Guarantee Rate Statistics and Product-Moment Correlation Analysis of the Optimal Deformation Allowance for Loess Tunnel in China

Abstract

The relevant value standards for reserved deformation of tunnels with different types of loess soil are not yet perfect. Through mathematical statistics, literature research, and other methods, 148 monitoring sections of loess tunnels with different soil types were investigated. Pearson, Kendall, and Spearman correlation coefficients were used to analyze the influence of surrounding rock grade, moisture content, and burial depth on the deformation law of tunnels, providing reference for the value of reserved deformation for loess tunnels. The research results indicate that: (1) The correlation between soil type and convergence around the tunnel is strong, followed by excavation span, and the correlation between moisture content and burial depth is weak. Therefore, the design of reserved deformation should focus on considering the soil type and excavation span. (2) For loess tunnel sections with class IV surrounding rock, the deformation allowance for cohesive loess and silty loess tunnel sections can be set within 15 cm, and for sandy loess, it can be set within 15–20 cm. (3) For the loess tunnel section with V-grade surrounding rock, the deformation margin can be determined as follows: viscous loess within 15–20 cm, silty loess within 15 cm, and sandy loess within 35–47.5 cm. (4) The moisture content is mainly distributed between 11% and 23%. It is worth noting that in sandy loess sections, the influence of moisture content on tunnel arch settlement is more significant compared to cohesive loess and silty loess sections. (5) The settlement of the arch crown is affected by changes in coverage depth and has discreteness. When the deformation margin is set to 16 cm, the assurance rate can reach 84.6%. Finally, the rationality of the proposed reserved deformation amount was verified through engineering practice. The research results can provide reference for related similar projects.

1. Introduction

Loess in China is predominantly found in northern and northwestern regions. This type of soil is characterized by its low bearing capacity and distinct properties, such as collapsibility, large pore spaces, looseness, and the development of vertical joints, all of which influence its stability and behavior under various conditions. Support failures and collapses frequently happen in tunnel and underground construction [1,2]. Tunnel construction involves several key processes, including excavation, support installation, stress management, and deformation control, with deformation allowance being a critical aspect of the overall process. This allowance refers to the reserved space between the initial and secondary tunnel linings, which is essential for accommodating the surrounding rock and the initial support structure. Properly defining the deformation allowance prevents encroachment by the surrounding rock, ensure the total deformation of surrounding rock, and mitigates surrounding rock pressure [3,4,5]. Apart from utilizing appropriate support parameters and construction methods, comprehending tunnel deformation patterns is essential. Based on the evolution process of surrounding rock, they summarized and improved the strategies for preventing and controlling collapse. Wu et al. [6] proposed a method to solve the large deformation of tunnels under compression, and derived its reliability and feasibility based on the theory of convergence constraint methods.

Research at home and abroad shows that tunnel deformation is usually related to factors such as surrounding rock grade, cover depth, section size, and excavation method. Tang et al. [7] analyzed the deformation law of the surrounding rock of the loess tunnel through on-site monitoring data and provided a reasonable reserved deformation amount.

Li et al. [8] studied the velocity distribution characteristics of tunnel arch surrounding rock through numerical simulation and proposed a new velocity field. By comparing existing analysis and experimental results, the accuracy of the proposed method was demonstrated. Investigating the deformation patterns of loess tunnels, identifying the key challenges in loess tunnel engineering, and improving management practices are essential steps in addressing safety concerns and ensuring the successful construction of loess tunnels. Li [9] suggested increasing the deformation allowance during biased loess tunnel construction to prevent the tunnel from exceeding its limits after stable support is in place. Ji [10] demonstrated through numerous engineering case studies that both the surrounding rock and the support structures experience uneven deformation across the entire tunnel ring. Uniformly reserving deformation allowance will lead to increased excavation and secondary lining backfill volumes, resulting in higher construction costs. This study uses the Tongzhai Tunnel on the Lanzhou-Chongqing Railway as a case study to investigate the deformation patterns of the surrounding rock and initial support. Through on-site monitoring, this study identifies the deformation characteristics and proposes the inclusion of a non-uniform deformation allowance to address these patterns. To ensure safety, the backfilling cost of secondary lining is effectively controlled. Xi et al. [11] defined the deformation and assurance rate of the initial support of the tunnel, investigated the cover depth’s impact on tunnel deformation, and provided deformation allowance values for deep and shallow buried large cross-section loess tunnels while maintaining a specific assurance rate. Li [12] conducted a study on the Zhengxi Railway and Baolan Passenger Specialized Tunnel Project, concluding that the span and moisture content were the primary factors influencing the overall arch subsidence. Jin et al. [13] investigated the setting of deformation allowance of shallow buried loess tunnels experiencing asymmetric stress at varying bias slopes. Through on-site monitoring and numerical simulation, it was found that the deformation allowance of shallowly buried measurements can be set uniformly. However, for deeply buried tunnels, deformation increases with the slope bias, and recommended deformation allowance values are provided for different slope biases. Additionally, Du et al. [14] studied the Gushan Tunnel of the Menghua Railway, analyzing the impact of each stage of bench excavation on deformation. Their research specifically investigates how each phase of bench excavation in soft rock tunnels affects the overall deformation. Tang et al. [15] relied on engineering practice to study the deformation and failure mechanism of loess tunnels under rainfall conditions through indoor model experiments, and summarized the stages of tunnel deformation and failure under rainfall conditions. Zhao et al. [16], He [17], and Liu et al. [18] counted the monitoring data of crown settlement and side wall convergence of thousands of tunnel sections in various strata, identified deformation patterns, and provided recommended deformation allowance values. He et al. [19] systematically analyzed the phenomenon and mechanism of loess liquefaction in China, analyzed new theories for improving loess liquefaction, and provided corresponding prevention and control methods. By reviewing domestic and international research, it becomes evident that the deformation of loess tunnels varies with soil types. Under the influence of different surrounding rock classes, tunnels in sandy loess sections experience more pronounced deformation compared to clayey loess and silty loess sections. Sandy loess tunnels are more sensitive to moisture content compared to tunnels in clayey loess and silty loess sections.

