# A Structural Calculation Model of Shield Tunnel Segment: Heterogeneous Equivalent Beam Model

Authors: Mingfeng Lei; Dayong Lin; Chenghua Shi; Jianjun Ma; Weichao Yang
Journal: Advances in Civil Engineering
 (2018)

---

## Abstract

A heterogeneous equivalent beam model (HEB model) of the shield tunnel segment structure is proposed based on a systematical analysis on the stress state of the cross section of segment joints. This model treats a noncontinuous segment structure as a continuous heterogeneous structure, on the basis of the principle of equivalent stress state on a section and equivalent conversion of the mechanical parameters. For a comprehensive demonstration of the proposed HEB model, an interpretative solution of equivalent mechanical parameters of the joint section is obtained through theoretical derivation, and a specific iterative computation flow is provided in accordance. Model validation and comparative analysis are also conducted for two industrial applications. It is found that the iterative process of calculation has good convergence, leading to reliable numerical results for all cases under consideration. Resulting simulations reveal that the proposed HEB model can reflect the effect of joints on overall rigidity of a segment structure. Compared with the computation results obtained using other models presented in the literature, there are smaller axial force deviation and larger bending moment deviation (up to 20% or higher), demonstrating that the model selection is important in design and computation of a segment structure of shield tunnels. The proposed model and analysis for model performance may provide useful reference for engineers in shield tunnel community.


---

## Body

## 1. Introduction

Shield technique has been widely applied in the construction of urban subway and underwater tunnels owing to its high level of mechanization, rapid construction, and minimal environmental impact [1, 2]. The design of the segment structure is of critical importance for the application of shield technique and project construction, in terms of both the safety of tunnel structure and construction cost. Numerous engineering reports show that segmental lining accounts for 30% to 50% of the total cost of a tunnel constructed by shield technique [3].Unlike traditional monolithic tunnel lining structures, segmental lining is a noncontinuous structure with vertical and transverse joints. The mechanical behavior of each segment is affected by structural and mechanical characteristics of joints [4]. Therefore, segmental lining is more complex, as both calculation and analysis of the stress state is much more difficult than monolithic lining. In the open literature, comprehensive studies on the calculation model for shield segment structure have been conducted.In 1978, International Tunneling Association (ITA) established a specialized team for collecting calculation models for shield tunnel segments. ITA supplemented updated data according to the industrial development. The published summary in 1988 shows that the design model for tunnel structure at that time covers continuum or noncontinuum models, elastic foundation beam models, convergence-constraint models, and empirical methods [5, 6].In recent decades, along with the massive development of urban rail transportation, studies on the calculation model of shield tunnel segment have made a great progress. Multiple structural calculation models, including uniform rigidity ring model (UR model), modified uniform rigidity ring model (MUR model), free hinge ring model, beam-spring model, and beam-joint model, have been proposed [7], as shown in Figure 1.Figure 1
Structural models of the shield segment [7]. (a) UR model. (b) MUR model. (c) Free hinge ring model. (d) Beam-spring model. (e) Beam-joint model.
(a)
(b)
(c)
(d)
(e)Ye et al. [8] comprehensively analyzed the performance of these calculation models for the shield tunnel segment. They found that the UR model completely ignores the effect of segmental joint, which is obviously different from the actual situation. This model results in higher internal force of the hard formation and relatively less deformation of the weak formation, leading to increased security risks [8]. The MUR model is carried out by reducing the segment stiffness and increasing bending moment coefficients, resulting in greater randomness and uncertainty [9–12]. The free hinge ring model takes the segment joint as a single-hinge structure and thus does not reflect the properties of force transfer on the joint section. In addition, this model is a nonstatic system, although it can provide a static solution with the support of a tunnel surrounding rock, the ground resistance of tunnel surrounding rock cannot be calculated, and therefore, this model is characterized by certain levels of limitations and uncertainty [13]. In fact, the beam-spring model and the beam-joint model can better reflect the effect of segment joint on the stiffness of the whole ring structure. However, the theoretical basis for selecting the spring coefficient has not been well investigated; thus the spring coefficient is estimated based on the model test or engineering experience, leading to a certain level of randomness in modelling results [12, 14, 15].Although the design, calculation, and modelling of shield tunnel segment structures have attracted considerable attentions, a variety of problems still need further investigation. Based on comprehensive analysis on the stress characteristics and the properties of force transfer on the joint between shield tunnel segments, the current study converts a noncontinuous segment structure into a continuous heterogeneous equivalent beam structure through equivalent conversion of mechanical parameters of the joint section. Mechanical parameters of the joint section are also deduced under different working conditions to establish a new calculation model for the shield tunnel segment. This model is significant, as it treats the noncontinuous shield segment joint as an equivalent continuous structure with only tiny changes on the mechanical and deformation characteristics; and most importantly, it is capable to capture the stress characteristics and the properties of force transfer on the joint comprehensively. The performance of this model has been demonstrated through two industrial applications from the literature. The proposed HEB model and the comprehensive analysis presented for the HEB model may provide useful reference for engineers and scholars in the community of shield tunnel construction.


## 2. Establishment and Basic Idea of HEB Model

Engineering practice and theoretical studies show that the section of a segment joint is used to maintain the balance of internal force and transfer forces, which is similar to the force-transferring property of reinforced concrete cross section [1]. Therefore, the segment joints and the partial structures in their local range can be regarded as a beam element. However, the cross-sectional parameters of a beam element (i.e., elastic modulus and geometrical parameters) are different from those of a segment section. Based on this idea, the HEB model with segment ring composed of the beam element is established, as shown in Figure 2. In this model, parameters of the section in the local range of joint are converted equivalently according to the actual stress state on the section of joint.Figure 2
Structural calculation model of shield tunnel segment with HEB model: (a) Structural calculation model; (b) Equivalent beam element of joint.
(a)
(b)


## 3. Analysis on Mechanical Properties of Equivalent Beam Element of Joint

### 3.1. Basic Assumptions

In the mechanical calculation of the shield segment structure, a shield segment is always taken as a bending element; that is, only its bending moment and axial force are considered. The finite element method is normally applied to calculate the distribution of internal forces on each cross section. Then, the damage phase method in which the shear properties are neglected is employed to check its bearing capacity. Studies on the beam-spring model showed that if the shear stiffness of a circular joint is set to be smaller, the bending moment of its main cross section turns out to be smaller accordingly. Therefore, for security considerations, the shear stiffness of circular joints is usually set to be infinite [16]. This means that mechanical properties of a joint equivalent beam element only rely on the cross-sectional parameter of its longitudinal joint. As such, the following hypotheses are applied:(1)
Shearing stiffness of bolt and friction of section can effectively resist shear force at the joint; that is, no obvious deformation is generated in both sides of the joint [17].(2)
Based on structural mechanics, only the effect of axial force and bending moment on the displacement of arch structure is considered (i.e., the effect of shear stiffness is ignored).(3)
The deformation of joint section complies with the plane section assumption in all planes and meets the deformation compatibility.(4)
The stress of the concrete at the end of a joint is distributed linearly in the form of trapezoid and triangle shapes, respectively, corresponding to fully contact (original joint section) and opening states (partial loss of contact) of a joint connection during service life.(5)
Compressive deformationδ0 of the compressed concrete at segment joint obeys the following equation [18–20]:(1)δ0=σxEc,where Ec is the elastic modulus of concrete, σ is the maximum compressive stress at the edge of concrete, and x is the effective height of compressed area of a joint section.


