Stability Analysis Method for Initial Support Structure of Tunnel in Swelling Loess

Abstract

The stability of tunnel is greatly affected by the changes in volume and strength of swelling loess caused by humidification. Quantitative reflection of the influence of humidification on stability in mechanical model has important engineering significance for the design of support structures. Based on the fact that parameters related to swelling and strength are functions of water content, the water content was introduced into the classical surrounding rock–support model, and a safety factor was employed as the index of stability. Thus, a stability analysis framework of swelling loess tunnel was established by considering the swelling and softening of swelling loess. In an engineering case, the theoretical models presented in this paper were introduced to investigate the change rules of surrounding rock deformation and tunnel safety factor, as well as the sensitivity of the safety factor of tunneling in swelling loess. Analysis results showed that the change rules of surrounding rock response curves and safety factors obtained by the models were consistent with the actual situation, and the predicted maximum radial displacement of free face was close to the measured value. These indicate that the models are applicable to the stability analysis of initial support of swelling loess tunnel.

1. Introduction

Swelling loess exhibits dual characteristics of loess and swelling soil. In humidification process, swelling loess is easy to cause large deformation, collapse, and initial support failure of tunnels, which seriously affects engineering stability [1]. For example, the JinggyouTunnel in Shanxi Province, China, suffered several collapse accidents during construction due to the swelling of loess.When the most severe collapse occurred, nearly 70 m of initial support were completely crushed, causing huge economic losses and delaying the construction period for more than one year [2,3]. The danger of swelling loess to tunnel engineering is increasingly highlighted with China’s railways and highways extending further to the west, which is covered by large areas of loess layer [4,5].

The deformation of swelling loess tunnel is characterized by high speed, large destructiveness, long duration, and difficulty in treatment. At present, the tunnel construction technology is relatively mature, but the design and construction technology of swelling loess tunnels still need to be improved. The hazards of swelling loess to tunnel construction are specifically manifested as loess expansion and collapse, tunnel subsidence, lining deformation and damage, and floor heave. The influence of temperature, time, permeability, water pressure, and other factors on loess swelling needs to be further studied.

Based on laboratory tests and the applications field of engineering practice, scholars have carried out a lot of research on swelling loess [6]. Yu explored the large deformation control technology of swelling loess [7]. Feng studied the large deformation characteristics of swelling loess tunnels through on-site monitoring and numerical simulation [8]. Zeng and Zhang [9,10] adopted the finite element method to simulate the swelling loess tunnel under the condition of rainfall infiltration. Li et al. [11] discussed the mechanical characteristics of the combined support of swelling loess tunnel based on the surrounding rock–support relationship.

Swelling loess shows typical swelling and softening characteristics during humidification. Its volume increases with the increase of water content, and its strength decreases with the increase of water content. When the stress state of initial support of swelling loess tunnel is analyzed, little consideration is given to the joint influence of swelling and softening. In addition, the joint influence of swelling and softening on the stability of support structure should be further discussed from the perspective of engineering safety.

In view of the universality and generality of the theory of surrounding rock–support relationship in the stability analysis of tunnel engineering [12,13], the stability analysis process of tunnel initial support during the humidification process is established by taking a safety factor as the stability index, and by considering the swelling and softening of loess after water absorption in this paper. The deformation law of swelling loess surrounding rock, the variation law of tunnel safety factor, and the sensitivity of safety factor during humidification are analyzed by combining with an engineering example and using this stability analysis process.

2. Response Curves of Surrounding Rocks

2.1. Introduction to Elastic-Plastic Solution of Circular Axisymmetric Tunnel

As shown in Figure 1, a circular tunnel with a radius of R₀ is excavated in acontinuous, homogeneous, and isotropic rockmass, which bears ground stress of P₀. The free face bears a uniform support pressure of P_S. Tunnel excavation results in the redistribution of radial stress ${\sigma }_r$ and circumferential stress ${\sigma }_{\mathsf{\theta }}$, i.e., the ${\sigma }_{\mathsf{\theta }}$ near the free face increases significantly while the ${\sigma }_r$ decreases significantly, which causes the surrounding rock within the R_P range around the tunnel to enter the plastic deformation stage. The solution of stress distribution and displacement distribution of the above tunnel based on elastoplastic theory is the classical circular axisymmetric tunnel model [14,15,16].

