Keywords

1 Introduction

With the application and popularization of computer technology in rock mass mechanics and geotechnical engineering, the application of numerical calculation in geotechnical engineering has developed rapidly [1]. However, because of the heterogeneity and discontinuity of rock materials, the rock mass parameters are generally difficult to determine, even if indoor tests and large field tests are often used in engineering, can not fully reflect the nature of the rock mass within the scope of the project, which causes the theoretical calculation value and the actual value is very different [2], based on the actual displacement can effectively determine the parameters of numerical calculation method. The establishment of a systematic and practical rock mass parameter inversion method has a good guiding significance for the tunnel design and construction [3], but also will produce significant economic benefits.

The elastic plasticity model was implemented using the FLAC3D program, which simulated the deformation and stress of mud shale during the construction process. (Huang et.al., 2009) the modeling and monitoring of a clay shale tunnel is studied [4]. This method can be used to study the effect of construction on the performance of mud shale tunnel. Combined with the actual engineering of Changling Tunnel of Guangdong Highway (Dengetal., 2011) [5], the influence of different sizes and positions of karst caves on the deformation of double-track tunnel was analyzed by FLAC3D. Since the cave is located above or below the left tunnel, the simulation indicates that the deformation characteristics of the surrounding rock are similar [6]. Based on the numerical simulation of the elastic response and EDZ behavior around the tunnel openings (in clay caused by excavation and ventilation), (Li et.al., 2013) found a larger radial elastic deformation tunnel of the roof and floor, and a significantly different stress distribution than SPHM [7]. (Li et al., 2013) focused on the numerical study of the mechanical response of excavation and ventilation around tunnels in clay rock. Throughout the tunnel, the patterns and expansion of the TPHM calculated for SPHM were clearly different from those calculated for SPHM. The entire failure process of the model was tracked by the 800 mm 800 mm 200 mm physical model (PM) (Zhao et al., 2015). Under high stress conditions, the deformation localization phenomenon in SR becomes severe [8], and different spatial positions exhibit different deformation features. A multiple grouting method is proposed to analyze the mechanical deformation behavior of surrounding rock using FLAC 3 D fast Lagrangian analysis) software (Gong et al., 2018).In the test site work, the grouting scheme reduced the maximum displacement from 300 t/h to 40 t/h, with no obvious deformation and abnormal stress in the tunnel.(Wang et al., 2019) Large deformation of long wall coal mine tunnel is studied. Stress-related elastic properties were not considered. The model provides a useful tool for assessing the safety edges of the underground tunnel. The numerical simulation and parameters of shield tunnel are introduced in detail, and the mechanism and characteristics of the site deformation caused by shield tunnel are systematically discussed (Li et al., 2020).The influence of construction parameters on the site deformation, including construction sequence, tunnel spacing and chamber pressure [9]. (Kabwe et al., 2020) present, a constitutive model of viscoelastic viscoplasticity (FDVP) with fractional energy estimation of compression delay deformation. Then, the constitutive equation is implemented in FLAC3D and applied to simulate the constitutive deformation of compression within the tunnel for validation using the built-in constitutive model. (Li et al., 2020) Objective To discuss the stress transfer mechanism of pile foundation and the influence of shield tunnel construction on pile stability. The structural deformation of shield tunnel lining mainly occurs near the pile foundation support area, mainly from the settlement deformation with small horizontal displacement. (Xu et al., 2021) a numerical method is proposed to study the damage evolution of secondary tunnel lining under the combination of preliminary supporting corrosion and surrounding rock creep deformation. The improved numerical method includes three main models: (1) tensile failure of corroded rock bolts; (2) yield of corroded supporting steel arch; and (3) elastic-plastic failure of secondary tunnel lining [10].

2 The Numerical Simulation

2.1 The Constitutive Model Employed in the FLAC3D

The constitutive model is actually a mathematical expression of the stress-strain relationship, which in a sense represents the general relationship between stress and strain. FLAC3D In order to meet the needs of all kinds of engineering research, a good simulation of the damage limit of 12 types of geotechnical constitutive model, including a zero model nulll, three elastic model (isotropic, isotropic and orthogonal anisotropic) and eight plastic models (Drucker-Prague (Drucker-Prager), Moore-coulen (Morh-Coulomb), strain hardening/softening, throughout joint, bilinear strain hardening/softening throughout joint, modified Cambridge and Hookbrown). Due to the use of finite difference method display scheme to handle field control differential equation, combined with the use of mixed discrete method, therefore, it can well simulate the material yield, plastic flow, softening large deformation, especially in the material elastic plasticity analysis, large deformation analysis and construction process simulation has unique advantages.

