Calculation Model and Influencing Factors of Surrounding Rock Loosening Pressure for Tunnel in Fold Zone

*is research aims to study the surrounding rock loosening pressure variation law of tunnel in the fold area. Based on the calculation method of surrounding rock loosening pressure for shallow tunnel, a new calculation model of the surrounding rock pressure was proposed for tunnel in the fold area; through this calculation model, the effects of tectonic stress (F), the angle (φ1) between tectonic stress and horizontal plane, tunnel buried depth (h), friction angle (θ), the multiple (k) between tectonic stress and rock mass gravity in the upper part of the tunnel, lateral pressure coefficient (λ), and tunnel midline offset (t) on tunnel surrounding rock loosening pressure in fold area are studied, respectively. Results show that in the anticline area, when φ1 increases, the vertical loosening pressure (q) decreases; when q> 0, the surrounding rock is in the elastic deformation stage, and q decreases monotonously as F increases; when q< 0, the rockmass is in the initial stage of failure, and as F continues to increase, the number of internal cracks increases, the rock mass reaches its ultimate bearing capacity and then fails completely, and q increases linearly in this process; q decreases with the increase of θ and k; the greater k is, the easier it is to reach its bearing limit; the horizontal loosening pressure (e) increased monotonously with the increase of h and λ. *e research process of surrounding rock loosening pressure of tunnel in the syncline area is similar to that of tunnel in the anticline area; q decreases with the increase of θ and λ; q monotonically increases with F increasing.


Introduction
For basic transportation construction in complex topography and geomorphic areas, tunnels have become an important part of basic transportation due to their unique advantages. Over the years, as the mileage of the tunnel is getting longer and longer, the geological conditions encountered become more and more complex. If the stress conditions of the surrounding rock of the tunnel are not found out, it will be very dangerous to the reckless construction of the tunnel. However, the surrounding rock loosening pressure is the pressure directly acting on the supporting structure by the gravity of loosened or collapsed rock mass. e quantity, property, and distribution law of surrounding rock loosening pressure play an important role in the design, construction, and operation of the supporting system [1][2][3][4][5][6][7]. If tunnel is not supported timely after excavation, there will be a transfixion sliding surface in surrounding rock. e rock mass will gradually collapse and move and then develop into a subsidence area eventually (as shown in Figure 1). Scholars have done a lot of work on the research on the loosening pressure of tunnel surrounding rock, such as, the upper bound theorem was one of the theories to analyze the surrounding rock pressure of tunnel [8], but its calculation principle and process were complicated, which was unfavorable to engineering application; the numerical method and empirical method were the common methods to analyze the surrounding rock pressure of shallow tunnel under unsymmetrical pressure [9]; the surrounding rock deformation curve and support characteristic curve could also be used to evaluate the stability of tunnel surrounding rock [10]. Sugimoto et al. considered the soil pressure caused by the active lateral displacement of the surface after tunnel excavation and the displacement caused by the deformation before the stability of the surrounding rock; they studied the soil pressure, displacement, and section force of the lining by changing the reaction coefficient of the roadbed and the grouting rate [11]. According to the calculation theory of surrounding rock pressure of shallow tunnel, linear failure criterion and nonlinear failure criterion were commonly used [12]. Based on the structural stress of shallow unsymmetrical loading tunnel with variable slope surface, Liu and Fang derived the calculation formula of the loosening surrounding rock pressure in the shallow bilateral bias tunnel [13]. Li et al. put forward the loose pressure calculation theory of deep buried double holes by modifying Protodyakonov equilibrium arch theory and studied the effects of clear distance of double holes, reinforcement coefficient, excavation span, and excavation height on surrounding rock stress [14]. Sugimoto et al. proposed an analysis model of tunnel lining frame structure based on nonlinear ground response curve. Considering passive earth pressure and active earth pressure, the model could better reflect the influence of different surrounding rock on lining earth pressure, lining displacement, and lining section force [15]. Based on two-linear-spring design, Shakeri et al. simulated the imperfect boundary conditions between surrounding rock and tunnel. e dynamic circumferential stress and solid displacement of tunnel surrounding rock were studied by using infinite Bessel series, Hankel series, Laplace transform, and the associated Durbin algorithm [16].
It is found that although there are abundant achievements on the surrounding rock loosening pressure of the tunnel, these results do not involve tunnel in fold area. However, tunnels are often built in the fold zone [17][18][19][20]. It is necessary and significant to study the surrounding rock pressure in the fold zone. erefore, this paper analyzes the calculation model and influencing factors of the loose pressure of the surrounding rock of the tunnel in the fold zone, providing a basis for the design and construction of tunnels with folds.