### 3.2. Positive Bending Moment and Small Eccentric Compression

#### 3.2.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and small eccentric compression is shown in Figure3(a); assuming the stress in the diagram is positive, the following equations can be obtained based on static equilibrium:(2)∑F=0,N=bh2σc,max+σc,min−nTb,∑M=0,M=bh212σc,max−σc,min+nTbh2−h0,where N and M are the axial force and bending moment of the section, respectively; σc,max and σc,min refer to the compressive stress outside and inside the joint section, respectively; h and b refer to the thickness and width of the segment, respectively; n is the number of bolts in this segment; Tb is the pulling force of a single bolt; and h0 is the distance from the centroid of bolt to the internal edge. The deformation compatibility and plane cross-sectional assumption can be applied to solve the strain at bolt:(3)εb=εc,min+h0hεc,max−εc,min,where εb is the strain at bolt in the joint section and εc,max and εc,min refer to the inside and outside edge strains of concrete in the joint section, respectively.Figure 3
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)A joint section consists of concrete, sealing rubber stripe, and force transferring cushion. Therefore, the elastic modulus of this composite section needs to be calculated based on the principle of equivalence. As shown in Figure4, its compressive deformation properties are(4)Nl1+l2EeqA=Nl1EcA+Nl2EpA,where Eeq is the equivalent compressive elastic modulus of the joint beam element; A is the cross-sectional area; l1 is the length of the concrete section for joint section calculation, and it takes twice the cover thickness of concrete for protection, which depends on the working environment and structure type; l2 is the thickness of the sealing rubber strip and force transferring cushion, and it can be taken as 5 mm in most circumstances; and Ep is the elastic modulus of the sealing rubber strip and force transferring cushion. Thus, the equivalent elastic modulus of the joint section can be obtained:(5)Eeq=EcEpl1+l2Ecl1+Epl2.Figure 4
Analysis of the equivalent elastic modulus on the joint section.Therefore, in (3), εc,max and εc,min can be expressed as(6)εc,max=σc,maxEeq,εc,min=σc,minEeq.With further initial pretightening force of bolt, there is(7)T0−Tb=εbK,where K is the tensile stiffness of bolt, K=EsAs in which Es is the elastic modulus of bolt and As is the sectional area of a single bolt, and T0 is the initial pretightening force of the bolt.Solving (3), (6), and (7) simultaneously,(8)Tb=T0−KEeqσc,min+h0hσc,max−σc,min.Then, considering (2) and (8)(9)σc,max=λ2λ4+λ1λ6λ3λ6+λ2λ5,σc,min=λ1λ5−λ3λ4λ3λ6+λ2λ5,where(10)λ1=nT0+N,λ2=hb2+nKEeq1−h0h,λ3=hb2+nKh0Eeqh,λ4=M−nT0h2−h0,λ5=bh212−h2−h0nKh0Eeqh,λ6=bh212+h2−h01−h0hnKEeq.


#### 3.2.2. Stress Characteristics of the Equivalent Section

A stress analysis chart of the equivalent section under positive bending moment and small eccentric compression can be established based on the equivalent principle of the stress state of cross section, as shown in Figure3(b). Based on the equilibrium relationship of force, there is(11)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,where N′ and M′ refer to the axial force and bending moment of the section of an equivalent beam element, respectively; σ′c,max and σ′c,min are outside and inside compressive stresses, respectively; and h′ and b′, respectively, refer to the height and calculation width of the equivalent beam section.Based on the equivalent principle of the stress state of cross section, the internal force and rotation of the equivalent beam section are the same as those of the original section:(12)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq,where E′eq is the equivalent elastic modulus under positive bending moment and small eccentric compression of equivalent beam element.For the total cross section compression, the height and width of the compressive area of an equivalent cross section are the same as those of the original cross section, that is,b′=b and h′=h. By solving (11) and (12) simultaneously, the following relationship is obtained:(13)E′eq=12MEeqbh2σc,max−σc,min,where E′eq is the elastic modulus of an equivalent beam, and it can be calculated from (9) and (13).


### 3.3. Positive Bending Moment and Large Eccentric Compression

#### 3.3.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and large eccentric compression is shown in Figure5(a). The axial force and bending moment can be expressed as(14)N=12σc,maxbx−nTb,M=112σc,maxbxh2−x3+nTbh2−h0.Figure 5
Mechanical characteristic of the joint section under positive bending moment and large eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to the assumption (4), the deformation at the edge of concrete is δc,max=x⋅θ. Similarly, the deformation at the bolt can be δb=h−h0−x⋅θ. Here, θ is the opening angle of the joint.Considering the initial pretightening force of bolt, axial force of bolt should meet the following relationship:(15)Tb−T0=δbLK,where L is the effective calculation length of the bolt.From the assumption (5), the deformation at the edge of the segment can be obtained:(16)δc,max=σc,maxEeq⋅x.Solving (14) and (16) simultaneously,(17)C1x2+C2x+C3=0,where(18)C1=λ2λ5,C2=λ3λ5−λ1λ6,C3=λ4λ5−λ1λ7,λ1=M−nT0h2−h0,λ2=−b6,λ3=bh4−nKEeqLh2−h0,λ4=nKEeqLh2−h0h−h0,λ5=N+nT0,λ6=b2+nKEeqL,λ7=−nKEeqLh−h0.Equation (17) shows the function of the height of the compressed area x. The height of the compressed area x under corresponding force can be obtained by substituting the relative parameter with the range of x, that is, x∈0,h. Then, the maximum compression at the edge of segment σc,max can be obtained(19)σc,max=λ1λ2x2+λ3x+λ4,orσc,max=λ5λ6x+λ7.


#### 3.3.2. Stress Characteristics of the Equivalent Section

A stress analysis diagram of the equivalent section under positive bending moment and large eccentric compression is shown in Figure5(b), which can be obtained from static equilibrium:(20)N′=12σ′c,maxb′x′−12σ′c,minb′h′−x′,M′=12σ′c,maxb′x′h′2−x′3+12σ′c,minb′h′−x′h′6+x′3.Given that the stress property of the equivalent section is the same as that of the original section:(21)N′=N,M′=M,σ′c,maxE′eqx′=σ′c,minE′eqh′−x′.Two conditions should be added to solve the equation system consisting of (20) and (21) They are (1) composition of forces in the tensile zone of the equivalent section is the same as the pulling force of bolt in the original section and (2) the width of the equivalent section is obtained according to the ratio of the area of bolt to the area of the internal reinforcement of segmental lining αcs. Thus,(22)nTb=12σ′c,minb′h′−x′,b′=αcsb.By solving (20), (21), and (22) simultaneously, the mechanical parameters of the equivalent section under positive bending moment and large eccentric compression can be obtained:(23)x′=Mλ1λ2−N/3,h′=λ1x′,E′eq=2nEeqTbxαcsbσc,maxh′−x′2,where(24)λ1=nTbN+nTb+1,λ2=N2+2nTb3.


### 3.4. Negative Bending Moment and Small Eccentric Compression

#### 3.4.1. Stress Characteristics of the Original Section

A stress analysis diagram of the segment joint section under negative bending moment and small eccentric compression is shown in Figure6(a). It can be obtained from the following static equilibrium:(25)N=bh2σc,max+σc,min−nTb,M=−bh212σc,max−σc,min+nTbh2−h0.Figure 6
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to deformation compatibility, the bolt strain is(26)εb=εc,min+h−h0hεc,max−εc,min.Solving (25) and (26) simultaneously, the following equation can be obtained:(27)σc,min=λ1λ5−λ2λ4λ2λ6+λ3λ5,σc,max=λ1λ6+λ4λ3λ2λ6+λ3λ5,where(28)λ1=nT0+N,λ2=12bh+nKEeq1−h0h,λ3=12bh+h0nKEeqh,λ4=nT0h2−h0−M,λ5=112bh2+nKEeq1−h0hh2−h0,λ6=112bh2−nKh0Eeqhh2−h0.


#### 3.4.2. Stress Characteristics of the Equivalent Section

Stress characteristics of the equivalent section under negative bending moment and small eccentric compression is shown in Figure6(b). From static equilibrium and equivalent principle, we obtain(29)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,(30)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq.In addition, considering the width and height of the equivalent sectionb′=b and h′=h, the following relationship can be obtained through (29) and (30):(31)E′eq=12MEeqbh2σc,max−σc,min.