In conclusion, domestic and foreign researchers have studied deformation allowance values for loess tunnels. However, there is a need for well-defined deformation allowance values for tunnels in different loess soil types, and there is a shortage of quantitative studies on this subject. Therefore, it is necessary to propose the design value of deformation allowance for tunnels in different loess soil types. Using data from 148 tunnel monitoring sections across various loess soil types, and considering factors like surrounding rock conditions, moisture content, and cover depth, this paper provides suggestions for the design of deformation allowance of tunnels in different loess soil types. This offers valuable guidance for future projects with similar designs.

This study conducted statistical analysis on the settlement data of the arch crown of loess tunnels in China, investigated the correlation of factors affecting tunnel settlement, and provided recommended values for reserved deformation of tunnels with different loess soil types. The rationality of the recommended values was verified through engineering practice. The research results will help improve the standard for determining the reserved deformation of loess tunnels and provide suggestions for similar projects.

2. Deformation Allowance of Loess Tunnel and Statistical Monitoring Samples

2.1. Surrounding Rock Deformation

The initial tunnel support serves as the primary means of protecting against unstable surrounding rock, while the deformation allowance plays an equally crucial role by effectively reducing the rock pressure exerted on the tunnel. The Constraint Convergence Method (CCM) is widely used to evaluate tunnel convergence and determine the necessary support requirements, taking into account the complex interaction between the surrounding rock and the support system. This method is particularly effective in analyzing these factors under different geological conditions and varying stress environments [20,21,22]. CCM establishes a relationship between the internal support force (Pi) and the radial displacement (u) of a circular tunnel, providing valuable insights into the tunnel’s structural behavior under different loading conditions. This relationship is visually represented by the ground reaction curve (GRC), which characterizes the interaction between the ground and the tunnel. In addition, the support reaction curve (SRC) illustrates the elastic relationship between the longitudinal support stress and the resulting tunnel convergence. Based on crucial, albeit potentially debated assumptions, tunnel support is presumed to not alter the surrounding rock’s response. The tunnel support system is expected to achieve equilibrium at the intersection of the GRC and the SRC [23].

The constitutive relationship of soft rock suggests that the use of flexible tunnel support allows the surrounding rock mass to transition into a plastic state, thereby accommodating the rock’s deformation under stress. Different from the load action, the huge plastic property of surrounding rock will be released in some form. The resultant force (P₀) from the tunnel excavation’s impact on surrounding rock mainly consists of three parts. Specifically, the process can be described as follows: (I) P₀ is initially released through deformation, which is governed by the elastic-plastic behavior of the soft rock; (II) P₀ is then supported by the intrinsic strength of the surrounding rock mass before it reaches a state of looseness; (III) however, these two factors alone are generally insufficient to ensure the stability of the surrounding rock, thus requiring the addition of external engineering support. Therefore, the support mechanism for tunnels in soft surrounding rock can be summarized as follows:

(1)$P_0=P_d+P_r+P_s$

$P_0$—Resultant force of surrounding rock movement to the free area after excavation of tunnel surrounding rock, including gravity, water force, expansion force, tectonic stress, etc.;

$P_d$—Engineering force is typically expressed as deformation, especially in the case of soft rock where plastic properties are primarily released through deformation;

$P_r$—Self-bearing capacity of surrounding rock;

$P_s$—Engineering support force.

In real-world construction, as illustrated in Figure 1, accurately quantifying the surrounding rock pressure presents a significant challenge due to the complex and variable nature of the geological conditions. Typically, when a certain level of displacement occurs, the actual rock pressure often exceeds the initial design assumptions for the support system. This discrepancy can lead to an insufficient support force, compromising the ability to maintain equilibrium between the surrounding rock and the support structure.

On the one hand, if support is applied rapidly following tunnel excavation, as depicted at point A in the figure, it requires a significant support force. However, this approach imposes excessive strain on the surrounding rock, as it undergoes only elastic deformation, which is not optimal. Without timely support, or in the absence of any support, the surrounding rock can experience uncontrolled deformation, potentially leading to instability. At the beginning, as the deformation of the surrounding rock increases, the required support force gradually decreases. In an ideal scenario, the support force could be reduced to a minimal level or even become negligible. However, at this critical stage, represented by point D in Figure 1, the surrounding rock may experience significant structural failure, including disintegration or complete collapse, which indicates a potential loss of stability. At this stage, the pressure exerted by the surrounding rock on the support is no longer the pressure from deformation, but rather the pressure resulting from the collapse of the surrounding rock, often referred to as the loosening pressure, which is primarily due to the weight of the disintegrated rock. As a result, anchor shotcrete support becomes unsuitable, and the only viable option is to employ the traditional construction method of cast-in-place concrete lining. On the other hand, the optimal location for support should be positioned to the left of point D, close to point C, as shown in the figure. At this location, the surrounding rock is capable of undergoing significant deformation, which helps to redistribute a larger portion of the rock mass pressure. Meanwhile, the support itself will experience minimal deformation pressure, ensuring more efficient load distribution and stability. This configuration ensures the stability of the surrounding rock. During construction, once the displacement and deformation of the first support stabilize, the second support is introduced when it reaches point C on the diagram. As construction progresses, both the surrounding rock and the support undergo creep deformation, at which point the support and deformation pressure will stabilize at PE. This process ensures that the support force maintains a sufficient safety margin, providing enhanced stability and reliability.

Given a specific amount of displacement in the tunnel surrounding rock and lining, as represented by the FG section in Figure 1, two treatment options can be considered. The first option involves allowing continued deformation of the surrounding rock, which is denoted as ∆u in Figure 1. However, the first approach is clearly impractical due to the significant encroachment of the lining on the building boundary. As an alternative, the second option involves increasing the support force, denoted as ∆P in the figure, which can help address the displacement issues without compromising the structural integrity of the building. Firstly, this approach requires further excavation of the tunnel while allowing the deformation to continue, effectively redistributing the load. Secondly, it necessitates an enhancement in the support strength of the tunnel to accommodate these changes. Therefore, the rational design of deformation allowances is a critical factor in ensuring the stability and safety of tunnel construction.

