A new approach to evaluate the effect of over-boring and standstill time on thrust force of shielded TBMs based on penetration rate under squeezing conditions

Abstract

One of the most substantial issues faced during the excavation of deep tunnels in weak geomechanical conditions is the squeezing phenomenon. Thrust force determination of the shielded tunnel boring machines (TBMs) in such conditions is of utmost importance. In this study, the Beheshtabad water conveyance tunnel in the central part of Iran is chosen as a case study. The creep parameters of Burger’s model (CVISC) are determined using laboratory creep tests on samples from the 19^th zone of the tunnel with high squeezing potential, and the results were verified using finite difference simulation of the test. A new approach based on the convergence-confinement method is used to calculate the required thrust force of a double-shield TBM using radial displacement profile (RDP) and ground reaction curve (GRC). The effect of various penetration rates on thrust reduction as well as the impact of different over-boring values and standstill times has been evaluated. The results show that in small values of over-boring, due to high overburden pressure and weak geomechanical conditions, required thrust values are incredibly high and the increment of penetration rate is almost ineffective, and the negative effect of standstill time is more evident. Higher penetration rates have a significant impact on thrust reduction in all cases which is a noticeable factor to avoid TBM jamming under squeezing conditions. In higher over-boring values, thrust reduction is positively conducted through increasing the penetration rate and the negative effect of standstill time is minor.

Introduction

While tunneling under squeezing conditions, myriads of challenges such as shield jamming, instability of excavation face, long-lasting delays, and heavy costs could be faced (Grandori 2006; Kovari and Staus 1996; Steiner 1996; Schubert 2000). In general, a high-stress regime together with weak (soft) ground conditions, which leads to a high squeezing potential with time-dependent creep behavior of the host rock during shield tunneling, takes up to 23% and 17% of the jamming cases, respectively (Xu et al. 2021). Different measures can be taken to avoid jamming problems, namely, over-boring operation, shield conicity and shield lubrication in TBMs to lower the frictional forces, and finally, choosing the optimal type of tunnel boring machine based on the required and available thrust force.

Thrust force evaluation is of high importance during tunneling operation, especially under adverse geological and geomechanical conditions. Underestimating or overestimating the thrust force in such conditions leads to irreparable damages to both the machine and personnel. In harsh conditions such as soft ground with squeezing potential, the risk of TBM jamming as a result of thrust miscalculation is very high. It is, therefore, essential to investigate the ground condition as well as the machine’s operational parameters in order to accurately evaluate the thrust force and to prevent such potential hazards. Evaluating the thrust force can be categorized into three main sections, namely, experimental, analytical, and numerical methods. Experimental methods target the statistical analysis carried out on machine parameters or geological parameters obtained from different projects and present diagrams to be used in similar projects (Farrokh and Rostami 2009; Shalabi 2005; Lombardi and Panciera 1997). Since analytical methods are based on closed-form solutions, they do not provide an exact estimation of thrust force. The most important reason could be neglecting the stress distribution along the tunnel in this category (Wittke et al. 2006; Farrokh et al. 2006). Numerical methods, however, calculate the stress distribution along the tunnel route through 3D simulation, and the thrust force can be easily calculated as the result (Amberg 2009; Ramoni and Anagnostou 2010; Hasanpour et al. 2014; Hasanpour et al. 2015; Barla et al. 2010; Lombardi and Panciera 1997).

Ramoni and Anagnostou (2010) performed a parametric study on a cylindrical tunnel located in the homogeneous ground exposed to hydrostatic stress using a two-dimensional axisymmetric model and presented dimensionless design diagrams to predict the thrust required to overcome friction in squeezing conditions. Based on the axisymmetric assumption, over-boring was considered uniform around the shields, and it was higher at the top of the shields. In a supplementary study, Nordas et al. (2023) proposed two novel thrust estimation methods considering the so-called “wall effect” in narrow fault zones under squeezing conditions. According to the authors, in fault zones interspersed between competent rock layers, the competent rock exhibits much smaller deformations, thus mobilizing shear stresses at the interfaces. Based on the results, both “priori modification of the nomogram input via an equivalent improved ground stiffness” and “posteriori correction of the nomogram output via an analytically determined coefficient” methods were determined to be an efficient tool for design.