2.2 FLAC3D Calculation Process

The required calculation object is discretized into a finite difference grid with hexahedral cells, and each hexahedral cell can be further divided into several constant strain tetrahedral cells. Now to present the calculation process with a tetrahedron. The tetraheon shown in Fig. 1, the number of nodes is 1 to 4, and the n th surface is the face of node n, shown in Fig. 2.

Fig. 1.
figure 1

The cube units are divided into five constant strain delta cone units

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Suppose that the velocity component at a point within this tetrahedron is viBy applying the Gaussian divergence theorem to the triangular cone shape unit, we can derive:

$$ \int_V {v_{i,j} dV} = \int_S {v_i n_j dS} $$
(1)

The integral in the formula represents the integral of the volume V and area S of the tetraheon cell, and njIs represents the outer normal vector of the tetrahedral surface.

For the tetrahedral units of a constant strain, viIs a linear distribution, and njBe unchanged on the same face. Then it can be obtained by summing over the upper formula:

$$ v_{i,j} { = } - \frac{1}{3V}\sum_{l = 1}^4 {v_i^l n_j^{(l)} } S^{(l)} $$
(2)

FLAC3D With the node as the calculation object, both the force and the mass are concentrated on the node, and then solved in the time domain through the equation of motion. The motion equation of the node is expressed as follows:

$$ \nu_i^l (t + \frac{\Delta t}{2}) = \nu_i^l (t - \frac{\Delta t}{2}) + \frac{F_i^l (t)}{m}\Delta t $$
(3)

\(F_i^l (t)\) Where: It is the unbalanced force component of time t l node in the i direction, and m is the concentrated mass of node i.

The unit strain increment form of a certain step is expressed as:

$$ \Delta e_{ij} = \frac{1}{2}\left| {\frac{\partial v_i^l }{{\partial x_j }} + \frac{\partial v_j^l }{{\partial x_i }}} \right|\Delta t $$
(4)

\(\Delta e_{ij}\) Where: it is the strain increment of the unit.

As the strain increment increases, the stress increment is obtained by the constitutive equation, and then the total stress is obtained by the superposition of the stress increment at each step. Then, the node imbalance force of the next step can be found by the virtual work principle, and the calculation of the next step can be entered.

This paper uses the most widely used Moore-Coulomb strength criterion in the rock mass calculation, and its yield function is:

$$ \left| \tau \right| = c +\sigma_n \tan \varphi $$
(5)

where τ is the shear stress on the yield surface; σn is the normal stress on the shear surface; the c is the impedance per material area; the φ is the internal friction angle of the rock mass, depending on the roughness of the shear surface. From Fig. 2 the ultimate stress circle is:

$$ f(\sigma_1 ,\sigma_2 ,\sigma_3 ) = \frac{1}{2}(\sigma_1 - \sigma_3 ) - \frac{1}{2}(\sigma_1 + \sigma_3 )\sin \phi - c\cos \phi = 0 $$
(6)

In e.g. σ1, σ2, σ3 Main stress.

Fig. 2.
figure 2

Ultimate stress circle

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3 Using the Template

This calculation only simulated the excavation and the lower steps, because the actual arch excavation lag many, displacement can be stable, in the process of parameter inversion is not simulation, at the same time the right hole excavation behind the left hole 50 m, also is not simulation, and because is V surrounding rock, using the core soil annular excavation.

Modeling of the model was carried out in ANSYS and then imported into FLAC3D. Other modeling, calculation, and analysis were all carried out in FLAC3D. The built model is shown in Fig. 3.

Fig. 3.
figure 3

FLAC3D Numerical model and mesh division

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4 Numerical Calculation

List a set of values (E = 0.3 GPa, K = 1.2) to calculate, the initial ground stress of the shallow buried section is mainly generated by the stress field, according to formula σxx = σyy = Kσzz. Due to rock mass fragmentation and more cracks, the Moore-Coulomb constitutive model is adopted.

The tunnel excavation is divided into three parts: the non-core soil part of the upper steps, the core soil part of the upper steps and the lower steps, and the whole process is written by FLAC3D built-in FISH. There are 3 cycle steps, 1 large cycle and two nested small cycles. In the simulation process, immediately after the tunnel excavation, the steel arch, grouting and the anchor are installed, without considering the construction gap. The built model is shown in Fig. 4.

Fig. 4.
figure 4

Form and support after 10 m of tunnel excavation

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Section surface subsidence, vault subsidence and surrounding convergence at 10 m were always monitored throughout the simulation, and these values were recorded in the storage documents.

4.1 Stress-Field Analysis

After tunnel excavation, due to the unloading effect of excavation, the stress redistribution occurs, the rock stress state around the tunnel changes, and the surrounding rock in the unfavorable stress state will be damaged. The built model is shown in Fig. 5.