2.2. Significance of Deformation Allowance

Deformation allowance refers to the designated space intentionally incorporated into engineering or construction projects to provide safe and stable flexibility, allowing for the natural movement or deformation of surrounding rock or structures without compromising their overall integrity. When determining the excavation dimensions across different rock layers, it is essential to allocate sufficient space for rock deformation, while also ensuring that the excavation meets the required construction limits and safety standards. Therefore, the deformation allowance refers to the amount of space deliberately left within the surrounding rock after excavation, ensuring sufficient flexibility to accommodate any potential deformation without compromising the stability of the structure. Monitoring the apparent displacement presents significant challenges, both in terms of difficulty and cost, due to the limitations of current construction technologies. In modern construction monitoring, deformation values are typically aligned with those of the support structure to ensure consistency and reliability in tracking structural changes. Due to limitations in measurement techniques and the interference caused by construction blasting, monitoring is generally initiated at a certain distance away from the excavation face to ensure accuracy and minimize the impact of these factors. As a result, the deformation values obtained from monitoring often reflect only a part of the deformation occurring in the support structure, failing to capture the full deformation process within both the surrounding rock and the support system. Before tunnel excavation, it is crucial to carefully design the tunnel deformation allowance. An excessively large allowance may lead to over-excavation, which significantly increases the volume of earth that needs to be removed during construction, resulting in higher costs for slag disposal and backfilling. If the allowance is set too small, it could restrict the initial support system’s intrusion and create safety risks by limiting the available space for secondary lining. A small intrusion limit may require only the replacement of the arch, potentially leading to construction delays, as illustrated in Figure 2. Therefore, it is essential to carefully determine an appropriate deformation allowance before rock excavation, and accurately predict the extent of deformations in loess tunnels to avoid such issues.

2.3. Setting of Deformation Allowance

Figure 3a,b shows the spatial position of the reserved deformation amount in the tunnel and the stress on the tunnel from different angles. Figure 3c represents the tunnel shield machine, and Figure 3d represents the stress on the tunnel support and the deformation trend of the surrounding rock and tunnel face during the forward excavation process of the shield machine. The deformation of surrounding rock refers to the shrinkage deformation of surrounding rock towards the tunnel direction due to the loss of restraint after tunnel excavation. The surrounding rock deformation is crown settlement and surrounding convergence in the monitoring items. Generally, the total deformation of surrounding rock after excavation µ includes advanced deformation before tunnel excavation µ_f. Deformation from the excavation of the tunnel face to the beginning of measurement work µ_i and accumulated deformation obtained during measurement µ_m, as shown in Figure 3. The most reliable and available is the deformation monitoring in part 3 [27,28,29]. The deformation allowance setting includes µ_m and µ_i two parts; when the surrounding rock deformation is stable, it just reaches the position of the design contour line, so the secondary lining construction can not only save materials, but also ensure the lining quality [30,31,32].

Considering the current support conditions and construction technology, the initial support will be applied after tunnel excavation. Due to the inherent characteristics of the surrounding rock, the tunnel is designed to accommodate displacement during operation. The cumulative displacement during initial support and subsequent stability represents the initial tunnel support deformation. Excavation span, tunnel support form, construction technology, surrounding rock properties, construction methods, and other factors impact the initial tunnel support deformation [33].

The reasonableness of the deformation allowance directly impacts the project’s economy and safety. The analysis of loess tunnel deformation and deformation allowances relies on field measurement data. By examining the variations in the surrounding rock’s grade, we are able to identify the deformation patterns and establish a range of appropriate deformation allowances. Based on varying rock levels, we can refer to specific guarantee rates (the percentage of the number of sections with initial support deformation less than the given value in the total number of statistical sections) to better align the deformation allowance with the project’s actual conditions. Furthermore, due to the complex interaction between tunnel support and the loess strata, as well as the various uncertain factors influencing the deformation of the initial support, the field-measured data may exhibit some degree of variability. To ensure that the deformation allowance aligns with site conditions, it is suitable to utilize a range value associated with a specific guarantee rate. When the corresponding guarantee rate of a given deformation allowance is greater than 80%, the deformation allowance is considered appropriate.

2.4. Basic Information of Monitoring Samples

This paper’s statistical samples consist of 148 monitoring sections from 29 tunnels, as indicated Table 1. Relevant instructions are as follows:

• (1). The data in this paper are sourced from published academic papers. Some information cannot be found, and it is replaced by “/” in the table.

• (2). In the table, the surrounding rock grade and cover depth pertain to the conditions at the monitoring section’s location.

• (3). According to the classification of surrounding rock according to the specifications [34], the higher the level of surrounding rock, the looser the soil, and the worse the properties of the surrounding rock. From the rock mass level, it can be seen that the rock mass levels of each tunnel monitoring section are classified into grade IV and grade V. The grade IV surrounding rock consists of a crushed stone-like structure, characterized by a loose arrangement of angular gravel and soil that is relatively soft. Due to the soft texture of the loess layer, which contains small and loose particles, most of the surrounding rock is classified as grade V, with grade IV surrounding rock being relatively scarce.

• (4). In terms of excavation methods, the bench method comprises the majority of the monitoring sections, while the CRD (Cross Diaphragm) and CD (Center Diaphragm) methods represent smaller proportions. So, even though the loess tunnel falls into the category of soil tunnel, the bench method does not perform as well as the CRD method and CD method in managing stratum deformation. However, due to its notable benefits in construction efficiency and cost-effectiveness, it remains the primary construction approach for loess tunnels.

• (5). Regarding the spatial distribution of tunnels, among the 29, 10 are in Shaanxi Province, 9 in Gansu Province, 4 in Henan Province, 3 in Shanxi Province, 1 in Ningxia Hui Autonomous Region, 1 in Qinghai Province, and 1 in Inner Mongolia Autonomous Region.

• (6). Due to the complexity of loess’s geological and hydrological conditions, tunnels vary significantly. Typically, a loess ground profile consists of Q1, Q2, Q3, and Q4 loess layers, each with distinct physical and mechanical properties in ascending order. There are 10 tunnels in the clayey loess, 13 in the sandy loess, and 7 in the silty loess sections.