### 3.5. Negative Bending Moment and Large Eccentric Compression

#### 3.5.1. Stress Characteristics of the Original Section

Stress characteristics of the original section under negative bending moment and large eccentric compression can be further divided into two circumstances, as shown in Figures7(a) and 7(b), according to the size of eccentricity.(1) The 1st Circumstance. As shown in Figure 7(a), based on static equilibrium, we know that(32)N=12σc,maxbx−nTb,M=−12σc,maxbxh2−x3+nTbh2−h0.From the plane section assumption, the deformation at bolt can be obtained as(33)εb=x−h0xεc,max.By solving (32) and (33) simultaneously, the following can be obtained:(34)λ3x2−λ4x−λ5x+λ2=0,where(35)λ1=N+nT0nT0h/2−h0−M,λ2=nKEeqλ1h2−h0−1,λ3=bλ16,λ4=b2+bhλ14,λ5=λ2h0.Figure 7
Mechanical characteristic of the joint section under hogging bending moment and large eccentricity pressing condition. (a) Original joint section with the first circumstance. (b) Original joint section with the second circumstance. (c) Equivalent joint section.
(a)
(b)
(c)Therefore, considering the range ofx, that is, x∈0,h, from (34), the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained:(36)σc,max=N+nT01/2bx+nK/Eeq1−h0/x,Tb=T0−σc,maxKEeq1−h0x.(2) The 2nd Circumstance. The corresponding stress analysis diagram is shown in Figure 7(b). Its static equilibrium conditions meet (32) and the deformation compatibility, that is,(37)δc,max=x⋅θ,δb=h0−x⋅θ.Solving (32) and (37) simultaneously, we obtain(38)λ3x2−λ4x+λ5=0,where(39)λ1=N+nT0M−nT0h/2−h0,λ2=nKEeqλ1h2−h0+1,λ3=bλ1,λ4=λ2+b2+bhλ14,λ5=λ2h0.Therefore, considering the range ofx, that is, x∈0,h0, the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained using (38).


#### 3.5.2. Stress Characteristics of the Equivalent Section

A diagram of the stress characteristics of the equivalent section under negative bending moment and large eccentric compression is shown in Figure7(c). Based on static equilibrium and equivalent principle, we obtain(40)σc,min=N′b′h′+6M′b′h′2,σc,max=N′b′h′−6M′b′h′2,(41)N′=N,M′=M,σc,maxEeqx=σ′c,max+σ′c,minE′eqh′.Similarly, solving (40) and (41) simultaneously, the following is obtained:(42)E′eqh′2=2NEeqxαcsbσc,max.The height of the equivalent section under positive bending moment and large eccentric conditions can be used to calculate the heighth′ of the equivalent section under negative bending moment and large eccentric compression, as shown in (23). Thus, the elastic modulus of the equivalent beam under negative bending moment and large eccentric conditions can be obtained:(43)E′eq=2NEeqxαcsbσc,maxh′2.


## 3.1. Basic Assumptions

In the mechanical calculation of the shield segment structure, a shield segment is always taken as a bending element; that is, only its bending moment and axial force are considered. The finite element method is normally applied to calculate the distribution of internal forces on each cross section. Then, the damage phase method in which the shear properties are neglected is employed to check its bearing capacity. Studies on the beam-spring model showed that if the shear stiffness of a circular joint is set to be smaller, the bending moment of its main cross section turns out to be smaller accordingly. Therefore, for security considerations, the shear stiffness of circular joints is usually set to be infinite [16]. This means that mechanical properties of a joint equivalent beam element only rely on the cross-sectional parameter of its longitudinal joint. As such, the following hypotheses are applied:(1)
Shearing stiffness of bolt and friction of section can effectively resist shear force at the joint; that is, no obvious deformation is generated in both sides of the joint [17].(2)
Based on structural mechanics, only the effect of axial force and bending moment on the displacement of arch structure is considered (i.e., the effect of shear stiffness is ignored).(3)
The deformation of joint section complies with the plane section assumption in all planes and meets the deformation compatibility.(4)
The stress of the concrete at the end of a joint is distributed linearly in the form of trapezoid and triangle shapes, respectively, corresponding to fully contact (original joint section) and opening states (partial loss of contact) of a joint connection during service life.(5)
Compressive deformationδ0 of the compressed concrete at segment joint obeys the following equation [18–20]:(1)δ0=σxEc,where Ec is the elastic modulus of concrete, σ is the maximum compressive stress at the edge of concrete, and x is the effective height of compressed area of a joint section.


## 3.2. Positive Bending Moment and Small Eccentric Compression

### 3.2.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and small eccentric compression is shown in Figure3(a); assuming the stress in the diagram is positive, the following equations can be obtained based on static equilibrium:(2)∑F=0,N=bh2σc,max+σc,min−nTb,∑M=0,M=bh212σc,max−σc,min+nTbh2−h0,where N and M are the axial force and bending moment of the section, respectively; σc,max and σc,min refer to the compressive stress outside and inside the joint section, respectively; h and b refer to the thickness and width of the segment, respectively; n is the number of bolts in this segment; Tb is the pulling force of a single bolt; and h0 is the distance from the centroid of bolt to the internal edge. The deformation compatibility and plane cross-sectional assumption can be applied to solve the strain at bolt:(3)εb=εc,min+h0hεc,max−εc,min,where εb is the strain at bolt in the joint section and εc,max and εc,min refer to the inside and outside edge strains of concrete in the joint section, respectively.Figure 3
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)A joint section consists of concrete, sealing rubber stripe, and force transferring cushion. Therefore, the elastic modulus of this composite section needs to be calculated based on the principle of equivalence. As shown in Figure4, its compressive deformation properties are(4)Nl1+l2EeqA=Nl1EcA+Nl2EpA,where Eeq is the equivalent compressive elastic modulus of the joint beam element; A is the cross-sectional area; l1 is the length of the concrete section for joint section calculation, and it takes twice the cover thickness of concrete for protection, which depends on the working environment and structure type; l2 is the thickness of the sealing rubber strip and force transferring cushion, and it can be taken as 5 mm in most circumstances; and Ep is the elastic modulus of the sealing rubber strip and force transferring cushion. Thus, the equivalent elastic modulus of the joint section can be obtained:(5)Eeq=EcEpl1+l2Ecl1+Epl2.Figure 4
Analysis of the equivalent elastic modulus on the joint section.Therefore, in (3), εc,max and εc,min can be expressed as(6)εc,max=σc,maxEeq,εc,min=σc,minEeq.With further initial pretightening force of bolt, there is(7)T0−Tb=εbK,where K is the tensile stiffness of bolt, K=EsAs in which Es is the elastic modulus of bolt and As is the sectional area of a single bolt, and T0 is the initial pretightening force of the bolt.Solving (3), (6), and (7) simultaneously,(8)Tb=T0−KEeqσc,min+h0hσc,max−σc,min.Then, considering (2) and (8)(9)σc,max=λ2λ4+λ1λ6λ3λ6+λ2λ5,σc,min=λ1λ5−λ3λ4λ3λ6+λ2λ5,where(10)λ1=nT0+N,λ2=hb2+nKEeq1−h0h,λ3=hb2+nKh0Eeqh,λ4=M−nT0h2−h0,λ5=bh212−h2−h0nKh0Eeqh,λ6=bh212+h2−h01−h0hnKEeq.


### 3.2.2. Stress Characteristics of the Equivalent Section

A stress analysis chart of the equivalent section under positive bending moment and small eccentric compression can be established based on the equivalent principle of the stress state of cross section, as shown in Figure3(b). Based on the equilibrium relationship of force, there is(11)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,where N′ and M′ refer to the axial force and bending moment of the section of an equivalent beam element, respectively; σ′c,max and σ′c,min are outside and inside compressive stresses, respectively; and h′ and b′, respectively, refer to the height and calculation width of the equivalent beam section.Based on the equivalent principle of the stress state of cross section, the internal force and rotation of the equivalent beam section are the same as those of the original section:(12)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq,where E′eq is the equivalent elastic modulus under positive bending moment and small eccentric compression of equivalent beam element.For the total cross section compression, the height and width of the compressive area of an equivalent cross section are the same as those of the original cross section, that is,b′=b and h′=h. By solving (11) and (12) simultaneously, the following relationship is obtained:(13)E′eq=12MEeqbh2σc,max−σc,min,where E′eq is the elastic modulus of an equivalent beam, and it can be calculated from (9) and (13).


