Stress and Displacement of Deep-Buried Composite Lining Tunnel under Different Contact Conditions

Stress and displacement of the composite lining are important factors to be considered during tunnel design. By the complex variable method, analytical solutions for stress and displacement of surrounding rock, primary support and secondary lining satisfying the interface continuity and boundary conditions under far-feld stresses are derived. Taking the railway composite lining tunnel as an example, the analytical distributions of stress and displacement along boundaries are given, which was in good agreement with the numerical solution calculated by fnite element software. Te results show that the maximum normal stress ratio (load sharing ratio) of the outer boundary between the secondary lining and the primary support is 0.74. Te radial displacement of the inner boundary of surrounding rock, primary support, and secondary lining change consistently. Te maximum settlement and uplift occur at the vault and bottom, respectively. Te tangential stress of secondary lining is compressive stress, while the tangential stress of primary support is tensile stress and compressive stress. Te maximum tangential stress of primary support and secondary lining is smaller than the allowable stress of concrete.


Introduction
Te complex variable function is the most efective method to solve deep-buried tunnel problems.Obtaining the conformal mapping function of the tunnel is the critical step when solving the mechanical response.Te conformal mapping functions of unlined tunnels can be obtained by diferent methods.Lu [1] and Wang [2] obtained the conformal mapping function of noncircular tunnels with a single lining.
For circular tunnels with single-layer lining, Wang and Li [14] obtained stress and displacement around a circular tunnel at great depth subjected to uniform internal pressure and unequal biaxial in situ stresses.Li et al. [15] deduced the elastic-plastic analytical solution for the stress and displacement of circular tunnel subjected to uniform internal and external pressure.Kargar [16] proposed analytical solutions around lined and unlined circular tunnels in viscous rock mass.Guo et al. [17] studied the stress, displacement, and stability of a deep lined circular pressure tunnel by combining the complex variable method with the Biot theory.For noncircular tunnels with single-layer lining, Kargar et al. [18,19] and Lu et al. [20] investigated analytical stress solutions of inverted U-shaped tunnels according to diferent contact conditions between rock mass and lining, respectively; analytical displacement solutions were also given by Lu et al. [21] and Wang et al. [22] in the isotropic and orthotropic rock mass, respectively.Li and Chen [23,24] and Liu et al. [25] obtained the analytical solutions for stress and displacement around horseshoe-shaped tunnels in the isotropic and orthotropic rock mass, respectively.Chen et al. [26] and Fang et al. [27] obtained analytic solutions of two circular and multiple noncircular tunnels at great depth considering the mutual interaction between linings and rock mass, respectively.
Zhou and Yang [28] and Zhou et al. [29,30] calculated the support loads of circular composite lining tunnels consisting of primary support and secondary lining in elastic and rheological rock mass, respectively.Li et al. [31] proposed the analytical stress solution of circular water conveyance tunnel with composite lining subjected to uniform internal pressure.Ramadan et al. [32], Maleska and Beben [33], and Embaby et al. [34] analysed numerically large-span culverts and soil-steel bridges.Maleska et al. [35], Maleska and Beben [36], Shen et al. [37], Jiang et al. [38], and Chen et al. [39] obtained numerical seismic response of soil-steel bridges, a shield tunnel, and subway stations.
For noncircular tunnels, it can be seen from the above literature that there are many theoretical research results for unlined tunnels and single-layer lining tunnels, while there are few research results for composite lining tunnels widely used in practice.In this paper, the analytical solutions of stress and displacement of surrounding rock, primary support, and secondary lining of the noncircular composite lining tunnel are derived by using the complex variable function method and verifed by fnite element software ANSYS, which provides a theoretical basis for safe and economical tunnel design.

Conformal Mapping Function of Composite Lining Tunnel
Figure 1(a) shows the load structure diagram of a composite lining tunnel with noncircular cross sections, where R, L 1 , and L 2 represent the area of surrounding rock, primary support, and secondary lining, respectively.According to the theory of elastic mechanics, deep-buried tunnels can be simplifed as a plane strain model with holes in infnite surrounding rock.Assuming that the buried depth is deep enough, the efect of gravity can be neglected.Te surrounding rock is subjected to far-feld stresses P and λP along Ox-and Oy-axes, respectively, and λ is the lateral pressure coefcient.Te conformal mapping function of the composite lining tunnel is where z � x + iy, ζ � ρ exp iθ , x and y are the rectangular coordinates in the physical z plane, and ρ and θ are the polar coordinates in the image ζ plane.Te three concentric circles with radius ρ � 1, R 1 , and R 2 in the image ζ plane are transformed into the outer boundary of primary support, the inner boundary of primary support (the outer boundary of secondary lining), and the inner boundary of secondary lining, respectively, as shown in Figure 1(b).m is the number of terms of the conformal mapping function.C k , R 1 , and R 2 are the coefcients related to the shape and size of the composite lining, which can be solved by Fan [40].

