# Analytical Solutions for the Mechanical Responses of Shallow Double-Arched Tunnel Subjected to Symmetric Loads

Authors: Song Yang; Rui Dai; Zubin Ai; Xiaochang Li; Chao Zhang; Xin Huang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Double-arched tunnel is a special kind of tunnel structure which is widely adopted for shallow rock tunnels. The internal forces within a double-arched tunnel are difficult to be determined due to its special geometry and complex interactions with surrounding rock masses. This paper presents a set of analytical solutions for determining the internal forces within the lining structure of shallow double-arched tunnel subjected to symmetric external loads. The double-arched tunnel is decomposed into three main parts, i.e., the arch rings, the side walls, and the middle wall. The force method and the elastic foundation beam model are employed for the arch rings and the side walls, respectively, while the middle wall is treated as a cantilever beam fixed at the bottom. Analytical solutions for the internal forces of the three parts are derived separately, which are continuous at the conjunctions of different parts. The derived analytical solutions are verified by comparing with the FEM simulation results. Finally, parametric studies are performed to investigate the influences of burial depth, elastic resistance coefficient, opening angle, and tunnel span on the internal forces based on which some recommendations are provided for the construction and design of the double-arched tunnels. The derived analytical solutions provide fast estimations for the internal forces and deformation of the double-arched lining structures, which will be a useful tool for design optimization.


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## Body

## 1. Introduction

The double-arched tunnel is a special kind of tunnel whose lining structures between two adjacent tunnels are connected and supported by the middle wall [1].Unlike single tunnels with a circular or straight wall arch cross section which have been profoundly investigated [2–9], due to its special geometry, the distribution of internal forces of double-arched tunnels has not yet been well understood. Yan et al. set up a comprehensive monitoring program at critical sections of a double-arched tunnel to measure the internal forces in both steel arches and secondary linings [10]. The monitoring and analysis results show that the forces of lining structures are directly related to multiple factors, including the construction procedures, geological conditions, and locations in the double-arched tunnel. They further showed that the arch vault is the most critical section, which was also observed by Yuan et al. [11]. Through in-site monitoring, Lai et al. found that the critical areas of double-arched tunnels are mainly located on the middle wall, side wall, and vault [12]. Zhang et al. carried out a model test for an unequally spanned double-arched tunnel to study the distribution of surrounding rock pressure and mechanical characteristics and found that the tunnel was in a small eccentric bending state, and thrust stress in the side walls increases with increasing span [13]. He et al. conducted model tests to investigate the distribution of internal forces of nonsymmetric double-arched tunnels in sand-cobble ground [14]. They showed that the internal force of the structure increased linearly with the increase in external load until the first crack appeared, after which the internal force redistributed but the structure still had a certain bearing capacity. Li et al. derived the mechanical responses of supporting structure for a shallow-buried double-arched tunnel under the China Great Wall based on reduced-scale model tests [15].Numerical simulations have also been conducted to analyse the stability and mechanical behaviours of double-arched lining structure. Yu and Yang analysed the internal force of the lining of the double-arched tunnel by using the loading structure method, considering the situations of both single-side and double-side bearings [16]. Zhang and He performed numerical simulations on the mechanical behaviour of middle wall of double-arched tunnel and found that the main factors affecting the loads of middle wall include the burial depth, the properties of surrounding rock, the construction method, and the eccentric loading condition [17]. Based on the coupled fluid-solid theorem, Li et al. investigated the stability of surrounding rock and internal forces of double-arched tunnel under seepage condition [18]. Mao et al. simulated the construction process of a double-arched tunnel in loess area considering the lateral recharge and fluid-solid coupling effect, which showed that the water disaster susceptible areas mainly are concentrated in the middle wall, arch springs of both tunnels, and tunnel face [19]. Liu et al. studied the acceleration response of a shallow-buried biased double-arched tunnel under the action of Wenchuan wave [20]. Based on both physical model testing and numerical simulation results, Min et al. reported that the presence of voids on the top of the middle wall may induce adverse effects on the lining, which are characterized by significant change of internal forces and increase in lining deformation [21]. Similarly, Zhang et al. showed that the internal forces change significantly at the area in the close vicinity of the void [22].The aforementioned studies are either costly or time-consuming. In contrast, analytical approach is a convenient, fast, and effective method for analysing the mechanical behaviour of both surrounding ground and tunnel lining [23–25]. Based on Airy’s stress function, Nagger and Hinchberger presented a closed-form solution for bending moment and thrust force of tunnel linings that can be idealized as an inner jointed segmental lining and an outer thick-walled cylinder embedded in a homogeneous infinite elastic soil or rock, accounting for the interaction between the lining and the surrounding ground [26]. Lee et al. adopted the force method to predict internal forces and displacements of a jointed circular segmental tunnel lining by treating the longitudinal joints as a set of linear rotational springs [27]. Wang et al. presented an analytical solution for circular segmental tunnel lining based on the state space method [28]. The curved Euler beam theory and the principle of minimum potential energy were adopted during the derivation of governing equations. Wang et al. further derived a unified analytical solution for the circular segmental tunnel lining irrespective of the layout of segmental assemblies between adjacent rings by modelling the interaction between the segments as a set of springs with three-directional resistances [29]. Talmon and Bezuijen presented an analytical model to analyse the longitudinal bending moment in the tunnel lining during the construction phase of bored tunnels, taking into account the influence of forces from the TBM, the loading of the tunnel lining by the grout in the liquid phase, and linear elastic properties of the tunnel lining and soil [30]. Zhao et al. employed the commonly accepted analytical solutions to calculate the lining forces and deformations for both shallow and deep tunnels [31]. Park et al. developed an analytical solution by considering the relationships between displacements and interaction forces at the soil-lining interface and including the slippage effect at the soil-lining interface using a spring-type flexibility coefficient [32]. Bobet presented closed-form solutions for the liner stresses of a deep circular tunnel excavated in transversely anisotropic rock [33]. Kargar et al. provided a semianalytical elastic plane-strain solution for stress field around a lined noncircular tunnel at great depth subjected to uniform ground load [34]. The analytical solution for an arbitrarily shaped tunnel excavated in anisotropic rock mass supported by an isotropic lining was presented by Wang et al. [35]. The equations and solutions of stresses and displacements in lining and surrounding rock mass were derived based on the complex variable method.Most of the research studies about analytical approach mentioned above mainly focus on the internal forces or stresses of single tunnel lining, especially for the circular tunnel due to the simplification and convenience in the derivation. Hu et al. presented an analytical solution for the determination of internal forces of a jointed segmental precast DOT (double-O-tube) lining by combining the unbalanced force transition method with the force method [23]. This is the only analytical work available for the double tunnel lining. However, it should be noted that the shape of the DOT lining is still relatively regular when comparing with the double-arched tunnel which is normally composed of arch rings, straight side walls, and middle wall. So far, no analytical solutions for the internal forces of double-arched tunnel have been reported in the literature.In this paper, analytical solutions for determining the internal forces of lining structure in shallow double-arched tunnels subjected to symmetric loading are presented. The double-arched tunnel is decomposed into three parts, including the arch rings, the side walls, and the middle walls. The force method is applied to determine the internal forces of arch rings. The elastic foundation beam model is employed for representing the side walls. The middle wall is treated as a cantilever beam fixed at the bottom. The internal forces are continuous at the conjunctions of the three different parts. The internal forces of a background project calculated using the proposed analytical solutions are compared with the values obtained by the numerical simulations. Finally, based on the obtained analytical solutions, parametric analyses are conducted to investigate the influences of some key influential factors on the internal forces of the double-arched tunnel lining.


## 2. Model Formulation

During the derivation, the following basic assumptions and methodologies are adopted:(1)
The double-arched tunnel is shallowly buried(2)
The middle wall of the double-arched tunnel is straight and can be taken as a cantilever beam while the side wall is assumed as an elastic foundation beam(3)
The bottom of the side wall is fixed in both horizontal and vertical directions but can rotate under the elastic resistance, while the bottom of the middle wall is fixed(4)
The inverted arch is not considered in the derivation of the analytical solution(5)
The analytical solutions for the internal forces along the tunnel arch, the middle wall, and the side walls are derived separately(6)
The force method and elastic foundation beam method are used for the derivation of internal forces along the tunnel arch and side wall, respectively(7)
The thicknesses of the tunnel lining and the middle wall are not considered during the derivation and the external dimensions of the lining are employed(8)
The tunnel lining is considered as linear elasticBased on the above assumptions, the original mechanical model presented in Figure1(a) could be simplified into Figure 1(b). The definitions of symbols presented in Figure 1 are given as follows: H is the burial depth at the crown of the double-arched tunnel; H1 is the height from the top of the middle wall to the ground surface; H2 is the height of the double-arched tunnel; B is the total width of the double-arched tunnel; R is the outer radius of the arch ring of left tunnel; α1, α2, and α3 can be considered as the growth angle of lateral earth pressure along depth of the corresponding three kind of pressures, which are given by tanα1=e2−e1/H2 and tanα2=e2′−e1′/f2, α3=q′/l2; a denotes the bottom point of the side wall in the left tunnel; h is the foot of left arch of left tunnel on the top of the side wall; d is the foot of right arch of left tunnel on the top of middle wall; l is the height of the side wall of left tunnel; l1 and l2 are the widths of left and right half arch rings of left tunnel, respectively; f1 and f2 are the height of left and right half arch rings of left tunnel; and ha denotes the bottom width of the left side wall. A prime is added to the same set of symbols to denote the corresponding quantities of the right tunnel. The external pressure from the surrounding rock mass is determined under the assumption that the double-arched tunnel is shallowly buried [36, 37].Figure 1
The model of double-arched tunnel: (a) the mechanical model; (b) the simplified mechanical model (explanations for each symbol are allied with the first place it appears in the paper).
(a)(b)The cut-off altitudeHp separating deep tunnel from shallow tunnel can be considered as follows:(1)Hp=2∼2.5hp,where hp is the load equivalent height, which can be calculated according to the approach proposed by [36, 37].As shown in Figure1(a), based on Hp and hp, the pressure of shallow double-arched tunnel can be determined according to two different conditions as shown in Table 1 [36].Table 1
The pressures on the double-arched tunnel.
Scenario1,H≤hpScenario2,hp<H≤HpNotesq=γHq=γH1−H/Bλtanθλ=tanβ−tanφ/tanβ1+tanβtanφ−tanθtanφtanθe1=γHtan245°−φ/2e1=λγHtanβ=tanϕ+tanφtan2φ+1/tanφ−tanθe2=γH+H2tan245°−φ/2e2=λγH+H2\q′=γH1−He1′=qtan245−φ/2e2′=q+q′tan245°−φ/2In Table1, γ is the unit weight of the surrounding rock mass, φ is the calculated friction angle of surrounding rock masses, λ is the lateral pressure coefficient, β is the angle between the failure surfaces and the horizontal for both sides of the tunnel, and θ is the friction angle of sliding surface, which can be determined according to Table 2.Table 2
Theθ values for different grades of surrounding rock [38].
Surrounding rock gradeI, II, IIIIVVVIθ0.9φ(0.7–0.9)φ(0.5–0.7)φ(0.3–0.5)φ


## 3. Model Derivation

In this section, the analytical solutions for the internal forces of arch ring, side wall, and middle wall are derived separately based on the simplified model given in Figure1(b). Since the structure configuration and pressure distribution are both symmetric, only the left part of the tunnel structure is employed in the following derivation. The internal forces of the middle wall can be obtained by the superposition of corresponding quantities from both arch rings.


