# Theoretical Study on Internal Forces of Primary Support of Tunnel by considering Time Effect

Authors: Zhuangzhi Yang; Yuqing Wang; Lin Song; Xinyu Wang
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

Understanding the internal force distribution and prejudging the weak part of tunnel supporting structure under complicated construction surroundings have become one of the important measures to ensure the safety of tunnel construction. Based on the initial parameter method and field-monitoring results of surrounding rock pressure, the theory of beam on elastic foundation is employed to derive the analytical solution of primary support internal forces. By combining the monitored data of rock pressure in one tunnel, solution of primary support internal force of a section is back analyzed and weak parts of the primary support are analyzed and evaluated. The results show that, after the tunnel excavation is completed, the internal forces of the primary support of the tunnel arch grow larger over time. Whenθ > 1.0, the internal forces change greatly. The internal force of the primary support decreased to different extents within 2 days after the primary support was applied and then gradually increased over time. Considering the situationality and changeability of the tunnel construction process, the analytical solution for the primary support internal forces of a multicenter arch tunnel proposed in this paper has a strong feasibility in tunnel construction. The conclusions obtained here could provide theoretical support for the design of supporting structure and the optimization of tunnel construction technology.


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## Body

## 1. Introduction

One basic concept of the New Austrian Tunnelling Method is to give full play to self-supporting capacity of the surrounding rock, and main safeguard to it is application of the primary support [1–3]. Tunnel excavation is a dynamic construction process. Given the complexity of the geological conditions of tunnel projects and the diversity of tunnel excavation methods, the distribution of internal forces in the concrete supporting structure shows its unique dynamic variation characteristics during tunnel construction. Its actual force state is usually different from that in design condition. Therefore, one of the key measures to ensure safe of the tunnel construction is to investigate distribution of internal forces of the supporting structure and identify the weak parts of the supporting structure in a timely and accurate manner.Since the creation of the New Austrian Tunnelling Method, the monitoring and measurement technology has been widely used in tunnel construction. The use of monitoring data to evaluate the safety of tunnel structures and the stability of surrounding rock has become an important means to ensure the safety of the tunnel [4–6].The back-analysis theory and technology for the tunnel was first introduced to obtain the actual material parameters [7, 8], by using displacements [8–11], stresses [12], or AE monitoring data [13, 14]. In recent years, introduction of new approaches, such as support vector machine [15], particle swarm optimization [16], evolution strategy algorithm [17], and genetic algorithms [18], have promoted the development of back-analysis theory and technology in geotechnical and underground engineering. Moreover, the back-analysis method has been used to investigate the internal forces of tunnel support. Wen et al. [19, 20] proposed analytical solution of internal force of initial support by using the measured touch stresses and steel arch stresses, respectively. On account of theory of the curved beam and straight beam on elastic foundation, Wang and Shao [21] theoretically deduced the inversion formula for internal forces of primary support of the horseshoe-shaped tunnel based on the measured surrounding rock pressure. He et al. [22] presented a novel back-analysis method based on radial displacement and contact stress to evaluate internal forces of the primary support. The initial parameter method [23], since proposed by Vlasov has been used to solve problems of beam column [24] and shell bending [25], while Sun et al. [26] and Wang and Shao [21] obtained internal force of lining and primary support of the tunnel, respectively.Based on the elastic foundation beam theory and the initial parameter method, the inversion formula to calculate the internal forces of the primary support of the multicenter circular arch tunnel is deduced considering the variation of the measured surrounding rock pressure and the elastic modulus of concrete of the primary support over time. Combined with the measured data of the surrounding rock pressure of the Qixiaying tunnel, the temporal and spatial distribution of the internal forces of the supporting structure is back analyzed. With these efforts, the key parts and time points after the application of the primary support are obtained, and the stability of the primary support is preliminarily determined.


## 2. Materials and Methods

The section of the Qixiaying tunnel is a multicenter arch section, and the tunnel is excavated with the two-bench method. Figure1 shows the dimensions of the tunnel section. Figure 2 shows a picture of the upper bench of the Qixiaying tunnel.Figure 1
Schematic diagram of the tunnel section (unit: cm).Figure 2
Picture of the Qixiaying tunnel.