The elastic-plastic analytical solutions of the above model can be obtained by combining the equilibrium differential equation, the geometric equation, the constitutive equation, the yield criterion, and the boundary conditions.

2.2. Response Curve of Surrounding Rock during Humidification

For a acircular axisymmetric tunnel, the swelling effect of surrounding rock can be equivalent to the additional ground stress, that is to say, the swelling force $P_{sw}$ can be introduced into the elastic-plastic analytical solution of circular axisymmetric tunnel model for the simulation analysis of swelling loss tunnel [11,12,17]. The mathematical relationship between the radial displacement and the support force at the free face of swelling loess tunnels can be obtained as follows:

$$u_{R0}=\frac{1+\nu }{E}(P_0+P_{sw}-P_s)R_0(P_s^{\mathrm{ep}}\le P_s\le P_0)$$

(1)

$$\begin{gathered} u_{R0}={\left[\frac{\left(P_0+P_{sw}+c\cot \varphi \right)\left(1-\sin \varphi \right)}{P_s+c\cot \varphi }\right]}^{\frac{1-\sin \varphi }{\sin \varphi }}\frac{(1+\nu )R_0}{E}\sin \varphi \left(P_0+P_{sw}+c\cot \varphi \right) \\ (0\le P_s<P_s^{\mathrm{ep}}) \end{gathered}$$

(2)

where, $u_{R0}$ is the radial displacement at the free face; $c$ is the cohesion; $\varphi$ is the internal friction angle; $E$ is the elasticity modulus; $\nu$ is the Poisson’s ratio; $P_s^{\mathrm{ep}}$ is the radial stress at the elastic-plastic junction of surrounding rock, which can be obtained by Equation (3) [11].

$$P_s^{\mathrm{ep}}=(P_0+P_{sw})(1-\sin \varphi )-c\cos \varphi$$

(3)

Previous studies [1,10] have shown that there is a significant linear relationship between the water content and swelling force of loess during the process from natural humidification to saturation, that is

$$P_{sw}=P_{\max }\frac{w-w_0}{w_1-w_0}$$

(4)

where, $w$ is the water content of loess corresponding to a swelling force of $P_{sw}$; $w_1$ and $w_0$ are initial water content and saturated water content, respectively; $P_{\max }$ is the swelling force corresponding to the final water content $w_1$.

In fact, during the humidification process, the swelling loess not only shows swelling characteristics, but also has typical softening characteristics, i.e., cohesion and internal friction angle gradually decrease with the increase of water content [10]. This decay relation can be found in Equations (5) and (6), whose parameters $a_1$, $a_2$, $a_3$, and $a_4$ can be fitted by test data.

$$\lg c=a_{1}w+a_2$$

(5)

$$\varphi =a_{3}w+a_4$$

(6)

The response curve of surrounding rocks of swelling loess tunnel during humidification can be obtained by substituting Equations (4)–(6) into Equations (1)–(3). This curve is a function of surrounding rock displacement with water content and support parameters, and can be used for describing the influence of humidification process on surrounding displacement.

3. Characteristic Equation of Support Structure

3.1. Characteristic Equation of Support Unit

With the development of the New Austrian Tunnelling Method (NATM), Rabcewicz et al. proposed the design method of anchor-shotcrete reinforcement for circular tunnels based on the combination arch principle. According to the shear-fracture principle of tunnel section, the mechanical expression of supporting units can be obtained [18,19,20]:

① Maximum support force of shotcrete:

$$P_{\mathrm{shot}}=\frac{{\tau }_{c}t{h}_c}{[R_0\cos (\pi /4-\varphi /2)\sin {\phi }_c]}$$

(7)

where, ${\tau }_c$ is the shear strength of concrete, generally 20~43% of the compressive strength; $t{h}_c$ is the thickness of the spray layer; ${\phi }_c$ is the shear failure angle of shotcrete, usually 30 degrees; $\varphi$ is the internal friction angle of rock and soil.