### 3.1. Analytical Solutions for the Arch Rings

The simplified models of the left arch ring for the force method are presented in Figure2. All the pressures exerted on the arch rings shown in Figure 2 are considered as active pressures, which are all positive. The redundant internal forces of the crown section of the arch ring induced by the active pressures are X1p, X2p, and X3p, respectively. The angular displacement of the arch foot turning outwards, the outward horizontal displacement, the upward vertical displacement of the right arch foot, and the downward vertical displacement of the left arch foot are positive.Figure 2
The simplified model for the left arch ring.βpl, upl, and vpl represent the angular, horizontal, and vertical displacements of the left arch foot, respectively, while βpr, upr, and vpr denote the same quantities of the right arch foot.According to the displacement coordination condition of the force method at the crown of the arch ring, the displacement equations are as follows:(2)X1pδ11+X2pδ12+X3pδ13+Δ1p+βpl+βpr=0X1pδ21+X2pδ22+X3pδ23+Δ2p+upl+upr+f1βpl+f2βpr=0X1pδ31+X2pδ32+X3pδ33+Δ3p+vpl+vpr+l2βpr−l1βpl=0,where δij=i,j=1,2,3 are the flexibility coefficients of the arch ring and ∆ip is the displacement induced by the active pressure in the direction of Xip at the cross section of the tunnel crown.By using the superposition principle, the displacements of the left and right arch feet are obtained as follows:(3)βpl=X1pβ1l+X2pβ2l+f1β1l+X3pβ3l−l1β1l+βploβpr=X1pβ1r+X2pβ2r+f2β1r+X3pβ3r+l2β1r+βproupl=X1pu1l+X2pu2l+f1u1l+X3pu3l−l1u1l+uploupr=X1pu1r+X2pu2r+f2u1r+X3pu3r+l2u1r+uprovpl=X1pv1l+X2pv2l+f1v1l+X3pv3l−l1v1l+vplovpr=X1pv1r+X2pv2r+f2v1r+X3pv3r+l2v1r+vpro,where βil, uil, and vil are the angular, horizontal, and vertical displacements of the left arch foot under the unit loads Xi=1i=1,2,3, respectively; βir, uir, and vir denote the same quantities of the right arch foot; βplo, uplo, and vplo are the angular, horizontal, and vertical displacements of the left arch foot under the active pressures; and βpro, upro, and vpro are the same quantities of the right arch foot.The reciprocal theorem of displacement gives the following relationships:(4)β2l=u1l,β3l=v1l,u3l=v2l,β2r=u1r,β3r=v1r,u3r=v2r.Substituting the displacements in equation (3) into equation (2), the following displacement equations containing the unknown X1p, X2p, and X3p are obtained:(5)a11X1p+a12X2p+a13X3p+a10=0a21X1p+a22X2p+a23X3p+a20=0a31X1p+a32X2p+a33X3p+a30=0,where the coefficients aij are determined by the following:(6)a11=δ11+β1l+β1ra12=δ12+β2l+β2r+f1β1l+f2β1ra13=δ13+β3l+β3r+l2β1r−l1β1la10=Δ1p+βplo+βproa21=δ21+u1l+u1r+f1β1l+f2β1ra22=δ22+u2l+u2r+f1u1l+f2u1r+f1β2l+f12β1l+f2β2r+f22β1ra23=δ23+u3l+u3r+l2u1r−l1u1l+f1β3l−f1l1β1l+f2β3r+f2l2β1ra20=Δ2p+uplo+upro+f1βplo+f2βproa31=δ31+v1l+v1r+l2β1r−l1β1la32=δ32+v2l+v2r+f1v1l+f2v1r+l2β2r+l2f2β1r−l1β2l−l1f1β1la33=δ33+v3l+v3r−l1v1l+l2v1r+l2β3r+l22β1r−l1β1l+l12β1la30=Δ3p+vplo+vpro+l2βpro−l1βplo.By solving equation (5), X1p, X2p, and X3p can be determined as follows:(7)X1p=Δ1ΔX2p=Δ2ΔX3p=Δ3Δ,where Δ=a11a12a13a21a22a23a31a32a33, Δ1=−a10a12a13a20a22a23a30a32a33, Δ2=−a11a10a13a21a20a23a31a30a33, and Δ3=−a11a12a10a21a22a20a31a32a30.The unknown coefficients for the arch ring can be solved based on the knowledge of structural mechanics. The basic structure of the arch ring under the unit loadsXi=1i=1,2,3 is shown in Figure 3. The internal forces of the arch ring corresponding to the unit loads X1=1, X2=1, and X3=1 are presented in the following equations, respectively:(8)M1=1,N1=0,Q1=0,(9)left half arch ring:M2=R1−cosφi,N2=cosφi,Q2=−sinφi,right half arch ring:M2=R1−cosφi,N2=cosφi,Q2=sinφi,(10)left half arch ring:M3=−Rsinφi,N3=sinφi,Q3=cosφi,right half arch ring:M3=Rsinφi,N3=−sinφi,Q3=cosφi.Figure 3
The basic structure of the arch ring under the unit load.
(a)(b)(c)(d)Since the effect of axial and shear forces on displacement is relatively small, usually only the effect of the bending moments is considered [27]. However, in order to fully consider the internal forces, the axial and shear forces are also involved. According to the structure mechanics, the flexibility coefficients δij for the arch ring can be calculated as follows:(11)δij=∑∫MiMjE′Ids+∑∫NiNjE′Αds+∑∫kQiQjGAds,where MiMj, NiNj, and QiQj are the bending moments, axial force, and shear force of the arch ring under the unit loads XiXj=1i,j=1,2,3 at the crown of the arch ring of the basic structure; E′ is the elastic modulus of the arch ring; G is the shear modulus of the arch ring; I is the polar moment of inertia of the arch ring’s cross section with respect to the longitudinal direction; A is the area of the cross section; k is the nonuniform coefficient of the shear stress, which is taken as 1.2 for the rectangular cross section in this paper; and s is the arch length.Substituting equations (8)–(10) into equation (11) for i, j = 1,2,3, respectively, the flexibility coefficients of the left tunnel can be obtained as follows:(12)δ11=REIφh+φdδ22=R3EI32φh+φd−2sinφh+sinφd+14sin2φh+sin2φd+REA+kRGA12φh+φd+REA−kRGA14sin2φh+sin2φdδ33=R3EI+REA12φh+φd−14sin2φh+sin2φd+kRGA12φh+φd+14sin2φh+sin2φdδ12=δ21=R2EIφh+φd−sinφh+sinφdδ13=δ31=R2EIcosφh−cosφdδ23=δ32=R3EIcosφh−cosφd+14cos2φd−cos2φh+kR4GA−R4EAcos2φh−cos2φd.Note thatφi ranges from 0 to φh for the left half arch ring and 0 to φd for the right half arch ring. The flexibility coefficients of the right tunnel can be obtained by replacing φh and φd by φd′ and φh′ in the same formulae of the left tunnel.The internal forces caused by active pressures can be determined by accumulating the pressures as shown in Figure4, namely, the vertical uniform pressure q, the vertical triangular pressure q′, and the lateral trapezoidal pressure e+Δe at both sides which can be considered as uniform pressures e=e1 for the left half arch ring and e=e1′ for the right half arch ring with the triangular pressure Δe=f1tanα1 for the left half arch ring and Δe=f2tanα2 for the right half ring.(13)Mipo=Miqo+Miq′o+Mieo+MiΔeo,Nipo=Niqo+Niq′o+Nieo+NiΔeo,Qipo=Qiqo+Qiq′o+Qieo+QiΔeo,where Mipo, Nipo, and Qipo denote the bending moment, axial force, and shear force under the active pressures, respectively.Figure 4
The force analysis diagrams of the basic structure of the arch ring under (a) vertical uniform pressure; (b) vertical triangular pressure; (c) lateral trapezoidal pressure.For the vertical uniform pressureq,(14)left half arch ring:Miqo=−12qR2sin2φi,Niqo=qRsin2φi,Qiqo=Rqsinφicosφi,right half arch ring:Miqo=−12qR2sin2φi,Niqo=qRsin2φi,Qiqo=−qRsinφicosφi.For the vertical triangular pressureq′,(15)Miq′o=−16R3tanα3⋅sin3φi,Niq′o=12R2tanα3⋅sin3φi,Qiq′o=−12R2tanα3⋅sin2φicosφi.For the lateral uniform pressuree,(16)lefthalfarchring:Mieo=−12e1R21−cosφi2,Nieo=−e1Rcosφi1−cosφi,Qieo=e1Rsinφi1−cosφi,righthalfarchring:Mieo=−12e1′R21−cosφi2,Nieo=−e1′Rcosφi1−cosφi,Qieo=−e1′Rsinφi1−cosφi.For the triangular pressure ∆e,(17)left half arch ring:MiΔeo=−16R3tanα11−cosφi3,NiΔeo=−12R2tanα11−cosφi2cosφi,QiΔeo=12R2tanα11−cosφi2sinφi,right half arch ring:MiΔeo=−16R3tanα21−cosφi3,NiΔeo=−12R2tanα21−cosφi2cosφi,QiΔeo=−12R2tanα21−cosφi2sinφi.The displacement coefficients ∆ip for the arch ring can be calculated as follows:(18)Δip=∑∫MiMpoE′Ids+∑∫NiNpoE′Αds+∑∫kQiQpoGAds.Substituting equation (14) and equations (8)–(10) into equation (18) yields the following displacement coefficients of the arch ring under the vertical uniform pressure q:(19)Δ1q=−R3q4EIφh+φd+R3q8EIsin2φh+sin2φdΔ2q=−1EI14qR4φh+φd−18qR4sin2φh+sin2φd−16qR4sin3φh+sin3φd+1EA−kGA13R2qsin3φh+sin3φdΔ3q=1EI12qR4cosφd−cosφh+16qR4cos3φh−cos3φd+1EAqR2cosφd−cosφh+13qR2cos3φh−cos3φd+kGA13qR2cos3φd−cos3φh.Using the same method, the corresponding displacement coefficients for other active pressures can be obtained as follows:(20)Δ1q′=R4tanα36EIcosφd−13cos3φd−23Δ2q′=R5tanα36EIcosφd−13cos3φd−23+14sin4φd+R3tanα38EAsin4φd−kR3tanα38GAsin4φdΔ3q′=−R5tanα36EI+R3tanα32EA38φd−14sin2φd+132sin4φd−kR3tanα32GA18φd−132sin4φd.Δ1e=−3R34EIe1φh+e1′φd−R3EIe1sinφh+e1′sinφd+R38EIe1sin2φh+e1′sin2φd,Δ2e=−R42EI54e1φh+e1′φd−4e1sinφh+e1′sinφd+34e1sin2φh+e1′sin2φd+13e1sin3φh+e1′sin3φd−R2EA12e1φh+e1′φd−e1sinφh+e1′sinφd+14e1sin2φh+e1′sin2φd+e1sin3φh+e1′sin3φd−kR2GA12e1φh+e1′φd−14e1sin2φ+e1′sin2φd−13e1sin3φh+e1′sin3φd,Δ3e=R42EIe11−cosφh33−e1′1−cosφd33+R2EA−kR2GA12e1cos2φh−e1′cos2φd+13e1′cos3φh−e1cos3φd,Δ1Δe=−R46EI52φhtanα1+φdtanα1−4tanα1sinφh+tanα2sinφd+34tanα1sin2φh+tanα2sin2φd+13tanα1sin3φh+tanα2sin3φdΔ2Δe=−R56EI358φhtanα1+φdtanα2−8tanα1sinφh+tanα2sinφd+74tanα1sin2φh+tanα2sin2φd+43tanα1sin3φh+tanα2sin3φd+132tanα1sin4φh+tanα2sin4φd−R32EA78φhtanα1+φdtanα2+12tanα1sin2φk+tanα2sin2φd+23tanα1sin3φh+tanα2sin3φd+132tanα1sin4φh+tanα2sin4φd−kR32GA58φhtanα1+φdtanα2−14tanα1sin2φh+tanα2sin2φd−23tanα1sin3φh+tanα2sin3φd−132tanα1sin4φh+tanα2sin4φdΔ3Δe=R56EIcosφdtanα2−cosφhtanα1+32cos2φhtanα1−cos2φdtanα2+cos3φdtanα2−cos3φhtanα1+14cos4φhtanα1−cos4φdtanα2+R32EA−kR32GA12cos2φhtanα1−cos2φdtanα2−23cos3φhtanα1−cos3φdtanα2+14cos4φhtanα1−cos4φdtanα2.The displacement coefficients under the active pressures are the superposition of the obtained coefficients, which can be expressed by the following:(21)Δ1p=Δ1q+Δ1q′+Δ1e+Δ1ΔeΔ2p=Δ2q+Δ2q′+Δ2e+Δ2ΔeΔ3p=Δ3q+Δ3q′+Δ3e+Δ3Δe.Accordingly, the displacement coefficients for the right tunnel can be obtained by utilizing the same method.After the coefficients for the arch ring have been obtained, the remaining unknown coefficients at the top of the side wall and middle wall could be calculated by the elastic beam model and the force analysis on the top of the middle wall, which are shown in Appendix A and B, respectively.Based on the obtained coefficients, the redundant forcesX1p, X2p, and X3p can be obtained by equation (7). Based on equation (7), the internal forces along the arch ring can be determined according to the static equilibrium condition, which are given by equations (22) and (23) for the left and right half arch rings, respectively. The bending moment Mia is positive when the intrados of the arch ring is in tension and the axial force Nia is positive when the arch ring is under compression, while the shear force Qia leading to the clockwise rotation of the arch section is taken as positive.(22)Mia=X1p+X2pyi−X3pxi+Mipo,Nia=X2pcosφi+X3psinφi+Nipo,Qia=−X2psinφi+X3pcosφi+Qipo,(23)Mia=X1p+X2pyi−X3pxi+Mipo,Nia=X2pcosφi+X3psinφi+Nipo,Qia=−X2psinφi+X3pcosφi+Qipo,where Mipo, Nipo, and Qipo are the internal forces for arbitrary positions of the arch ring subject to the active pressures and φi is the angle between the analysed cross section and crown of the arch ring, which is in the range of 0∼φh for the left half arch ring and 0∼φd for the right half arch ring.Because of symmetry and by using the same methods, the analytical solutions of internal forces for the right tunnel arch ring can be obtained accordingly.