### 2.1. Control Equation

Primary support of the tunnel excavation can be considered as a curved beam on elastic foundation, and Figure3(a) shows an arc section of the concrete primary support after excavation of the upper bench. Its inner and outer radiuses are r0 and r1, respectively. The thickness of the primary support is h=r1−r0, the cross-section area is A, and the moment of inertia is I. ri=r1+r0/2 is defined as the center radius, and Et is the elastic modulus of concrete.Figure 3
Circular beam on elastic foundation [21]. (a) Dimensions of the tunnel supporting structure. (b) Differential element.
(a)
(b)The differential elementridθ is shown in Figure 3(b), where ωθ,t is the radial displacement, K is the resistance coefficient of the surrounding rock, Mθ,t is the bending moment of the cross section, Qθ,t is the shear force, and Nθ,t is the axial force. All forces above are considered positive if they point in the direction shown in the figure. If the shear stress between the primary support and the surrounding rock is not taken into account and the second-order trace is omitted, the equilibrium equation of the differential element is as follows:(1)dQθ,t+Nθ,tdθ+Κωθ,tr1dθ=0,(2)dNθ,t−Qθ,tdθ=0,(3)dMθ,t−ridNθ,t=0.The radial displacement of the curved beam of the concrete foundation,ωθ,t, is related to the internal forces of the tunnel section as follows:(4)d2ωθ,tdθ2+ωθ,t=Mθ,tri2EtI+Nθ,triEtA.By combining equations (1)–(4), the control equation of the radial displacement ωθ,t can be derived:(5)d5ωθ,tdθ5+2d3ωθ,tdθ3+l2dωθ,tdθ=M0−N0riri2EtI,where l2=1+Kr1ri3/EtI+ri/EtA, M0=M0, and N0=N0.


### 2.2. Internal Forces of Primary Support

The general solution to the initial parameter of equation (5) is(6)ωθ,t=e−αθB1cosβθ+B2sinβθ+eαθB3cosβθ+B4sinβθ+M0−N0riri2EtIl2,where α=l−1/2, β=l+1/2, and B1, B2, B3, and B4 are the integral constants.According to equations (1)–(4) and equation (6), the solutions to the internal forces of the primary support are(7)Qθ,t=aMrie−αθβB−2αB1sinβθ+βB+1αB2cosβθ−aMrie−αθαB3+βB4sinβθ+βB−3αB4cosβθ,(8)Mθ,t=aMe−αθB1sinβθ−B2cosβθ−aMeαθB3sinβθ−B4cosβθ+bM+1M0−N0ri,(9)Nθ,t=aMrie−αθB1sinβθ−B2cosβθ−aMrieαθB3sinβθ−B4cosβθ+bMriM0−N0ri.The angular displacement of the curved beam sectionϕθ,t is given by(10)ϕθ,t=1re−αθ−αB1+βB2cosβθ−βB1+αB2sinβθ+1reαθαB3+βB4cosβθ+−βB3+αB4sinβθ,where aM=l2−1/1/EtA+ri2/EtI and bM=1−l2ri2/EtIl2/1/EtA+ri2/EtI.For equations (6)–(10), let θ=0, and the corresponding initial parameters can be expressed as follows:(11)ω0=ω0,ϕ0=ϕ0,Q0=Q0,M0=M0,N0=N0.By combining with equation (11), the integral constant ω0 described by the initial parameter constants Q0, M0, N0, ϕ0, and B1∼B4 is obtained. By substituting the expression of B1∼B4 into equations (6)–(10), the initial parameter formula for the radial displacement of the primary support ωθ,t with the internal forces Qθ,t, Mθ,t, and Nθ,t and angular displacement ϕθ,t is obtained.The matrix equation between the radial displacement and the internal forces of the primary support expressed by the initial parameter constants is(12)Wθ,t=Tθ,tW0+Pθ,t,where Wθ,t is the radial displacement and internal force array, of which the radius angle with the initial parameter point is θ, W0 is the radial displacement and internal force array of the known initial parameter point, and Tθ,t and Pθ,t are the corresponding coefficient matrix and column constant matrix, respectively. Wθ,t and W0 are expressed by Wθ=ωθ,Qθ,Mθ,Nθ,ϕθT and W0=ω0,Q0,M0,N0,ϕ0T, respectively.