Assumptions
According to the movement trend and stress characteristics of surrounding rock after the tunnel excavation in the fold zone, a simplified calculation model is established as shown in Figure 2. e calculated model takes the longitudinal length of the tunnel as 1 m as research object, which can transform the three-dimensional model into a two-dimensional plane model [21,22].
ere are two cases of the stability analysis of tunnel according to the relation between buried depth (h) and equivalent load height (h q ). When h > h q , the friction resistance of the sliding surface increases with the thickness of upper rock mass increasing. e influence of sliding surface resistance should be considered when analyzing the surrounding rock loosening pressure. If h ≤ h q , it can be ignored.

Simplified Calculation Model
e calculation models of tunnel in the fold zone are simplified as follows: (1) Based on the geological tectonics, the surface material above the anticline tunnel is loose and eroded to form a valley, so it is simplified as "V" shape ( Figure 2(a)). e surface material right above the oblique zone tunnel is tight and does not easily erode. On the contrary, the surface material on both sides erodes more easily. e shape of mountain is simplified as "∧" (Figure 2(b)). (2) e fracture surface formed in the rock mass is simplified into the oblique faces of AC and BD. e angle between AC (or BD) and x axis is β. (3) When the roof rock (EFGH) sinks, as it is blocked by both sides of the three-ridge rock body (AEC and BDF), it will drive the three-ridge rock down together, and when the whole rock declines, it will be hindered by the undisturbed rock. Similar to the force, the gravity of the roof rock is W 1 , the gravity of the three-ridge rock masses on both sides is W 2 , and the resistance of the sinking rock (EFGH) given by the three-ridge rock is T (T � T1 + t2). When the rock (EFGH) is sinking, the resistance given by undisturbed rock masses on both sides is N. e rock mass in the anticline zone is subjected to the tectonic stress (F) and F is at an angle (φ 1 ) to the horizontal. e rock in the oblique zone is subjected to a vertical downward tectonic stress (F). (4) e shear strength of AC and BD at the oblique plane depends on the inner friction angle (φ) and the cohesion force (c) of the sliding surface. In order to simplify the calculation, the equivalent inner friction angle φ 0 is introduced. e friction angle between the roof rock and the three-ridge rock is θ. e relation between θ and φ 0 is shown in Table 1. e values of specific frictional angle θ of various types of surrounding rocks and equivalent inner friction angle φ 0 are shown in

Loosening Rock Pressure Model of Tunnel in the Anticline Zone
e loosening rock pressure model of tunnel in the anticline zone can be divided into two situations: (1) the anticlinal axis parallel to tunnel centerline, as shown in Figure 2

Anticlinal Axis Parallel to Tunnel
Centerline. Based on the above assumptions, it can be seem from the corresponding mechanical principle that the total vertical pressure acting on the support structure is where W 1 is the gravity of the roof covering rock mass (EFHG), W 1 � 1/2c(h 1 + h 0 )B, B is the tunnel width, T1 is the entrainment force of two-sided three-ridge rock against the rock EFHG, and F is the crustal stress in the fold zone. Because T1 is unknown, we take the three-ridge rock BDF as a study object. Its force analysis is shown in Figure 4. e gravity of three-ridge rock BDF is W 2 : where c is the density of rock, h is the buried depth of tunnel, and β and α are shown in Figure 2(a). From the trigonometric function theorem, we obtain Equation (3) can be simplified as      Figure 4: Stress analysis of the triangular rock BDF and the apex rock EFHG.

Advances in Civil Engineering
By substituting equation (2) into equation (4), we obtain us, In equation (5) to equation (7), λ is coefficient of lateral pressure.
e entrainment force of two-sided three-ridge rock against the rock EFHG is T1, which can be shown as follows: where λ is a function of φ 0 , θ, and β, as shown in equation (6). φ 0 and θ are known quantities, and β is an unknown quantity. According to the calculation model in Figure 2(a), when the rock EFGH begins to slide, the position of the sliding surface is most likely at the maximum sliding force T; at the same time, it is at the sliding surface of two-sided three-ridge rock. erefore, we need to use the concept of extreme value to solve this problem. By letting dλ/dβ � 0, we can obtain Equation (9) can be simplified as We obtain where D � (φ 1 − φ 0 − θ)° en, the total surrounding rock pressure (P) is Finally, we can obtain the surrounding rock pressure (P) as e vertical surrounding rock loosening pressure is q, which can be shown as follows: e horizontal pressure at the top and bottom of the tunnel is e curves of the formation compression stress in the anticline zone with the angle φ 1 of horizontal plane and the vertical loosening pressure (q) of surrounding rock (the curve of φ 1 ∼ q) are shown in Figures 5(a)-5(f ). By analyzing the curve of φ 1 ∼ q, we can know that when φ 1 is increasing, the vertical loosening pressure of surrounding rock is in decline in different lateral pressure coefficients. When frictional angle θ � 7.5°, tunnel VI level of surrounding rock is in the surrounding rock loosening vertical pressure value maximum. When frictional angle θ � 73°, tunnel I level of surrounding rock is in the surrounding rock loosening vertical pressure value minimum.

