# The Stability Analysis of Tunnel Lining Structure with Seismic Excitation Based on the Energy Evaluation Principle

Authors: Yumin Wen; Chunlei Xin; Xi Zhang; Zeming Huang; Yusheng Shen; Wenjie Lei; Bo Gao
Journal: Shock and Vibration
 (2021)

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## Abstract

Earthquakes are vibrations induced by the rapid releases of large quantities of energy from the crustal movements. During seismic excitation, there are kinetic energy, damping energy, and strain energy acting on the tunnel structure. Based on the indexes of the total energy, releasable elastic strain energy, and dissipated energy, this paper proposes three energy evaluation criteria for the tunnel structure, which are applied to the optimization of the aseismic design of the cross-sectional shape and material property of the tunnel structure. It can be concluded that the peak values and accumulated values of elastic strain energy at the spandrel and arch springing are significantly larger than other positions, which indicates that the strengths of the spandrel and arch springing are the most influential factor for the seismic damage of the tunnel structure. Considering this factor, the width-to-height ratio of 1.33 and Poisson’s ratio of 0.3 are determined as the most optimal cross-sectional shape and material property, respectively. Furthermore, by analyzing the relationship between the internal energy and the input energy of the tunnel structure with seismic excitation and proposing an equation for the evaluation of the dynamic stability of tunnel structure, the stabilities of the tunnel structure with different PGAs are analyzed; it can be concluded that the larger the peak value of seismic wave acceleration, the longer the instable period and the greater the degree of dynamic instability. The derived equations can be used as references for the seismic analysis of the tunnel structure, and the conclusions of this paper can contribute to the aseismic design of tunnel lining.


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## Body

## 1. Introduction

Energy is an index that describes the state of the whole system [1], which includes the overall dynamic response [2, 3], structural parameters [4], and external excitation [5]. Therefore, the mechanism of the instability of structures with seismic excitation can be revealed based on the principle of energy [6].Originally, the convergence or divergence of the motion index based on the periodic load was proposed to analyze the dynamic response of structures [7]. However, there would be a jump phenomenon when applied to the nonlinear dynamic response analysis [8]. Afterwards, the quasi-static stiffness principle based on the tangent stiffness matrix was proposed [9, 10], which could evaluate the dynamic instability conditions of structures [11]. In terms of that principle, the motion equation was transformed to the differential equation of the first order [12], and its stability can be evaluated by the positive definite stability theory of the matrix in motion [13]. Moreover, in regard to the B-R principle, incremental dynamic analysis (IDA) curves were used to describe the relation of the strength and displacement [14], which could be effectively used for the dynamic stability analysis of structures. However, the IDA curves occasionally displayed nonmonotonic behaviors and discontinuous points [15]; sometimes, it was difficult to use them to identify the critical strength of structures [16, 17], which could influence the dynamic stability analysis of structures [18]; hence, there was no unified quantitative criterion to define the dynamic response of structures based on the B-R principle [19].Recently, the principle of energy conservation was used to describe the dynamic instability mechanism of structures with seismic excitation [20, 21], pointing out that the structures were stable when the input seismic energy was less than the critical value of the energy possessed by the structures [22]. However, it was difficult to determine the critical value of the energy possessed by complex structures with seismic excitation [23, 24].The stability of the tunnel structure with seismic excitation is closely related to the external vibration intensity and internal structural characteristics [25]; it is essential to adopt an appropriate principle to evaluate the stability of the tunnel structure with seismic excitation [26, 27]. From the perspective of the energy generated from the system and the internal tunnel structure induced by seismic excitation and considering the external vibration intensity and structural characteristics [28], the physical mechanism of seismic instability of complex deformation in the tunnel structure is researched in this paper.


## 2. The Methodology

### 2.1. Energy Balance Equation of Tunnel Structure

Only considering the single layer structure of tunnel lining, the multi-degree-of-freedom equation of motion can be expressed as follows [29]:(1)Mu¨t+Cu˙t+fu,t=−Mru˙gt,where M is the mass matrix, C is the damping matrix, fu,t is the internal force, r is the load impact matrix, u˙gt is the seismic ground acceleration vector, and u¨t, u˙t, and ut are the acceleration, velocity, and displacement vectors, respectively.Considering that the aseismic stability of the structure is closely related to the internal structural characteristics and external excitation load, integrating the terms in equation (1) over time, the energy equation in the form of work can be obtained as follows [30]:(2)∫0tu˙TMu¨tdt+∫0tu˙TCu˙tdt+∫0tu˙Tfu,tdt=−∫0tMru˙gtdt.Rewriting equation (2) into the energy balance equation, equation (3) can be obtained as follows [31]:(3)EKt+EDt+ESt=EIt,where EKt is the kinetic energy, EDt is the damping energy, ESt is the strain energy, and EIt is the input total energy. Each term can be expressed as the following equations, respectively:(4)EKt=∫0tu˙TMu¨tdt,(5)EDt=∫0tu˙TCu˙tdt,(6)ESt=∫0tu˙Tfu,tdt,(7)EIt=−∫0tMru˙gtdt.


### 2.2. Three Energy Evaluation Criteria of the Stability of Tunnel Structure

#### 2.2.1. Stability Evaluation Criterion Based on the Total Energy

When the structure is motivated by the external excitation [32], there are kinetic energy, damping energy, and strain energy existing inside the structure [33, 34]. The structure is dynamically stable when the total input energy is equal to the sum of the internal strain energy and dissipated energy [35]. The structure is still dynamically stable when the total input energy is greater than the sum of the internal strain energy and dissipated energy due to the fact that the redundant input energy will transform to other forms of energy to be dissipated [36, 37]. On the contrary, when the total input energy is less than the sum of the internal strain energy and dissipated energy, the structure is no longer stable and extradeformation will occur to offset the energy difference inside and outside the system. Therefore, the physical mechanism of structural dynamic instability can be attributed to the unbalance of the energy required by structural deformation and external input energy [38]. The state of the dynamic balance can be expressed as(8)Fextt=Fintt,where Fextt is the instantaneous external force vector and Fintt is the instantaneous internal force vector. Regarding the structural dynamic force, damping force, and the seismic force as the external force vector and regarding the restoring force generated by structural deformation as the internal force vector, the following equations can be obtained as(9)Fextt=−Mru˙gt−Mu¨t−Cu˙t,(10)Fintt=fu,t.The instantaneous work done by the instantaneous external force vector with seismic excitation can be expressed as(11)Wextt=−Mru˙gt−Mu¨t−Cu˙tTut.The work done by the restoring force vector of the internal deformation of the structure at a certain time can be expressed as(12)Wintt=∫0ufu,tdu.The difference between the instantaneous work done by the external and internal force vectors is denoted asWt, and equation (13) can be obtained as(13)Wt=−Mru˙gt−Mu¨t−Cu˙tTut−∫0ufu,tdu.From the perspective of thermodynamics, the work done by the internal and external forces during an adiabatic process actually represents the variation of the energy rather than the stored energy in the structure [39]. With seismic excitation, the work done by the internal and external forces of the structure is constantly varying, which can be expressed as(14)Eintrt=Wt=Wextt−Wintt.Therein, the input energy of the structure can be calculated by equation (7). The structure is instable when the input energy EIt is less than the internal energy Eintrt; it will absorb extra energy to offset the input energy so that the input energy can be equal to or greater than the intrinsic energy, which represents the dynamic stable state. Therefore, based on the energy evaluation principle, the structural stability can be expressed as(15)Eintrt≤EIt,stable,t>0,Eintrt>EIt,instable,t<0.Therefore, if the intrinsic energy curve of the structure exceeds the input energy curve, the structure is in a dynamic instable state; the dynamic stability equation can be expressed as(16)St=EIt−Eintrt,where the structure is dynamically stable when St≥0 and the structure is instable when St<0.


#### 2.2.2. Strength Failure Criterion Based on the Dissipated Energy

According to the law of thermodynamics, energy conversion is an essential feature of the physical process, and substance destruction is a phenomenon of instability induced by energy [40, 41]. According to equation (3), the accumulated elastic strain energy ESt can be divided into the elastic strain energy Eet and the plastic strain energy Ept, as shown in(17)ESt=Eet+Ept.In the process of element deformation, the elastic strain energy at the elastic stage continues to increase until the yield limit is reached, which means that the elastic strain energy of the element reaches the maximum [42]. When the yield limit is exceeded, the material will undergo an irrecoverable plastic deformation, and it reaches the maximum when the material is almost destroyed, which means that the accumulated plastic strain energy reached the maximum [43, 44]. With seismic excitation, the structure elements alternatively exhibit tension and compression; they will experience the cycle of loading, compaction, elasticity, plasticity, and unloading, which induces cyclic impact on the material and brings about crack expansion, crack penetration, and even damage of the structure [45].During seismic excitation, seismic energy is continuously input into the structural system over time; a part of seismic energy is dissipated by the structure through its damping and plastic deformation, and the other part of seismic energy is converted into the kinetic energy of the structure or absorbed in the form of elastic strain energy of the structure [46]. It can be concluded from equation (3) that the total input energy of the structure is converted to the kinetic energy, damping energy, elastic strain energy, and plastic strain energy. In view of that the elastic strain energy and plastic strain energy correspond to the deformation states of the structure, the stability of the structure can be evaluated by calculating the strain energy and analyzing the tendency of the strain energy [47, 48].(1)Strain Energy. The strain energy density can be calculated from the stress and strain of the structure elements, as shown in(18)w=∫σijεijdεij.There is a strain equation for the elastoplastic material, as shown in(19)εij=εije+εijp.The strain energy density is expressed as(20)w=12σijεije+σijεijp.Besides, the strain energy of the element can be expressed as(21)we=∫Vwdv,where w is the strain energy density of the element and we is the strain energy of the element.The total strain energy of the structure can be obtained by summing the strain energy of each element, as shown in(22)W=∑we.(2)Strength Failure Theory Based on the Dissipated Energy. Considering the deformation of the element with external force, assuming that there is no heat exchange with surrounding, and denoting the total input energy generated by the external force as U, the following equation can be obtained according to the first law of thermodynamics as(23)U=Up+Ue,where Up is the plastic strain energy and Ue is the releasable elastic strain energy. The plastic strain energy Up is used to form internal damage and plastic deformation, which result in the strength failure, and the release of the elastic strain energy Ue stored in the element is the underlying reason for the sudden failure of the tunnel element.The energy of each part of the element in the principal stress space can be expressed as follows:(24)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3,(25)Ue=12σ1ε1e+12σ2ε2e+12σ3ε3e,(26)Up=U−Ue.The energy damage quantity of the tunnel element can be defined as(27)ϖ=UpUc,where Uc is the critical value of energy dissipation when the element loses its strength, which is a material constant determined by the uniaxial tension test, uniaxial compression test, and pure shear test; the calculation method of Uc is same as that of Up. When ϖ=1, it denotes that the material loses its strength no matter what stress state the material is in, which can be expressed as(28)ϖ=UpUc=1.Substituting equation (28) into equations (25) and (26), the strength failure criterion based on the dissipated energy can be obtained as(29)∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3−12σ1ε1e−12σ2ε2e−12σ3ε3e=Uc.