### 2.3. Back Analysis of Internal Forces of Primary Support

Based on the Winkler hypothesis and the initial parameter method, the analytical formula for the internal forces of the supporting structure of the multicenter arch tunnel described by the surrounding rock pressure is derived.In Figure1, the tunnel structure is symmetrical about the center line of the lining. A and F are on the axis of symmetry, and B, C, and D are the intersection points of adjacent arches. Assuming that the tunnel load is symmetrical about the centreline of the tunnel and point A is set to be the starting point of the initial parameter of the primary support, then QA=Q0=0 and ϕ0=ϕ0=0. On that basis, the radial displacement and internal force array of point A is given by(13)WA=ωA,0,MA,NA,0T.To determine the internal force distribution of arc segmentsAB and BC of the lining after excavation of the upper bench, we only need to know the three initial parameter constants ωA, MA, and NA of the initial parameter point A in equation (13).Winkler [27] assumed that the pressure at any point on the foundation is proportional to the deformation of the point, i.e.,(14)Pθ,t=Kωθ,t.where Pθ,t is the contact pressure between the surrounding rock and the primary support of the tunnel.According to equations (13) and (14), the solutions for the initial parameter constants ωA, MA, and NA in equation (13) can be obtained from the three measured values of the surrounding rock pressure. The measuring points of the surrounding rock pressure can be arranged at any position in arc segments AB and BC. In Figure 1, the radius angle for the arc between measuring point 1/point 2 and initial parameter point A is θ1/θ2 and that for the arc between measuring point 3 and initial parameter point B is θ3.From equation (6)–(10),(15)Pθ1,t=KABω1,(16)Pθ2,t=KABω2,(17)Pθ3,t=KBGω3,where ω1, ω2, and ω3 are the radial displacements of the corresponding measuring points of surrounding rock pressure and KAB and KBC are the elastic resistance coefficients of the surrounding rock at the corresponding arc segments.According to equation (12), ω1 and ω2 are in the same arc segment and can be expressed directly by the initial parameter constant of the initial parameter point A. In addition, through the second iteration of point B, ω3 can also be given by the initial parameter constant of point A.By combining equations (15)–(17), the initial parameter constants ωA, MA, and NA of the initial parameter point A can be calculated. After that, point A is set to be the initial point, and equation (12) is applied segment by segment. In this way, the radial displacement and internal forces of each arc segment of the primary support are found.


## 2.1. Control Equation

Primary support of the tunnel excavation can be considered as a curved beam on elastic foundation, and Figure3(a) shows an arc section of the concrete primary support after excavation of the upper bench. Its inner and outer radiuses are r0 and r1, respectively. The thickness of the primary support is h=r1−r0, the cross-section area is A, and the moment of inertia is I. ri=r1+r0/2 is defined as the center radius, and Et is the elastic modulus of concrete.Figure 3
Circular beam on elastic foundation [21]. (a) Dimensions of the tunnel supporting structure. (b) Differential element.
(a)
(b)The differential elementridθ is shown in Figure 3(b), where ωθ,t is the radial displacement, K is the resistance coefficient of the surrounding rock, Mθ,t is the bending moment of the cross section, Qθ,t is the shear force, and Nθ,t is the axial force. All forces above are considered positive if they point in the direction shown in the figure. If the shear stress between the primary support and the surrounding rock is not taken into account and the second-order trace is omitted, the equilibrium equation of the differential element is as follows:(1)dQθ,t+Nθ,tdθ+Κωθ,tr1dθ=0,(2)dNθ,t−Qθ,tdθ=0,(3)dMθ,t−ridNθ,t=0.The radial displacement of the curved beam of the concrete foundation,ωθ,t, is related to the internal forces of the tunnel section as follows:(4)d2ωθ,tdθ2+ωθ,t=Mθ,tri2EtI+Nθ,triEtA.By combining equations (1)–(4), the control equation of the radial displacement ωθ,t can be derived:(5)d5ωθ,tdθ5+2d3ωθ,tdθ3+l2dωθ,tdθ=M0−N0riri2EtI,where l2=1+Kr1ri3/EtI+ri/EtA, M0=M0, and N0=N0.


