Introduction

Selection of the most economical and suitable support design contributes positively to an underground excavation project in design and construction processes. Empirical, analytical and numerical approaches are generally preferred methods when tunnels are projected by designers. In order to design tunnel supports, several researchers have suggested rock mass classification systems which have gained universal acceptance (i.e. Terzaghi 1946; Lauffer 1958; Rabcewicz 1964; Wickham et al. 1972; Barton et al. 1974; Bieniawski 1989; Palmström 1995; Marinos and Hoek 2000; Aydan et al. 2014). Due to their practicality, the Rock Mass Quality Index (Q; Barton et al. 1974) and rock mass rating (RMR; Bieniawski 1989) classifications are widely utilized systems by many geotechnical engineers. In spite of the fact that these techniques are advantageous instruments in preliminary support design, they do not provide essential information in strain and stress estimations. Therefore, numerical methods gain popularity in determination of stress distributions and deformations, and checking the lining design for the best remedy. Recent research has showed that if empirical techniques are integrated into numerical methods, more realistic outcomes in lining design will be acquired (i.e. Basarir 2006; Gurocak et al. 2007; Sopaci and Akgun 2008; Kaya et al. 2011; Aydin et al. 2014; Yalcin et al. 2015; Lin et al. 2017; Kanik and Gurocak 2018).

The Turkish General Directorate of Highways (TGDH) planned a project for reclamation of the Artvin–Hopa motorway (km: 6 + 500 – 13 + 787) owing to increasing traffic load and congestion. This project includes construction of the double-tubed Cankurtaran Tunnel situated between km: 7 + 980 – 13 + 208. The total length of the modified horseshoe-shaped tunnel is 5.228 km, and the tunnel is constructed using mechanical excavation and conventional drilling-blasting techniques subjected to existing rock mass behaviors and conditions. The planned, unsupported span and height are 12 and 9 m, respectively. The height of surcharge ranges between 80 and 410 m above the tunnel. The map showing the tunnel location is presented in Fig. 1, and the field view of the entrance and exit sections are given in Fig. 2a–b.

Fig. 1
figure 1

Location map of the study area

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Fig. 2
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(a) Pre-excavation view of the entrance, (b) exit sections and field view of the (c) A1 and A2, (d) B1, (e) B2, (f, g) C and (h) D geotechnical units

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In the present study, the Cankurtaran Tunnel was selected as an implementation site for the empirical, analytical and numerical tunnel support process. A geotechnical investigation followed by rock mass classification and numerical modeling was carried out. The performance of empirical support design was also controlled via finite element method (FEM) modeling. In this framework, the 2D FEM results were compared against 3D analysis results and assessed.

Study area

Based on the structural and lithological arguments, the Eastern Pontide Tectonic Assembly is divided into northern and southern zones (Ketin 1966; Guven 1993). The ages of geological units in the Eastern Pontide vary between Paleozoic and Cenozoic. The tunnel is located in the northern zone, and four formations crop out along the tunnel route (Capkinoglu 1981).

The Subasi Ridge Formation consists of andesitic pyroclastic rocks and intercalations of sandstone, limestone, marl and tuff (Fig. 2c). The Late Cretaceous-aged unit is generally located at the entrance section of the tunnel. The Late Cretaceous-Paleocene-aged Cankurtaran Formation cropped out in the inner section is defined by limestone and marl, and conformably overlies the Subasi Ridge Formation. While the lower and upper parts of the formation are thin-bedded (Fig. 2d), the middle part is thick-bedded (Fig. 2e). In the inner part of the tunnel, the Subasi and Cankurtaran Ridge Formations are juxtaposed along the NE–SW-trending inactive vertical faults F1 and F2. The Paleocene-aged Senkaya Ridge Formation is a sedimentary package including thin- to medium-layered marl and limestone with interbedded claystone (Fig. 2f and g). This unit is generally located towards the end of the inner part of the tunnel. The Eocene-aged Kabakoy Formation which is usually cropped out at the exit part of the tunnel consists of basalt-andesite and pyroclastic rocks (Fig. 2h), and unconformably overlies the Senkaya Ridge Formation. Near the village of Ciftekopru, the Senkaya Ridge and Kabakoy Formations are juxtaposed along the NE–SW-trending inactive vertical fault F3.