#### 2.2.3. Overall Failure Criterion Based on the Releasable Elastic Strain Energy

It can be concluded from equation (25) that the releasable strain energy stored by the element is related to the unloading elastic modulus Ei and Poisson’s ratio v after damaged. Considering the orthotropy of the element damage, the damage is directly related to the elasticity. Assuming that the compressive stress is positive, when the element is not damaged, it can be expressed as(30)Ue=12σiεie=12σ12E1+σ22E2+σ32E3−v1E1+1E2σ1σ2+1E1+1E3σ1σ3+1E2+1E3σ2σ3.In terms of the damaged structure, the conventional damage variableωi is introduced to consider the effect of damage on the unloading modulus Ei of the structure, as shown in(31)Ei=1−ωiE0,where E0 is the initial elastic modulus when the structure is not damaged.Assuming that Poisson’s ratio is not affected by the damage and substituting it into equation (30), then the following equation can be obtained as(32)Ue=12σiεie=12E0σ121−ω1+σ221−ω2+σ321−ω3−v11−ω1+11−ω2σ1σ2+11−ω1+11−ω3σ1σ3+11−ω2+11−ω3σ2σ3.Considering the average effect of the damage along the three directions of the principal stresses, equation (32) can be simplified as(33)E¯=E01−ω¯,v¯=v1−ω¯,ωi=ω¯,i=1,2,and3.The releasable elastic strain energy can be rewritten as(34)Ue=12E¯σ12+σ22+σ32−2v¯σ1σ2+σ1σ3+σ2σ3.Herein, the average valuesE¯,v¯,andω¯ can be determined by the unidirectional cyclic compression loading and unloading tests. It can also be taken as the initial elastic modulus E0 and Poisson’s ratio v for convenience; then, equation (34) can be rewritten as(35)Ue=12Eσ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3.Equation (35) is aimed at the linear unloading process in the nonlinear process of the element. Summing up the releasable elastic strain energy Uie of each element, the releasable elastic strain energy ∑Uie of the whole structure can be obtained. As the damage of the element increases with external action, the strength gradually decreases; when the releasable elastic strain energy Uie of an element reaches the surface energy U0 required by the element damage, Uie will be released in the form of elastic surface energy. When the damaged elements reach a certain critical number, the whole structure will be damaged.Based on the above releasable energy concept, the overall failure criterion of the structure element can be concluded that a part of the work done by the external force on the structure is converted into the dissipated energyUp, which makes the structure gradually loses its strength, and the other part is converted into the gradually increasing releasable elastic strain energy Ue. When Ue reaches the value of the surface energy U0 of the tunnel element, the strain energy Ue is released and the whole structure element is damaged. In the principal stress space, Ue is difficult to be released along the direction of the maximum compressive stress σ1, but is apt to be released along the direction of the minimum compressive stress or tensile stress σ3. The overall failure criteria of the element with compressive stress and tensile stress are given below.(1)Three-Directional Compressive Stress State (σ1>σ2>σ3≥0, Assuming That the Compressive Stress Is Positive). When the element is in a three-directional compressive stress state and the overall instability occurs, the releasable elastic strain energy will be released along the three directions of stresses. The quantity of the released energy is proportional to the releasable energy stored in the element, and the released strain energy Gi along each direction is distributed according to the difference value with the maximum principal stress, which can be expressed as(36)Gi=Kiσ1−σiUe,where G3 is the positive correlation coefficient that can be determined by the uniaxial compression test and Gc is the releasable elastic strain energy in the element.It can be known from equation (36) that the releasable elastic strain energy in the element is primarily released along the third principal stress direction. When the released strain energy along the third principal stress direction G3 reaches a critical value Gc, the releasable elastic strain energy will be released firstly along this direction, which can be expressed as(37)G3=K3σ1−σ3Ue=Gc,where Gc is the material constant of the element, which can be determined by the uniaxial compression test. When the tunnel element is in the uniaxial compression state, the element will be overall instable if the element undergoes strength failure. Assuming that σ1=σc when the element undergoes uniaxial compression strength failure, then σ2=σ3=0 and the following equation can be obtained as(38)Ue=σc22E0.Substituting equation (38) into equation (37), the following equation can be obtained as(39)Gc=K3σc32E0.Therefore, the instability criterion of the element in the three-directional compressive stress state can be obtained as(40)Ue=σc22E0σ1−σ3.It can be also expressed as(41)σ1−σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σc3.(2)Stress State with Both Tension and Compression (σ1>0,σ3<0). When at least one of the principal stresses is tensile stress, considering that the tensile stress of any magnitude will promote the energy release of the damaged element, the stored releasable strain energy Ue is distributed according to the principal stress value along the three directions of principal stress. Therefore, the energy release rate Gi of the element along the σi direction can be defined as(42)Gi=KiσiUe.The releasable elastic strain energy in the element is primarily released along the direction of the maximum principal tensile stress. Therefore, it is necessary to satisfy the following equation when the overall instability occurs:(43)G3=K3σ3Ue=Gt,where Gt is the critical strain energy release rate of the element with tensile stress, which is a material constant determined by the uniaxial tensile test.When the element is in uniaxial tension state, it will be damaged if the strength is lost. Assuming thatσ3=σt,σ1=σ2=0 and substituting them into equation (35), the following equation can be obtained as(44)Ue=σt22E0.Then, substituting it into equation (42), the following equation can be obtained as(45)Gt=K3σt32E0.Moreover, substituting equation (45) into equation (43), the following equation can be obtained as(46)σ3Ue=σt32E0.Eventually, substituting equation (46) into equation (35), the overall instability criterion in the stress state with both tension and compression can be obtained as(47)σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σt3.


## 2.1. Energy Balance Equation of Tunnel Structure

Only considering the single layer structure of tunnel lining, the multi-degree-of-freedom equation of motion can be expressed as follows [29]:(1)Mu¨t+Cu˙t+fu,t=−Mru˙gt,where M is the mass matrix, C is the damping matrix, fu,t is the internal force, r is the load impact matrix, u˙gt is the seismic ground acceleration vector, and u¨t, u˙t, and ut are the acceleration, velocity, and displacement vectors, respectively.Considering that the aseismic stability of the structure is closely related to the internal structural characteristics and external excitation load, integrating the terms in equation (1) over time, the energy equation in the form of work can be obtained as follows [30]:(2)∫0tu˙TMu¨tdt+∫0tu˙TCu˙tdt+∫0tu˙Tfu,tdt=−∫0tMru˙gtdt.Rewriting equation (2) into the energy balance equation, equation (3) can be obtained as follows [31]:(3)EKt+EDt+ESt=EIt,where EKt is the kinetic energy, EDt is the damping energy, ESt is the strain energy, and EIt is the input total energy. Each term can be expressed as the following equations, respectively:(4)EKt=∫0tu˙TMu¨tdt,(5)EDt=∫0tu˙TCu˙tdt,(6)ESt=∫0tu˙Tfu,tdt,(7)EIt=−∫0tMru˙gtdt.


## 2.2. Three Energy Evaluation Criteria of the Stability of Tunnel Structure

### 2.2.1. Stability Evaluation Criterion Based on the Total Energy

When the structure is motivated by the external excitation [32], there are kinetic energy, damping energy, and strain energy existing inside the structure [33, 34]. The structure is dynamically stable when the total input energy is equal to the sum of the internal strain energy and dissipated energy [35]. The structure is still dynamically stable when the total input energy is greater than the sum of the internal strain energy and dissipated energy due to the fact that the redundant input energy will transform to other forms of energy to be dissipated [36, 37]. On the contrary, when the total input energy is less than the sum of the internal strain energy and dissipated energy, the structure is no longer stable and extradeformation will occur to offset the energy difference inside and outside the system. Therefore, the physical mechanism of structural dynamic instability can be attributed to the unbalance of the energy required by structural deformation and external input energy [38]. The state of the dynamic balance can be expressed as(8)Fextt=Fintt,where Fextt is the instantaneous external force vector and Fintt is the instantaneous internal force vector. Regarding the structural dynamic force, damping force, and the seismic force as the external force vector and regarding the restoring force generated by structural deformation as the internal force vector, the following equations can be obtained as(9)Fextt=−Mru˙gt−Mu¨t−Cu˙t,(10)Fintt=fu,t.The instantaneous work done by the instantaneous external force vector with seismic excitation can be expressed as(11)Wextt=−Mru˙gt−Mu¨t−Cu˙tTut.The work done by the restoring force vector of the internal deformation of the structure at a certain time can be expressed as(12)Wintt=∫0ufu,tdu.The difference between the instantaneous work done by the external and internal force vectors is denoted asWt, and equation (13) can be obtained as(13)Wt=−Mru˙gt−Mu¨t−Cu˙tTut−∫0ufu,tdu.From the perspective of thermodynamics, the work done by the internal and external forces during an adiabatic process actually represents the variation of the energy rather than the stored energy in the structure [39]. With seismic excitation, the work done by the internal and external forces of the structure is constantly varying, which can be expressed as(14)Eintrt=Wt=Wextt−Wintt.Therein, the input energy of the structure can be calculated by equation (7). The structure is instable when the input energy EIt is less than the internal energy Eintrt; it will absorb extra energy to offset the input energy so that the input energy can be equal to or greater than the intrinsic energy, which represents the dynamic stable state. Therefore, based on the energy evaluation principle, the structural stability can be expressed as(15)Eintrt≤EIt,stable,t>0,Eintrt>EIt,instable,t<0.Therefore, if the intrinsic energy curve of the structure exceeds the input energy curve, the structure is in a dynamic instable state; the dynamic stability equation can be expressed as(16)St=EIt−Eintrt,where the structure is dynamically stable when St≥0 and the structure is instable when St<0.