## 2.2. Internal Forces of Primary Support

The general solution to the initial parameter of equation (5) is(6)ωθ,t=e−αθB1cosβθ+B2sinβθ+eαθB3cosβθ+B4sinβθ+M0−N0riri2EtIl2,where α=l−1/2, β=l+1/2, and B1, B2, B3, and B4 are the integral constants.According to equations (1)–(4) and equation (6), the solutions to the internal forces of the primary support are(7)Qθ,t=aMrie−αθβB−2αB1sinβθ+βB+1αB2cosβθ−aMrie−αθαB3+βB4sinβθ+βB−3αB4cosβθ,(8)Mθ,t=aMe−αθB1sinβθ−B2cosβθ−aMeαθB3sinβθ−B4cosβθ+bM+1M0−N0ri,(9)Nθ,t=aMrie−αθB1sinβθ−B2cosβθ−aMrieαθB3sinβθ−B4cosβθ+bMriM0−N0ri.The angular displacement of the curved beam sectionϕθ,t is given by(10)ϕθ,t=1re−αθ−αB1+βB2cosβθ−βB1+αB2sinβθ+1reαθαB3+βB4cosβθ+−βB3+αB4sinβθ,where aM=l2−1/1/EtA+ri2/EtI and bM=1−l2ri2/EtIl2/1/EtA+ri2/EtI.For equations (6)–(10), let θ=0, and the corresponding initial parameters can be expressed as follows:(11)ω0=ω0,ϕ0=ϕ0,Q0=Q0,M0=M0,N0=N0.By combining with equation (11), the integral constant ω0 described by the initial parameter constants Q0, M0, N0, ϕ0, and B1∼B4 is obtained. By substituting the expression of B1∼B4 into equations (6)–(10), the initial parameter formula for the radial displacement of the primary support ωθ,t with the internal forces Qθ,t, Mθ,t, and Nθ,t and angular displacement ϕθ,t is obtained.The matrix equation between the radial displacement and the internal forces of the primary support expressed by the initial parameter constants is(12)Wθ,t=Tθ,tW0+Pθ,t,where Wθ,t is the radial displacement and internal force array, of which the radius angle with the initial parameter point is θ, W0 is the radial displacement and internal force array of the known initial parameter point, and Tθ,t and Pθ,t are the corresponding coefficient matrix and column constant matrix, respectively. Wθ,t and W0 are expressed by Wθ=ωθ,Qθ,Mθ,Nθ,ϕθT and W0=ω0,Q0,M0,N0,ϕ0T, respectively.


## 2.3. Back Analysis of Internal Forces of Primary Support

Based on the Winkler hypothesis and the initial parameter method, the analytical formula for the internal forces of the supporting structure of the multicenter arch tunnel described by the surrounding rock pressure is derived.In Figure1, the tunnel structure is symmetrical about the center line of the lining. A and F are on the axis of symmetry, and B, C, and D are the intersection points of adjacent arches. Assuming that the tunnel load is symmetrical about the centreline of the tunnel and point A is set to be the starting point of the initial parameter of the primary support, then QA=Q0=0 and ϕ0=ϕ0=0. On that basis, the radial displacement and internal force array of point A is given by(13)WA=ωA,0,MA,NA,0T.To determine the internal force distribution of arc segmentsAB and BC of the lining after excavation of the upper bench, we only need to know the three initial parameter constants ωA, MA, and NA of the initial parameter point A in equation (13).Winkler [27] assumed that the pressure at any point on the foundation is proportional to the deformation of the point, i.e.,(14)Pθ,t=Kωθ,t.where Pθ,t is the contact pressure between the surrounding rock and the primary support of the tunnel.According to equations (13) and (14), the solutions for the initial parameter constants ωA, MA, and NA in equation (13) can be obtained from the three measured values of the surrounding rock pressure. The measuring points of the surrounding rock pressure can be arranged at any position in arc segments AB and BC. In Figure 1, the radius angle for the arc between measuring point 1/point 2 and initial parameter point A is θ1/θ2 and that for the arc between measuring point 3 and initial parameter point B is θ3.From equation (6)–(10),(15)Pθ1,t=KABω1,(16)Pθ2,t=KABω2,(17)Pθ3,t=KBGω3,where ω1, ω2, and ω3 are the radial displacements of the corresponding measuring points of surrounding rock pressure and KAB and KBC are the elastic resistance coefficients of the surrounding rock at the corresponding arc segments.According to equation (12), ω1 and ω2 are in the same arc segment and can be expressed directly by the initial parameter constant of the initial parameter point A. In addition, through the second iteration of point B, ω3 can also be given by the initial parameter constant of point A.By combining equations (15)–(17), the initial parameter constants ωA, MA, and NA of the initial parameter point A can be calculated. After that, point A is set to be the initial point, and equation (12) is applied segment by segment. In this way, the radial displacement and internal forces of each arc segment of the primary support are found.