Basic Equations
Te stress and displacement components of surrounding rock and composite lining are obtained by stress functions.Te stress functions of surrounding rock, primary support, and secondary lining and the stress and displacement components in both Cartesian and curvilinear coordinates are given.
3.1.Stress Functions.Te stress functions φ 1 (ζ) and ψ 1 (ζ) of surrounding rock can be expressed as follows: where constants B, B ′ , and C ′ are related to far-feld stress P and λP, where 10n + 6 unknown real coefcients a j , b j , d j , f j , g j , h j , p j , q j , s j , and t j can be determined by the interface continuity and boundary conditions.

Stresses and Displacements.
Te stress components are σ x , σ y , and τ xy ; the displacement components u x and u y in the Cartesian coordinates xoy can be expressed as follows: 2 Mathematical Problems in Engineering where the notation Te stress and displacement components in the curved coordinates are as follows: where σ ρ , σ θ , and τ ρθ are the radial, tangential, and shear stress components, respectively; u ρ and u θ are the radial and tangential displacements, respectively.Te stress functions φ (ζ) and ψ (ζ) in equations ( 7)-( 12) are replaced by φ 1 (ζ) and ψ 1 (ζ) for surrounding rock, φ 2 (ζ) and ψ 2 (ζ) for primary support, and φ 3 (ζ) and ψ 3 (ζ) for secondary lining.Te shear modulus G and constant κ in equations ( 9) and ( 12) are replaced by shear modulus G 1 and κ 1 of surrounding rock, G 2 and κ 2 of primary lining, and G 3 and κ 3 of secondary lining.Ten, the stress and displacement components of surrounding rock, primary support, and secondary lining can be obtained.
Te relationship between surface force and stress functions is as follows: where f x and f y are the surface force components in x and y directions, respectively.

Continuity and Boundary Conditions
Te surface between surrounding rock and primary support ζ(� exp i θ � σ) is assumed to be full contact.Te corresponding stress and displacement components are equal, respectively [18,19,25].From equations ( 9) and ( 13), the interface continuity conditions can be written as follows: Because the waterproof layer between primary support and secondary lining cannot bear the shear force, this interface (ζ � R 1 σ) is assumed to be slip contact.Te corresponding normal displacement and stress are equal, respectively, and the shear stress is equal to 0. From equation (12), the displacement continuity condition can be expressed as follows: From equations ( 11) and ( 12), the stress continuity condition can be expressed as follows: Te inner boundary of secondary lining (ζ � R 2 σ) is free; the radial stress and tangent stress are equal to 0. From equation ( 12), the stress boundary condition is as follows:

Solution Process of Stress Function
By applying series solution to solve unknown coefcients a j , b j , d j , f j , g j , h j , p j , q j , s j , and t j of stress functions, the stress and displacement components can be obtained.
Substituting stress functions into continuity and boundary conditions and equaling multipliers of the same order of variables, 2n + 1, 2n + 1, n + 1, n, 2n + 1, and 2n + 1 equations are obtained from equations ( 14)- (19), respectively.Tus, a total of 10n + 5 equations is obtained.But there is a total of 10n + 6 unknown coefcients, and an equation must be added.
Te displacement of the inner boundary of surrounding rock caused by tunnel excavation can be written as follows [25]: 4 Mathematical Problems in Engineering Te displacement at infnity caused by tunnel excavation is equal to 0, that is, the constant term in the right side of equation ( 22) is 0. Since the minimum positive exponent term of ϕ 0 ′ (σ) is σ 2 , the maximum negative exponent term of ω(σ)/ω ′ (σ) is σ − 2 .Te supplementary equation is expressed as follows:

Results and Discussion
Taking the double-line railway composite lining tunnel in grade IV surrounding rock as an example, the analytical distributions of stress and displacement along boundaries are given and compared with the numerical distributions.Te primary support is made of C25 concrete with a thickness of 0.25 m, and the secondary lining is made of C35 concrete with a thickness of 0.45 m.Te material parameters of surrounding rock, primary support, and secondary lining are as follows: Young's elastic modules: +0.0290ζ 4 + 0.0105ζ 5 − 0.0152ζ 6 + 0.0083ζ 7 − 0.0024ζ 8 − 0.0002ζ 9 + 0.0001ζ 10 − 0.0004ζ 11 + 0.0009ζ 12 and R 1 � 0.9632 and R 2 � 0.8975.Te composite lining after transformation is shown in Figure 2.