### 3.2. Analytical Solutions for the Side Wall

The internal forces of the arch rings are transferred continuously to the side wall through the arch foot, as shown in Figure5(a). For the side wall, the bending moment Mis is positive when the intrados of the side wall is in tension and the axial force Nis is positive when the side wall is under compression, while the shear force Qis leading to the clockwise rotation of the side wall is positive.Figure 5
The internal forces on the left side wall: (a) forces transferred to the top of the side wall; (b) the decomposition of the internal forces at the top of the side wall.
(a)(b)In general, the angle of the joint of the side wall and arch foot at the crown of the arch ring, namely,φh for the left tunnel and φh′ for the right one, are usually right angle and it is unnecessary to decompose the internal forces. Nevertheless, in order to embrace more general situations, φh and φh′ are considered to be arbitrary angles in the current derivation. The corresponding decompositions of the internal forces at the top of the side wall are shown in Figure 5(b).In the derivation of the analytical solutions, the side wall is assumed to be an elastic foundation beam. The internal forces induced by the unit bending moment, unit horizontal force, and vertical force exerted on the top of the side wall are first derived separately, which can refer to Appendix A. The real internal forces are obtained by applying the principle of superposition. For the left tunnel, the final solutions for the internal forces of the side wall are obtained as follows:(24)Mis=MhMM=1+−Qhsinφh+NhcosφhMH=1+Qhcosφh+NhsinφhMV=1+Me+MΔe,Qis=MhQM=1+−Qhsinφh+NhcosφhQH=1+Qhcosφh+Nhsinφh⋅QV=1+Qe+QΔe,Nis=Qhcosφh+Nhsinφh,where MM=1 and QM=1, MH=1 and QH=1, and MV=1 and QV=1 are the bending moments and shear forces of the side wall induced by the unit bending moment exerted on the top of the side wall, induced by the unit horizontal force exerted on the top of the sidewall, and induced by the unit vertical force exerted on the top of the sidewall, respectively. Me and Qe; M∆e and Q∆e are the bending moments and shear forces induced by the uniform pressure exerted on the side wall along the vertical direction and induced by the triangular pressure exerted on the side wall along the vertical direction, respectively. They are all related with the elastic resistance according to the elastic foundation beam method, with the derivation given in Appendix A.The analytical solutions of internal forces for the right tunnel side wall can be obtained in a similar way.


### 3.3. Analytical Solutions for the Middle Wall

The middle wall is assumed as a cantilever beam. The internal forces of the arch rings are transferred continuously to the middle wall through the arch feet connected with the middle wall. The bending momentMim causing the counterclockwise rotation of the middle wall is positive, the shear force Qim leading the middle wall to rotate clockwise is positive, and the compressive axial force Nim is positive. The forces at the top of the middle wall and their decomposition are shown in Figure 6(a).Figure 6
The forces and decompositions at the top of the middle wall.
(a)(b)(c)Based on Figure6, the analytical solutions of internal forces for the middle wall can be obtained as follows:(25)Mim=Md′−Md−Qdsinφd+Qd′sinφd′yi+Nd′cosφd′−Ndcosφdyi,Nim=Ndsinφd+Nd′sinφd′−Qdcosφd+Qd′cosφd′,Qim=Qdsinφd+Qd′sinφd′+Ndcosφd−Nd′cosφd′,Equations (22) to (25) constitute the complete analytical solutions of the internal forces of shallow double-arched tunnel for the left tunnel and middle wall. Detailed information on how to determine the unknown coefficients is presented in Appendix A and B.


### 3.4. Main Steps to Determine the Internal Forces of Double-Arched Tunnel

Figure7 summarizes the stepwise procedure of deducting the internal forces of double-arched tunnel using the analytical solutions derived in the current study:(1)
determination of the outer dimension and lining characteristics of a double-arched tunnel. The outer dimension of the tunnel should be determined according to the geometry design, including the outer radius of arch ring, the height of side and middle walls, the total width of tunnel, as well as the opening angle. The lining characteristics including the lining thickness and properties should also be determined.(2)
determination of the pressures on a double-arched tunnel. Based on the burial depth and geological conditions, the pressures on the double-arched tunnel can be derived according to the information listed in Figure1 and Table 1.(3)
determination of the flexibility coefficients and displacement coefficients. Based on the aforementioned methods and equations, the flexibility coefficients (equation (12)) and displacement coefficients (equations (19) to (21)) of arch ring can be determined, respectively. The displacement coefficients of the side wall and middle wall can be determined according to the method in Appendix A and B.(4)
determination of the internal forces of the arch ring from equations (22) and (23).(5)
determination of the internal forces of the side wall and middle wall from the equations (24) and (25), respectively.The derived analytical solutions in this section mainly focus on the double-arched tunnel subjected to symmetric external loads, while the nonsymmetric scenarios are more commonly encountered in engineering practice. The main differences between the two scenarios are the pressures exerted on the tunnel. For the nonsymmetric ones, the vertical pressures q in Figure 1 will be trapezoidal instead of rectangle; thus, another growth angle of vertical earth pressure α4 should be added [38], similar to the lateral earth pressure outside of double-arched tunnel. Therefore, the derivation process of displacement coefficients of the arch ring (equations (14) and (19)) should consider the effect of α4, which is the main difference in the derivation process between the two scenarios. In addition, for deep tunnels, such as the right tunnel, a larger lateral pressure outside the tunnel will be present, and another growth angle α5 of the lateral pressure may not equal to the present α1 in Figure 1. Nonetheless, the deduced process of the right-side tunnel is still similar to the left one. The remaining part of the deduction of the analytical solution for the nonsymmetric scenarios is the same with the aforementioned process. Overall, it is convenient to extend the present analytical solution to the more common nonsymmetric scenarios.


