Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method

The elastoplastic problem is a material nonlinear problem, and the strain depends not only on the current stress state but also on the entire loading history. In practical terms, the relationship between stress and strain can only be incremental, and the corresponding iterative solution can only be an incremental form of solution [3]. For the elastoplastic problem, the strain increment can be divided into elastic strain increment dɛ_e and plastic strain dɛ_p increment, i.e. (1) dε=dεe+dεp d\varepsilon = d{\varepsilon _e} + d{\varepsilon _p}

In general, the elastoplastic constitutive relationship of geomaterials includes the following four components: (1) Yield conditions and failure criteria to determine whether the material is plastically yielded or destroyed. (2) The law of hardening, indicating the change in yield conditions due to plastic strain. (3) The law of flow, which determines the direction of plastic strain. (4) Loading and unloading criteria, indicating the working state of the material.

Since the tunnel is a cylindrical object in the longitudinal direction, the size and shape of the tunnel section do not change with the length of the axis. It can be regarded as the external force acting in the longitudinal direction, and the two ends of the cylinder are regarded as fixed constraints. In this way, the original structure can be simplified into an infinite elastic body with a single hole, which is analysed according to the plane strain problem [4,5]. Since the shape and the external force of the elastic body do not change with the coordinate z, the displacement, strain and stress in the elastic body are both the coordinates z are independent of the coordinates. The geometric equation of the plane elasticity problem is (8) εx=∂ux∂xεy=∂uy∂yγxy=∂uy∂x+∂ux∂y \matrix{{{\varepsilon _x} = {{\partial {u_x}} \over {\partial x}}} \hfill \cr {{\varepsilon _y} = {{\partial {u_y}} \over {\partial y}}} \hfill \cr {{\gamma _{xy}} = {{\partial {u_y}} \over {\partial x}} + {{\partial {u_x}} \over {\partial y}}} \hfill \cr} (9) μ∇2ux+(λ+μ)∂∂x(∂ux∂x+∂uy∂y)=0μ∇2uy+(λ+μ)∂∂y(∂ux∂x+∂uy∂y)=0 \matrix{{\mu {\nabla ^2}{u_x} + (\lambda + \mu){\partial \over {\partial x}}\left({{{\partial {u_x}} \over {\partial x}} + {{\partial {u_y}} \over {\partial y}}} \right) = 0} \hfill \cr {\mu {\nabla ^2}{u_y} + (\lambda + \mu){\partial \over {\partial y}}\left({{{\partial {u_x}} \over {\partial x}} + {{\partial {u_y}} \over {\partial y}}} \right) = 0} \hfill \cr} where λ and υ are the Lame constants.

Introducing the function A = A(x, y), B = B(x, y), according to the equilibrium differential equation in Eq. (9), the following relationship can be obtained (10) U(x,y)=U(z+z¯2,(z−z¯2i))=U(z,z¯) U(x,y) = U\left({{{z + \bar z} \over 2},\left({{{z - \bar z} \over {2i}}} \right)} \right) = U(z,\bar z)

For z, the second integral (11) 2U(z,z¯)=z¯φ′f(z)+χ(z)+zφf1(z¯)+χ1(z¯) 2U(z,\bar z) = \bar z{\varphi '_f}(z) + \chi (z) + z{\varphi _{f1}}(\bar z) + {\chi _1}(\bar z)

Therefore, the stress function can be expressed as: (12) U(z,z¯)=Re[z¯φf(z)+χ(z)] U(z,\bar z) = {\mathop{\rm Re}\nolimits} [\bar z{\varphi _f}(z) + \chi (z)]

Complex variable function representation of stress component (13) ∂y−∂z+2iτxy=2[z¯φf″(z)+ψ′(z)]=2[z¯Φ′(z)+Ψ(z)]Φ(z)=dφf(z)dz=φf′(z)Ψ(z)=dψ(z)dz \matrix{{{\partial _y} - {\partial _z} + 2i{\tau _{xy}} = 2[\bar z{\varphi^{''}_f}(z) + \psi '(z)] = 2[\bar z\Phi '(z) + \Psi (z)]} \hfill \cr {\Phi (z) = {{d{\varphi _f}(z)} \over {dz}} = {{\varphi '}_f}(z)} \hfill \cr {\Psi (z) = {{d\psi (z)} \over {dz}}} \hfill \cr}