He et al. (2022) investigated the lining behavior of a cross-fault tunnel under squeezing condition in the construction-operation period. The monitoring data of the time-dependent deformation of the surrounding rock was used to establish a 3D FDM simulation. The suitable creep constitutive model was selected by a regression-comparison method, and the intelligent inversion model was adopted to obtain the basic parameters of the surrounding rock. According to the results, the radius of fault influence on squeezing intensity is about 1D from the fault. The vertical deformation of segment linings of the vault and inverted arch tends to be subsidence when the fault is above the tunnel; however, the trend is inverse when the fault is under the tunnel. Hasanpour et al. (2014) studied the thrust force of a double-shield TBM operating under squeezing condition using 3D numerical simulation. The thrust force was calculated based on longitudinal displacement profile (LDP) and longitudinal force profile (LFP) graphs, which were obtained based on five reference points along the shield. However, the deformations before the working face were not considered in this study. Zhang and Zhou (2017) investigated the jamming mechanism of a single-shield TBM under squeezing conditions. The Burger model and Drucker-Pruger criteria were used to study the plastic and time-dependent deformation of rock. The thrust force was evaluated using the linear relationships between the shield deformation and the forces acting on the shield after the closure of the annular gap between the machine and the tunnel. Aydan and Hasanpour (2019) utilized numerical simulation for Uluabat and T26 tunnels excavated using single-shield TBMs and presented intense jamming problems. Furthermore, 3D finite difference modeling of the tunnel was developed. The results extracted from numerical simulations were compared with the outcomes of the proposed method of Aydan et al. (Aydan et al. 1993 & Aydan et al. 1996). The numerical investigations were used for crosschecking the results from the method developed. Liu et al. (2023) used the finite difference method to simulate the jamming mechanism of a double-shield TBM at the Duoxiongla tunnel in China as a result of the squeezing potential of the ground. Based on the presented results, the method of injecting lubricants at the contact position between surrounding rocks and shields fails to release the jammed TBM and over-boring operation can widen the gap around the shield and reduce the risk of shield jamming. In a study done by Hou et al. (2023), a machine-learning shield jamming risk prediction method based on numerical samples and a random forest (RF) classifier was proposed. Seven influencing parameters, i.e., advance rate, overcut, elastic modulus, tensile strength, in situ stress, maximum thrust, and friction coefficient are considered, among which, elastic modulus, overcut, and in situ stress were determined to have the highest impact.

Creep models have been considered in various studies with respect to tunneling operations. However, the effect of creep rheological models on the required thrust force of tunnel boring machines has not been thoroughly investigated (Shalabi 2005; Weng et al. 2010; Fahimifar et al. 2010; Sharifzadeh et al. 2013; Graziani et al. 2007; Hoek and Guevara 2009; Hoek and Diederichs 2006; Kovari 1986; Sterpi and Gioda 2009; Sterpi 2007). Furthermore, most of the studies were carried out in a two-dimensional manner, and the effect of penetration rate, over-boring procedure, and standstill time on thrust reduction was rarely carried out. Besides, the high calculation costs of a 3D numerical simulation with complex interactions between the TBM shield and surrounding ground as well as advanced constitutive models accounting for time-dependent behavior of soft rock under squeezing conditions are among other challenges in analyzing TBM performance under such hazardous conditions. On the other hand, although some influential factors such as “face pressure,” “deformation before the working face,” and “penetration rate” are among the parameters having a great impact on thrust force and also on longitudinal displacement profile (LDP) and radial displacement profile (RDP), a minor number of studies considered them or simply referred to them without considering their effect during the thrust force calculations under the creep behavior of the host rock (Gertsch et al. 2007; Yagiz 2006; Yagiz et al. 2009; Armaghani et al. 2018; Samaei et al. 2020; Armaghani et al. 2017; Armaghani et al. 2019).

Based on the presented studies and the abovementioned shortcomings, this research presents a novel method to calculate the required thrust force of hard rock TBM with respect to different penetration rates. The calculations are also performed through taking the effect of over-boring and standstill time of the machine as well as the deformation before the working face into account. To do so, the time-dependent parameters of the rock mass were verified through the data obtained from creep laboratory tests and finite difference modeling (FDM) of the testing procedure. It is crucial to mention that the static modeling is conducted without considering the shield elements and subsequently the interface elements between the shield and the host rock to reduce the complexity of the model and to lower high calculation costs. According to the high importance of thrust prediction prior to tunneling in the mentioned challenging conditions, the aim of this study is to represent an authentic and reliable method to evaluate a more accurate thrust prediction necessary to excavate the tunnel, especially in hazardous squeezing conditions.

Project description and methodology

Geology of the case study

The Beheshtabad project is currently being investigated to transfer 1070 million m³ of water from the Beheshtabad River to the Zayandeh Roud River through a 65-km-long tunnel. The tunnel is partitioned into 79 geological zones according to prior investigations. The 19^th zone of the Beheshtabad tunnel is identified with a high potential of squeezing due to its weak geological conditions and high value of overburden. Since the tunnel will be excavated with a hard rock TBM in this zone, evaluating the required thrust force is a vital task in order to prohibit jamming potential.

The 19^th zone of the Beheshtabad tunnel is expanded from the 29^th to 37^th km of the tunnel with an over-burden of up to 790 m in some parts consisting of three geological units. The first geological unit (i) includes mylonitic limestone, and the second unit (i) consists of mylonitic limy sandstone along with micro-folds, micro-faults, and slaty cleavage. The third unit (iii) consists of marly limestone, metamorphic sandstone, metamorphic shale, and sandy limestone. The boring diameter of the tunnel is 7 m, taking the thickness of the segmental lining and backfilling into account. Figure 1 shows the longitudinal geological profile of the 19^th zone. Furthermore, Table 1 presents the intact geological parameters of the soft host rock in this zone, which are extracted based on the statistical analysis carried out on various laboratory tests on drilling samples.

Fig. 1

Longitudinal geological section of the 19^th zone in the Beheshtabad tunnel (Hashemi 2008)

Table 1 Intact geological parameters of the soft host rock in the 19^th zone of the Beheshtabad tunnel (Hashemi 2008)

The Mohr-Coulomb model parameters of the rock mass are calculated according to an approach presented by Hoek and Diederichs (2006). The least and the most favorable rock mass parameters are calculated on the basis of the intact rock characteristics, overburden, and GSI and are presented in Table 2.