② Maximum support force of bolt:

$$P_{\mathrm{bolt}}=T_{\mathrm{bf}}(\cos {\phi }_r-\cos {\mathsf{\theta }}_0)/(S_c{S}_1\cos {\phi }_{\mathrm{r}})$$

(8)

$${\phi }_{\mathrm{r}}=\pi /4-\varphi /2$$

(9)

$${\mathsf{\theta }}_0={\phi }_r+\ln [(R_0+t{h}_r)/R_0]/\tan {\phi }_r$$

(10)

$$t{h}_r=(R_0+l)[\sin (\frac{S_c}{2R_0})\tan (\frac{\pi }{4}+\frac{S_c}{2R_0})+\cos (\frac{S_c}{2R_0})-\frac{\sin (\frac{S_c}{2R_0})}{\cos (\frac{\pi }{4}+\frac{S_c}{2R_0})-\frac{S_c}{2R_0}}]$$

(11)

where, $l$ is the length of anchor rods; $S_c$ is the average circumferential spacing of anchor rods; $S_1$ is the average distance along the tunnel axis; $T_{\mathrm{bf}}$ is the ultimate failure load in the pull-out test of anchors; ${\mathsf{\theta }}_0$ is the maximum inclination angle of rock slip line; ${\phi }_r$ is the shear failure angle of rock; $t{h}_r$ is the thickness of the rock-bearing arch.

3.2. Characteristic Equation of Composite Support Structure

Composite support refers to the combined structure as a whole for tunnel support. In order to approximate analysis, the calculation of the ultimate support force of composite support structure does not consider the deformation coordination problem among support units. The value of ultimate support is directly obtained by adding the maximum support forces of various support units. Thus, the ultimate support force of the composite support is [18]:

$$P_{\mathrm{com},\lim }=P_{\mathrm{shot}}+P_{\mathrm{bolt}}$$

(12)

According to the overall stability criterion of parallel structure, the ultimate deformation of composite support structure is determined by the smallest allowable deformation support unit in each support unit. It is pointed out in the reference [18] that: the stiffness of shotcrete is the largest in each unit, soits allowable deformation is the smallest. Therefore, the ultimate deformation of a composite support structure is the deformation of shotcrete:

$$u_{\mathrm{com},\lim }=u_{\mathrm{shot},\lim }$$

(13)

where, the ultimate deformation $u_{\mathrm{shot},\lim }$ of shotcrete can be calculated from the stiffness $k_{\mathrm{shot}}$ and the maximum support force $P_{\mathrm{shot}}$ of shotcrete as follows:

$$u_{\mathrm{shot},\lim }=P_{\mathrm{shot}}/k_{\mathrm{shot}}$$

(14)

where, $k_{\mathrm{shot}}$ can be calculated by [11]:

$$k_{\mathrm{shot}}=\frac{E_c}{1+{\nu }_c}\frac{R_0^2-{(R_0^2-t{h}_c)}^2}{R_0^2[(1-2{\nu }_c)R_0^2+{(R_0^2-t{h}_c)}^2]}$$

(15)

where, $E_c$ is the elastic modulus of shotcrete;  ${\nu }_c$ is the Poisson ratio.

The precondition for the stability of initial support of tunnels is that each support unitis not destroyed. That is to say the composite support structure will be stable if its limit deformation does not exceed the limit deformation of concrete that $u_{\mathrm{shot},\lim }$ during the secondary stress release of surrounding rock. Therefore, the relationship between support resistance $P_{\mathrm{com},\lim }$ and its deformation can be obtained:

$$P_{\mathrm{com},\lim }=D{u}_{\mathrm{com},\lim }$$

(16)

where, $D$ is the integral stiffness of the composite support structure, i.e., the slope (support force/displacement) of the support characteristic curve in the relationship diagram between the surrounding rock and the support. $D$ can be determined by substituting Equations (12)–(14) into Equation (16).

3.3. Displacement Influenced by Distance to the Tunnel Face

The relationship between displacement influenced by the distance to the tunnel face $u_{\mathrm{sd}}$ and end-displacement $u_{\mathrm{maxR}0}$ is often illustrated by the experimental equations. As proposed by reference [15] this relationship can be expressed as:

$$u_{\mathrm{sd}}=u_{\max R0}{[1+\exp (\frac{-L}{1.1R_0})]}^{-1.7}$$

(17)

where, $L$ is the distance from the considered point (usually starting point support) to the tunnel face.