e Influence of F on q under the Condition of Different λ When Surrounding Rock Grades Are Different.
In order to study the influence of tectonic stress (F) on surrounding rock loosening pressure (q), now take φ 1 � 30°. e relationship of F in the anticline zone and surrounding rock Advances in Civil Engineering loosening pressure (q) is shown in Figures 6(a)-6(f ), in different surrounding rock levels. When q > 0, the surrounding rock is in elastic deformation stage. As the crustal stress increase, the surrounding rock loosening pressure decreases gradually. When q < 0, the surrounding rock has been damaged. As the crustal stress increases, q increases inversely. e function curves show that the tunnel is in I, II, and III surrounding rock levels, and the loosening pressure is less; when the tunnel is in IV, V, and VI surrounding rock levels, the loosening pressure is greater. erefore, it is necessary to strengthen the supporting structure when building a tunnel in IV, V, and VI surrounding rock levels with the influence of tectonic stress. Figure 7 shows the F-q curve. When q decreases gradually, the first stage is the compaction stage of the rock mass; as the ground stress increases, the pores and fissures in the rock mass are gradually compacted, and the loose pressure appears "upward convex" at this time. e second stage is the stage of elastic deformation of rock mass, when the relationship between ground stress and loosening pressure is nearly linear. As the ground stress continues to increase, it is about to enter the third stage; cracks in the rock mass begin to form and gradually expand. However, when q decreases to 0, the failure of rock mass begins, and the number of cracks in the rock mass continues to increase and run through. With the continuous increase of F, when the rock mass reaches the bearing limit, the rock mass is completely destroyed and q increases linearly in the opposite direction.

e Influence of θ on q under Different k Conditions.
In actual project, the surrounding rock level changes frequently and the vertical surrounding rock loose pressure is analyzed in response to the continuous change of the surrounding rock in different tectonic stresses. As can be seen from Figure 8, the tectonic stress F is taken as k times the gravity of the upper rock of tunnel (k � 0.05, 0.10, 0.15, 0.20, 0.25), and lateral pressure coefficient λ � 0.5. With the increase of friction angle, the surrounding rock level is changed from level VI to level I, and the vertical loosening pressure of surrounding rock decreases. As the coefficient k increases, the value of loosening pressure decreases. e larger k is, the lesser the loosening pressure is and the easier it is to reach the bearing limit of surrounding rock and the easier it is to destroy surrounding rock.

e Influence of h on e under Different λ Conditions.
Compared with conventional geology, the biggest difference of horizontal pressure in anticline area is the consideration of tectonic stress. In general, λ is in 0 to 1, but due to the influence of tectonic stress, λ>1. erefore, in the anticline area, λ is set to 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5, and then the relationship between the horizontal surrounding rock  loosening pressure with different λ parameters and the buried depth of the tunnel is studied. As shown in Figures 9(a) and 9(b), the horizontal loosening pressure of tunnel in oblique area increases with the depth of burial, and the horizontal surrounding rock loosening pressure increases with the increase of lateral pressure coefficient.