Table 2 Geological parameters of rock mass in the 19^th zone of the Beheshtabad tunnel (Hashemi 2008)

Creep parameters of the host rock

General definition

Subjected to specific loading conditions influenced by time at a constant temperature, the phenomenon of rock creep unfolds through a sequence of four well-established phases, namely, instantaneous elastic strain, primary creep, secondary creep, and tertiary creep, as documented in the literature (Lv et al. 2019; Kumar and Hajibeygi 2021; Paraskevopoulou and Diederichs 2018; Kabwe et al. 2020; Frenelus et al. 2022). The initial phase, instantaneous elastic strain, is characterized by its reversibility and prompt occurrence upon the application of a constant load. Following this, the primary creep phase typically demonstrates a decelerated rate of strain accumulation. Subsequently, the secondary creep phase represents a period of sustained strain rate devoid of fluctuations. The ultimate phase, tertiary creep, marks the culmination of this process, distinguished by an escalated rate of strain accumulation until the end of creep life. These distinctive creep behaviors are consistently discernible across various rock types, despite inherent differences in their mechanical strengths (Chen et al. 2018). A visual representation of the creep tendencies of rocks is provided in Fig. 2. It is worth noting that exceptions can arise for specific rock specimens contingent upon their exposure conditions. Particularly, at substantial depths, rocks invariably encounter high-stress conditions, potentially leading to a dual-phase creep phenomenon in certain hard rocks, often culminating in brittle failure (Wu et al. 2016). It should be noted that while predominantly exhibiting low rates of creep deformation, intact hard rocks nevertheless exhibit considerable time-dependent behavior across numerous scenarios (Malan 1999; Malan 2002). Furthermore, depending on the magnitude of confining pressures, soft rocks may undergo deformation without necessarily undergoing the complete spectrum of creep phases.

Fig. 2

The creep phases of rocks (Qian et al. 2022)

Constitutive model of creep

Burger’s viscoplastic creep model (CVISC) was adopted to simulate the time-dependent behavior of the rock mass during the 3D numerical simulation of the 19^th zone of the Beheshtabad tunnel (Itasca 2002). The conventional CVISC model, as commonly employed, consists of a classical Burgers model, which effectively captures both the transient and steady-state creep phases, and a Mohr-Coulomb (MC) frictional slider component (illustrated in Fig. 3). This model’s constitutive equations exhibit a combination of volumetric elastoplastic behavior and deviatoric elastoviscoplastic (EVP) behavior. Consequently, the strain rates within this composite model operate sequentially, with viscoelastic behaviors governed by the classical Burgers model and plastic behaviors dictated by the MC failure surface. The model proficiently accounts for creep strains observed during the transient and steady-state creep phases, representing these through the Kelvin and Maxwell rheological components, respectively. However, in scenarios of accelerated creep, the frictional slider component contributes to immediate plastic deformation, resulting in instantaneous yield and permanent strain rates. Notably, the plastic strains estimated during this accelerated creep process are predominantly influenced by stress levels rather than temporal considerations. Nonetheless, empirical investigations have indicated that over time, permanent strains tend to develop, particularly evident in the context of weak rocks subjected to elevated stress conditions. In situations such as tunnel excavation within weak rock formations experiencing substantial in situ stress, the resultant permanent strains exhibit a discernible time-dependent squeezing behavior, as demonstrated by Wang et al. (2017). As a consequence, it becomes imperative to accurately estimate time-dependent volumetric-plastic (VP) strains within the framework of the accelerated creep process to ensure reliable predictions of squeezing behavior.

Fig. 3

Schematic view of the CVISC model (Kabwe et al. 2020)

To extract the creep parameters of the CVISC model including Maxwell’s and Kelvin’s viscosity and shear modulus, a creep laboratory test was carried out on 10 borehole samples from the project site, namely, S(90), S(83-96), and S(29-30). To perform the test, a constant force equal to 40% of the uniaxial compressive strength (UCS) of the samples was applied to their top boundaries. Subsequently, the applied force was increased up to 60–70% of that of the UCS value for each sample to reach the secondary phase of creep until reaching the sample failure. The results of the laboratory experiments are presented in Table 3.

Table 3 CVISC parameters extracted from the creep laboratory test

Verification of creep parameters

The values representing group S(90) from Table 3 are chosen for the verification of creep parameters in the finite difference (FD) code. The strength properties of the sample are listed in Table 4. It should be noted that the laboratory creep tests for sample S(90) were carried out within approximately 22 days.

Table 4 Characteristics of sample S(90) during the creep tests

The 3D finite difference code to simulate the laboratory creep test was carried out with specifications mentioned in the above section, and CVISC parameters were adopted as a result of proper fitting between the numerical and experimental data (see Fig. 4). Further details regarding the verification of numerical simulation could be found in Mohammadzamani et al.’s (2019) study.

Fig. 4

Verification of creep parameters for the CVISC model in finite difference code (Mohammadzamani et al. 2019)

The maximum time-step for the FD code can be calculated through Eq. 1 in the CVISC model.

$${\Delta t}^{cr}\max =\min \left(\frac{\mu^K}{G^K},\frac{\mu^M}{G^M}\right)$$

(1)

Δt^cr_max is the maximum time-step in the code when the creep model is applied, μ^K and μ^M refer to Kelvin’s viscosity and Maxwell’s viscosity, respectively, and G^K and G^Mare indicative of Kelvin’s shear modulus and Maxwell’s modulus, respectively. In accordance with ground convergence and a number of primary analysis, the value of the maximum time-step is chosen equal to 0.01 min.