In the incremental finite element method of elastoplastic problem, the incremental elastoplastic constitutive relation is limited by the condition of ‘simple loading’. In practical engineering, many elastoplastic problems are often coupled with large deformation and nonlinearity. The simple loading theory does not conform to the actual structural stress process, and the most suitable solution for large deformation problems is the incremental load iterative method. The finite element method is an ideal choice for calculating the elastoplastic problem [6, 7]. When the unit is in loading, whether in an elastic state, a plastic state or a transition state (i.e. one part is in an elastic state and the other part is in a plastic state), and when the incremental load is small enough, the corresponding stress-strain relationship is (14) Δσ=DeΔεΔσ=DepΔεΔσ=D¯epΔε \matrix{{\Delta \sigma = {D_e}\Delta \varepsilon} \hfill \cr {\Delta \sigma = {D_{ep}}\Delta \varepsilon} \hfill \cr {\Delta \sigma = {{\bar D}_{ep}}\Delta \varepsilon} \hfill \cr} (1)

Explicit of elastoplastic matrix (15) Dp=1H′+3G(3Gσ¯)2SST {D_p} = {1 \over {H' + 3G}}{\left({{{3G} \over {\bar \sigma}}} \right)^2}S{S^T} (16) Dep=De−Dp {D_{ep}} = {D_e} - {D_p} where H′=dσ¯dε¯p=K′ H' = {{d\bar \sigma} \over {d{{\bar \varepsilon}_p}}} = K' , S represents the deviatoric stress tensor, G=[1−vv1−vvv1−v0001−2v200001−2v200001−2v2]. G = \left[ {\matrix{{1 - v} & {} & {} & {} & {} & {} \cr v & {1 - v} & {} & {} & {} & {} \cr v & v & {1 - v} & {} & {} & {} \cr 0 & 0 & 0 & {{{1 - 2v} \over 2}} & {} & {} \cr 0 & 0 & 0 & 0 & {{{1 - 2v} \over 2}} & {} \cr {} & 0 & 0 & 0 & 0 & {{{1 - 2v} \over 2}} \cr {} & {} & {} & {} & {} & {} \cr}} \right].

In order to effectively control the deformation of the tunnel, the effect of the retaining wall and the effect of the lime-soil compacted pile are studied. Taking the K75+750 section as an example, the specific two supporting schemes are shown in Figure 1. In the analysis of the supporting effect, the initial thickness of the C25 shotcrete is 25 cm, and the second lining is used as a safety reserve; the length of the anchor is 3.5 m and 1 m×1 m [8].

Fig. 1

The general large-scale structural analysis software ANSYS10.0 and continuous medium model are used for numerical simulation. For different supporting schemes, the surrounding rock deformation, stress, tunnel perimeter (vertical and horizontal deformation) and initial branch are studied in the tunnel excavation process. The force and safety factor are used to judge the safety of the retaining structure, and a reasonable supporting scheme is determined to ensure the stability of the tunnel.

On the premise of fully considering the geological environment near the tunnel and minimising the so-called ‘boundary effect’, the distance between the boundary of the calculation model and the centreline of the tunnel is 45 m, and the distance from the bottom boundary to the top of the tunnel is 30 m. Surface. According to the characteristics of the bias tunnel, the upper boundary is the free boundary, the left and right boundaries are horizontal constraints and the upper boundary is the vertical displacement constraint [9, 10]. There are 1876 quadrilateral elements and the total number of nodes is 1847. The calculation model is shown in Figure 2. The pre-support of the tunnel in the calculation considers the overall reinforcement effect of the surrounding rock, so that the reinforcement of the pre-support can improve the physical and mechanical parameters in the reinforcement area. The hardened area can be simplified to a solid unit. The tunnel adopts full-section excavation, and the calculated load release rate is 40%, i.e. the support bears 60% of the surrounding rock pressure.

Fig. 2

The initial support has the following characteristics: 20 cm thick C25 shotcrete, 3.5 m side wall anchor (1 m×1 m), plum-shaped arrangement, 4φ12 steel grille arch. The physical and mechanical parameters used in the calculation are shown in Table 1.

The non-retaining wall without the compacted pile makes the tunnel stable, and the structure meets the requirements of standard safety, but the settlement of the retaining wall foundation is too large, which brings engineering risks to the construction phase and the tunnel operation phase; the lime-soil compacted pile can make the retaining wall foundation settle Small, and the deformation of the key points of the hole wall is more uniform and reduced, and the horizontal constraint on the left side of the tunnel is strengthened, so that the overall safety factor of the slope is reduced, so the retaining wall + pipe shed is a reasonable construction plan.