4. Stability Analysis

4.1. The Theory for Surrounding Rock–Support Relationship

Figure 2 is a schematic diagram of the relationship between surrounding rock and support. The foundation and basic idea of the theory of surrounding rock–support is that: based on the elastoplastic theory, the displacement calculation formula of circular axisymmetric tunnels can be established by considering the constitutive model of surrounding rock; furthermore, the response curve of surrounding rock can be obtained; then, the coordinate of the intersection point between the response curve of surrounding rock and the characteristic curve of support structure can be figured out. This intersection coordinate is regarded as a condition that needs to be met when the surrounding rock and support structure reach equilibrium [13,14].

In this paper, the surrounding rock response curve, support characteristic curve, and radial deformation curve during the humidification process have been discussed in Section 2.2, Section 3.2, and Section 3.3, respectively.

4.2. Structural Safety Factor

According to the general calculation principle of safety factor, the safety factor of initial support in swelling loess tunnel is the ratio of structural resistance to load, i.e.,

$$F_s=\frac{P_{\mathrm{com},\lim }}{P_{\mathrm{eq}}}$$

(18)

where, $F_s$ is the safety factor; $P_{\mathrm{eq}}$ is the load on the support structure when the secondary radial stress of surrounding rock at the free face equals the resistance of the support structure, and it is also the support force at the intersection between the response curve of surrounding rock and the support characteristic curve.

4.3. Stability Analysis Framework

Based on the above analysis, the stability analysis framework is described as follows:

① According to Section 2.2, considering the swelling and softening of loess, the response curves of surrounding rocks under different humidity conditions are calculated and plotted.

② According to the formulas described in Section 3.1 and Section 3.2, the limit support force $P_{\mathrm{com},\lim }$ of the composite support structure is determined, and the key parameter $D$ of the support characteristic curve is determined. Then, the radial displacement $u_{\mathrm{sd}}$ of the free face at the starting position of tunnel support is determined by combining the formulas described in Section 3.3. Combining parameters $P_{\mathrm{com},\lim }$, $D$, and $u_{\mathrm{sd}}$, the support characteristic curve is drawn.

③ The coordinates at the intersection point of the two curves are determined by combining the response curve of surrounding rock with the characteristic curve of support structure, i.e., the displacement $u_{\mathrm{eq}}$ and the support force $P_{\mathrm{eq}}$ of the free face at equilibrium.

④The safety factor $F_s$ is calculated by substituting $P_{\mathrm{com},\lim }$ and $P_{\mathrm{eq}}$ into Equation (18).

⑤ If the safety factor $F_s$ is greater than 1, the tunnel is in a stable state, and the support design is reasonable. If the safety factor is less than 1, the tunnel is in an unstable state, and the support design is unreasonable, so the support parameters need to be adjusted.

5. Engineering Case Analysis

5.1. Background

A tunnel passes through swelling loess, which is 1.8 km long and has a maximum depth of 82 m. The landform of the tunnel is a loess beam and hill landform, with gullies developed in a “V” shape. Sandy loess is distributed on the surface, which is yellow-brown and slightly dense to moderately dense. There is swelling loess of upper tertiary (N) 20–60 m below the surface, which is red-brown, hard plastic, and compact. The geological section of the tunnel is shown in Figure 3. The swelling potential of loess is medium. The parameters of loess surrounding rocks are shown in

Table 1. Elastoplastic mechanical parameters of rock in swelling loess tunnel.

Parameter	Elastic Modulus	Poisson Ratio	Initial Water Content	Cohesion	Internal Friction Angle	Crustal Stress
Symbol	Ee/GPa	$\nu$	%	c/MPa	$\varphi$/°	P0/MPa
Datum value	0.6	0.35	15.28	0.61	26.34	2

, and the change rules of swelling force and strength parameters with water content during humidification, respectively, are shown in Figure 4 [10].

In this tunnel project, the support structure adopted a combined support method of shotcrete + steel arch + bolt. The mechanical parameters of each support unit are shown in

Table 2. Mechanical parameters of support units.