• (7). Typically, tunnel deformation allowances in loess areas consider factors such as cover depth, tunnel width, surrounding rock conditions, construction method, and moisture content variations. In the research, qualitative descriptions prevail, and quantitative analysis is infrequent.

Generally, typical loess in China consists of aeolian sediments [62,63,64], supported by the following characteristics. Loess soil particles in China transition from coarse to fine and from northwest to southeast. The mineral and chemical composition of loess exhibits relative consistency across a wide area, with more significant variations observed in the northwest region compared to the southeast. In Liu [65], China’s loess is classified into three regions from northwest to southeast: sandy loess, silty loess, and clayey soil (Figure 4). Moving from sandy loess to clayey loess, there is a decrease in sand content and porosity, along with an increase in clay content and cohesion. Tunnel mechanical properties and deformation behavior vary among different loess soil types. Hence, deformation allowance design for loess tunnels of varying soil types should also differ.

Compared to other weak surrounding rocks, loess stands out due to its distinctive engineering features, including vertical joints, macropores, and complex structure. This results in rapid initial-stage deformation characterized by significant, asymmetric, and spatially uneven changes in deformation speed, magnitude, and duration. The low bearing capacity of loess in the rock foundation leads to poor tunnel stability at the junctions of the arch foot, inverted arch, and side walls. If support is not applied in a timely manner or if the support stiffness is insufficient, the support structure may experience excessive settlement, which could ultimately result in tunnel collapse [66,67,68]. The low bearing capacity and weak self-stability of loess tunnel surrounding rock result in significant initial stage deformation rates during excavation. For instance, the Dayoushan and Qindong loess tunnels exhibit high initial deformation rates of 77 mm/d and 40 mm/d, respectively. The loess surrounding rock experiences prolonged deformation, marked by prominent creep characteristics. Analyzing arch crown settlement and horizontal convergence data from the Dayoushan tunnel, a significant deformation increase is observed 7–15 days after tunnel face excavation. Between days 10 and 15, the deformation rate slows down, eventually stabilizing approximately 30 days after excavation. Proper deformation allowance settings can enable the tunnel’s surrounding rock to undergo controlled deformation after excavation and support, thus releasing energy and reducing deformation pressure on the post-support structure. Deformation allowance is tied to final settlement, particularly in the case of 148 sections with grade IV and V surrounding rocks where crown settlement exceeds surrounding convergence. Therefore, the deformation allowance setting should prioritize arch crown settlement.

3. Analysis of Influence Factors on Deformation Allowance of Loess Tunnel

3.1. Analysis of Deformation Allowance Under Grade IV Surrounding Rock

The initial deformation of tunnel support is induced by the intricate interaction between loess and the uncertain factors involved. Hence, the field-measured data exhibit significant discreteness. If the deformation allowance, calculated based on the guaranteed rate, exceeds 80%, the tunnel can be considered to have suitable stability. However, tunnel deformation may vary due to regional differences in the quality of the loess soil. Therefore, the soil is divided into clayey loess, silty loess, and sandy loess for analysis. Considering that the crown settlement in the measured data exceeds surrounding convergence, the deformation allowance for the loess tunnel design should prioritize crown settlement. In the loess tunnel section with grade IV surrounding rock dominance, data from 68 monitoring sections for crown settlement and 46 monitoring sections for surrounding convergence were collected and analyzed. Analyzing Figure 5, when the vault’s deformation allowance is set at 10 cm, 15 cm, 20 cm, and 25 cm, the corresponding guarantee rates are 55.89%, 82.53%, 94.11%, and 100%, respectively. In clayey loess and silty loess sections, the tunnel deformation allowance is set to 15 cm, and in sandy loess sections, it is set to 20 cm. Based on the findings in Figure 6, setting the deformation allowance for both sides of the tunnel side wall at 40 mm, 80 mm, 120 mm, and 170 mm yields guarantee rates of 58.69%, 84.78%, 97.82%, and 100%, respectively. Furthermore, in sandy loess sections, the side wall deformation allowance for tunnel measurements can be set at 12 cm, taking into account soil quality. In clayey and silty loess sections, the tunnel deformation allowance can be set to 4 cm.

3.2. Analysis of Deformation Allowance Under Grade V Surrounding Rock

In loess tunnels with grade V surrounding rock, the vault settlement exceeding surrounding convergence is observed as well. A total of 80 crown settlements in a loess tunnel with V-grade surrounding rock and 60 surrounding convergence monitoring sections were statistically analyzed. As shown in Figure 7, adjusting the arch crown deformation allowance to 15 cm, 25 cm, 350 cm, and 45 cm corresponds to guarantee rates of 80%, 92.5%, 95%, and 100%, respectively. Based on the analysis in Figure 8, setting the deformation allowance for the tunnel’s side wall at 10 cm, 15 cm, 20 cm, and 27.5 cm yields guarantee rates of 85%, 93.3%, 96.67%, and 100%, respectively. Taking into account the distribution of soil types, the deformation allowance for the arch crown is set to 25 cm, while the deformation allowance for the side walls is set to 15 cm in the clayey loess tunnel section. In silty loess sections, one can set the tunnel crown deformation allowance to 15 cm and 10 cm on each side of the side wall. In sandy loess sections, one can set the arch crown deformation allowance to 35 cm and 20 cm on both sides of the side wall. Due to the varied geological conditions in loess regions, tunnels often traverse both collapsible and non-collapsible loess layers, alongside local areas containing fill material and recently deposited loess. Consequently, the deformation allowances derived from actual monitoring and statistical analysis exceed the specified values significantly. The appropriate deformation allowance should be determined by comprehensively considering the types and parameters of initial support, the timing of lining construction, and lining parameters while balancing safety and cost-effectiveness.