## 3.2.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and small eccentric compression is shown in Figure3(a); assuming the stress in the diagram is positive, the following equations can be obtained based on static equilibrium:(2)∑F=0,N=bh2σc,max+σc,min−nTb,∑M=0,M=bh212σc,max−σc,min+nTbh2−h0,where N and M are the axial force and bending moment of the section, respectively; σc,max and σc,min refer to the compressive stress outside and inside the joint section, respectively; h and b refer to the thickness and width of the segment, respectively; n is the number of bolts in this segment; Tb is the pulling force of a single bolt; and h0 is the distance from the centroid of bolt to the internal edge. The deformation compatibility and plane cross-sectional assumption can be applied to solve the strain at bolt:(3)εb=εc,min+h0hεc,max−εc,min,where εb is the strain at bolt in the joint section and εc,max and εc,min refer to the inside and outside edge strains of concrete in the joint section, respectively.Figure 3
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)A joint section consists of concrete, sealing rubber stripe, and force transferring cushion. Therefore, the elastic modulus of this composite section needs to be calculated based on the principle of equivalence. As shown in Figure4, its compressive deformation properties are(4)Nl1+l2EeqA=Nl1EcA+Nl2EpA,where Eeq is the equivalent compressive elastic modulus of the joint beam element; A is the cross-sectional area; l1 is the length of the concrete section for joint section calculation, and it takes twice the cover thickness of concrete for protection, which depends on the working environment and structure type; l2 is the thickness of the sealing rubber strip and force transferring cushion, and it can be taken as 5 mm in most circumstances; and Ep is the elastic modulus of the sealing rubber strip and force transferring cushion. Thus, the equivalent elastic modulus of the joint section can be obtained:(5)Eeq=EcEpl1+l2Ecl1+Epl2.Figure 4
Analysis of the equivalent elastic modulus on the joint section.Therefore, in (3), εc,max and εc,min can be expressed as(6)εc,max=σc,maxEeq,εc,min=σc,minEeq.With further initial pretightening force of bolt, there is(7)T0−Tb=εbK,where K is the tensile stiffness of bolt, K=EsAs in which Es is the elastic modulus of bolt and As is the sectional area of a single bolt, and T0 is the initial pretightening force of the bolt.Solving (3), (6), and (7) simultaneously,(8)Tb=T0−KEeqσc,min+h0hσc,max−σc,min.Then, considering (2) and (8)(9)σc,max=λ2λ4+λ1λ6λ3λ6+λ2λ5,σc,min=λ1λ5−λ3λ4λ3λ6+λ2λ5,where(10)λ1=nT0+N,λ2=hb2+nKEeq1−h0h,λ3=hb2+nKh0Eeqh,λ4=M−nT0h2−h0,λ5=bh212−h2−h0nKh0Eeqh,λ6=bh212+h2−h01−h0hnKEeq.


## 3.2.2. Stress Characteristics of the Equivalent Section

A stress analysis chart of the equivalent section under positive bending moment and small eccentric compression can be established based on the equivalent principle of the stress state of cross section, as shown in Figure3(b). Based on the equilibrium relationship of force, there is(11)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,where N′ and M′ refer to the axial force and bending moment of the section of an equivalent beam element, respectively; σ′c,max and σ′c,min are outside and inside compressive stresses, respectively; and h′ and b′, respectively, refer to the height and calculation width of the equivalent beam section.Based on the equivalent principle of the stress state of cross section, the internal force and rotation of the equivalent beam section are the same as those of the original section:(12)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq,where E′eq is the equivalent elastic modulus under positive bending moment and small eccentric compression of equivalent beam element.For the total cross section compression, the height and width of the compressive area of an equivalent cross section are the same as those of the original cross section, that is,b′=b and h′=h. By solving (11) and (12) simultaneously, the following relationship is obtained:(13)E′eq=12MEeqbh2σc,max−σc,min,where E′eq is the elastic modulus of an equivalent beam, and it can be calculated from (9) and (13).


## 3.3. Positive Bending Moment and Large Eccentric Compression

### 3.3.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and large eccentric compression is shown in Figure5(a). The axial force and bending moment can be expressed as(14)N=12σc,maxbx−nTb,M=112σc,maxbxh2−x3+nTbh2−h0.Figure 5
Mechanical characteristic of the joint section under positive bending moment and large eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to the assumption (4), the deformation at the edge of concrete is δc,max=x⋅θ. Similarly, the deformation at the bolt can be δb=h−h0−x⋅θ. Here, θ is the opening angle of the joint.Considering the initial pretightening force of bolt, axial force of bolt should meet the following relationship:(15)Tb−T0=δbLK,where L is the effective calculation length of the bolt.From the assumption (5), the deformation at the edge of the segment can be obtained:(16)δc,max=σc,maxEeq⋅x.Solving (14) and (16) simultaneously,(17)C1x2+C2x+C3=0,where(18)C1=λ2λ5,C2=λ3λ5−λ1λ6,C3=λ4λ5−λ1λ7,λ1=M−nT0h2−h0,λ2=−b6,λ3=bh4−nKEeqLh2−h0,λ4=nKEeqLh2−h0h−h0,λ5=N+nT0,λ6=b2+nKEeqL,λ7=−nKEeqLh−h0.Equation (17) shows the function of the height of the compressed area x. The height of the compressed area x under corresponding force can be obtained by substituting the relative parameter with the range of x, that is, x∈0,h. Then, the maximum compression at the edge of segment σc,max can be obtained(19)σc,max=λ1λ2x2+λ3x+λ4,orσc,max=λ5λ6x+λ7.


### 3.3.2. Stress Characteristics of the Equivalent Section

A stress analysis diagram of the equivalent section under positive bending moment and large eccentric compression is shown in Figure5(b), which can be obtained from static equilibrium:(20)N′=12σ′c,maxb′x′−12σ′c,minb′h′−x′,M′=12σ′c,maxb′x′h′2−x′3+12σ′c,minb′h′−x′h′6+x′3.Given that the stress property of the equivalent section is the same as that of the original section:(21)N′=N,M′=M,σ′c,maxE′eqx′=σ′c,minE′eqh′−x′.Two conditions should be added to solve the equation system consisting of (20) and (21) They are (1) composition of forces in the tensile zone of the equivalent section is the same as the pulling force of bolt in the original section and (2) the width of the equivalent section is obtained according to the ratio of the area of bolt to the area of the internal reinforcement of segmental lining αcs. Thus,(22)nTb=12σ′c,minb′h′−x′,b′=αcsb.By solving (20), (21), and (22) simultaneously, the mechanical parameters of the equivalent section under positive bending moment and large eccentric compression can be obtained:(23)x′=Mλ1λ2−N/3,h′=λ1x′,E′eq=2nEeqTbxαcsbσc,maxh′−x′2,where(24)λ1=nTbN+nTb+1,λ2=N2+2nTb3.


## 3.3.1. Stress Characteristics of the Original Section

A segment joint force diagram under positive bending moment and large eccentric compression is shown in Figure5(a). The axial force and bending moment can be expressed as(14)N=12σc,maxbx−nTb,M=112σc,maxbxh2−x3+nTbh2−h0.Figure 5
Mechanical characteristic of the joint section under positive bending moment and large eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to the assumption (4), the deformation at the edge of concrete is δc,max=x⋅θ. Similarly, the deformation at the bolt can be δb=h−h0−x⋅θ. Here, θ is the opening angle of the joint.Considering the initial pretightening force of bolt, axial force of bolt should meet the following relationship:(15)Tb−T0=δbLK,where L is the effective calculation length of the bolt.From the assumption (5), the deformation at the edge of the segment can be obtained:(16)δc,max=σc,maxEeq⋅x.Solving (14) and (16) simultaneously,(17)C1x2+C2x+C3=0,where(18)C1=λ2λ5,C2=λ3λ5−λ1λ6,C3=λ4λ5−λ1λ7,λ1=M−nT0h2−h0,λ2=−b6,λ3=bh4−nKEeqLh2−h0,λ4=nKEeqLh2−h0h−h0,λ5=N+nT0,λ6=b2+nKEeqL,λ7=−nKEeqLh−h0.Equation (17) shows the function of the height of the compressed area x. The height of the compressed area x under corresponding force can be obtained by substituting the relative parameter with the range of x, that is, x∈0,h. Then, the maximum compression at the edge of segment σc,max can be obtained(19)σc,max=λ1λ2x2+λ3x+λ4,orσc,max=λ5λ6x+λ7.


## 3.3.2. Stress Characteristics of the Equivalent Section

A stress analysis diagram of the equivalent section under positive bending moment and large eccentric compression is shown in Figure5(b), which can be obtained from static equilibrium:(20)N′=12σ′c,maxb′x′−12σ′c,minb′h′−x′,M′=12σ′c,maxb′x′h′2−x′3+12σ′c,minb′h′−x′h′6+x′3.Given that the stress property of the equivalent section is the same as that of the original section:(21)N′=N,M′=M,σ′c,maxE′eqx′=σ′c,minE′eqh′−x′.Two conditions should be added to solve the equation system consisting of (20) and (21) They are (1) composition of forces in the tensile zone of the equivalent section is the same as the pulling force of bolt in the original section and (2) the width of the equivalent section is obtained according to the ratio of the area of bolt to the area of the internal reinforcement of segmental lining αcs. Thus,(22)nTb=12σ′c,minb′h′−x′,b′=αcsb.By solving (20), (21), and (22) simultaneously, the mechanical parameters of the equivalent section under positive bending moment and large eccentric compression can be obtained:(23)x′=Mλ1λ2−N/3,h′=λ1x′,E′eq=2nEeqTbxαcsbσc,maxh′−x′2,where(24)λ1=nTbN+nTb+1,λ2=N2+2nTb3.