Parameter	Bolt Ultimate Load	Bolt Length	Average Circumferential Spacing of Bolt	Axial Average Spacing of Bolt	Equivalent Elastic Modulus of Concrete	Equivalent Shear Strength of Concrete	Poisson Ratio of Concrete	Shear Failure Angle of Concrete	Spray Layer Thickness
Symbol	$T_{\mathrm{bf}}$/MN	$l$/m	$S_c$/m	$S_1$/m	$E_c$/GPa	${\tau }_c$/MPa	${\nu }_c$	${\phi }_c$/°	$t{h}_c$/cm
Datum value	0.196	5	2	2	20	8	0.25	30	20
Research interval	0.1~0.3	1~7	0.5~3	0.5~3	10~40	4~16	0.25	30	10~40

. In view of the joint support system of shotcrete and steel arch, the support function of steel arch can be simulated by increasing the elastic modulus and compressive strength of concrete. Therefore, the equivalent elastic modulus and shear strength of concrete obtained by the method from reference [11] are shown in Table 2.

5.2. Analysis of Surrounding Rock–Support Relationship

According to the mechanical parameters of surrounding rocks, the response curve of loess surrounding rocks is drawn in the blue line, as shown in Figure 5. The ratio of support force to ground stress decreases gradually with the increase of radial displacement of free face. In the range of larger support force, the surrounding rock is in the elastic stage, and $u_{R0}$ increases linearly with the decrease of $P_s/P_0$. When the support force is less than or equal to 1.5602 MP, the surrounding rock enters the plastic deformation stage, and $u_{R0}$ increases rapidly with the decrease of $P_s/P_0$. When the support force is 0, the radial deformation of surrounding rock free face is as high as 4.456 cm.

The on-site monitoring data showed that the maximum radial displacement of the free face of the loess tunnel is 4.3 cm, which is slightly smaller than the theoretical prediction value from the surrounding rock response surface, i.e., 4.456 cm. On the other hand, as shown in the original design documents, the axial length without support during tunneling is 1.5 m. Therefore, Substituting $u_{\max R0}$ = 4.3 cm and $L$ = 1.5 m into Formula (17), the support starting position $u_{\mathrm{sd}}$ = 1.325 cm is calculated out. Then, based on the formulas described in Section 3.1 and Section 3.2, the characteristic parameters of the combined support structure are determined and listed in

Table 3. Characteristic parameters of the combined support structure.

${\mathit{P}}_{\mathbf{shot}}$/MPa	${\mathit{P}}_{\mathbf{bolt}}$/MPa	${\mathit{P}}_{\mathbf{com},\mathbf{lim}}$/MPa	${\mathit{u}}_{\mathbf{shot},\mathbf{lim}}$/mm	${\mathit{u}}_{\mathbf{com},\mathbf{lim}}$/mm	$\mathit{D}$  $/\mathbf{MPa}\cdot {\mathbf{m}}^{-1}$	${\mathit{k}}_{\mathbf{shot}}$  $/\mathbf{MPa}\cdot {\mathbf{m}}^{-1}$
1.1952	0.0703	1.2655	8.5	8.5	149.0016	140.7195

.

Furthermore, the support characteristic curve is drawn as the black line in Figure 5 according to the characteristic parameters $P_{\mathrm{com},\lim }$, $D$, and $u_{\mathrm{sd}}$. It can be seen from the graph that $P_s/P_0$ increases linearly with the increase of $u_{\mathrm{sd}}$. At $u_{R0}$ = 1.95 cm and $P_s/P_0$ = 0.2496, the support characteristic curve intersects with the response curve of surrounding rock, and the surrounding rock and support structure reach balance. At this time, the calculated support force $P_{\mathrm{eq}}$ is 0.4992 MPa. The calculated safety factor $F_s$ is 2.478 by substituting $P_{\mathrm{eq}}$ and $P_{\mathrm{com},\lim }$ into Formula (18), which indicates that the tunnel surrounding rock is in a stable state under this surrounding rock condition and supporting condition.