Deformations before the working face

More than 30% of the displacements occur before the working face when TBM is approaching (Panet 1993; Unlu and Gercek 2003). Therefore, the volume of rock bored with the machine exceeds the total tunnel volume, the ratio of which is called “ground loss.” The ground loss affects the deformations around the tunnel and in some cases, subsequently on the surface. The impact of these displacements is also considered in the numerical model of the current study with respect to the face pressure and penetration rate (for instance, if the penetration rate was 1 m/h, the mean values of the displacements in the first 1 m before the working face are determined and added to the tunnel diameter). This procedure is conducted due to the fact that the displacements usually start from zero in numerical simulations. However, a fraction of these deformations has already occurred when the boring machine is excavating the working face. It should be noted that these values have a direct effect on the “stress release” phenomenon which is in direct relationship with the normal forces acting on the shield.

Thrust force calculation

To calculate the required thrust force of the TBM, the stationary situation of the machine should be considered, meaning that, whether the TBM is under the excavation process or re-started after a standstill (the time during which the machine stops for maintenance purposes or installation of segments and backfill injection). The thrust calculation is explained in detail in the following sections.

Calculating the thrust force during the excavation process—dynamic mode

When the TBM is excavating the tunnel, the process is considered “dynamic” and the required thrust force is calculated through summation of applied forces acting on the shields (Mohammadzamani et al. 2019). In this study, a new method of thrust calculation based on the convergence-confinement method (Paraskevopoulou and Diederichs 2018; Oke et al. 2018; Vlachopoulos and Diederichs 2009) is presented considering various penetration rates (PR) of the machine. As an instance, if the PR is 1 m/h, the acting forces on the first 1 m of the shield are extracted in the first calculation hour. Afterwards, the acting forces on the second 1 m are extracted in the second calculation hour, and so on. The process of thrust calculation based on the mentioned method is presented in Fig. 5.

Fig. 5

Calculation of the thrust force based on the new suggested method (PR = 1 m/h)

After the summation of the total acting forces on the shield, the required thrust force will be calculated through Eq. 2.

$$\textrm{TH}=\left(\sum \textrm{F}+\textrm{W}\right)\bullet {\mu}_{\textrm{D}}$$

(2)

TH is the total thrust force of the TBM based on the penetration rate (PR), ∑F is the total applied forces acting on the shield, W is the machine’s weight, and μ_D is the dynamic friction coefficient which is considered equal to 0.25 in the current study. As mentioned earlier, the TBM shield is not simulated in the current model, and instead, the machine’s weight was considered to omit the complexity of interface-rock interactions and the high calculation costs. However, in case the shield was simulated, the weight effect should be neglected from Eq. 2.

Calculating the thrust force after re-starting the TBM—static mode

If the boring machine re-starts after a specific standstill time, the process is considered “static,” and the thrust calculation method is carried out similarly to the dynamic mode. The only difference is that, in this case, the value of standstill time is to be added to the FD code. For instance, if the penetration rate is 1 m/h and the machine re-starts after 3 h, the acting forces on the first 1 m of the shield are extracted after 1 + 3 = 4 h of calculation. Subsequently, the applied forces on the second 1 m of the machine must be extracted after 5 h of calculation, and so on. After the summation of the total forces, the required thrust force will be calculated through Eq. 3:

$$\textrm{TH}=\left(\sum \textrm{F}+\textrm{W}\right)\bullet {\mu}_{\textrm{S}}$$

(3)

in which TH is the total thrust force based on the penetration rate, ∑F is the total forces acting on the shield after a specific standstill time, W is the machine’s weight, and μ_S is the static friction coefficient which is considered equal to 0.4 in the current study.

Considering the shield-rock interactions in the simulation

As stated above, if the TBM shield (and the corresponding interface elements) is generated and considered inside the tunnel during the simulation, the applied forces on the shield must be extracted through interface nodes. In this case, the applied normal stresses on the interface nodes are extracted and multiplied into the node’s specific area to be converted into force values. Afterwards, the thrust force will be calculated based on Eqs. 2 or 3 depending on the stationary mode of the TBM.

One of the most important challenges when using interface element is the values of normal and shear stiffness which can be calculated through Eq. 4 (Itasca 2002).

$$\left(K+\left(4/3\right)\ast G\right)/{\varDelta Z}_{\textrm{min}}$$

(4)

where K is the bulk modulus, G is the shear modulus, and ΔZ_min is the length of the smallest adjacent element.

In this study, the calculated interface stiffness is equal to 15e9 Pa/m according to Eq. 4 which is a very high value and could result in unrealistic results which are far away from real case scenarios. Consequently, a parametric analysis has been carried out to evaluate the effect of interface stiffness on thrust force calculation. The results are presented in Fig. 6b. It should be noted that in this figure, different orders of Kn (normal stiffness of the interface) are applied in the model and the corresponding thrust based on the equations above is calculated considering various penetration rates.