## 3.4. Negative Bending Moment and Small Eccentric Compression

### 3.4.1. Stress Characteristics of the Original Section

A stress analysis diagram of the segment joint section under negative bending moment and small eccentric compression is shown in Figure6(a). It can be obtained from the following static equilibrium:(25)N=bh2σc,max+σc,min−nTb,M=−bh212σc,max−σc,min+nTbh2−h0.Figure 6
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to deformation compatibility, the bolt strain is(26)εb=εc,min+h−h0hεc,max−εc,min.Solving (25) and (26) simultaneously, the following equation can be obtained:(27)σc,min=λ1λ5−λ2λ4λ2λ6+λ3λ5,σc,max=λ1λ6+λ4λ3λ2λ6+λ3λ5,where(28)λ1=nT0+N,λ2=12bh+nKEeq1−h0h,λ3=12bh+h0nKEeqh,λ4=nT0h2−h0−M,λ5=112bh2+nKEeq1−h0hh2−h0,λ6=112bh2−nKh0Eeqhh2−h0.


### 3.4.2. Stress Characteristics of the Equivalent Section

Stress characteristics of the equivalent section under negative bending moment and small eccentric compression is shown in Figure6(b). From static equilibrium and equivalent principle, we obtain(29)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,(30)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq.In addition, considering the width and height of the equivalent sectionb′=b and h′=h, the following relationship can be obtained through (29) and (30):(31)E′eq=12MEeqbh2σc,max−σc,min.


## 3.4.1. Stress Characteristics of the Original Section

A stress analysis diagram of the segment joint section under negative bending moment and small eccentric compression is shown in Figure6(a). It can be obtained from the following static equilibrium:(25)N=bh2σc,max+σc,min−nTb,M=−bh212σc,max−σc,min+nTbh2−h0.Figure 6
Mechanical characteristic of the joint section under positive bending moment and small eccentricity pressing condition. (a) Original joint section. (b) Equivalent joint section.
(a)
(b)According to deformation compatibility, the bolt strain is(26)εb=εc,min+h−h0hεc,max−εc,min.Solving (25) and (26) simultaneously, the following equation can be obtained:(27)σc,min=λ1λ5−λ2λ4λ2λ6+λ3λ5,σc,max=λ1λ6+λ4λ3λ2λ6+λ3λ5,where(28)λ1=nT0+N,λ2=12bh+nKEeq1−h0h,λ3=12bh+h0nKEeqh,λ4=nT0h2−h0−M,λ5=112bh2+nKEeq1−h0hh2−h0,λ6=112bh2−nKh0Eeqhh2−h0.


## 3.4.2. Stress Characteristics of the Equivalent Section

Stress characteristics of the equivalent section under negative bending moment and small eccentric compression is shown in Figure6(b). From static equilibrium and equivalent principle, we obtain(29)N′=b′h′2σ′c,max+σ′c,min,M′=b′h′212σ′c,max−σ′c,min,(30)N′=N,M′=M,σc,max−σc,minEeq=σ′c,max−σ′c,minE′eq.In addition, considering the width and height of the equivalent sectionb′=b and h′=h, the following relationship can be obtained through (29) and (30):(31)E′eq=12MEeqbh2σc,max−σc,min.


## 3.5. Negative Bending Moment and Large Eccentric Compression

### 3.5.1. Stress Characteristics of the Original Section

Stress characteristics of the original section under negative bending moment and large eccentric compression can be further divided into two circumstances, as shown in Figures7(a) and 7(b), according to the size of eccentricity.(1) The 1st Circumstance. As shown in Figure 7(a), based on static equilibrium, we know that(32)N=12σc,maxbx−nTb,M=−12σc,maxbxh2−x3+nTbh2−h0.From the plane section assumption, the deformation at bolt can be obtained as(33)εb=x−h0xεc,max.By solving (32) and (33) simultaneously, the following can be obtained:(34)λ3x2−λ4x−λ5x+λ2=0,where(35)λ1=N+nT0nT0h/2−h0−M,λ2=nKEeqλ1h2−h0−1,λ3=bλ16,λ4=b2+bhλ14,λ5=λ2h0.Figure 7
Mechanical characteristic of the joint section under hogging bending moment and large eccentricity pressing condition. (a) Original joint section with the first circumstance. (b) Original joint section with the second circumstance. (c) Equivalent joint section.
(a)
(b)
(c)Therefore, considering the range ofx, that is, x∈0,h, from (34), the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained:(36)σc,max=N+nT01/2bx+nK/Eeq1−h0/x,Tb=T0−σc,maxKEeq1−h0x.(2) The 2nd Circumstance. The corresponding stress analysis diagram is shown in Figure 7(b). Its static equilibrium conditions meet (32) and the deformation compatibility, that is,(37)δc,max=x⋅θ,δb=h0−x⋅θ.Solving (32) and (37) simultaneously, we obtain(38)λ3x2−λ4x+λ5=0,where(39)λ1=N+nT0M−nT0h/2−h0,λ2=nKEeqλ1h2−h0+1,λ3=bλ1,λ4=λ2+b2+bhλ14,λ5=λ2h0.Therefore, considering the range ofx, that is, x∈0,h0, the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained using (38).


### 3.5.2. Stress Characteristics of the Equivalent Section

A diagram of the stress characteristics of the equivalent section under negative bending moment and large eccentric compression is shown in Figure7(c). Based on static equilibrium and equivalent principle, we obtain(40)σc,min=N′b′h′+6M′b′h′2,σc,max=N′b′h′−6M′b′h′2,(41)N′=N,M′=M,σc,maxEeqx=σ′c,max+σ′c,minE′eqh′.Similarly, solving (40) and (41) simultaneously, the following is obtained:(42)E′eqh′2=2NEeqxαcsbσc,max.The height of the equivalent section under positive bending moment and large eccentric conditions can be used to calculate the heighth′ of the equivalent section under negative bending moment and large eccentric compression, as shown in (23). Thus, the elastic modulus of the equivalent beam under negative bending moment and large eccentric conditions can be obtained:(43)E′eq=2NEeqxαcsbσc,maxh′2.


## 3.5.1. Stress Characteristics of the Original Section

Stress characteristics of the original section under negative bending moment and large eccentric compression can be further divided into two circumstances, as shown in Figures7(a) and 7(b), according to the size of eccentricity.(1) The 1st Circumstance. As shown in Figure 7(a), based on static equilibrium, we know that(32)N=12σc,maxbx−nTb,M=−12σc,maxbxh2−x3+nTbh2−h0.From the plane section assumption, the deformation at bolt can be obtained as(33)εb=x−h0xεc,max.By solving (32) and (33) simultaneously, the following can be obtained:(34)λ3x2−λ4x−λ5x+λ2=0,where(35)λ1=N+nT0nT0h/2−h0−M,λ2=nKEeqλ1h2−h0−1,λ3=bλ16,λ4=b2+bhλ14,λ5=λ2h0.Figure 7
Mechanical characteristic of the joint section under hogging bending moment and large eccentricity pressing condition. (a) Original joint section with the first circumstance. (b) Original joint section with the second circumstance. (c) Equivalent joint section.
(a)
(b)
(c)Therefore, considering the range ofx, that is, x∈0,h, from (34), the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained:(36)σc,max=N+nT01/2bx+nK/Eeq1−h0/x,Tb=T0−σc,maxKEeq1−h0x.(2) The 2nd Circumstance. The corresponding stress analysis diagram is shown in Figure 7(b). Its static equilibrium conditions meet (32) and the deformation compatibility, that is,(37)δc,max=x⋅θ,δb=h0−x⋅θ.Solving (32) and (37) simultaneously, we obtain(38)λ3x2−λ4x+λ5=0,where(39)λ1=N+nT0M−nT0h/2−h0,λ2=nKEeqλ1h2−h0+1,λ3=bλ1,λ4=λ2+b2+bhλ14,λ5=λ2h0.Therefore, considering the range ofx, that is, x∈0,h0, the height x, σc,max,andTb of the compressed area of the internal concrete of a segment under negative bending moment and large eccentric compression can be obtained using (38).