Fig. 5.
figure 5

Maximum main stress of 10 M in tunnel excavation

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In the secondary stress field formed after excavation, the main stress direction of the rock around the hole is obviously deflected. In general, the maximum main stress direction is parallel to the excavation wind surface, and the minimum main stress direction is perpendicular to the air surface. As shown in Fig. 4, the closer the surface, the greater the deflection from the surface. The stress of the lining is significantly higher than that of the surrounding rock, as shown in Fig. 5, indicating the majority of the lining support, the maximum main stress is the compression stress along the cutting direction of the tunnel excavation surface, the hole axis of the minimum main stress, the minimum main stress of the arch part ① the compression stress at the excavation stage, As the palm surface forward the compressive stress gradually into tensile stress; ② The minimum main stress after the excavation is the tensile stress, And gradually decreases as the palm surface moves forward; ③ The minimum main stress of the lower step is always the compressive stress, And gradually increased with the advance of the palm surface (the same trend as ②); This is because the arch lining gradually bears the main load after excavation, And manifested as longitudinal pressure and lateral tension. The built model is shown in Fig. 6.

Fig. 6.
figure 6

Vertical head view of the stress vector tunnel

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4.2 Analysis of the Plastic Areas

When calculating the Moore-Kulun constitutive model in FLAC3D, the yield of rock mass is judged by the following formula:

$$ f_x = \sigma_1 - \sigma_3 N_\varphi + 2c\sqrt {N_\varphi } $$
(7)

In formula: Nφ = （1 + sinφ)/(1 − sinφ）, σ1 Is the maximum principal stress, With σ 3 Is as the minimum principal stress, φ as the internal friction angle and c as cohesion.

Like fx ˂ 0 means the rock mass; fx0, indicating that no shear yield has occurred in the rock mass.

When the normal stress is tension stress, it is beyond the mechanical validity range of the Moore-Kulun criterion, and the minimum principal stress should not exceed the tensile mild σ of the rock masst Otherwise, the rock mass will appear in tension and yield, and the discrimination formula is:

$$ f_t = \sigma_3 - \sigma_t $$
(8)

like ft > 0, indicating the tensile yield of the rock mass; ft0, indicating that no tension yield occurs in the rock mass.

At the same time, because FLAC3D uses all the dynamic motion balance equations to solve the stress and strain problems, the output destruction area distribution data has the concept of relative time, which is divided into present (n) and past (p). Combined with the above two destruction modes, they are divided into the following five situations:

  • Class I (hear-n): indicates that a region is in a state of shear failure;

  • Class II (hear-p): means once went into shear yield state and now quit;

  • Class III (tension-n): indicates that a region is in a state of tensile failure;

  • Class IV (tension-p): means it has entered the tensile yield state and has now withdrawn;

  • Class V (none): indicates no damage.

Figure 7 shows the plastic zone marking when the tunnel excavation depth is 10 m. As can be seen from the figure, the plastic area occurs in the side wall and lower steps and excavation palm surface after the range of 3 m, and after the side wall surrounding rock with shear damage, and the bottom with stretching damage, in the arch near the plastic area, this is because of the timely excavation and anchor support, so that the stability of surrounding rock does not affect the development of deep rock strata. In the surrounding rock, the anchor rod strengthens some surrounding rock around the tunnel, increasing the strength of the rock mass, thus maintaining the stability of the surrounding rock.

Fig. 7.
figure 7

Plastic area after 10 m of tunnel excavation

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4.3 As for the Shift Field Analysis

After the tunnel excavation, the rock displacement, the overall displacement trend is to the hole, arch surrounding rock moved lower, the bottom around the rock to the above, Fig. 8 shows the vertical displacement cloud tunnel 10 m, Fig. 9 shows the 10 m tunnel 16 m profile vertical displacement cloud, from the Fig. 10 of surrounding rock because no support measures after excavation for upward uplift state, to about 15 mm..5.6

Fig. 8.
figure 8

Vertical displacement cloud map at 10 m of tunnel excavation (m)

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Fig. 9.
figure 9

Horizontal displacement cloud map at 10 m of tunnel excavation (m)

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Fig. 10.
figure 10

Top plate displacement curve during 10 M tunnel excavation

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5 Conclusion

For the study of tunnel surrounding rock stability, scholars and geotechnical engineers from various countries have done a lot of research, no matter which method, it aims to use the most economical and reliable scheme in the best time to ensure the safety of the construction and operation of the project. Currently only support with surrounding rock coupling is not reach quantitative level, not to mention other including stress field, displacement field and so on multiple field coupling, but the numerical calculation method and computer intelligence is increasing, the surrounding rock parameters can establish diversity, large parameter matrix is possible, therefore, can weaken the influence of some secondary factors, so as to reach the level of a half definite quantitative, using a variety of factors coupling mechanism between the accuracy of quantitative analysis, will provide a new analysis mode for rock mechanics and engineering research.