3.3. Analysis of Influence of Moisture Content on Deformation Allowance

Affected by the moisture content in the loess area, the mechanical properties of the loess body can rapidly deteriorate, potentially leading to instability and posing construction safety risks. In soil tunnel construction, uncontrolled initial support clearance displacement and unstable surrounding rock complicate the determination of the tunnel’s deformation allowance. Wang et al. [69] studied the settlement and deformation mechanism of shallow buried loess tunnel surrounding rock after the increase of surrounding rock moisture content caused by rainfall. Through a large amount of on-site monitoring data, it was shown that the stress on tunnel support increased after the increase of moisture content, and the settlement of surrounding rock continued to increase. They simultaneously provided corresponding rectification measures. Shao et al. [70] classified the geotechnical environment level and immersion level of loess tunnels, and provided a calculation and analysis method for the settlement deformation of loess foundation in tunnels. Considering the structural condition of loess, the Taisha formula was used to calculate the surrounding rock pressure of the tunnel, and the relationship between the surrounding rock pressure and the structural degree of loess was obtained. However, few scholars have provided corresponding reserved deformation values for the moisture content of tunnel surrounding rock under different loess soil types. Through an analysis of 57 monitoring sections in three soil types—clayey loess, silty loess, and sandy loess—the relationship between tunnel arch settlement and water content was examined. Deformation allowance recommendations are provided. In Figure 9, moisture content primarily falls within the range of 6% to 26%, with a notably high proportion concentrated between 11% and 23%. The influence of moisture content on loess tunnels in sandy loess sections is more significant than that in clayey loess and silty loess sections. With a 10 cm deformation allowance, the guarantee rate is 64.29%, while increasing it to 20 cm raises the guarantee rate to 94.64%. Taking soil type into account, crown settlement in clayey loess and silty loess sections typically remains below 100 mm due to moisture content influence, allowing a 10 cm deformation allowance. However, in sandy loess sections, extensive cumulative vault settlement occurs due to significant moisture content effects. Therefore, tunnels in sandy loess sections should have a deformation allowance of 20 cm.

3.4. Analysis of Influence of Cover Depth on Deformation Allowance

Analyzing the correlation between the loess tunnel’s cover depth and vault settlement allows for a reasonable prediction of the deformation allowance in loess areas, as demonstrated in Figure 10. At a cover depth of approximately 50 m, within a concentrated measurement area, a 16 cm deformation allowance yields an 84.6% guarantee rate. Using the Tongluochuan tunnel as an example, at cover depths of 28 m, 29 m, and 30 m, crown settlements measured were 92.97 mm, 212.97 mm, and 215.49 mm, respectively, indicating an unstable state in the section. When the cover depth of the Qindong tunnel is 28 m, the crown settlement measures 134.46 mm, while when the cover depth is 169 m, the crown settlement is 126.77 mm. It signifies the relationship between crown settlement and stratum properties, cover depth, support parameters, and construction methods. The connection between tunnel cover depth and crown settlement is not straightforward with a single parameter. Therefore, the deformation allowance for the tunnel can be determined through statistical measurements. Once the initial tunnel support is installed, the surrounding tunnel displacement stabilizes. For the control of deformation allowance, one can use the time of closed initial support as a reference. Due to data variations and the need to establish guarantee rates, deformation allowances for loess tunnels vary based on different cover depths.

4. Value Analysis of Deformation Allowance of Loess Tunnel

4.1. Three Methods for Calculating Correlation Coefficient

4.1.1. Pearson Correlation Coefficient

The Pearson correlation coefficient quantifies the linear relationship between two variables, X and Y, at a fixed distance, indicating their closeness. The value, typically denoted as “r” [71], falls within the range of −1 to 1, as calculated using Formula (2):

(2)$r_p={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(X_i-\overline{X}\right)\left(Y_i-\overline{Y}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{\left(Y_i-\overline{Y}\right)}^2}}}}$

With n as the sample size, X and Y represent values for two variables. When r > 0, it indicates a positive correlation; when r = 0, there is no correlation; when r < 0, it signifies a negative correlation. The higher the absolute value of r, the stronger the correlation.

4.1.2. Kendall Correlation Coefficient

The Kendall correlation coefficient, derived from ranking theory, finds widespread application across various fields [72]. X and Y, representing distinct signals, undergo cross-correlation, resulting in cross-correlation coefficients (x, y) within the [−1, 1] range. A correlation coefficient near 1 indicates a strong positive correlation, while near −1 signifies a strong negative correlation. A correlation coefficient of 0 indicates no correlation within the signal. The expression of correlation coefficient is shown in (3):

(3)$r_k={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(x_i-\frac{1}{n}{\displaystyle \sum _{j=1}^n{x}_j}\right)}\left(y_i-\frac{1}{n}{\displaystyle \sum _{j=1}^n{y}_j}\right)}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(x_i-\frac{1}{n}{\displaystyle \sum _{j=1}^n{x}_j}\right)}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{\left(y_i-\frac{1}{n}{\displaystyle \sum _{j=1}^n{y}_j}\right)}^2}}}}$

4.1.3. Spearman Correlation Coefficient

X and Y are two groups of data with independent and identical distribution. Each group of random variables, X and Y, contains N elements, represented as Xi and Yi for 1 < i < N, respectively. When X and Y are simultaneously sorted in either ascending or descending order, two sets of element sequences, denoted as x and y, are created. These sequences correspond to the rows of X_i in X and Yi in Y, respectively. The elements’ difference between sets x and y creates a row difference set, denoted as d, where $d_i=x_i-y_i$ for 1 < i < N. The Spearman correlation coefficient ($r_s$) between X and Y can be calculated using x, y, or d, as shown in Formula (4):

(4)$r_s={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\left(x_i-\overline{x}\right)}\left(y_i-\overline{y}\right)}{\sqrt{{\displaystyle \sum _{i=1}^N{\left(x_i-\overline{x}\right)}^2{\displaystyle \sum _{i=1}^N{\left(y_i-\overline{y}\right)}^2}}}}}$

In practical applications, the relationship between variables can be assessed as insignificant and calculated from the row difference set d:

(5)$r_s=1-{\displaystyle \frac{6{\displaystyle \sum _{i=1}^N{d}_i^2}}{N\left(N^2-1\right)}}$

The value range of Spearman correlation coefficient is [−1,1]. A higher absolute value of $r_s$ indicates a stronger correlation. A positive correlation is inferred when $r_s$ > 0, while a negative correlation is implied when $r_s$ < 0 in the context of the two discussed variable groups [73].