## 3.5.2. Stress Characteristics of the Equivalent Section

A diagram of the stress characteristics of the equivalent section under negative bending moment and large eccentric compression is shown in Figure7(c). Based on static equilibrium and equivalent principle, we obtain(40)σc,min=N′b′h′+6M′b′h′2,σc,max=N′b′h′−6M′b′h′2,(41)N′=N,M′=M,σc,maxEeqx=σ′c,max+σ′c,minE′eqh′.Similarly, solving (40) and (41) simultaneously, the following is obtained:(42)E′eqh′2=2NEeqxαcsbσc,max.The height of the equivalent section under positive bending moment and large eccentric conditions can be used to calculate the heighth′ of the equivalent section under negative bending moment and large eccentric compression, as shown in (23). Thus, the elastic modulus of the equivalent beam under negative bending moment and large eccentric conditions can be obtained:(43)E′eq=2NEeqxαcsbσc,maxh′2.


## 4. Calculation Flow of HEB Model for Shield Tunnel Segment

The derivation in Section3 shows that the key idea of the HEB model is that the stress characteristics of a segment joint is analyzed through equating a joint section to a continuous beam element. That is, a noncontinuous segment structure is changed into a continuous heterogeneous beam structure. Also, section parameters of the equivalent beam at the joint should be obtained through equivalent conversion. The internal force of a joint section and initial physical and mechanical parameters should be obtained before any structural calculation. In this study, the UR model is used to estimate the internal force, and iterative calculation is applied to achieve accurate solution, with calculation flow being shown in Figure 8.Figure 8
Calculation flow of the HEB model.


## 5. Model Validation and Applications

### 5.1. Case 1

To verify the reliability of the proposed model, a comparative analysis is conducted by comparing model simulation with the result from the literature [21]. In this case, a certain interval of the shield tunnel, that is, number 3 Guangzhou subway in China, is used as an example. The segment structure of this interval of shield is designed as a “3 + 2 + 1” block mode. The external diameter of the segment is 6.0 m, and its thickness is 0.3 m. The segment rings are connected through 2 high-tensile bolts (Class 8.8 M24) with initial pretightening force of 200 kN and C50 concrete. The stratum is a weathered stratum in the red beds of Cretaceous System; the depth of tunnel is 22.14 m, and the water level is 18.94 m. The main parameters are shown in Figure 9(a).Figure 9
Calculation model. (a) Geological condition. (b) Structural calculation model.
(a)
(b)


#### 5.1.1. Calculation Process

(1)
First, with basic model parameters, applying the UR model (Figure9(b)) to obtain the internal force of each joint section (Table 1). The load calculation takes both geological conditions and ground surcharge p0 = 20 kPa into consideration.
(2)
The mechanical equivalent parameter solution of a joint section deduced according to Section3 is used to solve the equivalent mechanical parameters of each section, as shown in Table 1.
(3)
The equivalent mechanical parameters obtained in Step 2 is used to update those in Step 1. Then, repeating Steps 1–3 to obtain accurate results of the equivalent mechanical parameters. Accordingly, the internal force of the segment produced by the HEB model is obtained finally.Table 1
Equivalent parameters on joints section.
Joints
Inner force evaluated by UR model
Force-bearing types
Equivalent parameters
M (kN·m)
N (kN)
h
′ (m)
b
′ (m)
E
eq (GPa)
B
67.99
1354.51
Positive bending moment and small eccentricity pressing
0.3
1.5
7.36
C
−76.74
1566.17
Hogging bending moment and small eccentricity pressing
0.3
1.5
13.05
D
31.94
2070.37
Positive bending moment and small eccentricity pressing
0.3
1.5
9.71The final result of each joint section calculated is shown in Table2. A very small difference in the calculation result of axial force is found between two models. However, the difference in bending moment is relatively large, with the maximum relative difference being about 8%.Table 2
Calculation results of inner force on joints section.
Joints
UR model
HEB model
Relative difference value (%)
M (kN·m)
N (kN)
M (kN·m)
N (kN)
M
N
B
67.99
1354.51
62.99
1355.62
7.94
0.08
C
−76.74
1566.17
−79.26
1566.39
3.28
0.01
D
31.94
2070.37
32.50
2045.25
1.75
1.23


#### 5.1.2. Comparative Analysis

The internal force on the vault section is used for comparative analysis, with results from the literature [21] and model simulation by this study being shown in Table 3. According to the principle of statistics, all kinds of methods for calculation are random events, and the mean value is closer to the true one. Thus, the mean values of simulation results and relative errors given by various methods are applied for comparative analysis.Table 3
Results comparison of inner force on crown section.
Calculation models
M (kN·m)
N (kN)
Mean value
Relative difference value (%)
M (kN·m)
N (kN)
M
N
UR model
76.94
1344.35
66.08
1347.79
16.43
0.26
MUR model
77.10
1344.22
16.68
0.26
Multihinge ring mode
38.40
1357.10
41.89
0.69
HEB model
71.89
1345.48
8.79
0.17
Note: the reduction coefficient in modified uniform rigidity model was selected asη=0.7.The following results can be seen:(1)
For both bending moment and axial force, gaps between the HEB model results and the mean values are smaller than those of other methods, indicating that the HEB model may produce results closer to the true ones.(2)
The bending moment calculated using various methods are considerably different, whereas the change in axial force is relatively small. This means all models listed in Table3 can meet the accuracy requirements of engineering applications in terms of axial force.(3)
Tables1 and 3 show that the presence of a joint section lowers the stiffness of the joint section, thus resulting in a lower bearing bending moment in a joint compared with that of the UR model. By contrast, this value in the segment section is higher than that of the UR model, which is consistent with previous theoretical studies and engineering practice [8]. Therefore, the current model can better reflect the effect of joints on the stiffness of segment structure, and the model result is reliable.


### 5.2. Case 2

Lei et al. [22] introduced the construction of a shield tunnel in Guangzhou metro line 3, China. The diameter of this tunnel is 5.4 m, constructed by earth pressure balance TBM, using precast reinforced concrete segments as inner lining (the diameter is 6 m). The main geological conditions of tunnel section are filled soil, mucky soil, fine sand layer, diluvium and alluvium layer, and intense weathering mud rock. The stratum condition and parameters for calculation are listed in Table 4 and Figure 10, respectively.Table 4
Calculation parameters of surrounding rock.
Layer name
γ (kN·m−3)
φ (°)
c (kPa)
Filled soil
17.3
28
12
Mucky soil, fine sand layer
18.5
18
6
Diluvium and alluvium layer
19.5
20
1
Intense weathering mud rock
21.6
24
200Figure 10
Tunnel depth and strata distribution for calculating.The relative literature and codes used in calculating the model parameters are shown in Table5. The calculated parameters are then applied to establish a load-structure calculation model, as shown in Figure 9(b). For comparative analysis, the internal forces of the structure are calculated by the UR model, MUR model (stiffness reduction coefficient is 0.7), and the proposed HEB model. For the HEB model, the calculation flow chart in Figure 8 is used for iterative calculation. The initial values are the same as those for a segment section. The iterative and convergence curves of the equivalent elastic modulus of all joints are shown as Figure 11. Results show that the proposed HEB model has good convergence in terms of the internal force calculation for shield segment structure. In addition, the proposed model can produce higher accuracy after 2-3 steps of iteration.Table 5
Summary of calculation parameters.
Parameters
Value
Illustrate
Vertical load,p1 (kPa)
223.32
Whole earth-pillar load
Horizontal load on the top of the tunnel,q1 (kPa)
111.66
Lateral pressure coefficient is 0.5
Horizontal load on the bottom of the tunnel,q2 (kPa)
170.16
Rock resistant coefficient,k (MPa·m−1)
100
[23]
Segment width,b (m)
1.5
/
Segment thickness,h (m)
0.3
Thickness of the reinforcement protective layer of segment,hc (cm)
5.0
Elastic modulus of bolt,Es (MPa)
2.06 × 105
Effective length of bolt,L (cm)
46.0
Cross-sectional area of bolt,Ag (mm2)
1017.88
Grade 8.8 M36
Thickness of transferring cushion,l2 (mm)
5.0
/
Total cross-sectional area of steel bar inner side of the segment,As (mm2)
3770.4
12Φ20
Number of circular bolt on each joint,n
2
/
Elastic modulus of segment concrete,Ec (MPa)
34.5 × 103
C50
Poisson’s ratio of segment concrete,μ
0.17
C50
Elastic modulus of transferring cushion,Ep (MPa)
0.4 × 103
/
Distance from the bolt centroid to the segment inner side,h0 (cm)
12.0
/
Effective length of equivalent beam element,l (m)
0.1
/
Tensile stiffness of bolt,K (N)
2.1 × 105
/
The ratio between the cross-sectional area of bolt and the cross section of steel bar inner side the segment,αcs
0.54
/
Equivalent elastic modulus of equivalent beam element under axial compressive condition,Eeq (MPa)
6.82 × 103
/
Initial pretightening force of bolt,T0 (kN)
200
Grade 8.8 M36Figure 11
Iteration process of the equivalent elastic modulus on joint elements.