## 3.1. Analytical Solutions for the Arch Rings

The simplified models of the left arch ring for the force method are presented in Figure2. All the pressures exerted on the arch rings shown in Figure 2 are considered as active pressures, which are all positive. The redundant internal forces of the crown section of the arch ring induced by the active pressures are X1p, X2p, and X3p, respectively. The angular displacement of the arch foot turning outwards, the outward horizontal displacement, the upward vertical displacement of the right arch foot, and the downward vertical displacement of the left arch foot are positive.Figure 2
The simplified model for the left arch ring.βpl, upl, and vpl represent the angular, horizontal, and vertical displacements of the left arch foot, respectively, while βpr, upr, and vpr denote the same quantities of the right arch foot.According to the displacement coordination condition of the force method at the crown of the arch ring, the displacement equations are as follows:(2)X1pδ11+X2pδ12+X3pδ13+Δ1p+βpl+βpr=0X1pδ21+X2pδ22+X3pδ23+Δ2p+upl+upr+f1βpl+f2βpr=0X1pδ31+X2pδ32+X3pδ33+Δ3p+vpl+vpr+l2βpr−l1βpl=0,where δij=i,j=1,2,3 are the flexibility coefficients of the arch ring and ∆ip is the displacement induced by the active pressure in the direction of Xip at the cross section of the tunnel crown.By using the superposition principle, the displacements of the left and right arch feet are obtained as follows:(3)βpl=X1pβ1l+X2pβ2l+f1β1l+X3pβ3l−l1β1l+βploβpr=X1pβ1r+X2pβ2r+f2β1r+X3pβ3r+l2β1r+βproupl=X1pu1l+X2pu2l+f1u1l+X3pu3l−l1u1l+uploupr=X1pu1r+X2pu2r+f2u1r+X3pu3r+l2u1r+uprovpl=X1pv1l+X2pv2l+f1v1l+X3pv3l−l1v1l+vplovpr=X1pv1r+X2pv2r+f2v1r+X3pv3r+l2v1r+vpro,where βil, uil, and vil are the angular, horizontal, and vertical displacements of the left arch foot under the unit loads Xi=1i=1,2,3, respectively; βir, uir, and vir denote the same quantities of the right arch foot; βplo, uplo, and vplo are the angular, horizontal, and vertical displacements of the left arch foot under the active pressures; and βpro, upro, and vpro are the same quantities of the right arch foot.The reciprocal theorem of displacement gives the following relationships:(4)β2l=u1l,β3l=v1l,u3l=v2l,β2r=u1r,β3r=v1r,u3r=v2r.Substituting the displacements in equation (3) into equation (2), the following displacement equations containing the unknown X1p, X2p, and X3p are obtained:(5)a11X1p+a12X2p+a13X3p+a10=0a21X1p+a22X2p+a23X3p+a20=0a31X1p+a32X2p+a33X3p+a30=0,where the coefficients aij are determined by the following:(6)a11=δ11+β1l+β1ra12=δ12+β2l+β2r+f1β1l+f2β1ra13=δ13+β3l+β3r+l2β1r−l1β1la10=Δ1p+βplo+βproa21=δ21+u1l+u1r+f1β1l+f2β1ra22=δ22+u2l+u2r+f1u1l+f2u1r+f1β2l+f12β1l+f2β2r+f22β1ra23=δ23+u3l+u3r+l2u1r−l1u1l+f1β3l−f1l1β1l+f2β3r+f2l2β1ra20=Δ2p+uplo+upro+f1βplo+f2βproa31=δ31+v1l+v1r+l2β1r−l1β1la32=δ32+v2l+v2r+f1v1l+f2v1r+l2β2r+l2f2β1r−l1β2l−l1f1β1la33=δ33+v3l+v3r−l1v1l+l2v1r+l2β3r+l22β1r−l1β1l+l12β1la30=Δ3p+vplo+vpro+l2βpro−l1βplo.By solving equation (5), X1p, X2p, and X3p can be determined as follows:(7)X1p=Δ1ΔX2p=Δ2ΔX3p=Δ3Δ,where Δ=a11a12a13a21a22a23a31a32a33, Δ1=−a10a12a13a20a22a23a30a32a33, Δ2=−a11a10a13a21a20a23a31a30a33, and Δ3=−a11a12a10a21a22a20a31a32a30.The unknown coefficients for the arch ring can be solved based on the knowledge of structural mechanics. The basic structure of the arch ring under the unit loadsXi=1i=1,2,3 is shown in Figure 3. The internal forces of the arch ring corresponding to the unit loads X1=1, X2=1, and X3=1 are presented in the following equations, respectively:(8)M1=1,N1=0,Q1=0,(9)left half arch ring:M2=R1−cosφi,N2=cosφi,Q2=−sinφi,right half arch ring:M2=R1−cosφi,N2=cosφi,Q2=sinφi,(10)left half arch ring:M3=−Rsinφi,N3=sinφi,Q3=cosφi,right half arch ring:M3=Rsinφi,N3=−sinφi,Q3=cosφi.Figure 3
The basic structure of the arch ring under the unit load.
(a)(b)(c)(d)Since the effect of axial and shear forces on displacement is relatively small, usually only the effect of the bending moments is considered [27]. However, in order to fully consider the internal forces, the axial and shear forces are also involved. According to the structure mechanics, the flexibility coefficients δij for the arch ring can be calculated as follows:(11)δij=∑∫MiMjE′Ids+∑∫NiNjE′Αds+∑∫kQiQjGAds,where MiMj, NiNj, and QiQj are the bending moments, axial force, and shear force of the arch ring under the unit loads XiXj=1i,j=1,2,3 at the crown of the arch ring of the basic structure; E′ is the elastic modulus of the arch ring; G is the shear modulus of the arch ring; I is the polar moment of inertia of the arch ring’s cross section with respect to the longitudinal direction; A is the area of the cross section; k is the nonuniform coefficient of the shear stress, which is taken as 1.2 for the rectangular cross section in this paper; and s is the arch length.Substituting equations (8)–(10) into equation (11) for i, j = 1,2,3, respectively, the flexibility coefficients of the left tunnel can be obtained as follows:(12)δ11=REIφh+φdδ22=R3EI32φh+φd−2sinφh+sinφd+14sin2φh+sin2φd+REA+kRGA12φh+φd+REA−kRGA14sin2φh+sin2φdδ33=R3EI+REA12φh+φd−14sin2φh+sin2φd+kRGA12φh+φd+14sin2φh+sin2φdδ12=δ21=R2EIφh+φd−sinφh+sinφdδ13=δ31=R2EIcosφh−cosφdδ23=δ32=R3EIcosφh−cosφd+14cos2φd−cos2φh+kR4GA−R4EAcos2φh−cos2φd.Note thatφi ranges from 0 to φh for the left half arch ring and 0 to φd for the right half arch ring. The flexibility coefficients of the right tunnel can be obtained by replacing φh and φd by φd′ and φh′ in the same formulae of the left tunnel.The internal forces caused by active pressures can be determined by accumulating the pressures as shown in Figure4, namely, the vertical uniform pressure q, the vertical triangular pressure q′, and the lateral trapezoidal pressure e+Δe at both sides which can be considered as uniform pressures e=e1 for the left half arch ring and e=e1′ for the right half arch ring with the triangular pressure Δe=f1tanα1 for the left half arch ring and Δe=f2tanα2 for the right half ring.(13)Mipo=Miqo+Miq′o+Mieo+MiΔeo,Nipo=Niqo+Niq′o+Nieo+NiΔeo,Qipo=Qiqo+Qiq′o+Qieo+QiΔeo,where Mipo, Nipo, and Qipo denote the bending moment, axial force, and shear force under the active pressures, respectively.Figure 4
The force analysis diagrams of the basic structure of the arch ring under (a) vertical uniform pressure; (b) vertical triangular pressure; (c) lateral trapezoidal pressure.For the vertical uniform pressureq,(14)left half arch ring:Miqo=−12qR2sin2φi,Niqo=qRsin2φi,Qiqo=Rqsinφicosφi,right half arch ring:Miqo=−12qR2sin2φi,Niqo=qRsin2φi,Qiqo=−qRsinφicosφi.For the vertical triangular pressureq′,(15)Miq′o=−16R3tanα3⋅sin3φi,Niq′o=12R2tanα3⋅sin3φi,Qiq′o=−12R2tanα3⋅sin2φicosφi.For the lateral uniform pressuree,(16)lefthalfarchring:Mieo=−12e1R21−cosφi2,Nieo=−e1Rcosφi1−cosφi,Qieo=e1Rsinφi1−cosφi,righthalfarchring:Mieo=−12e1′R21−cosφi2,Nieo=−e1′Rcosφi1−cosφi,Qieo=−e1′Rsinφi1−cosφi.For the triangular pressure ∆e,(17)left half arch ring:MiΔeo=−16R3tanα11−cosφi3,NiΔeo=−12R2tanα11−cosφi2cosφi,QiΔeo=12R2tanα11−cosφi2sinφi,right half arch ring:MiΔeo=−16R3tanα21−cosφi3,NiΔeo=−12R2tanα21−cosφi2cosφi,QiΔeo=−12R2tanα21−cosφi2sinφi.The displacement coefficients ∆ip for the arch ring can be calculated as follows:(18)Δip=∑∫MiMpoE′Ids+∑∫NiNpoE′Αds+∑∫kQiQpoGAds.Substituting equation (14) and equations (8)–(10) into equation (18) yields the following displacement coefficients of the arch ring under the vertical uniform pressure q:(19)Δ1q=−R3q4EIφh+φd+R3q8EIsin2φh+sin2φdΔ2q=−1EI14qR4φh+φd−18qR4sin2φh+sin2φd−16qR4sin3φh+sin3φd+1EA−kGA13R2qsin3φh+sin3φdΔ3q=1EI12qR4cosφd−cosφh+16qR4cos3φh−cos3φd+1EAqR2cosφd−cosφh+13qR2cos3φh−cos3φd+kGA13qR2cos3φd−cos3φh.Using the same method, the corresponding displacement coefficients for other active pressures can be obtained as follows:(20)Δ1q′=R4tanα36EIcosφd−13cos3φd−23Δ2q′=R5tanα36EIcosφd−13cos3φd−23+14sin4φd+R3tanα38EAsin4φd−kR3tanα38GAsin4φdΔ3q′=−R5tanα36EI+R3tanα32EA38φd−14sin2φd+132sin4φd−kR3tanα32GA18φd−132sin4φd.Δ1e=−3R34EIe1φh+e1′φd−R3EIe1sinφh+e1′sinφd+R38EIe1sin2φh+e1′sin2φd,Δ2e=−R42EI54e1φh+e1′φd−4e1sinφh+e1′sinφd+34e1sin2φh+e1′sin2φd+13e1sin3φh+e1′sin3φd−R2EA12e1φh+e1′φd−e1sinφh+e1′sinφd+14e1sin2φh+e1′sin2φd+e1sin3φh+e1′sin3φd−kR2GA12e1φh+e1′φd−14e1sin2φ+e1′sin2φd−13e1sin3φh+e1′sin3φd,Δ3e=R42EIe11−cosφh33−e1′1−cosφd33+R2EA−kR2GA12e1cos2φh−e1′cos2φd+13e1′cos3φh−e1cos3φd,Δ1Δe=−R46EI52φhtanα1+φdtanα1−4tanα1sinφh+tanα2sinφd+34tanα1sin2φh+tanα2sin2φd+13tanα1sin3φh+tanα2sin3φdΔ2Δe=−R56EI358φhtanα1+φdtanα2−8tanα1sinφh+tanα2sinφd+74tanα1sin2φh+tanα2sin2φd+43tanα1sin3φh+tanα2sin3φd+132tanα1sin4φh+tanα2sin4φd−R32EA78φhtanα1+φdtanα2+12tanα1sin2φk+tanα2sin2φd+23tanα1sin3φh+tanα2sin3φd+132tanα1sin4φh+tanα2sin4φd−kR32GA58φhtanα1+φdtanα2−14tanα1sin2φh+tanα2sin2φd−23tanα1sin3φh+tanα2sin3φd−132tanα1sin4φh+tanα2sin4φdΔ3Δe=R56EIcosφdtanα2−cosφhtanα1+32cos2φhtanα1−cos2φdtanα2+cos3φdtanα2−cos3φhtanα1+14cos4φhtanα1−cos4φdtanα2+R32EA−kR32GA12cos2φhtanα1−cos2φdtanα2−23cos3φhtanα1−cos3φdtanα2+14cos4φhtanα1−cos4φdtanα2.The displacement coefficients under the active pressures are the superposition of the obtained coefficients, which can be expressed by the following:(21)Δ1p=Δ1q+Δ1q′+Δ1e+Δ1ΔeΔ2p=Δ2q+Δ2q′+Δ2e+Δ2ΔeΔ3p=Δ3q+Δ3q′+Δ3e+Δ3Δe.Accordingly, the displacement coefficients for the right tunnel can be obtained by utilizing the same method.After the coefficients for the arch ring have been obtained, the remaining unknown coefficients at the top of the side wall and middle wall could be calculated by the elastic beam model and the force analysis on the top of the middle wall, which are shown in Appendix A and B, respectively.Based on the obtained coefficients, the redundant forcesX1p, X2p, and X3p can be obtained by equation (7). Based on equation (7), the internal forces along the arch ring can be determined according to the static equilibrium condition, which are given by equations (22) and (23) for the left and right half arch rings, respectively. The bending moment Mia is positive when the intrados of the arch ring is in tension and the axial force Nia is positive when the arch ring is under compression, while the shear force Qia leading to the clockwise rotation of the arch section is taken as positive.(22)Mia=X1p+X2pyi−X3pxi+Mipo,Nia=X2pcosφi+X3psinφi+Nipo,Qia=−X2psinφi+X3pcosφi+Qipo,(23)Mia=X1p+X2pyi−X3pxi+Mipo,Nia=X2pcosφi+X3psinφi+Nipo,Qia=−X2psinφi+X3pcosφi+Qipo,where Mipo, Nipo, and Qipo are the internal forces for arbitrary positions of the arch ring subject to the active pressures and φi is the angle between the analysed cross section and crown of the arch ring, which is in the range of 0∼φh for the left half arch ring and 0∼φd for the right half arch ring.Because of symmetry and by using the same methods, the analytical solutions of internal forces for the right tunnel arch ring can be obtained accordingly.