### 2.2.2. Strength Failure Criterion Based on the Dissipated Energy

According to the law of thermodynamics, energy conversion is an essential feature of the physical process, and substance destruction is a phenomenon of instability induced by energy [40, 41]. According to equation (3), the accumulated elastic strain energy ESt can be divided into the elastic strain energy Eet and the plastic strain energy Ept, as shown in(17)ESt=Eet+Ept.In the process of element deformation, the elastic strain energy at the elastic stage continues to increase until the yield limit is reached, which means that the elastic strain energy of the element reaches the maximum [42]. When the yield limit is exceeded, the material will undergo an irrecoverable plastic deformation, and it reaches the maximum when the material is almost destroyed, which means that the accumulated plastic strain energy reached the maximum [43, 44]. With seismic excitation, the structure elements alternatively exhibit tension and compression; they will experience the cycle of loading, compaction, elasticity, plasticity, and unloading, which induces cyclic impact on the material and brings about crack expansion, crack penetration, and even damage of the structure [45].During seismic excitation, seismic energy is continuously input into the structural system over time; a part of seismic energy is dissipated by the structure through its damping and plastic deformation, and the other part of seismic energy is converted into the kinetic energy of the structure or absorbed in the form of elastic strain energy of the structure [46]. It can be concluded from equation (3) that the total input energy of the structure is converted to the kinetic energy, damping energy, elastic strain energy, and plastic strain energy. In view of that the elastic strain energy and plastic strain energy correspond to the deformation states of the structure, the stability of the structure can be evaluated by calculating the strain energy and analyzing the tendency of the strain energy [47, 48].(1)Strain Energy. The strain energy density can be calculated from the stress and strain of the structure elements, as shown in(18)w=∫σijεijdεij.There is a strain equation for the elastoplastic material, as shown in(19)εij=εije+εijp.The strain energy density is expressed as(20)w=12σijεije+σijεijp.Besides, the strain energy of the element can be expressed as(21)we=∫Vwdv,where w is the strain energy density of the element and we is the strain energy of the element.The total strain energy of the structure can be obtained by summing the strain energy of each element, as shown in(22)W=∑we.(2)Strength Failure Theory Based on the Dissipated Energy. Considering the deformation of the element with external force, assuming that there is no heat exchange with surrounding, and denoting the total input energy generated by the external force as U, the following equation can be obtained according to the first law of thermodynamics as(23)U=Up+Ue,where Up is the plastic strain energy and Ue is the releasable elastic strain energy. The plastic strain energy Up is used to form internal damage and plastic deformation, which result in the strength failure, and the release of the elastic strain energy Ue stored in the element is the underlying reason for the sudden failure of the tunnel element.The energy of each part of the element in the principal stress space can be expressed as follows:(24)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3,(25)Ue=12σ1ε1e+12σ2ε2e+12σ3ε3e,(26)Up=U−Ue.The energy damage quantity of the tunnel element can be defined as(27)ϖ=UpUc,where Uc is the critical value of energy dissipation when the element loses its strength, which is a material constant determined by the uniaxial tension test, uniaxial compression test, and pure shear test; the calculation method of Uc is same as that of Up. When ϖ=1, it denotes that the material loses its strength no matter what stress state the material is in, which can be expressed as(28)ϖ=UpUc=1.Substituting equation (28) into equations (25) and (26), the strength failure criterion based on the dissipated energy can be obtained as(29)∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3−12σ1ε1e−12σ2ε2e−12σ3ε3e=Uc.


### 2.2.3. Overall Failure Criterion Based on the Releasable Elastic Strain Energy

It can be concluded from equation (25) that the releasable strain energy stored by the element is related to the unloading elastic modulus Ei and Poisson’s ratio v after damaged. Considering the orthotropy of the element damage, the damage is directly related to the elasticity. Assuming that the compressive stress is positive, when the element is not damaged, it can be expressed as(30)Ue=12σiεie=12σ12E1+σ22E2+σ32E3−v1E1+1E2σ1σ2+1E1+1E3σ1σ3+1E2+1E3σ2σ3.In terms of the damaged structure, the conventional damage variableωi is introduced to consider the effect of damage on the unloading modulus Ei of the structure, as shown in(31)Ei=1−ωiE0,where E0 is the initial elastic modulus when the structure is not damaged.Assuming that Poisson’s ratio is not affected by the damage and substituting it into equation (30), then the following equation can be obtained as(32)Ue=12σiεie=12E0σ121−ω1+σ221−ω2+σ321−ω3−v11−ω1+11−ω2σ1σ2+11−ω1+11−ω3σ1σ3+11−ω2+11−ω3σ2σ3.Considering the average effect of the damage along the three directions of the principal stresses, equation (32) can be simplified as(33)E¯=E01−ω¯,v¯=v1−ω¯,ωi=ω¯,i=1,2,and3.The releasable elastic strain energy can be rewritten as(34)Ue=12E¯σ12+σ22+σ32−2v¯σ1σ2+σ1σ3+σ2σ3.Herein, the average valuesE¯,v¯,andω¯ can be determined by the unidirectional cyclic compression loading and unloading tests. It can also be taken as the initial elastic modulus E0 and Poisson’s ratio v for convenience; then, equation (34) can be rewritten as(35)Ue=12Eσ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3.Equation (35) is aimed at the linear unloading process in the nonlinear process of the element. Summing up the releasable elastic strain energy Uie of each element, the releasable elastic strain energy ∑Uie of the whole structure can be obtained. As the damage of the element increases with external action, the strength gradually decreases; when the releasable elastic strain energy Uie of an element reaches the surface energy U0 required by the element damage, Uie will be released in the form of elastic surface energy. When the damaged elements reach a certain critical number, the whole structure will be damaged.Based on the above releasable energy concept, the overall failure criterion of the structure element can be concluded that a part of the work done by the external force on the structure is converted into the dissipated energyUp, which makes the structure gradually loses its strength, and the other part is converted into the gradually increasing releasable elastic strain energy Ue. When Ue reaches the value of the surface energy U0 of the tunnel element, the strain energy Ue is released and the whole structure element is damaged. In the principal stress space, Ue is difficult to be released along the direction of the maximum compressive stress σ1, but is apt to be released along the direction of the minimum compressive stress or tensile stress σ3. The overall failure criteria of the element with compressive stress and tensile stress are given below.(1)Three-Directional Compressive Stress State (σ1>σ2>σ3≥0, Assuming That the Compressive Stress Is Positive). When the element is in a three-directional compressive stress state and the overall instability occurs, the releasable elastic strain energy will be released along the three directions of stresses. The quantity of the released energy is proportional to the releasable energy stored in the element, and the released strain energy Gi along each direction is distributed according to the difference value with the maximum principal stress, which can be expressed as(36)Gi=Kiσ1−σiUe,where G3 is the positive correlation coefficient that can be determined by the uniaxial compression test and Gc is the releasable elastic strain energy in the element.It can be known from equation (36) that the releasable elastic strain energy in the element is primarily released along the third principal stress direction. When the released strain energy along the third principal stress direction G3 reaches a critical value Gc, the releasable elastic strain energy will be released firstly along this direction, which can be expressed as(37)G3=K3σ1−σ3Ue=Gc,where Gc is the material constant of the element, which can be determined by the uniaxial compression test. When the tunnel element is in the uniaxial compression state, the element will be overall instable if the element undergoes strength failure. Assuming that σ1=σc when the element undergoes uniaxial compression strength failure, then σ2=σ3=0 and the following equation can be obtained as(38)Ue=σc22E0.Substituting equation (38) into equation (37), the following equation can be obtained as(39)Gc=K3σc32E0.Therefore, the instability criterion of the element in the three-directional compressive stress state can be obtained as(40)Ue=σc22E0σ1−σ3.It can be also expressed as(41)σ1−σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σc3.(2)Stress State with Both Tension and Compression (σ1>0,σ3<0). When at least one of the principal stresses is tensile stress, considering that the tensile stress of any magnitude will promote the energy release of the damaged element, the stored releasable strain energy Ue is distributed according to the principal stress value along the three directions of principal stress. Therefore, the energy release rate Gi of the element along the σi direction can be defined as(42)Gi=KiσiUe.The releasable elastic strain energy in the element is primarily released along the direction of the maximum principal tensile stress. Therefore, it is necessary to satisfy the following equation when the overall instability occurs:(43)G3=K3σ3Ue=Gt,where Gt is the critical strain energy release rate of the element with tensile stress, which is a material constant determined by the uniaxial tensile test.When the element is in uniaxial tension state, it will be damaged if the strength is lost. Assuming thatσ3=σt,σ1=σ2=0 and substituting them into equation (35), the following equation can be obtained as(44)Ue=σt22E0.Then, substituting it into equation (42), the following equation can be obtained as(45)Gt=K3σt32E0.Moreover, substituting equation (45) into equation (43), the following equation can be obtained as(46)σ3Ue=σt32E0.Eventually, substituting equation (46) into equation (35), the overall instability criterion in the stress state with both tension and compression can be obtained as(47)σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σt3.