Fig. 6

a An instance of shield machine and corresponding interface elements considered in the simulated model and b calculated thrust force based on different orders of stiffness (over-boring = 15 cm)

According to Fig. 6, it is evident that the calculated thrust force highly depends on the value of interface stiffness. As seen in this figure, the required thrust force is equal to 400 MN if the value of stiffness is 15e9 Pa/m (as calculated based on Eq. 4) which is a far-fetched value compared with real-case projects. Therefore, the adoption of the right value for the interface stiffness through a back-analysis method is inevitable in such cases. The back analysis methodology could be conducted based on the machine’s weight or operative thrust force during the machine excavation. On the other hand, using such high values of stiffness will also result in zone-overlapping errors during the calculation of the FD code, causing more complexities for the numerical integration. Although various researchers have used the conventional method of using the interface element in tunneling cases taking into account the sensitivity of the interface elements as well as verification of the interactions to get more reliable results (Fargnoli et al. 2013; Liu et al. 2014; Pang et al. 2005; Kasper and Meschke 2004), a number of researchers have used non-interface models (also called the “fictitious-shield” models) to avoid further complexities and to omit the uncertainties of such method (Ring and Comulada 2018; Jin et al. 2022; Liu et al. 2021; Mohammadzamani et al. 2023; Do et al. 2014; Mollon et al. 2009; Shen 2009; Zhang et al. 2016; Epel et al. 2021; Le et al. 2022).

New approach of calculating the thrust force of TBM

In order to prevent the potential challenges arising as a result of interface integration into the FD code as stated above, a new method is to be utilized for the thrust calculation of the TBM in the current squeezing condition. To do so, the required thrust force of a double-shield TBM in the Beheshtabad tunnel is calculated using a novel approach employing the convergence-confinement method, omitting the interface complexities (TBM shield is not modeled), and calculating the frictional forces using the ground reaction curve (GRC) and radial displacement profile (RDP). The process of thrust calculation is similar to the “Calculating the thrust force during the excavation process—dynamic mode” and “Calculating the thrust force after re-starting the TBM—static mode” sections. The only difference here is that the potential applied normal stresses acting on the virtual shield are extracted from GRC diagrams and converted into force values. To plot the GRC diagrams in the tunnel based on various calculation times (since the time-dependent creep model is applied), a model in which no TBM shield is considered was simulated, as shown in Fig. 7. The segmental lining as well as backfilling material are considered in the simulation, as shown in this figure.

Fig. 7

The schematic cross-section of the 3D model in the FD code (not in scale)

As it was mentioned earlier, the deformation before the working face is calculated and applied to the numerical model (tunnel diameter) to generate more reliable results.

Face pressure, applied through the cutter-head via the thrust cylinders, limits the working face deformation and subsequently reduces the tunnel deformation. Therefore, the thrust force required to overcome the friction will be found lower. If the face pressure is calculated and underestimated, the required thrust force will be predicted significantly higher than the real value. Determining the authentic value of the face pressure is, therefore, one of the crucial factors for the precise evaluation of the required thrust force. In order to calculate the face pressure, the normal forces applied to the face should be determined. According to the geological conditions of the rock, the machine parameters, and the tunnel’s cross-section, the value of disc penetration into the working face is estimated to be 5 mm per revolution. Assuming the pressure distribution to be uniform around the disc cutters (Rostami and Ozdemir 1993), the face pressure is calculated equal to 75 kPa. This value is achieved by considering the number of disc-cutters, the geomechanical parameters of the host rock, the penetration rate, and the Colorado School of Mines (CSM) method. The process of thrust calculation is as follows.

The radial displacement profiles (RDP) and ground reaction curves (GRC) are to be plotted along the tunnel route in locations where the TBM is supposed to be located (up to 12 m from the working face). To do so, 8 reference points are considered on tunnel circumference (see Fig. 8). and GRC diagrams are plotted for each of these points along 12 m of the tunnel route (equal to the length of the TBM shield in the Beheshtabad project).

Fig. 8

Reference points considered around the tunnel for GRC calculation

On the other hand, RDP diagrams are to be plotted along the same 12 m to investigate whether the rock has reached the shields during finite difference modeling or not. The cross-section of “virtual” double-shield TBM is illustrated in Fig. 9a. Besides, the TBM specifications of the studied tunnel are shown in Table 5, and the mechanical parameters of lining and backfilling are summarized in Table 6. The values of radial distance between the tunnel and TBM shields are shown in Fig. 9b. It should be noted that in order to evaluate the effect of different over-boring values, since no TBM shield is modeled in the current simulation, the over-boring above the shield is simply assumed with a different value in each case (according to Fig. 9a) and the contact between rock and shield is determined by judging the obtained deformations from RDP diagrams. This will result in much lower calculation costs (in terms of time and re-modeling for each over-boring value).

Fig. 9

a Schematic cross-section of virtual double-shield TBM in the current study (values in mm). b Radial distances between the host rock and TBM shields (values in cm)

Table 5 Beheshtabad DS-TBM specifications

Table 6 Mechanical properties of tunnel components

The reference points, which are in direct contact with the shields during the calculation, are identified through RDP diagrams considering the radial distances in Fig. 9b, and therefore, the values of applied normal stress on the shields in a reference point are extracted from the corresponding GRC diagram. The stress values are, then, multiplied into the shield’s specific area to be converted into force values, and subsequently, thrust calculation will be conducted as explained in the “Thrust force calculation” section. It should be noted that if the shield’s length (considering the value of penetration rate) is equal to L and its diameter is equal to d, the values of stress, before calculating the thrust force, must be multiplied into $\frac{\uppi dL}{8}$ in which πdL is the shield’s area and $\frac{1}{8}$ is considered relative to the 8 reference points around the shield.