## 3.2. Analytical Solutions for the Side Wall

The internal forces of the arch rings are transferred continuously to the side wall through the arch foot, as shown in Figure5(a). For the side wall, the bending moment Mis is positive when the intrados of the side wall is in tension and the axial force Nis is positive when the side wall is under compression, while the shear force Qis leading to the clockwise rotation of the side wall is positive.Figure 5
The internal forces on the left side wall: (a) forces transferred to the top of the side wall; (b) the decomposition of the internal forces at the top of the side wall.
(a)(b)In general, the angle of the joint of the side wall and arch foot at the crown of the arch ring, namely,φh for the left tunnel and φh′ for the right one, are usually right angle and it is unnecessary to decompose the internal forces. Nevertheless, in order to embrace more general situations, φh and φh′ are considered to be arbitrary angles in the current derivation. The corresponding decompositions of the internal forces at the top of the side wall are shown in Figure 5(b).In the derivation of the analytical solutions, the side wall is assumed to be an elastic foundation beam. The internal forces induced by the unit bending moment, unit horizontal force, and vertical force exerted on the top of the side wall are first derived separately, which can refer to Appendix A. The real internal forces are obtained by applying the principle of superposition. For the left tunnel, the final solutions for the internal forces of the side wall are obtained as follows:(24)Mis=MhMM=1+−Qhsinφh+NhcosφhMH=1+Qhcosφh+NhsinφhMV=1+Me+MΔe,Qis=MhQM=1+−Qhsinφh+NhcosφhQH=1+Qhcosφh+Nhsinφh⋅QV=1+Qe+QΔe,Nis=Qhcosφh+Nhsinφh,where MM=1 and QM=1, MH=1 and QH=1, and MV=1 and QV=1 are the bending moments and shear forces of the side wall induced by the unit bending moment exerted on the top of the side wall, induced by the unit horizontal force exerted on the top of the sidewall, and induced by the unit vertical force exerted on the top of the sidewall, respectively. Me and Qe; M∆e and Q∆e are the bending moments and shear forces induced by the uniform pressure exerted on the side wall along the vertical direction and induced by the triangular pressure exerted on the side wall along the vertical direction, respectively. They are all related with the elastic resistance according to the elastic foundation beam method, with the derivation given in Appendix A.The analytical solutions of internal forces for the right tunnel side wall can be obtained in a similar way.


## 3.3. Analytical Solutions for the Middle Wall

The middle wall is assumed as a cantilever beam. The internal forces of the arch rings are transferred continuously to the middle wall through the arch feet connected with the middle wall. The bending momentMim causing the counterclockwise rotation of the middle wall is positive, the shear force Qim leading the middle wall to rotate clockwise is positive, and the compressive axial force Nim is positive. The forces at the top of the middle wall and their decomposition are shown in Figure 6(a).Figure 6
The forces and decompositions at the top of the middle wall.
(a)(b)(c)Based on Figure6, the analytical solutions of internal forces for the middle wall can be obtained as follows:(25)Mim=Md′−Md−Qdsinφd+Qd′sinφd′yi+Nd′cosφd′−Ndcosφdyi,Nim=Ndsinφd+Nd′sinφd′−Qdcosφd+Qd′cosφd′,Qim=Qdsinφd+Qd′sinφd′+Ndcosφd−Nd′cosφd′,Equations (22) to (25) constitute the complete analytical solutions of the internal forces of shallow double-arched tunnel for the left tunnel and middle wall. Detailed information on how to determine the unknown coefficients is presented in Appendix A and B.


## 3.4. Main Steps to Determine the Internal Forces of Double-Arched Tunnel

Figure7 summarizes the stepwise procedure of deducting the internal forces of double-arched tunnel using the analytical solutions derived in the current study:(1)
determination of the outer dimension and lining characteristics of a double-arched tunnel. The outer dimension of the tunnel should be determined according to the geometry design, including the outer radius of arch ring, the height of side and middle walls, the total width of tunnel, as well as the opening angle. The lining characteristics including the lining thickness and properties should also be determined.(2)
determination of the pressures on a double-arched tunnel. Based on the burial depth and geological conditions, the pressures on the double-arched tunnel can be derived according to the information listed in Figure1 and Table 1.(3)
determination of the flexibility coefficients and displacement coefficients. Based on the aforementioned methods and equations, the flexibility coefficients (equation (12)) and displacement coefficients (equations (19) to (21)) of arch ring can be determined, respectively. The displacement coefficients of the side wall and middle wall can be determined according to the method in Appendix A and B.(4)
determination of the internal forces of the arch ring from equations (22) and (23).(5)
determination of the internal forces of the side wall and middle wall from the equations (24) and (25), respectively.The derived analytical solutions in this section mainly focus on the double-arched tunnel subjected to symmetric external loads, while the nonsymmetric scenarios are more commonly encountered in engineering practice. The main differences between the two scenarios are the pressures exerted on the tunnel. For the nonsymmetric ones, the vertical pressures q in Figure 1 will be trapezoidal instead of rectangle; thus, another growth angle of vertical earth pressure α4 should be added [38], similar to the lateral earth pressure outside of double-arched tunnel. Therefore, the derivation process of displacement coefficients of the arch ring (equations (14) and (19)) should consider the effect of α4, which is the main difference in the derivation process between the two scenarios. In addition, for deep tunnels, such as the right tunnel, a larger lateral pressure outside the tunnel will be present, and another growth angle α5 of the lateral pressure may not equal to the present α1 in Figure 1. Nonetheless, the deduced process of the right-side tunnel is still similar to the left one. The remaining part of the deduction of the analytical solution for the nonsymmetric scenarios is the same with the aforementioned process. Overall, it is convenient to extend the present analytical solution to the more common nonsymmetric scenarios.