## 2.2.1. Stability Evaluation Criterion Based on the Total Energy

When the structure is motivated by the external excitation [32], there are kinetic energy, damping energy, and strain energy existing inside the structure [33, 34]. The structure is dynamically stable when the total input energy is equal to the sum of the internal strain energy and dissipated energy [35]. The structure is still dynamically stable when the total input energy is greater than the sum of the internal strain energy and dissipated energy due to the fact that the redundant input energy will transform to other forms of energy to be dissipated [36, 37]. On the contrary, when the total input energy is less than the sum of the internal strain energy and dissipated energy, the structure is no longer stable and extradeformation will occur to offset the energy difference inside and outside the system. Therefore, the physical mechanism of structural dynamic instability can be attributed to the unbalance of the energy required by structural deformation and external input energy [38]. The state of the dynamic balance can be expressed as(8)Fextt=Fintt,where Fextt is the instantaneous external force vector and Fintt is the instantaneous internal force vector. Regarding the structural dynamic force, damping force, and the seismic force as the external force vector and regarding the restoring force generated by structural deformation as the internal force vector, the following equations can be obtained as(9)Fextt=−Mru˙gt−Mu¨t−Cu˙t,(10)Fintt=fu,t.The instantaneous work done by the instantaneous external force vector with seismic excitation can be expressed as(11)Wextt=−Mru˙gt−Mu¨t−Cu˙tTut.The work done by the restoring force vector of the internal deformation of the structure at a certain time can be expressed as(12)Wintt=∫0ufu,tdu.The difference between the instantaneous work done by the external and internal force vectors is denoted asWt, and equation (13) can be obtained as(13)Wt=−Mru˙gt−Mu¨t−Cu˙tTut−∫0ufu,tdu.From the perspective of thermodynamics, the work done by the internal and external forces during an adiabatic process actually represents the variation of the energy rather than the stored energy in the structure [39]. With seismic excitation, the work done by the internal and external forces of the structure is constantly varying, which can be expressed as(14)Eintrt=Wt=Wextt−Wintt.Therein, the input energy of the structure can be calculated by equation (7). The structure is instable when the input energy EIt is less than the internal energy Eintrt; it will absorb extra energy to offset the input energy so that the input energy can be equal to or greater than the intrinsic energy, which represents the dynamic stable state. Therefore, based on the energy evaluation principle, the structural stability can be expressed as(15)Eintrt≤EIt,stable,t>0,Eintrt>EIt,instable,t<0.Therefore, if the intrinsic energy curve of the structure exceeds the input energy curve, the structure is in a dynamic instable state; the dynamic stability equation can be expressed as(16)St=EIt−Eintrt,where the structure is dynamically stable when St≥0 and the structure is instable when St<0.


## 2.2.2. Strength Failure Criterion Based on the Dissipated Energy

According to the law of thermodynamics, energy conversion is an essential feature of the physical process, and substance destruction is a phenomenon of instability induced by energy [40, 41]. According to equation (3), the accumulated elastic strain energy ESt can be divided into the elastic strain energy Eet and the plastic strain energy Ept, as shown in(17)ESt=Eet+Ept.In the process of element deformation, the elastic strain energy at the elastic stage continues to increase until the yield limit is reached, which means that the elastic strain energy of the element reaches the maximum [42]. When the yield limit is exceeded, the material will undergo an irrecoverable plastic deformation, and it reaches the maximum when the material is almost destroyed, which means that the accumulated plastic strain energy reached the maximum [43, 44]. With seismic excitation, the structure elements alternatively exhibit tension and compression; they will experience the cycle of loading, compaction, elasticity, plasticity, and unloading, which induces cyclic impact on the material and brings about crack expansion, crack penetration, and even damage of the structure [45].During seismic excitation, seismic energy is continuously input into the structural system over time; a part of seismic energy is dissipated by the structure through its damping and plastic deformation, and the other part of seismic energy is converted into the kinetic energy of the structure or absorbed in the form of elastic strain energy of the structure [46]. It can be concluded from equation (3) that the total input energy of the structure is converted to the kinetic energy, damping energy, elastic strain energy, and plastic strain energy. In view of that the elastic strain energy and plastic strain energy correspond to the deformation states of the structure, the stability of the structure can be evaluated by calculating the strain energy and analyzing the tendency of the strain energy [47, 48].(1)Strain Energy. The strain energy density can be calculated from the stress and strain of the structure elements, as shown in(18)w=∫σijεijdεij.There is a strain equation for the elastoplastic material, as shown in(19)εij=εije+εijp.The strain energy density is expressed as(20)w=12σijεije+σijεijp.Besides, the strain energy of the element can be expressed as(21)we=∫Vwdv,where w is the strain energy density of the element and we is the strain energy of the element.The total strain energy of the structure can be obtained by summing the strain energy of each element, as shown in(22)W=∑we.(2)Strength Failure Theory Based on the Dissipated Energy. Considering the deformation of the element with external force, assuming that there is no heat exchange with surrounding, and denoting the total input energy generated by the external force as U, the following equation can be obtained according to the first law of thermodynamics as(23)U=Up+Ue,where Up is the plastic strain energy and Ue is the releasable elastic strain energy. The plastic strain energy Up is used to form internal damage and plastic deformation, which result in the strength failure, and the release of the elastic strain energy Ue stored in the element is the underlying reason for the sudden failure of the tunnel element.The energy of each part of the element in the principal stress space can be expressed as follows:(24)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3,(25)Ue=12σ1ε1e+12σ2ε2e+12σ3ε3e,(26)Up=U−Ue.The energy damage quantity of the tunnel element can be defined as(27)ϖ=UpUc,where Uc is the critical value of energy dissipation when the element loses its strength, which is a material constant determined by the uniaxial tension test, uniaxial compression test, and pure shear test; the calculation method of Uc is same as that of Up. When ϖ=1, it denotes that the material loses its strength no matter what stress state the material is in, which can be expressed as(28)ϖ=UpUc=1.Substituting equation (28) into equations (25) and (26), the strength failure criterion based on the dissipated energy can be obtained as(29)∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3−12σ1ε1e−12σ2ε2e−12σ3ε3e=Uc.


## 2.2.3. Overall Failure Criterion Based on the Releasable Elastic Strain Energy

It can be concluded from equation (25) that the releasable strain energy stored by the element is related to the unloading elastic modulus Ei and Poisson’s ratio v after damaged. Considering the orthotropy of the element damage, the damage is directly related to the elasticity. Assuming that the compressive stress is positive, when the element is not damaged, it can be expressed as(30)Ue=12σiεie=12σ12E1+σ22E2+σ32E3−v1E1+1E2σ1σ2+1E1+1E3σ1σ3+1E2+1E3σ2σ3.In terms of the damaged structure, the conventional damage variableωi is introduced to consider the effect of damage on the unloading modulus Ei of the structure, as shown in(31)Ei=1−ωiE0,where E0 is the initial elastic modulus when the structure is not damaged.Assuming that Poisson’s ratio is not affected by the damage and substituting it into equation (30), then the following equation can be obtained as(32)Ue=12σiεie=12E0σ121−ω1+σ221−ω2+σ321−ω3−v11−ω1+11−ω2σ1σ2+11−ω1+11−ω3σ1σ3+11−ω2+11−ω3σ2σ3.Considering the average effect of the damage along the three directions of the principal stresses, equation (32) can be simplified as(33)E¯=E01−ω¯,v¯=v1−ω¯,ωi=ω¯,i=1,2,and3.The releasable elastic strain energy can be rewritten as(34)Ue=12E¯σ12+σ22+σ32−2v¯σ1σ2+σ1σ3+σ2σ3.Herein, the average valuesE¯,v¯,andω¯ can be determined by the unidirectional cyclic compression loading and unloading tests. It can also be taken as the initial elastic modulus E0 and Poisson’s ratio v for convenience; then, equation (34) can be rewritten as(35)Ue=12Eσ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3.Equation (35) is aimed at the linear unloading process in the nonlinear process of the element. Summing up the releasable elastic strain energy Uie of each element, the releasable elastic strain energy ∑Uie of the whole structure can be obtained. As the damage of the element increases with external action, the strength gradually decreases; when the releasable elastic strain energy Uie of an element reaches the surface energy U0 required by the element damage, Uie will be released in the form of elastic surface energy. When the damaged elements reach a certain critical number, the whole structure will be damaged.Based on the above releasable energy concept, the overall failure criterion of the structure element can be concluded that a part of the work done by the external force on the structure is converted into the dissipated energyUp, which makes the structure gradually loses its strength, and the other part is converted into the gradually increasing releasable elastic strain energy Ue. When Ue reaches the value of the surface energy U0 of the tunnel element, the strain energy Ue is released and the whole structure element is damaged. In the principal stress space, Ue is difficult to be released along the direction of the maximum compressive stress σ1, but is apt to be released along the direction of the minimum compressive stress or tensile stress σ3. The overall failure criteria of the element with compressive stress and tensile stress are given below.(1)Three-Directional Compressive Stress State (σ1>σ2>σ3≥0, Assuming That the Compressive Stress Is Positive). When the element is in a three-directional compressive stress state and the overall instability occurs, the releasable elastic strain energy will be released along the three directions of stresses. The quantity of the released energy is proportional to the releasable energy stored in the element, and the released strain energy Gi along each direction is distributed according to the difference value with the maximum principal stress, which can be expressed as(36)Gi=Kiσ1−σiUe,where G3 is the positive correlation coefficient that can be determined by the uniaxial compression test and Gc is the releasable elastic strain energy in the element.It can be known from equation (36) that the releasable elastic strain energy in the element is primarily released along the third principal stress direction. When the released strain energy along the third principal stress direction G3 reaches a critical value Gc, the releasable elastic strain energy will be released firstly along this direction, which can be expressed as(37)G3=K3σ1−σ3Ue=Gc,where Gc is the material constant of the element, which can be determined by the uniaxial compression test. When the tunnel element is in the uniaxial compression state, the element will be overall instable if the element undergoes strength failure. Assuming that σ1=σc when the element undergoes uniaxial compression strength failure, then σ2=σ3=0 and the following equation can be obtained as(38)Ue=σc22E0.Substituting equation (38) into equation (37), the following equation can be obtained as(39)Gc=K3σc32E0.Therefore, the instability criterion of the element in the three-directional compressive stress state can be obtained as(40)Ue=σc22E0σ1−σ3.It can be also expressed as(41)σ1−σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σc3.(2)Stress State with Both Tension and Compression (σ1>0,σ3<0). When at least one of the principal stresses is tensile stress, considering that the tensile stress of any magnitude will promote the energy release of the damaged element, the stored releasable strain energy Ue is distributed according to the principal stress value along the three directions of principal stress. Therefore, the energy release rate Gi of the element along the σi direction can be defined as(42)Gi=KiσiUe.The releasable elastic strain energy in the element is primarily released along the direction of the maximum principal tensile stress. Therefore, it is necessary to satisfy the following equation when the overall instability occurs:(43)G3=K3σ3Ue=Gt,where Gt is the critical strain energy release rate of the element with tensile stress, which is a material constant determined by the uniaxial tensile test.When the element is in uniaxial tension state, it will be damaged if the strength is lost. Assuming thatσ3=σt,σ1=σ2=0 and substituting them into equation (35), the following equation can be obtained as(44)Ue=σt22E0.Then, substituting it into equation (42), the following equation can be obtained as(45)Gt=K3σt32E0.Moreover, substituting equation (45) into equation (43), the following equation can be obtained as(46)σ3Ue=σt32E0.Eventually, substituting equation (46) into equation (35), the overall instability criterion in the stress state with both tension and compression can be obtained as(47)σ3σ12+σ22+σ32−2vσ1σ2+σ1σ3+σ2σ3=σt3.