5.3. Tunnel Stability Analysis under Humidifying Conditions

As shown in Figure 6, based on the variation rule of swelling force and strength parameters with water content during humidification, the response curves of surrounding rocks under different water content conditions and the support characteristic curves under the original design conditions are plotted to analyze the stability variation rule during humidification, respectively. Response curves of surrounding rock move in a whole direction away from the coordinate axis with the increase of water content, indicating that the swelling and softening will significantly increase the displacement of free face of tunnel under a same support force. With the increase of water content, the linear part of the response curve of surrounding rock is shortened, i.e., the elastic deformation is reduced. This is because the strength parameter decreases with the increase of water content, which makes the surrounding rock more likely to enter the plastic deformation stage. With the increase of water content, the equilibrium point (intersection) between the response curve of surrounding rock and the support characteristic curve also moves away from the coordinate axis, which indicates that under the same support conditions, the swelling and softening will cause the deformation and stress of the support structure to increase. It can be seen from the above that the stability of surrounding rocks significantly decreases during humidification.

According to Formula (18), the safety factors of tunnel surrounding rock–support structure under different water content conditions are calculated, and the corresponding calculation results are listed in

Table 4. Calculation results of safety factors.

$\mathit{w}$/%	${\mathit{u}}_{\mathbf{eq}}$/cm	${\mathit{P}}_{\mathbf{eq}}$/MPa	${\mathit{P}}_{\mathbf{com},\mathbf{lim}}$/MPa	${\mathit{F}}_{\mathit{s}}$
15.28	1.95	0.4992	1.237	2.48
18.42	2.05	0.6484	1.237	1.91
20.56	2.15	0.7978	1.237	1.55
21.68	2.20	0.8724	1.237	1.42
24.64	2.30	1.0216	1.237	1.21

. In this table, $u_{\mathrm{eq}}$ refers to the radial displacement of free face at the intersection of the response curve of surrounding rock and the support characteristic curve, i.e., the radial displacement at the equilibrium point. It can be seen from the table that with the increase of water content, the radial displacement of free face at the equilibrium point increases, the support force at the equilibrium state increases, while the safety factor decreases.

Based on step ③ in Section 4.3, the $u_{\mathrm{eq}}$ and $P_{\mathrm{eq}}$ is calculated by combining the response curves of surrounding rocks and support characteristic curves under different water content. Then the corresponding safety factor $F_s$ is calculated according to step ④. Thus, the water content-safety coefficient curve can be drawn, as shown in the red line in Figure 7. It can be seen from the diagram that with the increase of water content, the safety coefficient decreases, and the rate of safety coefficient decreases greater in the range of less water content, while the rate of safety coefficient decreases in the range of higher water content.

In rock–support relationship, a reasonable support starting point $u_{\mathrm{sd}}$ can not only fully exert the resistance of support units, but also satisfy the stress release during excavation. According to Formula (17), the value of $u_{\mathrm{sd}}$ can be adjusted by changing the parameter $L$ which is the space between the support starting position and the tunnel face (i.e., one excavation cycle). In Figure 7, the water content-safety factors under $L$ = 0.5 m and 2.5 m conditions are further plotted. Comparing the three curves in Figure 7, it can be seen that the water content-safety coefficient curve moves away from the coordinate axis with the increase of L. This is because the swelling loess surrounding rock is a kind of soft rock, which the more deformation is released, the less the corresponding support structure bears, so the support structure is not easy to be destroyed.

5.4. Sensitivity Analysis of Safety Factor

5.4.1. Sensitivity Analysis Methods

In practical engineering, when the location of the tunnel is determined, the mechanical parameters of surrounding rock can be taken as known conditions. At this time, engineers or designers need to select the appropriate combination of support parameters so that the safety factor of the tunnel can meet the specifications during the humidification process. Therefore, exploring the sensitivity of safety factor to support parameters and water content of surrounding rock is of great engineering significance for guiding actual support design and guaranteeing tunnel safety.

Sensitivity analysis can identify key input variables in a model and quantify the effect of the uncertainty of input variables on model output. In the study interval of support parameters (see Table 2) and water content (see Figure 4), the sensitivity analysis is performed by using the LHS sampling technique and calculating the partial rank correlation coefficients (PRCCs) for safety factor against the support parameters and water content. The PRCCs can be a sensitivity index for quantitative comparison among the input parameters. The significance of a non-zero value of a PRCC is tested by computing the p-value [21,22,23]. It needs to be explained that the selection of research interval refers to highway tunnel design specifications and commonly used soft rock support parameters, and the sample capacity of LHS technique in this paper is 1000.