## 3. Results and Discussion

Based on the results above, the internal forces of the primary support of a multicenter arch tunnel are obtained. For the primary support of the tunnel per unit axial length, the geometric parameters and calculation parameters are shown in Tables1 and 2.Table 1
Geometric parameters of primary support.
Arc segment
Inner radiusr0(cm)
Radius angleθ (rad)
AB
820.0
π/4
BC
586.7
π/5
BD
586.7
π/3Table 2
Tunnel section parameters.
Parameter
Value
Primary support thicknessh (cm)
20
Cross-section areaA (cm2)
2000
Elastic resistance coefficient of surrounding rockK(MPa/m)
480


### 3.1. Elastic Modulus of Concrete of Primary Support

The left tunnel line K196 + 520 is located at a mileage section, with a strength grade of C20 used in the primary support. The variation trend of the elastic modulus of the concrete with age is defined in the literature as follows [28]:(18)Et=1.05Eγ−ηζ−η0.8,where γ=ζexp−19.73/t0.6841, η = 0.2819, ζ = 0.8246, E is the standard value of the elastic modulus of concrete 28 d, and E = 23 GPa.


### 3.2. Surrounding Rock Pressure

At the construction site, TXR-2020 vibrating wire earth pressure cell (as shown in Figure4) manufactured by Jiangsu Haiyan Engineering Material and Instrument Co., Ltd. was used to measure the surrounding rock pressure of the cross section of the left Qixiaying tunnel line K196 + 520. The measuring points in the section are shown in Figure 1.Figure 4
Measuring points of surrounding rock pressure.Figure5 shows the measured data for the surrounding rock pressure of the three monitoring points. The solid lines are the results by fitting the measured data. The specific expressions are shown in Table 3.Figure 5
Measured rock pressure.Table 3
Surrounding rock pressure.
Measuring point
Positionθ (rad)
Surrounding rock pressureP (t) (kPa)
1
0.09
89.05 (exp (−0.001123t) − exp (−3.328t))
2
0.61
43.51 (exp (−0.01061t) − exp (−0.6016t))
3
0.50
20.15 (exp (−0.00316t) − exp (−0.5015t))By substituting the data in Table3 into equations (15)–(17), the expressions for ωA, MA, and NA are obtained. By combining equations (6)–(9), the distribution of the internal forces of the primary support for the left Qixiaying tunnel line K196 + 520 and the variation trend of the internal forces with time are obtained.