## 3. Numerical Simulation

The finite element method (FEM) was employed to conduct the numerical simulation, which had an advantage in solving the simulation of the structure composed of components with different properties. In this series of numerical simulation, stratum-structure model and solid elements were adopted to simulate lining structure and surrounding rock, and contact elements were adopted to simulate the slip between lining structure and surrounding rock. Besides, considering the dynamic parameters of the materials, both lining structure and surrounding rock were set as the elastoplastic constitutive model.


### 3.1. Tunnel Structures with Cross Sections in Different Ratios of Width to Height

#### 3.1.1. Finite Element Calculation Model

This paper uses local damping for calculation, which was set as 0.1571, and the critical damping ratio is taken as 5% according to the engineering experience. In regard to the boundary conditions, the bottom surface of the model was set as the viscoelastic boundary, and the side surfaces were set as the free field boundary.Five types of ratios of width to height were established, which were set as 0.68, 0.76, 0.92, 1.33, and 1.52, as shown in Figure1. Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), as shown in Figure 2(a), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting the solid elements, the Mohr–Coulomb model was used to simulate surrounding rock and initial support, while the elastic constitutive model was used to simulate the lining structure. The physical and mechanical parameters of surrounding rock, initial support, and lining structure were derived from the actual working conditions, and the material of the surrounding rock was strongly weathered sandstone, and the material of the lining structure was C30 concrete, and the specific values of the parameters are listed in Table 1.Figure 1
Cross sections in different ratios of width to height. (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)Figure 2
Calculation model of numerical simulation. (a) Overall model. (b) Monitoring section. (c) Monitoring points.
(a)(b)(c)Table 1
Physical and mechanical parameters of surrounding rock, initial support, and lining structure.
MaterialDensity (kg·m−3)Elastic modulus (GPa)Poisson’s ratio (μ)Cohesion (MPa)Friction angle (°)Surrounding rock250030.30.004435Lining structure2500300.2/0.3/0.4——Initial support2200200.34.435


#### 3.1.2. Design of Operating Conditions

Analyzing the dynamic response characteristics of tunnel structures in different structural shapes with seismic excitation from the perspective of energy, the optimal tunnel structure shape was researched, and the operating conditions are listed in Table2.Table 2
Operating conditions for the dynamic response analysis of tunnel structures in different shapes.
Operating conditionWidth-to-height ratioLining Poisson’s ratioTunnel form10.680.2Two-lane railway tunnel20.7630.9241.33Two-lane highway tunnel51.52Four-lane highway tunnel


#### 3.1.3. Monitoring Point Arrangement

A monitoring section was set up at a distance of 20 m from the tunnel portal, and six monitoring points were set at the crown, left spandrel, right spandrel, left arch springing, right arch springing, and invert, as shown in Figures2(b) and 2(c).


### 3.2. Tunnel Structures with Linings in Different Poisson’s Ratios

Adopting the cross-sectional shape with the width-to-height ratio of 0.92 (shown in Figure1(c)), the tunnel model with the lining structure in Poisson’s ratio of 0.2, 0.3, and 0.4 was established. Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting the solid elements, the Mohr–Coulomb model was used to simulate surrounding rock and initial support, while the elastic constitutive model was used to simulate the lining structure.


### 3.3. Seismic Wave Processing

The Wolong seismic wave in the “5.12” Wenchuan earthquake was selected as the input seismic wave. The original seismic wave was an acceleration wave with a duration of 180 s, whose peak acceleration was 953 cm·s−2 and occurred at 33.01 s. Considering that the energy of the seismic wave was concentrated in the range of 0–18 Hz, the original seismic wave was processed by baseline correction and the components greater than 18 Hz were filtered out. Furthermore, the time period of 27.1 s–42.1 s was selected for the dynamic calculation, which included the peak acceleration and most of the energy, then it was multiplied by the reduction coefficient to simulate the seismic response with 0.4 g PGA. The acceleration time history curve and velocity time history curve obtained after processing are shown in Figure 3. Therein, the velocity wave was input from the bottom of the model after being converted into the stress form.Figure 3
Time history curves of the input seismic wave. (a) Acceleration time history curve. (b) Velocity time history curve.
(a)(b)


### 3.4. Tunnel Structures with Different PGAs

Adopting the operating condition with the width-to-height ratio of 1.33 in Section3.1 and Poisson’s ratio of 0.3 in Section 3.2, seismic waves with PGAs of 0.15 g, 0.3 g, and 0.4 g obtained by being multiplied by different reduction coefficients were input, which were corresponding to seismic fortification intensities of 7, 8, and 9, respectively. The stability of tunnel structures with different PGAs was researched from the perspective of energy.Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting solid elements, the surrounding rock, initial support, and lining structure were all simulated by the Mohr–Coulomb model.


## 3.1. Tunnel Structures with Cross Sections in Different Ratios of Width to Height

### 3.1.1. Finite Element Calculation Model

This paper uses local damping for calculation, which was set as 0.1571, and the critical damping ratio is taken as 5% according to the engineering experience. In regard to the boundary conditions, the bottom surface of the model was set as the viscoelastic boundary, and the side surfaces were set as the free field boundary.Five types of ratios of width to height were established, which were set as 0.68, 0.76, 0.92, 1.33, and 1.52, as shown in Figure1. Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), as shown in Figure 2(a), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting the solid elements, the Mohr–Coulomb model was used to simulate surrounding rock and initial support, while the elastic constitutive model was used to simulate the lining structure. The physical and mechanical parameters of surrounding rock, initial support, and lining structure were derived from the actual working conditions, and the material of the surrounding rock was strongly weathered sandstone, and the material of the lining structure was C30 concrete, and the specific values of the parameters are listed in Table 1.Figure 1
Cross sections in different ratios of width to height. (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)Figure 2
Calculation model of numerical simulation. (a) Overall model. (b) Monitoring section. (c) Monitoring points.
(a)(b)(c)Table 1
Physical and mechanical parameters of surrounding rock, initial support, and lining structure.
MaterialDensity (kg·m−3)Elastic modulus (GPa)Poisson’s ratio (μ)Cohesion (MPa)Friction angle (°)Surrounding rock250030.30.004435Lining structure2500300.2/0.3/0.4——Initial support2200200.34.435


### 3.1.2. Design of Operating Conditions

Analyzing the dynamic response characteristics of tunnel structures in different structural shapes with seismic excitation from the perspective of energy, the optimal tunnel structure shape was researched, and the operating conditions are listed in Table2.Table 2
Operating conditions for the dynamic response analysis of tunnel structures in different shapes.
Operating conditionWidth-to-height ratioLining Poisson’s ratioTunnel form10.680.2Two-lane railway tunnel20.7630.9241.33Two-lane highway tunnel51.52Four-lane highway tunnel


### 3.1.3. Monitoring Point Arrangement

A monitoring section was set up at a distance of 20 m from the tunnel portal, and six monitoring points were set at the crown, left spandrel, right spandrel, left arch springing, right arch springing, and invert, as shown in Figures2(b) and 2(c).


## 3.1.1. Finite Element Calculation Model

This paper uses local damping for calculation, which was set as 0.1571, and the critical damping ratio is taken as 5% according to the engineering experience. In regard to the boundary conditions, the bottom surface of the model was set as the viscoelastic boundary, and the side surfaces were set as the free field boundary.Five types of ratios of width to height were established, which were set as 0.68, 0.76, 0.92, 1.33, and 1.52, as shown in Figure1. Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), as shown in Figure 2(a), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting the solid elements, the Mohr–Coulomb model was used to simulate surrounding rock and initial support, while the elastic constitutive model was used to simulate the lining structure. The physical and mechanical parameters of surrounding rock, initial support, and lining structure were derived from the actual working conditions, and the material of the surrounding rock was strongly weathered sandstone, and the material of the lining structure was C30 concrete, and the specific values of the parameters are listed in Table 1.Figure 1
Cross sections in different ratios of width to height. (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)Figure 2
Calculation model of numerical simulation. (a) Overall model. (b) Monitoring section. (c) Monitoring points.
(a)(b)(c)Table 1
Physical and mechanical parameters of surrounding rock, initial support, and lining structure.
MaterialDensity (kg·m−3)Elastic modulus (GPa)Poisson’s ratio (μ)Cohesion (MPa)Friction angle (°)Surrounding rock250030.30.004435Lining structure2500300.2/0.3/0.4——Initial support2200200.34.435


## 3.1.2. Design of Operating Conditions

Analyzing the dynamic response characteristics of tunnel structures in different structural shapes with seismic excitation from the perspective of energy, the optimal tunnel structure shape was researched, and the operating conditions are listed in Table2.Table 2
Operating conditions for the dynamic response analysis of tunnel structures in different shapes.
Operating conditionWidth-to-height ratioLining Poisson’s ratioTunnel form10.680.2Two-lane railway tunnel20.7630.9241.33Two-lane highway tunnel51.52Four-lane highway tunnel


## 3.1.3. Monitoring Point Arrangement

A monitoring section was set up at a distance of 20 m from the tunnel portal, and six monitoring points were set at the crown, left spandrel, right spandrel, left arch springing, right arch springing, and invert, as shown in Figures2(b) and 2(c).


## 3.2. Tunnel Structures with Linings in Different Poisson’s Ratios

Adopting the cross-sectional shape with the width-to-height ratio of 0.92 (shown in Figure1(c)), the tunnel model with the lining structure in Poisson’s ratio of 0.2, 0.3, and 0.4 was established. Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting the solid elements, the Mohr–Coulomb model was used to simulate surrounding rock and initial support, while the elastic constitutive model was used to simulate the lining structure.