### 5.3. Analysis on Results

Figures12 and 13 show the iteration process of the HEB model and the internal force distribution curve fluctuates associated with the update in mechanical parameters of a joint. With the increase of the iteration step, the simulation results approach a stable level gradually. Thus, this method has good convergence. The distribution laws of the final calculation results by three models are similar, coinciding with the theoretical studies and practical experience [8]. This indicates that the proposed HEB model is reliable.Figure 12
Iteration results of the HEB model. (a) Bending moment. (b) Axial force.
(a)
(b)Figure 13
Calculation results by three models. (a) Bending moment. (b) Axial force.The internal forces of joints are shown in Table6 and Figure 14. Note that the values of bending moment produced by the HEB model fall in between those calculated by the other two methods. In addition, the existing research shows that the UR model ignores the effect of joint and artificially increases the stiffness of structure; thus, the calculation result of bending moment is slightly high. By contrast, the MUR model may result in lower values as expected [8]. Therefore, the HEB model can better reflect the effect of joint on the stiffness of lining structure than the other two models.Table 6
Final calculation results of inner force on joint sections calculated by the 3 models.
Models
Joints
Inner force
M (kN·m)
N (kN)
UR model
B
80.03
477.34
C
67.12
692.69
D
19.78
831.50
MUR model
B
72.96
485.87
C
56.95
695.20
D
15.95
838.90
HEB model
B
71.79
484.00
C
61.32
694.42
D
19.11
835.54Figure 14
Comparison of inner force on joint sections. (a) Bending moment. (b) Axial force.
(a)
(b)Further analysis shows that the presence of joints has the greatest impact on the distribution of bending moment. The maximum difference can be as high as 24%. Its impact on axial force is relatively small, with 5% difference, which is consistent with the practical situation [12, 14, 15].


## 5.1. Case 1

To verify the reliability of the proposed model, a comparative analysis is conducted by comparing model simulation with the result from the literature [21]. In this case, a certain interval of the shield tunnel, that is, number 3 Guangzhou subway in China, is used as an example. The segment structure of this interval of shield is designed as a “3 + 2 + 1” block mode. The external diameter of the segment is 6.0 m, and its thickness is 0.3 m. The segment rings are connected through 2 high-tensile bolts (Class 8.8 M24) with initial pretightening force of 200 kN and C50 concrete. The stratum is a weathered stratum in the red beds of Cretaceous System; the depth of tunnel is 22.14 m, and the water level is 18.94 m. The main parameters are shown in Figure 9(a).Figure 9
Calculation model. (a) Geological condition. (b) Structural calculation model.
(a)
(b)


### 5.1.1. Calculation Process

(1)
First, with basic model parameters, applying the UR model (Figure9(b)) to obtain the internal force of each joint section (Table 1). The load calculation takes both geological conditions and ground surcharge p0 = 20 kPa into consideration.
(2)
The mechanical equivalent parameter solution of a joint section deduced according to Section3 is used to solve the equivalent mechanical parameters of each section, as shown in Table 1.
(3)
The equivalent mechanical parameters obtained in Step 2 is used to update those in Step 1. Then, repeating Steps 1–3 to obtain accurate results of the equivalent mechanical parameters. Accordingly, the internal force of the segment produced by the HEB model is obtained finally.Table 1
Equivalent parameters on joints section.
Joints
Inner force evaluated by UR model
Force-bearing types
Equivalent parameters
M (kN·m)
N (kN)
h
′ (m)
b
′ (m)
E
eq (GPa)
B
67.99
1354.51
Positive bending moment and small eccentricity pressing
0.3
1.5
7.36
C
−76.74
1566.17
Hogging bending moment and small eccentricity pressing
0.3
1.5
13.05
D
31.94
2070.37
Positive bending moment and small eccentricity pressing
0.3
1.5
9.71The final result of each joint section calculated is shown in Table2. A very small difference in the calculation result of axial force is found between two models. However, the difference in bending moment is relatively large, with the maximum relative difference being about 8%.Table 2
Calculation results of inner force on joints section.
Joints
UR model
HEB model
Relative difference value (%)
M (kN·m)
N (kN)
M (kN·m)
N (kN)
M
N
B
67.99
1354.51
62.99
1355.62
7.94
0.08
C
−76.74
1566.17
−79.26
1566.39
3.28
0.01
D
31.94
2070.37
32.50
2045.25
1.75
1.23


### 5.1.2. Comparative Analysis

The internal force on the vault section is used for comparative analysis, with results from the literature [21] and model simulation by this study being shown in Table 3. According to the principle of statistics, all kinds of methods for calculation are random events, and the mean value is closer to the true one. Thus, the mean values of simulation results and relative errors given by various methods are applied for comparative analysis.Table 3
Results comparison of inner force on crown section.
Calculation models
M (kN·m)
N (kN)
Mean value
Relative difference value (%)
M (kN·m)
N (kN)
M
N
UR model
76.94
1344.35
66.08
1347.79
16.43
0.26
MUR model
77.10
1344.22
16.68
0.26
Multihinge ring mode
38.40
1357.10
41.89
0.69
HEB model
71.89
1345.48
8.79
0.17
Note: the reduction coefficient in modified uniform rigidity model was selected asη=0.7.The following results can be seen:(1)
For both bending moment and axial force, gaps between the HEB model results and the mean values are smaller than those of other methods, indicating that the HEB model may produce results closer to the true ones.(2)
The bending moment calculated using various methods are considerably different, whereas the change in axial force is relatively small. This means all models listed in Table3 can meet the accuracy requirements of engineering applications in terms of axial force.(3)
Tables1 and 3 show that the presence of a joint section lowers the stiffness of the joint section, thus resulting in a lower bearing bending moment in a joint compared with that of the UR model. By contrast, this value in the segment section is higher than that of the UR model, which is consistent with previous theoretical studies and engineering practice [8]. Therefore, the current model can better reflect the effect of joints on the stiffness of segment structure, and the model result is reliable.


## 5.1.1. Calculation Process

(1)
First, with basic model parameters, applying the UR model (Figure9(b)) to obtain the internal force of each joint section (Table 1). The load calculation takes both geological conditions and ground surcharge p0 = 20 kPa into consideration.
(2)
The mechanical equivalent parameter solution of a joint section deduced according to Section3 is used to solve the equivalent mechanical parameters of each section, as shown in Table 1.
(3)
The equivalent mechanical parameters obtained in Step 2 is used to update those in Step 1. Then, repeating Steps 1–3 to obtain accurate results of the equivalent mechanical parameters. Accordingly, the internal force of the segment produced by the HEB model is obtained finally.Table 1
Equivalent parameters on joints section.
Joints
Inner force evaluated by UR model
Force-bearing types
Equivalent parameters
M (kN·m)
N (kN)
h
′ (m)
b
′ (m)
E
eq (GPa)
B
67.99
1354.51
Positive bending moment and small eccentricity pressing
0.3
1.5
7.36
C
−76.74
1566.17
Hogging bending moment and small eccentricity pressing
0.3
1.5
13.05
D
31.94
2070.37
Positive bending moment and small eccentricity pressing
0.3
1.5
9.71The final result of each joint section calculated is shown in Table2. A very small difference in the calculation result of axial force is found between two models. However, the difference in bending moment is relatively large, with the maximum relative difference being about 8%.Table 2
Calculation results of inner force on joints section.
Joints
UR model
HEB model
Relative difference value (%)
M (kN·m)
N (kN)
M (kN·m)
N (kN)
M
N
B
67.99
1354.51
62.99
1355.62
7.94
0.08
C
−76.74
1566.17
−79.26
1566.39
3.28
0.01
D
31.94
2070.37
32.50
2045.25
1.75
1.23