Comparison of Analytical Solution with Numerical
Solution.In order to verify the correctness of the above analytical solution, the fnite element software ANSYS is used for numerical simulation, and the analytical solution of stress and displacement is compared with the numerical solution.Te composite lining adopts the transformed shape.Because the structure and load are symmetrical, the left half structure is adopted.Te size of the plane strain model is 180 m × 360 m, far greater than the size of the tunnel.Te upper, left, and lower boundaries are free, and normal constraints are imposed on the right boundary.Te upper and lower boundaries apply pressure of 0.148762 Mpa, and the left boundary applies pressure of 0.0297524 Mpa.Te above measures ensure that the model, boundary conditions, and loads in the numerical solution are consistent with those in the analytical solution, thus ensuring the comparability of the two solutions.Because the model is large and the mesh is dense, the fnite element mesh of surrounding rock near the tunnel is shown in Figure 3(a), with a total of 24516 units and 49573 nodes.Te primary support grid is shown in Figure 3(b), with 1140 elements and 2286 nodes.Te secondary lining grid is shown in Figure 3(c), with 1484 elements and 3333 nodes.Te contact surfaces of surrounding rock, primary support, and secondary lining are full contact and slip contact, respectively.Slip contact is realized by establishing contact pairs, which are composed of target surface and contact surface.Te contact type is nonseparation contact, and the friction coefcient is zero.
Figures 4(a) and 4(b) show the contours of stress σ x and displacements u x around the tunnel, respectively.It can be seen that stress σ x changes obviously in the composite lining and gradually approaches the applied load of 0.148762 Mpa at a distance from the tunnel in surrounding rock.Te displacement u x changes consistently.Te changes of stress and displacement of surrounding rock are mainly concentrated near the tunnel.
It is convenient to read the calculation results under the rectangular coordinate system in the fnite element software ANSYS.In order to clearly see the diference between the numerical solution and the analytical solution, as an example, Figures 5(a) and 5(b) show the numerical solution and the analytical solution of the stress and displacement of the inner boundary of secondary lining (ρ � 0.8975).A positive angle α turns counterclockwise from the positive xaxis to the positive y-axis (Figure 1(a)).Te analytical solution is calculated by taking n � 100 in this study.From Figure 5(a), it can be seen that the numerical solution and the analytical solution of stresses σ x is zero, satisfying the boundary condition.From Figure 5(b), it can be seen that the numerical solution and the analytical solution of the displacement u L 2 y are almost identical.While, there is a small rigid body translation for the displacement u L 2 x because the fnite element model has no constraints in the vertical direction.When α � 0 °and 180 °, the displacement u L 2 y is zero, satisfying the symmetry condition.

Analytical Distributions of Stress and Displacement along
Boundaries.Te analytical distributions of stress and displacement along boundaries are given, in order to verify the continuity conditions and boundary conditions and fnd the maximum stress and displacement.
On the interface between surrounding rock and primary support (ρ �1), the radial stresses σ R ρ , σ θ of the inner boundary of surrounding rock and the outer boundary of primary support are shown in Figures 6(a) and 6(b), respectively.It can be seen that the corresponding stress and displacement are equal to satisfy the full contact condition.Te radial stress is all compressive, and the maximum value of 0.146 MPa occurs at point A (Figure 2).
On the interface between primary support and secondary lining (ρ � 0.9632), the radial stresses σ  From these results, it can be seen that the slip contact condition is satisfed.Te radial stress is compressive stress, and the maximum value of 0.108 MPa occurs at point B (Figure 2).Te load sharing ratio concerned in the design, that is, the maximum normal stress ratio of the outer boundary between the secondary lining and the primary support is 0.74.Mathematical Problems in Engineering displacement change trend is consistent.Te maximum settlement, uplift, and peripheral displacement occur at the vault, bottom, and waist, respectively.On the vertical axis, the shear stress and tangential displacement are equal to zero, satisfying the symmetry condition.