## 4. Model Validation

### 4.1. Model Overview

As shown in Figure8(a), the adopted double-arched tunnel model referred to the project introduced by [10]. Since the dimension of the side wall was not clearly defined in [10], for simplicity, the height of the side wall is roughly set to be 3m according to the length scale as presented in [10]. The thickness of the sprayed concrete and the secondary lining is 25 cm and 50 cm, respectively.Figure 7
The flow chart of the determining procedures.The quality of rock mass mainly reflects the pressure exerted on the tunnel lining, which is usually evaluated by the rock mass classification schemes. Over the past decades, a number of rock mass classification schemes have been proposed [39], in which the rock masses are classified into different grades according to their compactness and mechanical properties. For example, the RMR method divides the rock mass into five grades, i.e., grades I ∼IV, of which the grade I denotes the rock mass with the highest quality, while the grade V represents the rock mass with the lowest quality. Based on the data available in [10] and according to the design code of China [38], the K89 + 729 section is investigated herein and surrounding rock mass is classified as grade V by the RMR method. The main physical and mechanical properties of the grade V surrounding rock masses are listed in Table 3.Table 3
The main physical and mechanical properties of the grade V surrounding rock [10].
Cohesion (MPa)Internal friction angle (°)Elastic modulus (GPa)Unit weight (kN/m3)0.07–0.127–290.8–1.927.5–31.5


### 4.2. Input Parameters in the Analytical Solutions

Although the exactly burial depth of the K89 + 729 section was not given in [10], the contact pressures (q) between the surrounding rock and the primary lining at the crown of the arch ring were presented, which equaled to 0.3 MPa. Therefore, the equivalent burial depth (H) of the tunnel section can be back calculated based on Protodyakonov’s theory, i.e., H = q/γ = 9.5 m, which is smaller than the load equivalent height hq = 14.24 m. Then, the detailed information about the ground condition and the secondary lining input in the analytical solutions can be derived as shown in Table 4.Table 4
Input information in the analytical solutions.
ItemValuesSurrounding rockEquivalent depthH(m)9.5Unit weightγ (kN/m3)27.5–31.5Elastic modulusE (GPa)0.8–1.9Poisson’s ratioμ0.35–0.45Elastic resistance coefficientK(MPa/m)100–200Calculated friction angleφ(°)40–50Friction angle of sliding surfaceθ (°)(0.5–0.7)φLining structureElastic modulusE′ (GPa)31.75Shear modulusG (GPa)13.65Poisson’s ratioμ′0.2Unit weightγ′(kN/m3)23


### 4.3. FEM Numerical Modelling

In order to accurately capture the key characteristics of the elastic foundation beam and the analytical solutions, the load-structure method is adopted in the 2D FEM model. The external pressures exerted on the double-arched tunnel can be calculated using the corresponding values in Table4 based on the method introduced in Table 1. The determined external pressures will be imported into the load-structure method-based FEM simulations.All the FEM analyses are carried out under the plane-strain conditions. The numerical model is presented in Figure8(b), in which the primary lining structure is not considered and the secondary lining is simulated by the beam elements according to the dimensions in Figure 8(a). The extrados and bottom of both side walls are restricted by unidirectional compression springs with a stiffness K = 200 MPa/m, which models the elastic resistances from the surrounding rock masses. The bottoms of the side and middle walls are also fixed in both horizontal and vertical directions. The rotation of the middle wall is also inhibited. It should be noted that the FEM model is specified to have the same loading configuration and assumptions of material properties as those adopted for analytical derivations. Therefore, the dynamic ground-structure interaction and plastic yielding of lining structure are not considered.


### 4.4. Comparison of Results

Figure9 compares the internal forces derived from analytical solutions and numerical simulations. The variational trend and the distributive pattern of the internal forces of both methods are very close to each other. The discrepancies between the two are overall small and acceptable, except that the bending moment predicted by the analytical solutions deviates increasingly more from the numerical simulation data when approaching the middle wall. This may be because in the analytical solution the elastic resistance coefficient kd between the top of the middle wall and the arch ring is set to the elastic modulus E′ of the lining structure in the analytical solutions. The simplification of beam element of the numerical model at the top of the middle wall may also deviate from the real situation, which may also account for the discrepancies. Further investigations may be required to determine the real contact characteristics at the top of the middle wall for double-arched tunnel.Figure 8
The adopted model of double-arched tunnel: (a) the cross section of the model (adapted from [10]); (b) the 2D FEM model.
(a)(b)Despite the discrepancies, the analytical solutions and the numerical solutions are overall in good agreement. It shows that the shear forces are smaller than the axial forces with the maximum shear force located at the conjunction point between the arch ring and the side wall, which is also an inflection point of the shear force distribution. Such an obvious inflection point may be because we only apply the elastic resistance to the side wall. The middle wall only bears the axial forces without presence of bending moment and shear forces in accordance with the symmetry condition of the lining structure and surrounding pressure during the derivation of the analytical solutions.The distribution law of internal forces can be derived from Figure9. Specifically, the bending moment in the side wall is positive and increases from the bottom upwards but then decreases when approaching the arch foot. The maximum bending moment of the lining which is in the upper side along the side wall may be caused by the constraint to deformation provided by the external elastic resistance. This is confirmed by Figure 10, which shows that the upper side of the side wall experiences a larger outside deformation than the lower side. The maximum positive bending moment of the arch ring lies near the crown of the arch, and its magnitude is larger than that of the maximum negative bending moment, which is located at the point with an angle about 60° measured from arch crown to the half ring near both side walls. This may also be verified by the deformation shown in Figure 10, of which the larger deformation is present in the crown and outside haunch of both tunnels. The results indicate that both the intrados of the crown and extrados of the haunch of the arch ring are in larger tension. These characteristics should be noted during the design and construction of the tunnel lining structure. Generally, the axial force shows a relatively uniform distribution along the arch ring, while the axial force near the side wall is slightly larger than that near the crown of the arch ring. Thus, the side wall is in larger compression. The smaller axial force near the arch crown may be caused by the larger vertical pressure at the crown, which helps to reduce the axial force in the quasi-horizontal direction. The maximum values of shear force of arch rings in Figure 9(b) mainly appeared near the zero point of bending moment with the nonshear point located close to the place where the maximum bending moment is present, consistent with the general distribution law of forces within beam elements. Furthermore, the large shear force at the bottom of the side wall should be taken into consideration during the design and construction.Figure 9
Comparison of the internal forces obtained from the analytical solutions and numerical simulations: (a) bending moment and axial force; (b) shear force.
(a)(b)


## 4.1. Model Overview

As shown in Figure8(a), the adopted double-arched tunnel model referred to the project introduced by [10]. Since the dimension of the side wall was not clearly defined in [10], for simplicity, the height of the side wall is roughly set to be 3m according to the length scale as presented in [10]. The thickness of the sprayed concrete and the secondary lining is 25 cm and 50 cm, respectively.Figure 7
The flow chart of the determining procedures.The quality of rock mass mainly reflects the pressure exerted on the tunnel lining, which is usually evaluated by the rock mass classification schemes. Over the past decades, a number of rock mass classification schemes have been proposed [39], in which the rock masses are classified into different grades according to their compactness and mechanical properties. For example, the RMR method divides the rock mass into five grades, i.e., grades I ∼IV, of which the grade I denotes the rock mass with the highest quality, while the grade V represents the rock mass with the lowest quality. Based on the data available in [10] and according to the design code of China [38], the K89 + 729 section is investigated herein and surrounding rock mass is classified as grade V by the RMR method. The main physical and mechanical properties of the grade V surrounding rock masses are listed in Table 3.Table 3
The main physical and mechanical properties of the grade V surrounding rock [10].
Cohesion (MPa)Internal friction angle (°)Elastic modulus (GPa)Unit weight (kN/m3)0.07–0.127–290.8–1.927.5–31.5


## 4.2. Input Parameters in the Analytical Solutions

Although the exactly burial depth of the K89 + 729 section was not given in [10], the contact pressures (q) between the surrounding rock and the primary lining at the crown of the arch ring were presented, which equaled to 0.3 MPa. Therefore, the equivalent burial depth (H) of the tunnel section can be back calculated based on Protodyakonov’s theory, i.e., H = q/γ = 9.5 m, which is smaller than the load equivalent height hq = 14.24 m. Then, the detailed information about the ground condition and the secondary lining input in the analytical solutions can be derived as shown in Table 4.Table 4
Input information in the analytical solutions.
ItemValuesSurrounding rockEquivalent depthH(m)9.5Unit weightγ (kN/m3)27.5–31.5Elastic modulusE (GPa)0.8–1.9Poisson’s ratioμ0.35–0.45Elastic resistance coefficientK(MPa/m)100–200Calculated friction angleφ(°)40–50Friction angle of sliding surfaceθ (°)(0.5–0.7)φLining structureElastic modulusE′ (GPa)31.75Shear modulusG (GPa)13.65Poisson’s ratioμ′0.2Unit weightγ′(kN/m3)23


## 4.3. FEM Numerical Modelling

In order to accurately capture the key characteristics of the elastic foundation beam and the analytical solutions, the load-structure method is adopted in the 2D FEM model. The external pressures exerted on the double-arched tunnel can be calculated using the corresponding values in Table4 based on the method introduced in Table 1. The determined external pressures will be imported into the load-structure method-based FEM simulations.All the FEM analyses are carried out under the plane-strain conditions. The numerical model is presented in Figure8(b), in which the primary lining structure is not considered and the secondary lining is simulated by the beam elements according to the dimensions in Figure 8(a). The extrados and bottom of both side walls are restricted by unidirectional compression springs with a stiffness K = 200 MPa/m, which models the elastic resistances from the surrounding rock masses. The bottoms of the side and middle walls are also fixed in both horizontal and vertical directions. The rotation of the middle wall is also inhibited. It should be noted that the FEM model is specified to have the same loading configuration and assumptions of material properties as those adopted for analytical derivations. Therefore, the dynamic ground-structure interaction and plastic yielding of lining structure are not considered.