## 5.1.2. Comparative Analysis

The internal force on the vault section is used for comparative analysis, with results from the literature [21] and model simulation by this study being shown in Table 3. According to the principle of statistics, all kinds of methods for calculation are random events, and the mean value is closer to the true one. Thus, the mean values of simulation results and relative errors given by various methods are applied for comparative analysis.Table 3
Results comparison of inner force on crown section.
Calculation models
M (kN·m)
N (kN)
Mean value
Relative difference value (%)
M (kN·m)
N (kN)
M
N
UR model
76.94
1344.35
66.08
1347.79
16.43
0.26
MUR model
77.10
1344.22
16.68
0.26
Multihinge ring mode
38.40
1357.10
41.89
0.69
HEB model
71.89
1345.48
8.79
0.17
Note: the reduction coefficient in modified uniform rigidity model was selected asη=0.7.The following results can be seen:(1)
For both bending moment and axial force, gaps between the HEB model results and the mean values are smaller than those of other methods, indicating that the HEB model may produce results closer to the true ones.(2)
The bending moment calculated using various methods are considerably different, whereas the change in axial force is relatively small. This means all models listed in Table3 can meet the accuracy requirements of engineering applications in terms of axial force.(3)
Tables1 and 3 show that the presence of a joint section lowers the stiffness of the joint section, thus resulting in a lower bearing bending moment in a joint compared with that of the UR model. By contrast, this value in the segment section is higher than that of the UR model, which is consistent with previous theoretical studies and engineering practice [8]. Therefore, the current model can better reflect the effect of joints on the stiffness of segment structure, and the model result is reliable.


## 5.2. Case 2

Lei et al. [22] introduced the construction of a shield tunnel in Guangzhou metro line 3, China. The diameter of this tunnel is 5.4 m, constructed by earth pressure balance TBM, using precast reinforced concrete segments as inner lining (the diameter is 6 m). The main geological conditions of tunnel section are filled soil, mucky soil, fine sand layer, diluvium and alluvium layer, and intense weathering mud rock. The stratum condition and parameters for calculation are listed in Table 4 and Figure 10, respectively.Table 4
Calculation parameters of surrounding rock.
Layer name
γ (kN·m−3)
φ (°)
c (kPa)
Filled soil
17.3
28
12
Mucky soil, fine sand layer
18.5
18
6
Diluvium and alluvium layer
19.5
20
1
Intense weathering mud rock
21.6
24
200Figure 10
Tunnel depth and strata distribution for calculating.The relative literature and codes used in calculating the model parameters are shown in Table5. The calculated parameters are then applied to establish a load-structure calculation model, as shown in Figure 9(b). For comparative analysis, the internal forces of the structure are calculated by the UR model, MUR model (stiffness reduction coefficient is 0.7), and the proposed HEB model. For the HEB model, the calculation flow chart in Figure 8 is used for iterative calculation. The initial values are the same as those for a segment section. The iterative and convergence curves of the equivalent elastic modulus of all joints are shown as Figure 11. Results show that the proposed HEB model has good convergence in terms of the internal force calculation for shield segment structure. In addition, the proposed model can produce higher accuracy after 2-3 steps of iteration.Table 5
Summary of calculation parameters.
Parameters
Value
Illustrate
Vertical load,p1 (kPa)
223.32
Whole earth-pillar load
Horizontal load on the top of the tunnel,q1 (kPa)
111.66
Lateral pressure coefficient is 0.5
Horizontal load on the bottom of the tunnel,q2 (kPa)
170.16
Rock resistant coefficient,k (MPa·m−1)
100
[23]
Segment width,b (m)
1.5
/
Segment thickness,h (m)
0.3
Thickness of the reinforcement protective layer of segment,hc (cm)
5.0
Elastic modulus of bolt,Es (MPa)
2.06 × 105
Effective length of bolt,L (cm)
46.0
Cross-sectional area of bolt,Ag (mm2)
1017.88
Grade 8.8 M36
Thickness of transferring cushion,l2 (mm)
5.0
/
Total cross-sectional area of steel bar inner side of the segment,As (mm2)
3770.4
12Φ20
Number of circular bolt on each joint,n
2
/
Elastic modulus of segment concrete,Ec (MPa)
34.5 × 103
C50
Poisson’s ratio of segment concrete,μ
0.17
C50
Elastic modulus of transferring cushion,Ep (MPa)
0.4 × 103
/
Distance from the bolt centroid to the segment inner side,h0 (cm)
12.0
/
Effective length of equivalent beam element,l (m)
0.1
/
Tensile stiffness of bolt,K (N)
2.1 × 105
/
The ratio between the cross-sectional area of bolt and the cross section of steel bar inner side the segment,αcs
0.54
/
Equivalent elastic modulus of equivalent beam element under axial compressive condition,Eeq (MPa)
6.82 × 103
/
Initial pretightening force of bolt,T0 (kN)
200
Grade 8.8 M36Figure 11
Iteration process of the equivalent elastic modulus on joint elements.


## 5.3. Analysis on Results

Figures12 and 13 show the iteration process of the HEB model and the internal force distribution curve fluctuates associated with the update in mechanical parameters of a joint. With the increase of the iteration step, the simulation results approach a stable level gradually. Thus, this method has good convergence. The distribution laws of the final calculation results by three models are similar, coinciding with the theoretical studies and practical experience [8]. This indicates that the proposed HEB model is reliable.Figure 12
Iteration results of the HEB model. (a) Bending moment. (b) Axial force.
(a)
(b)Figure 13
Calculation results by three models. (a) Bending moment. (b) Axial force.The internal forces of joints are shown in Table6 and Figure 14. Note that the values of bending moment produced by the HEB model fall in between those calculated by the other two methods. In addition, the existing research shows that the UR model ignores the effect of joint and artificially increases the stiffness of structure; thus, the calculation result of bending moment is slightly high. By contrast, the MUR model may result in lower values as expected [8]. Therefore, the HEB model can better reflect the effect of joint on the stiffness of lining structure than the other two models.Table 6
Final calculation results of inner force on joint sections calculated by the 3 models.
Models
Joints
Inner force
M (kN·m)
N (kN)
UR model
B
80.03
477.34
C
67.12
692.69
D
19.78
831.50
MUR model
B
72.96
485.87
C
56.95
695.20
D
15.95
838.90
HEB model
B
71.79
484.00
C
61.32
694.42
D
19.11
835.54Figure 14
Comparison of inner force on joint sections. (a) Bending moment. (b) Axial force.
(a)
(b)Further analysis shows that the presence of joints has the greatest impact on the distribution of bending moment. The maximum difference can be as high as 24%. Its impact on axial force is relatively small, with 5% difference, which is consistent with the practical situation [12, 14, 15].


## 6. Conclusions

Starting from-force transferring properties of segment joint, this paper converts a noncontinuous segment structure into a continuous heterogeneous structure through equivalent conversion of mechanical parameters of a joint section. The solution of equivalent mechanical parameters of joint section is deduced and obtained according to material mechanics, thus avoiding the uncertainty and complexity of segment calculation induced by joint. An iterative calculation flow of the proposed HEB model is also provided.Model validation and applications show that the axial forces of a segment structure obtained by all models are slightly different, and all models can meet the accuracy requirement of engineering application. However, for bending moment, simulation results are considerably different; particularly in the joint section, where the maximum deviation of bending moment obtained by the proposed HEB model can be up to 20% and more. Therefore, the selection of calculation model is very important in design analysis. Comparative analysis of different segment structure calculation models shows that the proposed heterogeneous equivalent model can better reflect the effects of joint on the stiffness of a segment. The proposed model provides reliable results with better convergence. Thus, the proposed model may provide useful suggestions for designers, engineers, and scholars who focus on the designing of shield tunnel, assessment of segment joints’ stability and durability, maintenance of shield tunnel subway, and investment estimation of subway operation.



---

*Source: 9637838-2018-08-26.xml*