Verification of GRC diagrams in the FD code

In order to plot GRC diagrams at the considered reference points, GRC diagrams attained from the FD code must be verified to make sure the results are reliable. To do so, a case example of a GRC diagram based on an analytical solution from Harrison and Hudson (2000) is chosen. Ground specifications in this solution are shown in Table 7, and the resulting GRC diagrams for the crown, invert, and sidewall of the tunnel are illustrated in Fig. 10.

Table 7 Ground specifications in the analytical solution (Harrison and Hudson 2000)

Fig. 10

GRC diagrams in the analytical solution (Harrison and Hudson 2000)

To plot the representative GRC curves in the FD code after the tunnel excavation, the nodal forces on the tunnel periphery are extracted by applying a FISH function. Afterwards, normal forces with a similar value are applied in each representative node, and the mean value of nodal forces based on the node’s specific area is considered the initial mean stress. Therefore, the first point of the ground reaction curve with “zero” displacement is generated. To generate further points of the GRC curve, the value of normal forces at each node is decreased using a “reduction factor,” and the calculation is further continued until the equilibrium is reached. The resulting GRC diagrams using the FD model are compared with the analytical solution in Fig. 11. As seen in this figure, there is a good agreement between the GRC diagrams in the numerical simulation and analytical solution which allows for further analysis of the verified numerical model.

Fig. 11

Verification of GRC diagrams in the FD code through analytical solution

To present an example, four cross-sections of the tunnel with a distance of 1, 11, 12, and 13 m (where the first segment is installed) from the working face of the tunnel are chosen, and the corresponding RDP diagram is plotted for each section through 16 h of numerical calculation (creep time) and presented in Fig. 12. It should be noted that in this figure, the top of the diagrams (90°) is indicative of the tunnel crown.

Fig. 12

RDP diagrams in cross-sections with a distance of a 1 m, b 11 m, c 12 m, and d 13 m (where the first segment is installed) to the working face after 16 h of creep time

As seen in Fig. 12, more radial deformation is observed as more creep time is induced in the model as well as more longitudinal distance from the working face. The segmental lining (Fig. 12d) has a restricting role in which the radial deformation is equal from the first hour of calculation, meaning that the host rock reaches the segments from the first creep time. This is indicative of the unstable and weak mechanical properties of the soft rock under the squeezing pressure at this section of the tunnel which requires accurate stabilizing measures. The supporting role of segmental lining, however, is more evident in the RDP diagrams 12 m from the face (close to the first installed segment, see Fig. 12c) when some upheaval deformation is still present at the bottom of the tunnel as well as the RDP at 11 m to the face (see Fig. 12b) where the radial deformations are lower compared to one near the working face. When comparing the RDP diagrams at the face and at the end of the TBM tail (13 m from the face) with the corresponding GRC diagrams (plotted for the reference point C at these locations), the same trend could be observed where the displacements are limited after 15 cm of deformation (equal to the annular gap between the first segment and the host rock). The resulting GRC diagrams are shown in Fig. 13.

Fig. 13

GRC diagrams plotted for the reference point C with the distance of a 1 m and b 13 m to face (where the segmental lining is located) after 16 h of creep time

Results and discussions

Five values of over-boring have been considered for the studied case, and the required thrust force of the double-shield TBM based on various penetration rates (PR ranging from 1 to 6 m/h) has been calculated. It should be noted that the simulation is carried out in ground conditions, presented in Table 2, applying the creep parameters listed in Table 3. It should be noted that the considered penetration rates are chosen for the time interval of 1 h. Thus, the values of PR would be lower if the time interval is increased to days or weeks. On the other hand, the values of PR are considered an “instant penetration rate.” Although some values such as 5 m/h or 6 m/h are not technically feasible, they are considered to evaluate the effect of PR increment on thrust reduction and easier interpretation of the results.

In the first case, a 3 cm over-boring is chosen for the tunnel, and according to Fig. 9b, the radial distance between the tunnel’s reference points and TBM components is specified. Using RDP diagrams plotted along 12 m of the excavated area of the tunnel (where the virtual TBM is located), the sections in which the shield gets in contact with the host rock are determined. Figure 14 shows the contact percentage between rock and TBM shield based on various penetration rates in two cases of dynamic and static mode.

Fig. 14

Contact between TBM shield and host rock based on various penetration rates in a no standstill time, i.e., static mode, and b after 4 h of standstill time, i.e., dynamic mode (over-boring = 3 cm)

As it is evident in Fig. 14, almost all parts of the TBM are in contact with rock in 3 cm of over-boring regardless of the penetration rate or standstill time. In other words, the annular gap between rock and shield components is immediately closed in this case, and almost all components of the TBM (95.83%, as seen in the figure) are in direct contact with the surrounding rock. This could result in a high risk of machine jamming since the applied pressure on TBM components can immediately increase from the first moments of calculation, and therefore, the required thrust to overcome these frictional forces would be extremely high. To evaluate this, the required thrust force of DS-TBM in 3 cm of over-boring is plotted in Fig. 15.