Te radial stress σ
Figure 9 shows the tangential stress of the inner boundary of surrounding rock (ρ �1), the inner and outer boundaries of primary support (ρ � 1 and ρ � 0.9632), and the inner and outer boundaries of secondary lining (ρ � 0.9632 and ρ � 0.8975), respectively.Te tangential stress σ R θ value of the inner boundary of surrounding rock is the smallest.For composite lining, the maximum compressive stress and tensile stress occur at the inner boundary of primary support.Te maximum compressive stress is 1.425 MPa at point C (Figure 2), and the maximum tensile stress is 0.472 MPa at the bottom.Te maximum values of compressive stress and tensile stress are much smaller than the allowable compressive design strength of 13 MPa and tensile design strength of 1.3 MPa of C25 shotcrete [41].Te tangential normal stress σ L 2 θ of secondary lining is compressive stress, and the maximum compressive stress is 1.338 MPa at point D (Figure 2) of the inner boundary of secondary lining, which is much smaller than the allowable compressive stress of 13 MPa of C35 concrete [41].Terefore, the thickness of primary support and secondary lining can be appropriately reduced.Mathematical Problems in Engineering

Conclusion
According to the complex variable function method, the analytical solutions for stress and displacement of composite lining tunnel are obtained, and compared with the numerical solutions obtained by fnite element software, the results are in good agreement.Te main conclusions are as follows: (1) On the contact surface between surrounding rock and primary support, the corresponding radial stress, shear stress, radial displacement, and tangential displacement of the inner boundary of surrounding rock and the outer boundary of primary support are equal, respectively, which satisfy the full contact condition.On the contact surface between primary support and secondary lining, the corresponding radial stress and displacement of the inner boundary of primary support and the outer boundary of secondary lining are equal, respectively, and the shear stress is zero, which satisfes the slip contact condition.Te radial stress and shear stress of the inner boundary of secondary lining are both zero, which satisfes the stress boundary condition.Te shear stress and horizontal displacement on the vertical axis are zero, satisfying the symmetry condition.(2) Te maximum normal stress ratio (load sharing ratio) of the outer boundary between the secondary lining and the primary support is 0.74.Te maximum compressive and tensile tangential stress occur at the inner boundary of primary support and the maximum compressive tangential stress of secondary lining occurs at the arch foot of inner boundary.Since they are far less than the allowable stress of concrete, the thickness of primary support and secondary lining can be appropriately reduced from an economic point of view.(3) Te radial displacement change trend of surrounding rock, primary support, and secondary lining is consistent.Te maximum settlement, uplift, and peripheral displacement occur at the vault, bottom, and waist, respectively.Terefore, during the design and construction of the tunnel, attention should be paid to these locations.
Subsequently, we will combine the on-site monitoring of specifc projects to verify the applicability of our solutions, so as to better guide the safe and economical tunnel design.

Mathematical Problems in Engineering
and the functions φ 0 (ζ) and ψ 0 (ζ) are analytic outside the unit circle in the ζ plane and can be written as follows: Te stress functions ϕ 2 (ζ) and ψ 2 (ζ) of primary support are analytic in the annular region with radius R 1 ≤ ρ ≤ 1 in the ζ image plane.Te stress functions ϕ 3 (ζ) and ψ 3 (ζ) of secondary lining are analytic in the annular region with radius R 2 ≤ ρ ≤ R 1 in the ζ image plane.Tey can be written as the following series expansions: are introduced for simple expression.ϕ(ζ) and ψ(ζ) are the stress functions.Constant κ � 3 − 4μ for plane strain problem, the shear modulus G � E/2(1 + μ), and E and μ is Young's modulus and Poisson's ratio, respectively.

Figure 1 :
Figure 1: Conformal mapping of the composite lining tunnel.(a) Schematic diagram of the tunnel in the physical z plane.(b) Tree concentric circles in the image ζ plane.

2 θ
of the inner boundary of primary support and the outer boundary of secondary lining are shown in Figures 7(a) and 7(b), respectively.Te corresponding radial stress is equal, while the shear stress is zero.Te corresponding radial displacement is equal, while Mathematical Problems in Engineering the corresponding tangential displacement is not equal.

Figure 6 :
Figure 6: Distributions on the interface between surrounding rock and primary support.(a) Stresses.(b) Displacements.

Figure 7 :Figure 8 :
Figure 7: Distributions on the interface between primary support and secondary lining.(a) Stresses.(b) Displacements.