4.2. Correlation Analysis of Influencing Factors of Arch Settlement

Using Python (v.3.13), we applied the three correlation coefficient methods to classify and calculate data from 148 monitoring sections. Since the crown settlement (CS) is greater than that of the surrounding convergence, it is based on the amount of crown settlement. The settlement of the vault is assessed and verified using three correlation coefficient calculation methods based on factors like soil type (ST), surrounding rock grade (SRG), excavation span (ES), excavation method (EM), moisture content (MC), and cover depth (CD), each evaluating the impact of these factors. Finally, based on the strength of correlation, Figure 11 recommends the deformation allowance range.

Quantitative analysis of parameter correlations in different sections of the loess tunnel is conducted through Pearson, Kendall, and Spearman correlation coefficient heatmaps. This analysis effectively reveals the complex relationships between tunnel deformation under multi-parameter conditions and the influencing factors, thereby helping to identify the key factors that control tunnel deformation. Among them, the arch crown subsidence exhibits significant deformation and is thus designated as a critical indicator. The correlation matrix between arch crown subsidence and factors influencing tunnel deformation is calculated using Pearson, Kendall, and Spearman methods, creating a heatmap that visually depicts their relationship. Figure 12a–c depicts thermal coefficient matrices for Pearson, Kendall, and Spearman correlations between crown settlement and tunnel deformation factors. The darker the color of the thermal diagram, the stronger the correlation. As shown in Figure 12d, the influencing factors on crown settlement, ranked from strong to weak, are ST, ES, EM, SRG, MC, and CD.

The thermal diagram and Pearson, Kendall, and Spearman correlation coefficient matrices provide statistical insights into crown settlement and its various influencing factors. This approach helps identify key factors controlling tunnel deformation and determine a reasonable deformation allowance range. Figure 13 displays the calculation results, and Figure 14 confirms the consistency of the three calculation methods. ST has the greatest impact and the strongest correlation with vault settlement. However, after the ES, MC and CD have little influence and weak correlation. Therefore, in designing the deformation allowance for the loess tunnel, it is essential to account for ST and ES influences and establish appropriate allowances based on these factors.

4.3. Suggestions on the Value of Deformation Allowance of Loess Tunnel

Designing the deformation allowance is a crucial step in tunnel design. Given the unique characteristics of loess, establishing a reasonable deformation allowance range is essential for loess tunnel design and construction. Table 2 in the specifications provides the deformation allowance. Statistical analysis of various loess tunnels reveals that the arch crown settlement is greater than surrounding convergence. Using correlation calculations and considering different factors like soil type, excavation span, surrounding rock levels, moisture content, and cover depth, all three analysis methods consistently identify soil type and excavation span as influential factors with strong correlations. In clay loess and silty loess sections, where moisture content ranges from 14.3% to 22.59%, arch crown settlement typically remains below 100 mm. Hence, a deformation allowance of 10 cm is suitable. In sandy loess sections, moisture content typically falls between 10.3% and 26.5%, with arch crown settlement accumulating within the range of 75 mm to 200 mm. Hence, a deformation allowance of up to 20 cm can be chosen. Considering the cover depth, a deformation allowance of 12 cm is appropriate for depths ranging from 0 to 50 m. For cover depths greater than 50 m, a deformation allowance of 16 cm provides an 84.6% guarantee rate. Based on soil type, surrounding rock levels, cover depth, and moisture content, a suggested range of deformation allowances is provided. Refer to Table 3 for the recommended values:

Based on the statistical and analytical results presented above, and taking into account the complex and variable geological conditions in loess areas—where tunnels are typically located in both collapsible and non-collapsible loess strata, as well as in areas with heterogeneous fill and newly accumulated loess—the deformation allowance derived from actual monitoring and statistical analysis significantly exceeds the specified value. Based on the summary and analysis, and taking into account factors like cover depth, moisture content, and surrounding rock grade, and referring to the specifications, the recommended deformation allowance values for loess tunnels are provided in Table 4. The appropriate deformation allowance should be determined by considering factors like initial support type and parameters, lining construction timing, and lining parameters in a comprehensive manner. Both safety and cost-effectiveness must be taken into account. Thus, for diverse loess tunnels in different regions, a comprehensive evaluation of various factors is necessary, and the proposed values should be further verified and analyzed through field measurements for validity.

5. Example Analysis of Deformation Allowance of Loess Tunnel

5.1. Project Overview

To simplify the analysis, consider a two-way, four-lane, separated loess tunnel as an example. Study the temporal distribution patterns of crown settlement and clearance convergence in the tunnel. Utilize measured data to establish a statistical analysis and prediction model for construction deformation, leading to a determination of a reasonable deformation allowance. The tunnel section excavation span B = 12.82 m and height H = 10.32 m, representing a typical large-section tunnel with collapsible loess grade V surrounding rock. The tunnel site is situated in a loess hilly ridge landform, with abundant gullies and deep valleys. The tunnel’s cover depth (h) varies between 6.0 m and 78.0 m. The upper section consists of 20.0 m to 25.0 m of Upper Pleistocene Q3al + pl new loess, which includes a mixture of grade IV self-weight collapsible loess and loess-like silty loess. The lower section is composed of Middle Pleistocene Q2al + pl silty soil, characterized by consistent soil quality (Figure 15). The tunnel entrance and exit are situated in a steep, high-slope section of the loess beam gully. Of these, the left tunnel entrance and exit have a cover depth of 6.0 m each, while the right tunnel entrance has a cover depth of 1.5 m, and the exit is 5.0 m deep. The CRD method is used for tunnel excavation. At the tunnel entrance and exit, the surface layer of the mountain consists of a thin, loose silt layer with a moisture content ranging from 8.6% to 20.2%. The tunnel site’s groundwater primarily comprises loose rock pore water, with a cover depth exceeding 40.0 m. It is primarily replenished by atmospheric precipitation and surface water infiltration. Settlement monitoring points (point C) are placed at the tunnel crown, and precise leveling is employed for measurements to understand the peripheral displacement variation during tunnel construction. The surrounding convergence measurement points are arranged at the position with the largest span of each section, and a horizontal convergence measurement baseline (AB horizontal measurement line) is buried using digital convergence measurement (Qiu et al. 2021).

