of Diversion Tunnel Lining Structure. Appl. Sci. 2025, 15, 9210. https://doi.org/10.3390/app15169210

AMA Style

Xie H, Wang H, Zou X, Chen Y, Liu Z, Yang L, Liu K. Research on Cracking Mechanism and Crack Extension of Diversion Tunnel Lining Structure. Applied Sciences. 2025; 15(16):9210. https://doi.org/10.3390/app15169210

Chicago/Turabian Style

Xie, Hui, Haoran Wang, Xingtong Zou, Yongcan Chen, Zhaowei Liu, Liyi Yang, and Kang Liu. 2025. "Research on Cracking Mechanism and Crack Extension of Diversion Tunnel Lining Structure" Applied Sciences 15, no. 16: 9210. https://doi.org/10.3390/app15169210

APA Style

Xie, H., Wang, H., Zou, X., Chen, Y., Liu, Z., Yang, L., & Liu, K. (2025). Research on Cracking Mechanism and Crack Extension of Diversion Tunnel Lining Structure. Applied Sciences, 15(16), 9210. https://doi.org/10.3390/app15169210

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Article metric data becomes available approximately 24 hours after publication online.
Back to TopTop