## 4.4. Comparison of Results

Figure9 compares the internal forces derived from analytical solutions and numerical simulations. The variational trend and the distributive pattern of the internal forces of both methods are very close to each other. The discrepancies between the two are overall small and acceptable, except that the bending moment predicted by the analytical solutions deviates increasingly more from the numerical simulation data when approaching the middle wall. This may be because in the analytical solution the elastic resistance coefficient kd between the top of the middle wall and the arch ring is set to the elastic modulus E′ of the lining structure in the analytical solutions. The simplification of beam element of the numerical model at the top of the middle wall may also deviate from the real situation, which may also account for the discrepancies. Further investigations may be required to determine the real contact characteristics at the top of the middle wall for double-arched tunnel.Figure 8
The adopted model of double-arched tunnel: (a) the cross section of the model (adapted from [10]); (b) the 2D FEM model.
(a)(b)Despite the discrepancies, the analytical solutions and the numerical solutions are overall in good agreement. It shows that the shear forces are smaller than the axial forces with the maximum shear force located at the conjunction point between the arch ring and the side wall, which is also an inflection point of the shear force distribution. Such an obvious inflection point may be because we only apply the elastic resistance to the side wall. The middle wall only bears the axial forces without presence of bending moment and shear forces in accordance with the symmetry condition of the lining structure and surrounding pressure during the derivation of the analytical solutions.The distribution law of internal forces can be derived from Figure9. Specifically, the bending moment in the side wall is positive and increases from the bottom upwards but then decreases when approaching the arch foot. The maximum bending moment of the lining which is in the upper side along the side wall may be caused by the constraint to deformation provided by the external elastic resistance. This is confirmed by Figure 10, which shows that the upper side of the side wall experiences a larger outside deformation than the lower side. The maximum positive bending moment of the arch ring lies near the crown of the arch, and its magnitude is larger than that of the maximum negative bending moment, which is located at the point with an angle about 60° measured from arch crown to the half ring near both side walls. This may also be verified by the deformation shown in Figure 10, of which the larger deformation is present in the crown and outside haunch of both tunnels. The results indicate that both the intrados of the crown and extrados of the haunch of the arch ring are in larger tension. These characteristics should be noted during the design and construction of the tunnel lining structure. Generally, the axial force shows a relatively uniform distribution along the arch ring, while the axial force near the side wall is slightly larger than that near the crown of the arch ring. Thus, the side wall is in larger compression. The smaller axial force near the arch crown may be caused by the larger vertical pressure at the crown, which helps to reduce the axial force in the quasi-horizontal direction. The maximum values of shear force of arch rings in Figure 9(b) mainly appeared near the zero point of bending moment with the nonshear point located close to the place where the maximum bending moment is present, consistent with the general distribution law of forces within beam elements. Furthermore, the large shear force at the bottom of the side wall should be taken into consideration during the design and construction.Figure 9
Comparison of the internal forces obtained from the analytical solutions and numerical simulations: (a) bending moment and axial force; (b) shear force.
(a)(b)


## 5. Parametric Analysis

During the derivation of the analytical solutions, most parameters can be deduced according to the mechanical properties and geometrical characteristics of the lining structure. However, there are still some factors that cannot be derived directly, such as the real equivalent depthH and the elastic resistance coefficient K of the elastic foundation beam, which may affect the internal force distributions. Furthermore, the opening angle φd of the double-arched tunnel is also a main geometrical parameter of the lining structure that may affect the internal force. Besides, to accommodate the increasing transportation demand, the mountain tunnels tend to have large dimensions, while the difference in tunnel span may also change the loading states of the lining structure. Therefore, it is necessary to explore how these factors affect the internal forces and provide useful guidance for the design optimization. In this section, the four factors are varied systematically in the benchmark case presented in the section of Model Validation to investigate their influences on the internal forces. Due to the symmetry of the absolute values, only the internal forces at the left half of the double-arched tunnel are analysed in the following subsections. Note that the shear force on the right half is opposite in sign to that of the left half, while the bending moment and axial force are identical at both sides.


### 5.1. The Effect of the Burial Depth

The equivalent depthH is varied between 5m and 10m at an increment of 1m. Figures 11–13 show the internal forces along the lining structure under different H values. The variation trend and the locations of the maximum and minimum internal forces are independent of H, but the absolute magnitude of the internal forces increases consistently with increasing H.Figure 10
The deformation vector graph of the double-arched tunnel by the FEM analysis.Figure 11
Variations of bending moment with (H): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 12
Variations of axial force with (H): (a) along the arch ring; (b) along the side wall; (c) along the middle wall
(a)(b)(c)As shown in Figure11, the largest positive and negative bending moment are in the side wall about l = 0.9 m and the arch ring at location about β = 125°, respectively. The maximum axial force is in the middle wall as shown in Figure 12. The largest shear force occurs at the bottom and top of the side wall as shown in Figure 13. The variations of the largest magnitudes of the internal forces with increasing H are shown in Figure 14, which all show linear correlations. Figure 14 indicates that the internal forces could be significantly affected by the burial depth as expected.Figure 13
Variations of shear force with (H): (a) along the arch ring; (b) along the side wall.
(a)(b)


### 5.2. The Effect of Elastic Resistance Coefficient

The elastic resistance coefficientK = 200 MPa/m was adopted in the analytical analysis and FEM analysis presented above for the grade V rock masses. In order to embrace the elastic resistance characteristics of surrounding rock masses with different grades, a wide range of K (K = 100, 200, 500, 1200, 1800, and 2800 MPa/m) is adopted in this section which may represent the grade I∼V rock masses. Figures 15–17 show the internal forces along the lining structure under different elastic resistance coefficients.Figure 14
Variations of largest magnitudes of internal forces with (H): (a) bending moment; (b) axial force; (c) shear force.
(a)(b)(c)Figure 15
Variations of bending moment with (K): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 16
Variations of axial force with (K): (a) along the arch ring; (b) along the side wall; (c) along the middle wall.
(a)(b)(c)The magnitude of bending moment along the arch ring decreases with the increasingK except the section near the top of the side wall which becomes larger as K increases. The bending moment along the side wall increases as K increases from 100 to 200 MPa/m. When K is greater than 500 MPa/m, the bending moment initially increases with the increasing K for the top fifth of the side wall and then decreases as K increases for the remaining section of the side wall. The axial force along the arch ring initially decreases with increasing K but becomes approximately constant as K reaches 1200 MPa/m. The axial force along the side wall increases and the axial force along the middle wall decreases with increasing K, but their changes in magnitude are negligible. The magnitude of the shear force along the arch ring decreases as K increases for β in the range of 0° to about 120°, while as β becomes greater than 120°, the magnitude of shear force increases with increasing K. The magnitude of shear force along the side wall decreases in the top three tenths of the side wall and increases for the remaining section for which K increases in the range of 100–200 MPa/m. When K is greater than 500 MPa/m, the magnitude of shear force decreases with increasing K for the top one tenth and bottom four tenths and increases for the remaining section of side wall. Overall, the elastic resistance coefficient has more influence on the internal forces along the side wall. This may be because the elastic resistance is only applied to the side wall but is not present over the arch ring. The situation when elastic resistance is present over the entire lining ring needs further exploration in the future.


### 5.3. The Effect of Opening Angle

The opening angleφd is one of the main geometrical parameters determining the dimension of the double-arched tunnel, which is measured from the arch crown to the top of the middle wall as shown in Figure 2. The analysis of the opening angle serves as an important reference for designing and optimizing the cross section of the double-arched tunnel.The arch ring radiusR is assumed to be a constant. In such a case, the tunnel width B and height of middle wall hd would change in accordance with the change of φd. The adopted values in the analysis are shown in Table 5. The results of the internal forces obtained for different cases are shown in Figures 18–20.Table 5
Geometrical parameters corresponding to different opening anglesφd.
φd (°)f2 (m)hd (m)B(m)351.118.0419.35451.807.3521.00552.626.5322.38653.555.6023.45754.564.5924.18Figure 17
Variations of shear force with (K): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 18
Variations of bending moment withφd: (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 19
Variations of axial force withφd: (a) along the arch ring; (b) along the side wall, and (c) along the middle wall.
(a)(b)(c)It can be found that for both side walls and middle wall, the bending moment and axial force increase asφd increases. The magnitude of shear force of the side wall also increases with increasing φd except in the range about l = 0.6–1.0 m. The axial force of the arch ring also increases with the increase in opening angle. The variation of bending moment and shear force of the arch ring with φd is more complex. A greater change in the internal forces happens near the conjunction region between the arch ring and the top of the middle wall. Since the conjunction region between the arch ring and the top of middle wall is a special location closely related to φd where faults may exist and seepage may usually take place [40], it is of key concern for construction and design. Therefore, the influence of φd on the internal force at this location is explored in more details. Figure 21 shows the variation of the internal forces at the conjunction region between the arch ring and the top of the middle wall under different φd. As can be seen from Figure 21, in the range of opening angles considered, the bending moment increases initially to about 630 kN m and then decreases as φd further increases. In contrast, the shear force decreases initially to zero and reverses direction as φd further increases. The axial force increases almost linearly with increasing φd. Besides, Figure 19(a) shows that when φd exceeds 55°, the axial force at the conjunction region will be the largest along the arch ring. Considering that the conjunction region is the most difficult part for construction and the quality of lining installation may be the lowest in comparison to other parts of the lining structure, it is practically safer to maintain the axial force at the conjunction region at a low level. Therefore, it is suggested that φd should not exceed 55°.Figure 20
Variations of shear force withφd: (a) along the arch ring; (b) along the side wall.
(a)(b)


### 5.4. The effect of tunnel span

The tunnel span of double-arched tunnel is represented by the total width B and is mainly decided by the radius of arch ring R and thickness of middle wall. In order to analyse the effect of tunnel span on the internal forces, the height of the side wall and middle wall and the opening angle in Figure8(a) are not changed, and the burial depth remains 9.5 m, while the outer radius of arch ring increases from 5 m to 9 m at an increment of 1 m, and the outer radius 6.15 m in Figure 8(a) is still maintained. Neglecting the thickness of the middle wall in the analysis, the total width of the double-arched tunnel increases from 18.83 m to 33.89 m accordingly. The results of the internal forces obtained for different cases are shown in Figures 22–24.A larger tunnel span means a larger area of tunnel lining subjected to the vertical pressure on the top of the tunnel. Figures22–24 show that the variation trends of the internal forces with tunnel span are very similar to those with burial depth as shown in the Figures 11–13. As R increases, the absolute magnitude of axial force increases, while most of the absolute magnitude of the bending moment and shear force also increase except that near the inflection point. The point of the largest positive bending moment and the zero point of shear force of side wall move towards the top of side wall from about 1.2 m to 0.6 m as R increases. The largest negative bending moment is present in the arch ring at locations of β = 125° to β = 116° as R increases. These positions are different from those in the analysis of burial depth H.The largest positive of bending moment is present in the side wall aboutI = 0.6 m to I = 1.2 m except that at R = 5 m, the largest positive of bending moment is located in the arch ring at location about β = 56°. The maximum axial force is in the middle wall as shown in Figure 23. The largest shear force occurs at the bottom and top of the side wall as shown in Figure 24. As shown in Figure 25, nonlinear correlations between the largest internal forces and R are observed, which are different from the linear correlations as between the largest internal forces and burial depth. This indicates that the tunnel span is more influential to the internal forces than the burial depth (Figure 25).