5.2. Analysis of Assurance Rate of Deformation Allowance for Crown Settlement

The statistical range of the crown settlement V_max is (−17.1−201.1) mm. Tunnel construction deformations exhibit inherent randomness due to the influence of numerous factors. Assuming the data follow a normal distribution, a 95% confidence interval can be derived from the sample for statistical analysis. This interval allows for the determination of the upper and lower confidence bounds for deformation at a specified confidence level. Based on the statistical analysis of the data in Figure 16a, the 95% confidence interval of the crown settlement V_max is [−51.53, −65.11]. Sections where crown settlement remains below 90.0 mm are primarily found in the tunnel with the cover depth h ≤ 3(H + B). For tunnel cover depth h > 3(H + B), the crown settlement exhibits an upward trend, as depicted in Figure 16b. This is primarily due to the increase in tunnel cover depth, which results in higher soil moisture content (15.0–23.0%), reduced soil stability, and significant overlying load, leading to substantial deformation. With a deformation allowance of 10 cm for crown settlement, the assurance rate is 85.8%. This rate increases to 94.9% when the allowance is increased to 15 cm. Based on the analysis above, a deformation allowance can be considered reasonable if it results in a guarantee rate greater than 80%.

5.3. Analysis on Assurance Rate of Deformation Allowance of Surrounding Convergence

The statistical range of surrounding convergence deformation U_max is (−12.1−122.0) mm. Based on the statistical analysis of measured data, the 95% confidence interval of U_max is [−35.08, −43.39]. For tunnel cover depths h ≤ 3(H + B), surrounding convergence deformation remains below 75.0 mm, while sections exceeding this threshold are primarily found in tunnels with cover depths h > 3(H + B), as illustrated in Figure 17b. In the aforementioned analysis, the data from field measurements exhibit relatively high variability due to the intricate interplay between loess and tunnel support systems, as well as various uncertain factors impacting the initial support deformation. Hence, the deformation allowance should vary based on the vault settlement. Referring to Figure 17, the guarantee rate for crown settlement is 94.9% and 98.9% when deformation allowances are set to 15 cm and 20 cm, respectively. When the deformation allowance is set at 15 cm, the assurance rate has reached 100% in contrast. Hence, the figure analysis compares crown settlement with surrounding convergence deformation (Figure 16 and Figure 17). In the same test section, crown settlement deformation surpasses surrounding convergence deformation, emphasizing the need for deformation allowance in the tunnel to prioritize crown settlement impact. The suggested deformation allowance range for grade V surrounding rock in a loess tunnel is (−10 to −15) cm. Based on the preceding analysis, it is recommended to set the deformation allowance for tunnel vaults in grade V surrounding rock within the silty loess section to be less than 15 cm. Therefore, the validity and precision of the previous analysis are confirmed.

6. Discussion

This study used statistical analysis methods to investigate and collect settlement data of 148 loess tunnel sections. Three correlation coefficients were used to analyze the deformation patterns of loess tunnels under different factors, and recommended values for reserved deformation of loess tunnels with different soil types were provided. At the same time, the rationality of the recommended values was verified through engineering practice. The research results aim to improve the relevant standards for reserved deformation of loess tunnels. However, the recommended values for reserved deformation given in this article are only applicable to loess tunnels. Proposing reserved deformation amounts applicable to other surrounding rock tunnels is the focus of our future work.

7. Conclusions

This paper’s research and analysis help the construction team establish the allowable deformation range for loess tunnel design. The main conclusions are as follows:

• (1). Through the three correlation analysis methods of Pearson, Kendall, and Spearman, it is revealed that soil type strongly affects arch crown settlement, followed by excavation span, while moisture content and cover depth have a weaker impact. There-fore, deformation allowance design should prioritize soil type and excavation span.

• (2). In grade IV surrounding rock sections of loess tunnels, for clayey loess and silty loess, vault deformation allowance can be limited to 10 cm, and tunnel side wall deformation allowance to 4 cm. For sandy loess arches, consider a 15–20 cm crown deformation allowance and an 8–12 cm allowance at the tunnel side wall measurement.

• (3). In grade V loess tunnel sections with clayey loess, consider a 15–25 cm deformation allowance at the crown and 10–15 cm at both sides of the tunnel wall. In silty loess sections, limit deformation to 15 cm at the tunnel crown and 10 cm on both tunnel side walls. For sandy loess sections, allow for 35–47.5 cm deformation at the tunnel crown and 15–20 cm on both tunnel side walls.

• (4). Moisture content ranges from 6% to 26%, with crown settlement falling between 11% and 23%. Notably, moisture content has a more pronounced impact on sandy loess tunnels compared to cohesive and silty loess sections. Therefore, the deformation allowance for tunnels in sandy loess sections should exceed that in clayey loess and silty loess sections.

• (5). Finally, in line with the specifications, a recommended deformation allowance value is suggested. The deformation allowance suggested by the real monitoring data and statistical analysis greatly exceeds the standard value. Therefore, considering the specifications, it is advisable to raise the deformation allowance in the loess tunnel section.

Author Contributions

X.Q.: conceptualization, formal analysis, writing—original draft, software. J.Q.: funding acquisition, methodology, visualization. J.L.: resources, supervision, project administration. Y.L.: data curation, investigation, validation. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

We also acknowledge the editor for the valuable suggestions.

Conflicts of Interest

Author Yuhang Liu was employed by the company China Railway First Bureau Group Urban Investment Operation Management Co. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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.

Figures and Tables

Figure 1. The support principle based on the GRC and SRC [24,25,26].

Figure 2. Unreasonable design of deformation allowance.

Figure 3. Deformation composition of tunnel surrounding rock.

Figure 4. Distribution of loess soil types in China.