Fig. 15

Required thrust force of DS-TBM based on various penetration rates in two cases of dynamic and static mode (over-boring = 3 cm)

As it was mentioned before, the range of required thrust force of DS-TBM when the over-boring is 3 cm is extremely high especially when TBM re-starts after a standstill time. Increasing the PR, for instance, from 1 to 2 m/h and from 1 to 3 m/h in dynamic mode (no standstill time) decreases the thrust force up to 40.42% and 47.8%, respectively, which shows the importance of PR increment even in low values of over-boring. There is a nonlinear relationship between thrust force and PR and as it is evident, increasing PR to 2 m/h has the highest effect on the thrust reduction compared to other PR values. The effect of PR increment on the thrust reduction is more evident in static mode, especially when PR is risen from 1 m/h to 2 m/h. For instance, after 4 h of standstill time, the thrust reduction is approximately 49.2% when PR is risen from 1 to 2 m/h. As seen, there is a noticeable gap between the diagrams of static mode (with standstill time) and the dynamic mode which is attributed to the utilization of static friction coefficient and creep behavior of rock. In other words, after the passage of a longer standstill time, the rock applies higher pressure on the TBM shield since the tunnel is under a high amount of overburden pressure, and therefore, the thrust force values increase dramatically in comparison with the values in dynamic mode in which no standstill time is applied.

Raising the value of over-boring to 6 cm, the required thrust force can be evaluated through RDP and GRC diagrams at the tunnel. Figure 16 illustrates the percentage of contact between rock and shields in this case.

Fig. 16

Contact between TBM shield and host rock based on various penetration rates in a no standstill time, i.e., static mode, and b after 4 h of standstill time, i.e., dynamic mode (over-boring = 6 cm)

According to Fig. 16a, the cutter-head is fully in contact with the surrounding ground in all of the penetration rates. The front shield, however, is in full contact only at a penetration rate of 1 m/h, and only 97.5% in contact with other penetration rates. The contact percentage at the rear shield is 89.55%, 87.46%, and 85.37% in penetration rates of 1 m/h to 3 m/h, respectively and it is 83.28% for other penetration rates. Generally, it could be concluded that in lower values of over-boring (in this case, 3 cm and 6 cm), increasing the penetration rate does not have a noticeable effect on the reduction of contact forces, especially in components which are closer to the working face such as cutter-head and front shield (due to the conical shape of the TBM components).

In the penetration rate of 1 m/h, 94.79% of the shield is in contact with the rock in all the standstill time values, and it is 93.75% for the penetration rate of 2 m/h. In other words, for the penetration rates of 1 m/h and 2 m/h, the standstill time increment does not have any effect on the rock-TBM contact (since the closure immediately occurs due to the low value of over-boring) and generally, penetration rate increment simply leads to a minimum contact reduction. As the penetration rate is increased, the percentage of contact between rock and TBM decreases in a way that, in PR = 6 m/h, the percentage of contact is 91.66%, 92.71%, 92.71%, and 93.75%, respectively, for the standstill times of 1 to 4 h. Although the contact percentage between rock and shield in this over-boring is not so different from the values in 3 cm over-boring, the required thrust force is affected and reduced. This is evident in Fig. 17.

Fig. 17

Required thrust force of DS-TBM based on various penetration rates in two cases of dynamic and static mode (over-boring = 6cm)

According to Fig. 17, the value of thrust force in the penetration rate of 1 m/h is equal to 710.81 MN which has undergone a considerable reduction compared to 3 cm of over-boring (965.94 MN). As the penetration rate is risen from 1 m/h to 6 m/h, the required thrust force reduces with a percentage of 40.45%, 47.93%, 65.79%, 82.24%, and 94.51%, respectively, which is almost in the range of thrust reduction for the 3 cm over-boring. Although the thrust force undergoes almost the same reduction domain as 3 cm over-boring, the values of thrust force in 6 cm over-boring have started from a considerably lower amount and thus, highlighting the importance of the over-boring. For instance, the value of thrust experiences a reduction of 26.41% in over-boring of 6 cm compared to over-boring of 3 cm in a similar condition (dynamic mode). In general, when Figs. 16 and 17 are compared, it is clear that in the dynamic mode, the thrust force decreases with a higher gradient when PR is risen from 1 to 2 m/h which is due to lower contact forces between rock and shields. Thrust variation is negligible when PR increases from 2 to 3 m/h since the contact percentage does not vary at this range, and furthermore, thrust force decreases with a lower gradient when PR rises from 3 to 4 m/h since the contact percentage changes with a minor amount. In the static mode, however, increasing the PR from 1 to 2 m/h leads to a 40% reduction of thrust force after 4 h of standstill time which confirms the practicality of double-shield TBMs in squeezing grounds (in terms of their high penetration rates). Increasing the PR to higher values than 2 m/h, however, does not have a considerable effect on thrust reduction (providing that it rises to the feasible value of 4 m/h). In other words, increasing the penetration rate causes the thrust force to decrease with a lower gradient and the effect of PR increment becomes clearer as the standstill time rises.

The values of 9 cm, 12 cm, and 15 cm are also chosen for the over-boring, and the calculations have been carried out to evaluate the impact. Figure 18 shows the percentage of contact between rock and TBM components based on different penetration rates in the mentioned values of over-boring when the standstill time varies from 0 to 4 h. These diagrams have been plotted based on RDP graphs.