5.4.2. Sensitivity Analysis Results

Figure 8 shows a scatter plot of sensitivity analysis for each parameter, in which each dot represents the safety factor for a specific sampled value. For each subplot, these dots qualitatively describe the influence of the corresponding parameters on the safety factor with simultaneous variation of other parameters. The safety factor increases significantly with the increase of $t{h}_c$ and ${\tau }_c$, and decreases slightly with the increase of $E_c$ and $S_c$.

Figure 9 is a histogram of partial rank correlation coefficients for each parameter.

Table 5. Partial rank correlation coefficient and p value for safety factor against input parameters.

Parameter	$\mathit{w}$	${\mathit{E}}_{\mathit{c}}$	$\mathit{t}{\mathit{h}}_{\mathit{c}}$	${\mathit{\tau }}_{\mathit{c}}$	$\mathit{l}$	${\mathit{S}}_{\mathit{c}}$	${\mathit{S}}_1$	${\mathit{T}}_{\mathbf{bf}}$
Partial rank correlation coefficient	−0.846	−0.473	0.911	0.944	0.125	−0.493	−0.369	0.249
The value of p	0	0	0	0	0	0	0	0

shows partial rank correlation coefficients and p values. PRCC can be used to quantitatively reflect the effect of input parameters on output parameters. PRCC is used to illustrate the correlation between input variables (support parameters and water content) and output variables (safety factors); A positive value indicates a positive correlation, and a negative value indicates a negative correlation. The magnitude of PRCC absolute value is used to quantify the sensitivity of the output variable to the input variable. The greater the absolute value, the more sensitive it is. From the results of Figure 9 and Table 5, it can be seen that the influence of each parameter on safety factor is ${\tau }_c$, $t{h}_c$, $w$, $S_c$, $E_c$, $S_1$, $T_{\mathrm{bf}}$, and $l$ from large to small. Only the absolute value of PRCC for the safety factor against the three input parameters $t{h}_c$, ${\tau }_c$, and $w$ is greater than 0.75, which means that the safety factor is relatively sensitive to the shear strength of concrete, the thickness of the spray layer, and the water content, and relatively insensitive to other parameters.

In Table 5, the PRCC value for the safety factor against the parameter $E_c$ is −0.473, and its absolute value is less than 0.75. It indicates that the equivalent elastic modulus of concrete has little influence on the safety factor in the range of 10 GPa to 40 GPa. For parameter $E_c$ is negative, from the point of view of mathematics, $F_s$ decreases with the increase of $E_c$. From the point of view of engineering, this means that under certain conditions, flexible support is more conducive to improving the stability of the initial support of swelling loess tunnels.

6. Conclusions

In this paper, the stability analysis of swelling loess tunnels during humidification was studied from the perspective of mathematical models. Main conclusions are drawn as below.

(1) The joint influence of swelling and softening was introduced into the surrounding rock-support relationship so that the response curve of surrounding rock of loess tunnels during humidification was established. Then a simple and practical stability analysis framework of initial support was established by taking the safety factor as a stability index.

(2) In an engineering case, the models in this paper were employed to analyze the change rules of surrounding rock and tunnel safety factor during the humidification of swelling loess. The result indicated that: with the increase of water content, the response curve of surrounding rock moved toward the direction away from the coordinate axis, and the safety factor decreased gradually, which is in accordance with the actual situation; The maximum radial displacement of the free face without support predicted by the model is 4.4456 cm, which is similar to the actual monitoring value of 4.3 cm. So the applicability of the theoretical models in this paper was verified in this engineering case.

(3) The sensitivity analysis was carried out, and the result indicated that the influence of key parameters on the safety factor from large to small is ${\tau }_c$, $t{h}_c$, $w$, $S_c$, $E_c$, $S_1$, $T_{\mathrm{bf}}$, and $l$. The safety factor is sensitive to the shear strength of concrete, the thickness of the spray layer, and the water content, which is helpful for the design work of tunneling in the swelling loess.