## 5.1. The Effect of the Burial Depth

The equivalent depthH is varied between 5m and 10m at an increment of 1m. Figures 11–13 show the internal forces along the lining structure under different H values. The variation trend and the locations of the maximum and minimum internal forces are independent of H, but the absolute magnitude of the internal forces increases consistently with increasing H.Figure 10
The deformation vector graph of the double-arched tunnel by the FEM analysis.Figure 11
Variations of bending moment with (H): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 12
Variations of axial force with (H): (a) along the arch ring; (b) along the side wall; (c) along the middle wall
(a)(b)(c)As shown in Figure11, the largest positive and negative bending moment are in the side wall about l = 0.9 m and the arch ring at location about β = 125°, respectively. The maximum axial force is in the middle wall as shown in Figure 12. The largest shear force occurs at the bottom and top of the side wall as shown in Figure 13. The variations of the largest magnitudes of the internal forces with increasing H are shown in Figure 14, which all show linear correlations. Figure 14 indicates that the internal forces could be significantly affected by the burial depth as expected.Figure 13
Variations of shear force with (H): (a) along the arch ring; (b) along the side wall.
(a)(b)


## 5.2. The Effect of Elastic Resistance Coefficient

The elastic resistance coefficientK = 200 MPa/m was adopted in the analytical analysis and FEM analysis presented above for the grade V rock masses. In order to embrace the elastic resistance characteristics of surrounding rock masses with different grades, a wide range of K (K = 100, 200, 500, 1200, 1800, and 2800 MPa/m) is adopted in this section which may represent the grade I∼V rock masses. Figures 15–17 show the internal forces along the lining structure under different elastic resistance coefficients.Figure 14
Variations of largest magnitudes of internal forces with (H): (a) bending moment; (b) axial force; (c) shear force.
(a)(b)(c)Figure 15
Variations of bending moment with (K): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 16
Variations of axial force with (K): (a) along the arch ring; (b) along the side wall; (c) along the middle wall.
(a)(b)(c)The magnitude of bending moment along the arch ring decreases with the increasingK except the section near the top of the side wall which becomes larger as K increases. The bending moment along the side wall increases as K increases from 100 to 200 MPa/m. When K is greater than 500 MPa/m, the bending moment initially increases with the increasing K for the top fifth of the side wall and then decreases as K increases for the remaining section of the side wall. The axial force along the arch ring initially decreases with increasing K but becomes approximately constant as K reaches 1200 MPa/m. The axial force along the side wall increases and the axial force along the middle wall decreases with increasing K, but their changes in magnitude are negligible. The magnitude of the shear force along the arch ring decreases as K increases for β in the range of 0° to about 120°, while as β becomes greater than 120°, the magnitude of shear force increases with increasing K. The magnitude of shear force along the side wall decreases in the top three tenths of the side wall and increases for the remaining section for which K increases in the range of 100–200 MPa/m. When K is greater than 500 MPa/m, the magnitude of shear force decreases with increasing K for the top one tenth and bottom four tenths and increases for the remaining section of side wall. Overall, the elastic resistance coefficient has more influence on the internal forces along the side wall. This may be because the elastic resistance is only applied to the side wall but is not present over the arch ring. The situation when elastic resistance is present over the entire lining ring needs further exploration in the future.


## 5.3. The Effect of Opening Angle

The opening angleφd is one of the main geometrical parameters determining the dimension of the double-arched tunnel, which is measured from the arch crown to the top of the middle wall as shown in Figure 2. The analysis of the opening angle serves as an important reference for designing and optimizing the cross section of the double-arched tunnel.The arch ring radiusR is assumed to be a constant. In such a case, the tunnel width B and height of middle wall hd would change in accordance with the change of φd. The adopted values in the analysis are shown in Table 5. The results of the internal forces obtained for different cases are shown in Figures 18–20.Table 5
Geometrical parameters corresponding to different opening anglesφd.
φd (°)f2 (m)hd (m)B(m)351.118.0419.35451.807.3521.00552.626.5322.38653.555.6023.45754.564.5924.18Figure 17
Variations of shear force with (K): (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 18
Variations of bending moment withφd: (a) along the arch ring; (b) along the side wall.
(a)(b)Figure 19
Variations of axial force withφd: (a) along the arch ring; (b) along the side wall, and (c) along the middle wall.
(a)(b)(c)It can be found that for both side walls and middle wall, the bending moment and axial force increase asφd increases. The magnitude of shear force of the side wall also increases with increasing φd except in the range about l = 0.6–1.0 m. The axial force of the arch ring also increases with the increase in opening angle. The variation of bending moment and shear force of the arch ring with φd is more complex. A greater change in the internal forces happens near the conjunction region between the arch ring and the top of the middle wall. Since the conjunction region between the arch ring and the top of middle wall is a special location closely related to φd where faults may exist and seepage may usually take place [40], it is of key concern for construction and design. Therefore, the influence of φd on the internal force at this location is explored in more details. Figure 21 shows the variation of the internal forces at the conjunction region between the arch ring and the top of the middle wall under different φd. As can be seen from Figure 21, in the range of opening angles considered, the bending moment increases initially to about 630 kN m and then decreases as φd further increases. In contrast, the shear force decreases initially to zero and reverses direction as φd further increases. The axial force increases almost linearly with increasing φd. Besides, Figure 19(a) shows that when φd exceeds 55°, the axial force at the conjunction region will be the largest along the arch ring. Considering that the conjunction region is the most difficult part for construction and the quality of lining installation may be the lowest in comparison to other parts of the lining structure, it is practically safer to maintain the axial force at the conjunction region at a low level. Therefore, it is suggested that φd should not exceed 55°.Figure 20
Variations of shear force withφd: (a) along the arch ring; (b) along the side wall.
(a)(b)


## 5.4. The effect of tunnel span

The tunnel span of double-arched tunnel is represented by the total width B and is mainly decided by the radius of arch ring R and thickness of middle wall. In order to analyse the effect of tunnel span on the internal forces, the height of the side wall and middle wall and the opening angle in Figure8(a) are not changed, and the burial depth remains 9.5 m, while the outer radius of arch ring increases from 5 m to 9 m at an increment of 1 m, and the outer radius 6.15 m in Figure 8(a) is still maintained. Neglecting the thickness of the middle wall in the analysis, the total width of the double-arched tunnel increases from 18.83 m to 33.89 m accordingly. The results of the internal forces obtained for different cases are shown in Figures 22–24.A larger tunnel span means a larger area of tunnel lining subjected to the vertical pressure on the top of the tunnel. Figures22–24 show that the variation trends of the internal forces with tunnel span are very similar to those with burial depth as shown in the Figures 11–13. As R increases, the absolute magnitude of axial force increases, while most of the absolute magnitude of the bending moment and shear force also increase except that near the inflection point. The point of the largest positive bending moment and the zero point of shear force of side wall move towards the top of side wall from about 1.2 m to 0.6 m as R increases. The largest negative bending moment is present in the arch ring at locations of β = 125° to β = 116° as R increases. These positions are different from those in the analysis of burial depth H.The largest positive of bending moment is present in the side wall aboutI = 0.6 m to I = 1.2 m except that at R = 5 m, the largest positive of bending moment is located in the arch ring at location about β = 56°. The maximum axial force is in the middle wall as shown in Figure 23. The largest shear force occurs at the bottom and top of the side wall as shown in Figure 24. As shown in Figure 25, nonlinear correlations between the largest internal forces and R are observed, which are different from the linear correlations as between the largest internal forces and burial depth. This indicates that the tunnel span is more influential to the internal forces than the burial depth (Figure 25).


## 6. Conclusions

This paper derived the analytical solutions for the internal forces of shallow double-arched tunnel based on the force method and elastic foundation beam model. The internal forces predicted by the proposed analytical solutions were in good agreement with the results of numerical simulations. The influences of different factors on the internal forces were investigated using the newly derived solutions. Based on the calculated and numerical simulation data, some conclusions are drawn as follows:(1)
The variation trend of the internal forces in different burial depths is similar, but the absolute values increase as the burial depth increases.(2)
The elastic resistance coefficient of the surrounding rock has a notable effect on the internal force of the side wall. In general, the internal forces decrease with increasing elastic resistance coefficient, which indicates a better performance of lining structure surrounded by rock masses with higher quality.(3)
The internal forces of the lining structure can be significantly affected by the opening angle of the double-arched tunnel, especially at the conjunction region between the arch ring and the top of the middle wall. Considering the reduced construction quality at the conjunction region, it is recommended that the open angle should lie in the range of 35°–55° for the safety of the lining structure.(4)
Overall, the internal forces increase with increasing tunnel span. The correlations between the largest internal forces with tunnel span are nonlinear(5)
The results of the internal forces along the lining structure mainly show that the intrados of the crown and extrados of the haunch of the arch ring experience the largest tensile forces, while the side and middle walls experience larger compressive forces than other regions. These zones require special attention during the construction and design.The derived analytical solutions provide fast estimations for the internal forces and deformation of double-arched lining structures, which does not rely on either costly comprehensive field monitoring or time-consuming sophisticated numerical modelling. However, it should be noted that both the analytical solutions and numerical simulations conducted in this paper are based on the load-structure method but do not consider the dynamic interaction between tunnel structure and surrounding rock masses. Furthermore, it is likely that the double-arched tunnels will lie in the sloping region which induces unsymmetrical pressures. Both deserve further investigations in the future.



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*Source: 8867536-2021-04-26.xml*