Fig. 18

Contact percentage between rock and TBM components with various penetration rates when standstill time varies from 0 to 4 h (indicated with red color) for a over-boring = 9 cm, b over-boring = 12 cm, and c over-boring = 15 cm

According to Fig. 18a, when there is no standstill time (i.e., dynamic mode), 87.5% of the cutter-head is in contact with the rock in all the penetration rates and the front shield is in contact with values of 92.5%, 90%, 85%, 82.5%, 80%, and 80%, respectively, as the penetration rate increases from 1 to 6 m/h. The rear shield, however, is in lower contact with the rock when the PR is increased. The percentage of contact between the rock and the rear shield is 8.28%, 79.1%, 66.56%, 56.11%, 49.84%, and 45.75%, respectively, as PR rises from 1 to 6 m/h. After 1 h of standstill time (i.e., static mode), it is shown that the entire cutter-head gets in contact with the rock, and the percentage of contact between the rock and the rest of the shield increases. In general, the more the standstill time passes, the more components of the TBM get in contact with the rock, as expected. However, the effect of PR increment is evident, meaning that even though more shield components get in contact with the rock, the contact percentage is lower for higher penetration rates.

Increasing the value of over-boring has a significant impact on the reduction of contact forces between rocks and shields. As it is shown, even after 4 h of standstill time, only 62.5% of the cutter-head gets in contact with the rock for 15 cm of over-boring (Fig. 18c). In this case, the annular gap between the rock and shield is in its maximum value, causing the rock to reach the shield after a longer time and thus, the required thrust force to overcome frictional forces would be lower. As it is evident in Fig. 18c, the percentage of contact between cutter-head and rock in dynamic mode is 37.5% which is much lower than previous over-boring values. The contact between the rear shield and the rock is also considerably lower than in previous cases. The contact percentage for the rear shield in dynamic mode with 15 cm of over-boring is 51.93%, 33.12%, 26.85%, 26.85%, 26.85%, and 26.85%, respectively as the penetration rate is risen from 1 to 6 m/h. Increasing the standstill time still has a minor effect on the contact percentage between the rock and rear shield compared with previous over-boring values (except for the penetration rate of 1 m/h).

To evaluate the effect of over-boring and standstill time on the required thrust force of DS-TBM, the values have been calculated through FD calculation and presented in Table 8.

Table 8 The required thrust force of DS-TBM based on various penetration rates and over-borings when the standstill time varies from 0 to 4 h (values in MN)

As seen in Table 8, increasing the over-boring value has a dramatic effect on thrust reduction, especially for higher penetration rates. For instance, in the dynamic mode, when the over-boring (O-B) is risen from 9 to 12 cm, thrust reduction is approximately 38% while it is 51% when O-B rises from 9 to 15 cm. In other words, the thrust reduction has higher values with a noticeable range when the over-boring is risen, and this range is even more noticeable for higher penetration rates. Therefore, it can be concluded that higher penetration rates are more effective when higher over-boring is induced, and the lower the over-boring, the less effective the penetration rate, and subsequently, the negative effect of standstill time would be more evident.

Conclusions

Thrust force requirements are of high importance when facing adverse geological conditions such as high squeezing potential in deep mechanized tunnel. Calculation of the required thrust force to overcome the frictional forces between the host ground and TBM shield is, therefore, one of the crucial tasks in such conditions. If the TBM shield is modeled inside the tunnel, the interface element must be considered around the shield to account for the frictional contact between the surrounding ground and the excavation machine. According to a parametric analysis in the current study, the interface element is highly sensitive to its shear and normal stiffness (Ks and Kn, respectively), and if the necessary measures such as back-analysis for the stiffness determination is ignored, the calculated thrust force in this case could be highly over-estimated. On the other hand, underestimating the interface stiffness could result in zone-overlapping and numerical instability. A new method of calculating the thrust force is, therefore, proposed in the current research.

The required thrust force in this study (Beheshtabad tunnel in the central part of Iran) is calculated based on various penetration rates through the utilization of ground reaction curve (GRC) as well as radial displacement profile (RDP) as a derivation of the convergence-confinement method. The impact of different over-boring values (3 to 15 cm) as well as various penetration rates (1 to 6 m/h) whether the TBM is excavating without any stoppage (i.e., dynamic mode in the current study) or re-starts after a certain standstill time (i.e., static mode in the current study) is also investigated. In this case, there is no need to simulate the TBM shield inside the tunnel to remove the complexities of rock-shield interactions, and the applied forces on the virtual shield can be calculated through stress values extracted from GRC diagrams.

The superiority of this method is that it is able to calculate the required thrust force under any type of geological conditions as well as TBM specifications with time-dependent conditions. Since the TBM (and therefore the interface element) is not simulated inside the model, the calculation costs would be lower, and the calculated results will be more reliable. Therefore, the effect of the machine’s geometry on thrust can be evaluated without the need for re-modeling. The following results are achieved:

• Almost all components of the TBM (95.83%) are in contact with surrounding rock in 3 cm of over-boring regardless of penetration rate increment. It can be concluded that the penetration rate is almost ineffective in over-borings smaller than 3 cm to reduce the contact forces between rock and shield, in the current study. The values of thrust force are noticeably high in 3 cm over-boring due to the smaller gap between rock and shield and the high amount of overburden pressure.

• Increasing the value of over-boring to 6 cm causes the TBM components to get in lesser contact with rock compared to 3 cm over-boring. However, the range of thrust reduction in this case is almost the same as the reduction ranges in 3 cm over-boring and the values are still high which can cause the jamming of the TBM. The reason can be due to the extreme creep condition of surrounding rock which is more evident in smaller over-borings.

• Increasing the over-boring to maximum values of 12 cm and 15 cm is the case where the TBM is in minimum contact with the rock, especially for higher penetration rates. For instance, in higher penetration rates, the thrust reduction is more than 40% when over-boring rises from 3 to 15 cm.