### 3.3. Analysis of Calculation Results

Figure6(a) shows the temporal and spatial distribution of the calculated shear force of the primary support. As can be seen from Figure 6(a), for the θ ≤ 0.74 arc segment, the shear force of the concrete shows no obvious change with time and space.Figure 6
Primary support shear force (kN). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.48.
(a)
(b)
(c)Figure6(b) shows the spatial variation curve of the primary support shear force at t = 28 d. From this figure, it can be known that when θ ≤ 0.74, the shear force of the concrete shows no obvious change with time and space. When θ > 0.74, the shear force changes greatly with time and space. It converts from positive to negative at θ = 1.2 and reaches a maximum value of 41.75 kN at θ = π/2. Figure 6(c) shows the variation of the shear force with time when θ = 1.484. It can be seen that the shear force decreases and then increases after the primary support is applied. This trend shall be related to the growth trend of the concrete modulus of the primary support.Figure7(a) shows the temporal and spatial distribution of the calculated bending moment of the primary support concrete. As can be seen from Figure 7(a), for the θ ≤ 0.84 arc segment, the bending moment of the concrete shows no obvious change with time and space. Figure 7(b) shows the spatial variation curve of the primary support bending moment at t = 28 d. It can be seen that at the θ ≤ 0.84 arc segment, the bending moment of the concrete shows little change. When θ = 1.2, the bending moment reaches its positive maximum 2.58 kN·m, and when θ = π/2, it reaches its negative maximum 35.50 kN·m. Figure 7(c) shows the variation of the bending moment with time when θ = 1.484. It can be seen that the bending moment decreases and then increases after the primary support is applied and reaches the maximum value of 18.32 kN·m at t = 28 d.Figure 7
Primary support bending moment (kN·m). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.484.
(a)
(b)
(c)Figure8(a) shows the temporal and spatial distribution of the calculated axial force of the primary support. As can be seen from Figure 8(a), for the θ ≤ 0.98 arc segment, the axial force of the concrete shows no obvious change with time and space. Figure 8(b) shows the spatial variation curve of the primary support axial force at t = 28 d. It can be seen that when θ = 0.88, the axial force of the concrete is 2.36 kN. With the increase of θ, the axial force reaches its negative maximum 155.37 kN when θ = 1.38 and its positive maximum 214.48 kN when θ = π/2. Figure 8(c) shows the variation of the axial force with time when θ = 1.31. It can be seen that the axial force decreases and then increases after the primary support is applied. This trend is consistent with that of the concrete modulus and surrounding rock of the primary support at an early stage.Figure 8
Primary support axial force (kN). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.31.
(a)
(b)
(c)


### 3.4. Limitation: Unsymmetrical Loading Tunnel

The theoretical inversion model for the internal force of the primary support proposed in this paper applies to the study on the primary support without a closed loop and can estimate the weak part of the primary support. In this model, the surrounding rock pressure on the primary support is bilaterally symmetrical. However, limited by the geological and topographical conditions and route options, the tunnel sections in tunnel entrance, mountain areas, and river valleys are mostly unsymmetrically loaded [29, 30]. Compared with conventional tunnels, the support structure of bias tunnels is more complicated. Therefore, the applicability of the proposed theoretical model in bias tunnels should be further investigated.


## 3.1. Elastic Modulus of Concrete of Primary Support

The left tunnel line K196 + 520 is located at a mileage section, with a strength grade of C20 used in the primary support. The variation trend of the elastic modulus of the concrete with age is defined in the literature as follows [28]:(18)Et=1.05Eγ−ηζ−η0.8,where γ=ζexp−19.73/t0.6841, η = 0.2819, ζ = 0.8246, E is the standard value of the elastic modulus of concrete 28 d, and E = 23 GPa.


## 3.2. Surrounding Rock Pressure

At the construction site, TXR-2020 vibrating wire earth pressure cell (as shown in Figure4) manufactured by Jiangsu Haiyan Engineering Material and Instrument Co., Ltd. was used to measure the surrounding rock pressure of the cross section of the left Qixiaying tunnel line K196 + 520. The measuring points in the section are shown in Figure 1.Figure 4
Measuring points of surrounding rock pressure.Figure5 shows the measured data for the surrounding rock pressure of the three monitoring points. The solid lines are the results by fitting the measured data. The specific expressions are shown in Table 3.Figure 5
Measured rock pressure.Table 3
Surrounding rock pressure.
Measuring point
Positionθ (rad)
Surrounding rock pressureP (t) (kPa)
1
0.09
89.05 (exp (−0.001123t) − exp (−3.328t))
2
0.61
43.51 (exp (−0.01061t) − exp (−0.6016t))
3
0.50
20.15 (exp (−0.00316t) − exp (−0.5015t))By substituting the data in Table3 into equations (15)–(17), the expressions for ωA, MA, and NA are obtained. By combining equations (6)–(9), the distribution of the internal forces of the primary support for the left Qixiaying tunnel line K196 + 520 and the variation trend of the internal forces with time are obtained.