Figure 5. Analysis of deformation allowance caused by crown settlement of grade IV surrounding rock.

Figure 6. Analysis of deformation allowance caused by surrounding convergence of grade IV surrounding rock.

Figure 7. Analysis of deformation allowance caused by crown settlement of grade V surrounding rock.

Figure 8. Analysis of deformation allowance caused by surrounding convergence of grade V surrounding rock.

Figure 9. Analysis of deformation allowance under the influence of moisture content.

Figure 10. Analysis of deformation allowance under the influence of cover depth.

Figure 11. Correlation calculation process of different influencing factors.

Figure 12. Thermodynamic diagram of three correlation coefficients.

Figure 13. Correlation analysis of influence of different factors on crown settlement.

Figure 14. Relevant statistics of calculation results of three correlation coefficients.

Figure 15. Geological longitudinal section of the entrance and exit section of loess tunnel [74].

Figure 16. Analysis of assurance rate of deformation allowance for crown settlement of different test sections.

Figure 17. Analysis of assurance rate of deformation allowance for peripheral.

Table 1

Loess tunnel monitoring samples.

Geographical Position	Tunnel Name	Soil Type	Excavation Span/m	Surrounding Rock Grade	Moisture Content/%	Excavation Method	Reference
Shaanxi Baoji	Yangshan tunnel	Clayey loess	11.2	IV	15–17.9	Step method	[35]
Gansu Qingyang	Yulinzi tunnel	Silty loess	/	IV	20.2	Step method	[36]
HenanSanmenxia	HeJiazhuang tunnel	Clayey loess	15.2	IV	15.4	Step method	[37]
HenanSanmenxia	Hanguguan tunnel	Sandy loess	15.5	IV	13.2	CRD method	[38]
HenanSanmenxia	Pandong tunnel	Sandy loess	/	IV	8.7–14.7	CRD method + Step method	[39]
ShaanxiWeinan	Tongluochua-n tunnel	Sandy loess	15.2	IV	/	CRD method + CD method	[40]
Gansu Lanzhou	Wangjiagou tunnel	Silty loess	15.2	IV	23	Step method	[41]
Gansu Lanzhou	Guchengling tunnel	Silty loess, Sandy loess	/	IV, V	/	Step method	[42]
Gansu Dingxi	Sujiachuan tunnel	Sandy loess	14.72	IV	10.3	Step method	[43]
Ningxia Guyuan	Pangwan tunnel	Silty loess	13.36	V	25.8	Step method	[44]
Shanxi Taiyuan	Shanxi Yangqu No.1 tunnel	Silty loess	17.23	V	22.59	CRD method	[45]
Shanxi Xinzhou	Shata tunnel	Clayey loess	10.86	V	20	Step method	[46]
Shanxi Shuozhou	Yuanyanghui tunnel	Clayey loess	10.75	V	/	Reserved core soil excavation method	[47]
Shaanxi Yulin	Dunliang tunnel	Sandy loess	16.26	V	9	Step method	[48]
ShaanxiXi’an	Tangjiayuan tunnel	Clayey loess	17.25	V	/	CD method	[49]
Qinghai Xining	Dayushan tunnel	Sandy loess	10.75	V	6.8–13.4	CRD method	[50]
ShaanxiWeinan	Gaoqiao tunnel	Sandy loess	15.2	IV, V	10.3	Step method	[51]
HenanSanmenxia	Zhangmao tunnel	Clayey loess	15.2	V	23	Step method	[52]
Shaanxi Yan’an	Makeyi tunnel	Clayey loess	/	V	/	Step method	[53]
Gansu Lanzhou	Bingcaoling tunnel	Silty loess	10.86	V	/	Step method	[54]
Gansu Qingyang	Shiyangling tunnel	Silty loess	/	V	26.5	Step method	[55]
Gansu Dingxi	Beiershilipu tunnel	Sandy loess	/	V	12–22	Step method	[56]
Inner Mongolia Hohhot	Yaogou tunnel	Sandy loess	11.4	V	/	Step method	[57]
Gansu Tianshui	Wharf tunnel	Sandy loess	10.5	V	23	Step method	[58]
Gansu Dingxi	Xipo tunnel	Sandy loess	14.72	V	/	Step method	[59]
ShaanxiWeinan	Qindong tunnel	Sandy loess	15.2	IV, V	6–25	Step method+ Double side heading method + CRD method	[38]
Shaanxi Xianyang	Lingshang tunnel	Clayey loess	15.2	V	13.3–21.5	Step method	[60]
Shaanxi Xianyang	Ganbao tunnel	Clayey loess	/	V	13.3–21.5	Step method	[61]
Shaanxi Xianyang	Yongping tunnel	Clayey loess	/	V	13.3–21.5	Step method	[61]

Table 2

Specification deformation allowance (JTGD).

Lane type (Excavation Span)Surrounding Rock Grade	Two Lane (<14 m)	Three Lane (>14 m)
Grade IV surrounding rock	5–8 cm	8–12 cm
Grade V surrounding rock	8–12 cm	12–15 cm

Table 3

Suggestions on the value of deformation allowance.

	Grade IV Surrounding Rock		Grade V Surrounding Rock
	Vault/cm	Side Wall/cm	Vault/cm	Side Wall/cm
Clayey loess	<10	<4	15–25	10–15
Silty loess	<10	<4	<15	<10
Sandy loess	15–20	8–12	35–47.5	15–20

Table 4

Recommended values of deformation allowance in reference to specifications.

	Clayey Loess		Silty Loess		Sandy Loess
	Lane Type (Excavation Span)		Lane Type (Excavation Span)		Lane Type (Excavation Span)
	Two Lane (<14 m)	Three Lane (>14 m)	Two Lane (<14 m)	Three Lane (>14 m)	Two Lane (<14 m)	Three Lane (>14 m)
Grade IV surrounding rock	8–12 cm	12–15 cm	8–12 cm	12–15 cm	12–15 cm	15–20 cm
Grade V surrounding rock	<15 cm	15–25 cm	<12 cm	12–15 cm	15–35 cm	35–47.5 cm

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