## 3.3. Seismic Wave Processing

The Wolong seismic wave in the “5.12” Wenchuan earthquake was selected as the input seismic wave. The original seismic wave was an acceleration wave with a duration of 180 s, whose peak acceleration was 953 cm·s−2 and occurred at 33.01 s. Considering that the energy of the seismic wave was concentrated in the range of 0–18 Hz, the original seismic wave was processed by baseline correction and the components greater than 18 Hz were filtered out. Furthermore, the time period of 27.1 s–42.1 s was selected for the dynamic calculation, which included the peak acceleration and most of the energy, then it was multiplied by the reduction coefficient to simulate the seismic response with 0.4 g PGA. The acceleration time history curve and velocity time history curve obtained after processing are shown in Figure 3. Therein, the velocity wave was input from the bottom of the model after being converted into the stress form.Figure 3
Time history curves of the input seismic wave. (a) Acceleration time history curve. (b) Velocity time history curve.
(a)(b)


## 3.4. Tunnel Structures with Different PGAs

Adopting the operating condition with the width-to-height ratio of 1.33 in Section3.1 and Poisson’s ratio of 0.3 in Section 3.2, seismic waves with PGAs of 0.15 g, 0.3 g, and 0.4 g obtained by being multiplied by different reduction coefficients were input, which were corresponding to seismic fortification intensities of 7, 8, and 9, respectively. The stability of tunnel structures with different PGAs was researched from the perspective of energy.Considering the Saint-Venant principle, the boundary dimension was set as 50 m (length) × 40 m (width) × 50 m (height), and the calculation model was composed of 33286 nodes and 29280 elements in total. Furthermore, adopting solid elements, the surrounding rock, initial support, and lining structure were all simulated by the Mohr–Coulomb model.


## 4. Analysis of Numerical Simulation Results

### 4.1. Seismic Responses of Tunnel Structures with Different Ratios of Width to Height

#### 4.1.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to five types of ratios of width to height with the operating condition of 15 s Wolong wave and Poisson’s ratio of 0.2, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 4.Figure 4
Peak elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)The peak values of elastic strain energy at the invert and crown are obviously smaller than those at the spandrel and arch springing, which is due to the fact that the seismic extrusion effect of surrounding rock on both sides of the tunnel structure is stronger than that on the top and bottom of the tunnel structure. As for the crown, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest. In terms of the invert, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and both the peak elastic strain energy of the crown and invert decreases with the increase of the width-to-height ratio.In regard to the left spandrel, the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.68 is the largest; what is different is that the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. The seismic response of the right spandrel is similar to that of the left spandrel; the only difference is that the peak elastic strain energy of the right spandrel reaches the maximum at the operating condition of the width-to-height ratio of 1.52.Furthermore, the peak elastic strain energy of the arch springing is generally larger than that of other positions, which indicates that the energy generated by seismic excitation mostly concentrates at the arch springing. Besides, the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is smaller than 1, and it reaches the maximum at the operating condition of the width-to-height ratio of 0.92, and the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52.In summary, comparing the operating conditions of different ratios of width to height at different positions of the tunnel structure, one can see that the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, which indicates that the cross-sectional shape can effectively reduce the elastic strain energy concentrating on the tunnel structure with seismic excitation, and the shape of the width-to-height ratio of 1.33 is the optimal tunnel cross-sectional shape.


#### 4.1.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure was plotted in Figure5. With the increase of the width-to-height ratio, the accumulated elastic strain energy of the crown and invert gradually decreases, whereas the accumulated elastic strain energy of the spandrel and arch springing gradually increases, and the strain state with seismic excitation gradually transferred from the crown and invert alternatively expanding and contracting to the spandrel and arch springing alternatively expanding and contracting.Figure 5
Accumulated elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)As for the crown, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest, and the accumulated elastic strain energy gradually decreases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. In terms of the invert, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52, which indicates that the seismic performance of the shape of the width-to-height ratio of 1.33 is better than that of the width-to-height ratio of 1.52.The accumulated elastic strain energy of the spandrel and arch springing is obviously greater than that of the invert and crown. The accumulated elastic strain energy of the left spandrel, left arch springing, and right arch springing with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the accumulated elastic strain energy of the right spandrel is also relatively small. Therefore, it can be concluded that the cross section in the width-to-height ratio of 1.33 has the best seismic performance.


### 4.2. Seismic Responses of Tunnel Structures with Linings in Different Poisson’s Ratios

#### 4.2.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to three types of Poisson’s ratios with the operating condition of 15 s Wolong wave and the width-to-height ratio of 0.92, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 6.Figure 6
Peak elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)With the operating conditions of three different Poisson’s ratios, the peak values of elastic strain energy at the invert and crown are all smaller than that at the spandrel and arch springing. The peak elastic strain energy at the crown, invert, and left spandrel slightly decreases with the operating condition of Poisson’s ratio of 0.3, and the peak elastic strain energy at other positions increases, whereas the seismic response with the operating condition of Poisson’s ratio of 0.4 is on the contrary. Considering that the peak elastic strain energy at the arch springing is the largest, it is determined that Poisson’s ratio of 0.4 is most optimal Poisson’s ratio.


#### 4.2.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure is plotted in Figure7. Different from the distribution trend of the peak elastic strain energy, the accumulated elastic strain energy at the spandrel is greater than that of arch springing, which indicates that the cyclic deformation of spandrel is greater than that of arch springing. With the increase of Poisson’s ratio, the accumulated elastic strain energy of the crown increases greatly, indicating that the increase of Poisson’s ratio is unfavorable to the stability of the crown.Figure 7
Accumulated elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)The variations of the positions except the crown are stable when Poisson’s ratio is 0.3, and the increase and decrease of the accumulated elastic strain energy are relatively small. When Poisson’s ratio is 0.4, the accumulated elastic strain energy at the crown increases significantly, and the accumulated elastic strain energy at the right spandrel also increases. The maximum value of the accumulated elastic strain energy with the operating conditions of three Poisson’s ratios all occurs at the right spandrel, and when Poisson’s ratio is 0.3, it is less than the value when Poisson’s ratio is 0.4. In summary, Poisson’s ratio of 0.3 is most optimal Poisson’s ratio for the aseismic design.


### 4.3. Seismic Responses of Tunnel Structures with Different PGAs

Analyzing the total seismic input energy, kinetic energy, damping energy, and strain energy in the energy balance equation, the energy responses of tunnel structures with different PGAs are researched, and a structural dynamic stability function based on the energy is established to evaluate the safety and stability of the tunnel structure with seismic excitation.With seismic excitation, the tunnel structure is in an energy field, and the effect of the earthquake on the structure is a process of transferring and transformation of energy. According to the derivation in Section2.2, when the total input energy is greater than or equal to the energy generated inside the structure with seismic excitation, the structure can maintain a dynamic balance state, which can be expressed as(48)EI≥ES+EP+ED+EK,where EI is the total input energy, ES is the elastic strain energy, EP is the plastic strain energy, ED is the damping energy, EK is the kinetic energy, and the sum of the elastic strain energy ES and the plastic strain energy EP is called the strain energy EN.


#### 4.3.1. Energy Response of Dynamic Process

Inputting the seismic wave with the peak acceleration of 0.15 g, 0.3 g, and 0.4 g, for most of the time, the energy of the lining structure has the trend of kinetic energy > damping energy > strain energy. Besides, the energy increases with the increase of the peak acceleration of the input seismic wave. In order to observe the variation of energy with the increase of the peak acceleration of the input seismic wave, the maximum and secondary maximum of each energy are listed in Table3.Table 3
The maximum and secondary maximum of each energy (unit: 1010 J).
Operating condition (g)Total input energyEIKinetic energyEKDamping energyEDStrain energyEN0.15Maximum32.918.77.13.3Secondary maximum19.110.34.13.00.3Maximum131.874.828.513.1Secondary maximum77.242.116.111.70.4Maximum364.613350.723.3Secondary maximum214.27330.221.5The kinetic energy, damping energy, and strain energy all increase significantly with the increase of the peak acceleration of input seismic wave. For example, when the PGA increases from 0.15 g to 0.3 g, the kinetic energy, damping energy, and strain energy increase by 300%, 302%, and 297%, respectively. When the PGA increases from 0.3 g to 0.4 g, the kinetic energy, damping energy, and strain energy all increase by 78%.Considering that the proportions of each energy of the lining structure in the total input energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g are 56.8%, 56.7%, and 36.5% for the kinetic energy, 21.6%, 21.6%, and 13.9% for the damping energy, and 10.0%, 9.9%, and 6.4% for the strain energy, the proportions of each energy decreases significantly when the peak acceleration of the input seismic wave increases to 0.4 g, which is due to the fact that the energy is dissipated by the instability of the tunnel structure.At the end of the dynamic calculation, the accumulated value of each energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g is shown in Figure8.Figure 8
Accumulated values of the energy with different PGAs.The kinetic energy has a rapider growth trend during the dynamic process, whereas the damping energy and elastic strain energy have a much slower growth trend and the plastic strain energy is very small throughout the whole process, which indicates that the lining structure has little plastic deformation. In addition, with the increase of PGA, the proportions of each energy decreases whereas the total input energy increases, which indicates that the larger the PGA, the more likely the energy of the structure will be lost due to the instability.


#### 4.3.2. Energy Dynamic Stability Equation of Lining Structure

In order to observe clearly whether the tunnel structure is instable and the corresponding period of instability, the dynamic stability equation of the lining structure can be obtained as(49)St=EIt−Eintrt.The internal energyEintrt can be obtained by equation (14), and the dynamic stability equation curves with different PGAs are shown in Figure 9, where the red lines in the diagrams represent the critical value of tunnel stability.Figure 9
Dynamic stability equation curves with different PGAs (unit: J). (a) 0.15 g; (b) 0.3 g; (c) 0.4 g.
(a)(b)(c)It is known from equation (16) that the structure is stable when St≥0 and is instable when St<0, and the value of St can be used to evaluate the stability of the lining structure. As shown in Figure 9, the lining structure is stable for most of the time, and it is instable only at about 2 s, 3.2 s, 6.5 s, 8 s, and 11.2 s. The instability of the lining structure is the most serious at about 11.2 s with all the three PGAs, which is consistent with the seismic performance in the actual event, while the minimum values of St for the PGAs of 0.15 g, 0.3 g, and 0.4 g are −2.79 × 1010, −1.30 × 1011, and −2.43 × 1011, respectively. It can be concluded that the proposed equation can accurately evaluate the stability of the tunnel structure, and the instability degree of the tunnel structure can be represented by the specific value. Furthermore, the larger the peak value of seismic wave acceleration, the longer period of the tunnel structure is instable and the greater the degree of dynamic instability.