## 3.3. Analysis of Calculation Results

Figure6(a) shows the temporal and spatial distribution of the calculated shear force of the primary support. As can be seen from Figure 6(a), for the θ ≤ 0.74 arc segment, the shear force of the concrete shows no obvious change with time and space.Figure 6
Primary support shear force (kN). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.48.
(a)
(b)
(c)Figure6(b) shows the spatial variation curve of the primary support shear force at t = 28 d. From this figure, it can be known that when θ ≤ 0.74, the shear force of the concrete shows no obvious change with time and space. When θ > 0.74, the shear force changes greatly with time and space. It converts from positive to negative at θ = 1.2 and reaches a maximum value of 41.75 kN at θ = π/2. Figure 6(c) shows the variation of the shear force with time when θ = 1.484. It can be seen that the shear force decreases and then increases after the primary support is applied. This trend shall be related to the growth trend of the concrete modulus of the primary support.Figure7(a) shows the temporal and spatial distribution of the calculated bending moment of the primary support concrete. As can be seen from Figure 7(a), for the θ ≤ 0.84 arc segment, the bending moment of the concrete shows no obvious change with time and space. Figure 7(b) shows the spatial variation curve of the primary support bending moment at t = 28 d. It can be seen that at the θ ≤ 0.84 arc segment, the bending moment of the concrete shows little change. When θ = 1.2, the bending moment reaches its positive maximum 2.58 kN·m, and when θ = π/2, it reaches its negative maximum 35.50 kN·m. Figure 7(c) shows the variation of the bending moment with time when θ = 1.484. It can be seen that the bending moment decreases and then increases after the primary support is applied and reaches the maximum value of 18.32 kN·m at t = 28 d.Figure 7
Primary support bending moment (kN·m). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.484.
(a)
(b)
(c)Figure8(a) shows the temporal and spatial distribution of the calculated axial force of the primary support. As can be seen from Figure 8(a), for the θ ≤ 0.98 arc segment, the axial force of the concrete shows no obvious change with time and space. Figure 8(b) shows the spatial variation curve of the primary support axial force at t = 28 d. It can be seen that when θ = 0.88, the axial force of the concrete is 2.36 kN. With the increase of θ, the axial force reaches its negative maximum 155.37 kN when θ = 1.38 and its positive maximum 214.48 kN when θ = π/2. Figure 8(c) shows the variation of the axial force with time when θ = 1.31. It can be seen that the axial force decreases and then increases after the primary support is applied. This trend is consistent with that of the concrete modulus and surrounding rock of the primary support at an early stage.Figure 8
Primary support axial force (kN). (a) Temporal-spatial distribution. (b)t = 28 d. (c) θ = 1.31.
(a)
(b)
(c)


## 3.4. Limitation: Unsymmetrical Loading Tunnel

The theoretical inversion model for the internal force of the primary support proposed in this paper applies to the study on the primary support without a closed loop and can estimate the weak part of the primary support. In this model, the surrounding rock pressure on the primary support is bilaterally symmetrical. However, limited by the geological and topographical conditions and route options, the tunnel sections in tunnel entrance, mountain areas, and river valleys are mostly unsymmetrically loaded [29, 30]. Compared with conventional tunnels, the support structure of bias tunnels is more complicated. Therefore, the applicability of the proposed theoretical model in bias tunnels should be further investigated.


## 4. Conclusions

Based on the elastic foundation beam theory and the initial parameter method, the inversion formula to calculate the internal forces of the primary support of the multicenter circular arch tunnel is deduced, which takes into account the variation of the surrounding rock pressure and the elastic modulus of concrete of the primary support with time. Combined with the measured data of the surrounding rock pressure of a tunnel, the temporal-spatial distribution of the internal forces of the primary support is back analyzed.The following conclusions could be drawn:(1)
After the tunnel excavation is completed, the internal forces of the primary support of the tunnel arch grow larger over time.(2)
Whenθ > 1.0, the internal forces change greatly. In particular, the axial force becomes positive when θ > 1.5.(3)
The internal force of the primary support decreased to different extents within 2 days after the primary support was applied, and then gradually increased over time.(4)
The weak part of the primary support of the tunnel constructed with the bench cut method is at the arch foot. On-site observation should be strengthened during construction, and the feet-lock bolt can be applied as necessary.The analytical expression of the internal force of the primary support concrete proposed in this paper enables fast and real-time determination of the internal force state and stress distribution of the primary support and locates the weak part of the primary support of the tunnel constructed with the bench cut method, so as to provide a basis for safety decision-making in the construction process.



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*Source: 4254178-2020-04-27.xml*