Anticlinal Axis Intersecting Tunnel Centerline.
For practical projects, the anticlinal axis is usually not parallel to tunnel centerline. So, there will be a certain angle (Δφ) between anticlinal axis and tunnel centerline, as shown in Figure 3. Based on the results that anticlinal axis is parallel to tunnel centerline, the unparallel calculation results can be directly obtained as follows. e total vertical surrounding rock pressure (P) is From the trigonometric function theorem: After simplification, we obtain
In equation (23), In equation (24), λ 1 is the lateral pressure coefficient. e entrainment force T1 of the two-sided three-ridge rock to the rock (EFGH) is e total surrounding rock pressure (P) is obtained as e vertical surrounding rock loosening pressure (q) is e horizontal pressure at the top (e1) and bottom (e2) of the tunnel is shown in the following equation: ere are two ideal situations. In practice, the tunnel is usually not built directly under the V-shaped tip, but the middle line and crimp middle of the tunnel will undergo some deviation, as shown in Figure 10. e difference of deviation (t) will directly affect the value of the surrounding rock loosening pressure. Next, based on the model of anticlinal axis parallel to tunnel centerline, the relationship between the surrounding rock displacement and the surrounding rock loosening pressure is discussed. e gravity of the rock mass at the top of tunnel is e vertical surrounding rock loosening pressure (q) is Take h1 � 14 m, h0 � 12 m, B � 5 m, c � 19 kN/m 3 , α � 20°, φ 1 � 30°, and F � 1000 kN. q can be obtained as (31) en, we make a � 1/Bc tan 2 αλ, b � 2/Bh 1 cλ tan α, c � −1/2c tan α, d � −1/Bch 2 1 λ, e � −2/BF sin φ 1 . So, q � −a · sin θ · t 2 +(b · sin θ + c)t +(d · sin θ + e). (32) In equation (32), a, b, c, d, and e are pressure reduction coefficients. e q-t curves at different surrounding rock levels are shown in Figures 11(a)-11(f ). It can be seen that q quadratically increases with the increase of t. When the variation range of offset is [−B/2, B/2], the change rate of loose pressure of surrounding rock is larger; the variation range of offset is (−∞, −B/2]∪[B/2, ∞); the variation trend of loose pressure of surrounding rock is small. When the tunnel offset is 0, the process of the surrounding rock level changes from VI to I, and the loosening pressure of surrounding rock decreases gradually. e reason is that the better the grade of surrounding rock, the easier it is to form a pressure arch above the tunnel to resist the external stress and reduce the stress around the tunnel. e horizontal pressure at the top and bottom of the tunnel is From equation (33), it can be seen that with t monotonically increasing, the bottom horizontal surrounding rock pressure is also monotonically decreasing.

Calculation Models of Surrounding Rock Loosening Pressure of Tunnel in the Syncline Zone
Loosening rock pressure model of tunnel in the syncline zone can also be divided into two situations: (1) the synclinal axis parallel to tunnel centerline, as shown in Figure 2(b); (2) the synclinal axis intersecting the tunnel centerline, as shown in Figure 12.

Synclinal Axis Parallel to Tunnel
Centerline. Based on the above assumptions, according to the corresponding mechanical principles, the total vertical pressure acting on the support structure is In equation (34), As T1 is unknown, the three-ridge BDF is taken as the research object. e force analysis is shown in Figure 13. e resistance of the three-ridge rock mass to the sinking rock mass is T. Make λ 2 � 1 tan α − tan β · tan β − tan ϕ 0 1 + tan β tan ϕ 0 − tan θ + tan ϕ 0 tan θ . So, In equation (37), λ is lateral pressure coefficient. en, the resistance force T1 of the two-sided three-ridge rock mass on rock mass (EFGH) is Δφ y x Figure 12: Surrounding rock loosening rock pressure model of tunnel in the syncline zone.
(c) Figure 13: Rock mass force analysis of BDF and EFHG.
And tan β can be calculated as Finally, the total surrounding rock pressure (P) is calculated as Vertical surrounding rock loosening pressure is Take the single line tunnel section for example. e soil bulk density is c � 19 kN/m 3 , h1 � 14 m, and h0 � 12 m. Net width of the tunnel is B � 5 m, lateral pressure coefficient is λ 2 � (1.0 ∼ 1.5), and tectonic stress is F � 1000 kN. e relation curves between the friction angle (θ) and the vertical surrounding rock loosening pressure (q) are shown in Figure 14. It can be seen from the figures that with increase of the friction angle (θ),q gradually decreases, and with the increase of the lateral pressure coefficient, q decreases too. erefore, the higher the surrounding rock level and the smaller the friction angle (θ) are, the greater the vertical surrounding rock loosening pressure is.
It can be seen from equation (42) that the vertical surrounding rock loosening pressure (q) is a function of the tectonic stress (F), and q shows a monotonic increasing trend with the increase of tectonic stress. e horizontal pressure at the top (e1) and bottom (e2) of the tunnel is e 1 � ch 1 λ 2 , e 2 � chλ 2 . (43) As the function relation between the buried depth (h) and the horizontal surrounding rock loosening pressure (e) is similar to that of the anticline zone, only the lateral pressure coefficient is different in it, so it is not analyzed again.

Synclinal Axis Intersecting Tunnel Centerline.
In practice, the synclinal axis intersects the tunnel centerline with a certain angle (Δφ), as shown in Figure 7. Based on the calculation results that synclinal axis is parallel to the tunnel centerline, the corresponding calculation results can be obtained as follows. e total vertical surrounding rock pressure (P) is e resistant force of the three-ridge rock mass to the sinking rock mass (EFHG) is T.