## 4.1. Seismic Responses of Tunnel Structures with Different Ratios of Width to Height

### 4.1.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to five types of ratios of width to height with the operating condition of 15 s Wolong wave and Poisson’s ratio of 0.2, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 4.Figure 4
Peak elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)The peak values of elastic strain energy at the invert and crown are obviously smaller than those at the spandrel and arch springing, which is due to the fact that the seismic extrusion effect of surrounding rock on both sides of the tunnel structure is stronger than that on the top and bottom of the tunnel structure. As for the crown, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest. In terms of the invert, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and both the peak elastic strain energy of the crown and invert decreases with the increase of the width-to-height ratio.In regard to the left spandrel, the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.68 is the largest; what is different is that the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. The seismic response of the right spandrel is similar to that of the left spandrel; the only difference is that the peak elastic strain energy of the right spandrel reaches the maximum at the operating condition of the width-to-height ratio of 1.52.Furthermore, the peak elastic strain energy of the arch springing is generally larger than that of other positions, which indicates that the energy generated by seismic excitation mostly concentrates at the arch springing. Besides, the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is smaller than 1, and it reaches the maximum at the operating condition of the width-to-height ratio of 0.92, and the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52.In summary, comparing the operating conditions of different ratios of width to height at different positions of the tunnel structure, one can see that the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, which indicates that the cross-sectional shape can effectively reduce the elastic strain energy concentrating on the tunnel structure with seismic excitation, and the shape of the width-to-height ratio of 1.33 is the optimal tunnel cross-sectional shape.


### 4.1.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure was plotted in Figure5. With the increase of the width-to-height ratio, the accumulated elastic strain energy of the crown and invert gradually decreases, whereas the accumulated elastic strain energy of the spandrel and arch springing gradually increases, and the strain state with seismic excitation gradually transferred from the crown and invert alternatively expanding and contracting to the spandrel and arch springing alternatively expanding and contracting.Figure 5
Accumulated elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)As for the crown, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest, and the accumulated elastic strain energy gradually decreases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. In terms of the invert, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52, which indicates that the seismic performance of the shape of the width-to-height ratio of 1.33 is better than that of the width-to-height ratio of 1.52.The accumulated elastic strain energy of the spandrel and arch springing is obviously greater than that of the invert and crown. The accumulated elastic strain energy of the left spandrel, left arch springing, and right arch springing with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the accumulated elastic strain energy of the right spandrel is also relatively small. Therefore, it can be concluded that the cross section in the width-to-height ratio of 1.33 has the best seismic performance.


## 4.1.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to five types of ratios of width to height with the operating condition of 15 s Wolong wave and Poisson’s ratio of 0.2, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 4.Figure 4
Peak elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)The peak values of elastic strain energy at the invert and crown are obviously smaller than those at the spandrel and arch springing, which is due to the fact that the seismic extrusion effect of surrounding rock on both sides of the tunnel structure is stronger than that on the top and bottom of the tunnel structure. As for the crown, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest. In terms of the invert, the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and both the peak elastic strain energy of the crown and invert decreases with the increase of the width-to-height ratio.In regard to the left spandrel, the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the peak elastic strain energy with the operating condition of the width-to-height ratio of 0.68 is the largest; what is different is that the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. The seismic response of the right spandrel is similar to that of the left spandrel; the only difference is that the peak elastic strain energy of the right spandrel reaches the maximum at the operating condition of the width-to-height ratio of 1.52.Furthermore, the peak elastic strain energy of the arch springing is generally larger than that of other positions, which indicates that the energy generated by seismic excitation mostly concentrates at the arch springing. Besides, the peak elastic strain energy increases with the increase of the width-to-height ratio when the width-to-height ratio is smaller than 1, and it reaches the maximum at the operating condition of the width-to-height ratio of 0.92, and the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52.In summary, comparing the operating conditions of different ratios of width to height at different positions of the tunnel structure, one can see that the peak elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is the smallest, which indicates that the cross-sectional shape can effectively reduce the elastic strain energy concentrating on the tunnel structure with seismic excitation, and the shape of the width-to-height ratio of 1.33 is the optimal tunnel cross-sectional shape.


## 4.1.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure was plotted in Figure5. With the increase of the width-to-height ratio, the accumulated elastic strain energy of the crown and invert gradually decreases, whereas the accumulated elastic strain energy of the spandrel and arch springing gradually increases, and the strain state with seismic excitation gradually transferred from the crown and invert alternatively expanding and contracting to the spandrel and arch springing alternatively expanding and contracting.Figure 5
Accumulated elastic strain energy diagrams of tunnel structures with different ratios of width to height (unit: J). (a) 0.68; (b) 0.76; (c) 0.92; (d) 1.33; (e) 1.52.
(a)(b)(c)(d)(e)As for the crown, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.76 is the largest, and the accumulated elastic strain energy gradually decreases with the increase of the width-to-height ratio when the width-to-height ratio is greater than 1. In terms of the invert, the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 0.92 is the largest, and the accumulated elastic strain energy with the operating condition of the width-to-height ratio of 1.33 is smaller than that of the width-to-height ratio of 1.52, which indicates that the seismic performance of the shape of the width-to-height ratio of 1.33 is better than that of the width-to-height ratio of 1.52.The accumulated elastic strain energy of the spandrel and arch springing is obviously greater than that of the invert and crown. The accumulated elastic strain energy of the left spandrel, left arch springing, and right arch springing with the operating condition of the width-to-height ratio of 1.33 is the smallest, while the accumulated elastic strain energy of the right spandrel is also relatively small. Therefore, it can be concluded that the cross section in the width-to-height ratio of 1.33 has the best seismic performance.


## 4.2. Seismic Responses of Tunnel Structures with Linings in Different Poisson’s Ratios

### 4.2.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to three types of Poisson’s ratios with the operating condition of 15 s Wolong wave and the width-to-height ratio of 0.92, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 6.Figure 6
Peak elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)With the operating conditions of three different Poisson’s ratios, the peak values of elastic strain energy at the invert and crown are all smaller than that at the spandrel and arch springing. The peak elastic strain energy at the crown, invert, and left spandrel slightly decreases with the operating condition of Poisson’s ratio of 0.3, and the peak elastic strain energy at other positions increases, whereas the seismic response with the operating condition of Poisson’s ratio of 0.4 is on the contrary. Considering that the peak elastic strain energy at the arch springing is the largest, it is determined that Poisson’s ratio of 0.4 is most optimal Poisson’s ratio.


### 4.2.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure is plotted in Figure7. Different from the distribution trend of the peak elastic strain energy, the accumulated elastic strain energy at the spandrel is greater than that of arch springing, which indicates that the cyclic deformation of spandrel is greater than that of arch springing. With the increase of Poisson’s ratio, the accumulated elastic strain energy of the crown increases greatly, indicating that the increase of Poisson’s ratio is unfavorable to the stability of the crown.Figure 7
Accumulated elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)The variations of the positions except the crown are stable when Poisson’s ratio is 0.3, and the increase and decrease of the accumulated elastic strain energy are relatively small. When Poisson’s ratio is 0.4, the accumulated elastic strain energy at the crown increases significantly, and the accumulated elastic strain energy at the right spandrel also increases. The maximum value of the accumulated elastic strain energy with the operating conditions of three Poisson’s ratios all occurs at the right spandrel, and when Poisson’s ratio is 0.3, it is less than the value when Poisson’s ratio is 0.4. In summary, Poisson’s ratio of 0.3 is most optimal Poisson’s ratio for the aseismic design.


## 4.2.1. Analysis of Peak Elastic Strain Energy

Extracting three principal stresses of the measuring points at the crown, spandrel, arch springing, and invert corresponding to three types of Poisson’s ratios with the operating condition of 15 s Wolong wave and the width-to-height ratio of 0.92, the peak elastic strain energy diagram can be plotted by calculating the elastic strain energy of the elements at the measuring points through equation (21), as illustrated in Figure 6.Figure 6
Peak elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)With the operating conditions of three different Poisson’s ratios, the peak values of elastic strain energy at the invert and crown are all smaller than that at the spandrel and arch springing. The peak elastic strain energy at the crown, invert, and left spandrel slightly decreases with the operating condition of Poisson’s ratio of 0.3, and the peak elastic strain energy at other positions increases, whereas the seismic response with the operating condition of Poisson’s ratio of 0.4 is on the contrary. Considering that the peak elastic strain energy at the arch springing is the largest, it is determined that Poisson’s ratio of 0.4 is most optimal Poisson’s ratio.


## 4.2.2. Analysis of Accumulated Elastic Strain Energy

The accumulated elastic strain energy after inputting the 15 s seismic wave at the six monitoring points of the tunnel structure is plotted in Figure7. Different from the distribution trend of the peak elastic strain energy, the accumulated elastic strain energy at the spandrel is greater than that of arch springing, which indicates that the cyclic deformation of spandrel is greater than that of arch springing. With the increase of Poisson’s ratio, the accumulated elastic strain energy of the crown increases greatly, indicating that the increase of Poisson’s ratio is unfavorable to the stability of the crown.Figure 7
Accumulated elastic strain energy diagrams of tunnel structures with different Poisson’s ratios (unit: J). (a) 0.2; (b) 0.3; (c) 0.4.
(a)(b)(c)The variations of the positions except the crown are stable when Poisson’s ratio is 0.3, and the increase and decrease of the accumulated elastic strain energy are relatively small. When Poisson’s ratio is 0.4, the accumulated elastic strain energy at the crown increases significantly, and the accumulated elastic strain energy at the right spandrel also increases. The maximum value of the accumulated elastic strain energy with the operating conditions of three Poisson’s ratios all occurs at the right spandrel, and when Poisson’s ratio is 0.3, it is less than the value when Poisson’s ratio is 0.4. In summary, Poisson’s ratio of 0.3 is most optimal Poisson’s ratio for the aseismic design.


## 4.3. Seismic Responses of Tunnel Structures with Different PGAs

Analyzing the total seismic input energy, kinetic energy, damping energy, and strain energy in the energy balance equation, the energy responses of tunnel structures with different PGAs are researched, and a structural dynamic stability function based on the energy is established to evaluate the safety and stability of the tunnel structure with seismic excitation.With seismic excitation, the tunnel structure is in an energy field, and the effect of the earthquake on the structure is a process of transferring and transformation of energy. According to the derivation in Section2.2, when the total input energy is greater than or equal to the energy generated inside the structure with seismic excitation, the structure can maintain a dynamic balance state, which can be expressed as(48)EI≥ES+EP+ED+EK,where EI is the total input energy, ES is the elastic strain energy, EP is the plastic strain energy, ED is the damping energy, EK is the kinetic energy, and the sum of the elastic strain energy ES and the plastic strain energy EP is called the strain energy EN.


### 4.3.1. Energy Response of Dynamic Process

Inputting the seismic wave with the peak acceleration of 0.15 g, 0.3 g, and 0.4 g, for most of the time, the energy of the lining structure has the trend of kinetic energy > damping energy > strain energy. Besides, the energy increases with the increase of the peak acceleration of the input seismic wave. In order to observe the variation of energy with the increase of the peak acceleration of the input seismic wave, the maximum and secondary maximum of each energy are listed in Table3.Table 3
The maximum and secondary maximum of each energy (unit: 1010 J).
Operating condition (g)Total input energyEIKinetic energyEKDamping energyEDStrain energyEN0.15Maximum32.918.77.13.3Secondary maximum19.110.34.13.00.3Maximum131.874.828.513.1Secondary maximum77.242.116.111.70.4Maximum364.613350.723.3Secondary maximum214.27330.221.5The kinetic energy, damping energy, and strain energy all increase significantly with the increase of the peak acceleration of input seismic wave. For example, when the PGA increases from 0.15 g to 0.3 g, the kinetic energy, damping energy, and strain energy increase by 300%, 302%, and 297%, respectively. When the PGA increases from 0.3 g to 0.4 g, the kinetic energy, damping energy, and strain energy all increase by 78%.Considering that the proportions of each energy of the lining structure in the total input energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g are 56.8%, 56.7%, and 36.5% for the kinetic energy, 21.6%, 21.6%, and 13.9% for the damping energy, and 10.0%, 9.9%, and 6.4% for the strain energy, the proportions of each energy decreases significantly when the peak acceleration of the input seismic wave increases to 0.4 g, which is due to the fact that the energy is dissipated by the instability of the tunnel structure.At the end of the dynamic calculation, the accumulated value of each energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g is shown in Figure8.Figure 8
Accumulated values of the energy with different PGAs.The kinetic energy has a rapider growth trend during the dynamic process, whereas the damping energy and elastic strain energy have a much slower growth trend and the plastic strain energy is very small throughout the whole process, which indicates that the lining structure has little plastic deformation. In addition, with the increase of PGA, the proportions of each energy decreases whereas the total input energy increases, which indicates that the larger the PGA, the more likely the energy of the structure will be lost due to the instability.


### 4.3.2. Energy Dynamic Stability Equation of Lining Structure

In order to observe clearly whether the tunnel structure is instable and the corresponding period of instability, the dynamic stability equation of the lining structure can be obtained as(49)St=EIt−Eintrt.The internal energyEintrt can be obtained by equation (14), and the dynamic stability equation curves with different PGAs are shown in Figure 9, where the red lines in the diagrams represent the critical value of tunnel stability.Figure 9
Dynamic stability equation curves with different PGAs (unit: J). (a) 0.15 g; (b) 0.3 g; (c) 0.4 g.
(a)(b)(c)It is known from equation (16) that the structure is stable when St≥0 and is instable when St<0, and the value of St can be used to evaluate the stability of the lining structure. As shown in Figure 9, the lining structure is stable for most of the time, and it is instable only at about 2 s, 3.2 s, 6.5 s, 8 s, and 11.2 s. The instability of the lining structure is the most serious at about 11.2 s with all the three PGAs, which is consistent with the seismic performance in the actual event, while the minimum values of St for the PGAs of 0.15 g, 0.3 g, and 0.4 g are −2.79 × 1010, −1.30 × 1011, and −2.43 × 1011, respectively. It can be concluded that the proposed equation can accurately evaluate the stability of the tunnel structure, and the instability degree of the tunnel structure can be represented by the specific value. Furthermore, the larger the peak value of seismic wave acceleration, the longer period of the tunnel structure is instable and the greater the degree of dynamic instability.


## 4.3.1. Energy Response of Dynamic Process

Inputting the seismic wave with the peak acceleration of 0.15 g, 0.3 g, and 0.4 g, for most of the time, the energy of the lining structure has the trend of kinetic energy > damping energy > strain energy. Besides, the energy increases with the increase of the peak acceleration of the input seismic wave. In order to observe the variation of energy with the increase of the peak acceleration of the input seismic wave, the maximum and secondary maximum of each energy are listed in Table3.Table 3
The maximum and secondary maximum of each energy (unit: 1010 J).
Operating condition (g)Total input energyEIKinetic energyEKDamping energyEDStrain energyEN0.15Maximum32.918.77.13.3Secondary maximum19.110.34.13.00.3Maximum131.874.828.513.1Secondary maximum77.242.116.111.70.4Maximum364.613350.723.3Secondary maximum214.27330.221.5The kinetic energy, damping energy, and strain energy all increase significantly with the increase of the peak acceleration of input seismic wave. For example, when the PGA increases from 0.15 g to 0.3 g, the kinetic energy, damping energy, and strain energy increase by 300%, 302%, and 297%, respectively. When the PGA increases from 0.3 g to 0.4 g, the kinetic energy, damping energy, and strain energy all increase by 78%.Considering that the proportions of each energy of the lining structure in the total input energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g are 56.8%, 56.7%, and 36.5% for the kinetic energy, 21.6%, 21.6%, and 13.9% for the damping energy, and 10.0%, 9.9%, and 6.4% for the strain energy, the proportions of each energy decreases significantly when the peak acceleration of the input seismic wave increases to 0.4 g, which is due to the fact that the energy is dissipated by the instability of the tunnel structure.At the end of the dynamic calculation, the accumulated value of each energy with the PGAs of 0.15 g, 0.3 g, and 0.4 g is shown in Figure8.Figure 8
Accumulated values of the energy with different PGAs.The kinetic energy has a rapider growth trend during the dynamic process, whereas the damping energy and elastic strain energy have a much slower growth trend and the plastic strain energy is very small throughout the whole process, which indicates that the lining structure has little plastic deformation. In addition, with the increase of PGA, the proportions of each energy decreases whereas the total input energy increases, which indicates that the larger the PGA, the more likely the energy of the structure will be lost due to the instability.


## 4.3.2. Energy Dynamic Stability Equation of Lining Structure

In order to observe clearly whether the tunnel structure is instable and the corresponding period of instability, the dynamic stability equation of the lining structure can be obtained as(49)St=EIt−Eintrt.The internal energyEintrt can be obtained by equation (14), and the dynamic stability equation curves with different PGAs are shown in Figure 9, where the red lines in the diagrams represent the critical value of tunnel stability.Figure 9
Dynamic stability equation curves with different PGAs (unit: J). (a) 0.15 g; (b) 0.3 g; (c) 0.4 g.
(a)(b)(c)It is known from equation (16) that the structure is stable when St≥0 and is instable when St<0, and the value of St can be used to evaluate the stability of the lining structure. As shown in Figure 9, the lining structure is stable for most of the time, and it is instable only at about 2 s, 3.2 s, 6.5 s, 8 s, and 11.2 s. The instability of the lining structure is the most serious at about 11.2 s with all the three PGAs, which is consistent with the seismic performance in the actual event, while the minimum values of St for the PGAs of 0.15 g, 0.3 g, and 0.4 g are −2.79 × 1010, −1.30 × 1011, and −2.43 × 1011, respectively. It can be concluded that the proposed equation can accurately evaluate the stability of the tunnel structure, and the instability degree of the tunnel structure can be represented by the specific value. Furthermore, the larger the peak value of seismic wave acceleration, the longer period of the tunnel structure is instable and the greater the degree of dynamic instability.


## 5. Conclusions

Beginning from the energy balance equation of the tunnel structure, this paper analyzed the relationship between the internal energy and the input energy of tunnel structure with seismic excitation and proposed equations for the evaluation of the dynamic stability of the tunnel structure. Based on the indexes of the releasable elastic strain energy and dissipated energy, the strength failure criterion of the tunnel structure was proposed, which was applied to the optimization of the aseismic design of the cross-sectional shape and material property of the tunnel structure. Besides, the stability of the tunnel structure with different PGAs based on the energy dynamic stability equation was analyzed. The conclusions are summarized as follows:(1)
The energy balance equation with seismic excitation is established for the tunnel structure. Further, the structure stability evaluation index is proposed from the perspective of the total energy balance of the system, and the tunnel strength failure criterion is proposed from the perspective of irreversible dissipation energy and releasable elastic strain energy.(2)
The releasable elastic strain energy index is used to optimize the cross-sectional shape of the tunnel structure. It is concluded that the peak values and accumulated values of elastic strain energy at the spandrel and arch springing are significantly larger than other positions, which indicates that the strengths of the spandrel and arch springing are the most influential factor for the seismic damage of the tunnel structure. Considering this factor, the width-to-height ratio of 1.33 is determined as the most optimal cross-sectional shape.(3)
The releasable elastic strain energy index is used to optimize the material property of the tunnel structure. It is shown that the peak elastic strain energy of the arch springing is the largest, while the accumulated elastic strain energy of the spandrel is the largest. Comparing the variations of the elastic strain energy with different operating conditions, Poisson’s ratio of 0.3 is determined as the most optimal material property.(4)
The seismic responses of the total input energy, kinetic energy, damping energy, and strain energy of the tunnel structure with different PGAs are analyzed. From the perspective of the total energy balance of the system and applying the energy dynamic stability equation derived by analyzing the relationship between the internal energy and the total input energy, the stability of the tunnel structure is analyzed. It can be concluded that the larger the peak value of seismic wave acceleration, the longer period of the tunnel structure is instable and the greater the degree of dynamic instability.



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*Source: 9995682-2021-07-20.xml*