In this study, for convenience, the andesitic pyroclastites in the Subasi Ridge Formation were named as “unit A1” and as “unit A2” for sedimentary intercalations (Fig. 2c). The thin-bedded and thick-bedded levels of the Cankurtaran Formation were designated as “unit B1” and “unit B2”, respectively (Fig. 2d and e). Furthermore, the Senkaya Ridge Formation was symbolized as “unit C” and as “unit D” for the Kabakoy Formation (Fig. 2f-h). Figure 3 shows the generalized geological map of the study area. The rock masses in the portal sections and fault zones (F1, F2 and F3) were also delineated as separate segments in this study (Fig. 4).

Fig. 3
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Simplified geological map of close vicinity of the study area (modified from Capkinoglu 1981)

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Fig. 4
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Geological cross section showing the borehole locations, segments, chainages and geotechnical units along the tunnel alignment

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The geological cross section was prepared by the help of the drilling, seismic survey and field studies data, and the tunnel route was divided into 15 segments based on the structural, lithological and geotechnical properties. The tunnel segments are shown in Fig. 4.

Method

Geotechnical studies

Geotechnical characteristic of the rock units located along the Cankurtaran Tunnel alignment were determined performing surface, subsurface and laboratory studies. The field work consisted of mapping, drilling, scan-line survey, water pressure test, seismic investigation and geotechnical description.

Because of the dense vegetation and steep morphology, seven drillings (total length of 607 m) have been opened under the control of the TGDH (Figs. 3 and 4) to monitor the ground properties at the tunnel depth, to characterize the joint features and groundwater level, core sampling, and water pressure tests (Lugeon test). Three of the drillings (BH 1–3) were placed at the inlet section, one borehole (BH 7) at the outlet part and another three (BH 4–6) towards the center of the tunnel line.

The water pressure tests suggested by Lugeon (1933) were performed to identify the permeability properties of the tunnel ground and, therefore, to collect data on the water inflow potency into the tunnel. Along the tunnel alignment, the groundwater depth varied between 3.4 and 17.6 m. The Lugeon tests were carried out below the groundwater level in all boreholes at 5-m intervals. The Lugeon experiment results showed that the rock mass permeability varied from very low to moderate for “unit A1 [0.92–5.74 Lugeon units (Lu)]” and “unit A2 (0.38–6.72 Lu)” considering the classification suggested by Quinones-Rozo (2010). On the other hand, the permeability of rock mass varied from very low to low for “unit B1 (0.00–1.08 Lu)” and “unit B2 (0.32–4.17 Lu).” However, the permeability of rock mass is very low (0.00–0.32 Lu) for “unit C” and low (1.02–5.00 Lu) for “unit D.”

In other words, based on the average Lugeon values of the rock masses presented in Table 1, unit A1, unit A2, unit B2 and unit D exhibit low permeability, and unit B1 and unit C are impermeable considering the classification of Lugeon (1933). A general decrease in permeability with depth was observed due to the reduction in fracture aperture and filling of joints with clay particles during weathering. These data indicate that the geotechnical units are generally dry, and no sudden water flows were expected during the tunnel excavation. On the other hand, because the fault zone materials located along the tunnel consist of finer grain size and angular shape of fractured grains, these zones were assumed to be more permeable than non-faulted rock masses. Therefore, difficulties due to the risk of sudden water flow are possible during the excavation of fault zones.

Table 1 Physico-mechanical and elastic properties of rock materials, rock quality designation (RQD) and Lugeon values
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Laboratory experiments were performed on the core samples compiled from the drillings and rock blocks based on the techniques suggested by ISRM (2007) to define the physical, mechanical and elastic characteristics of the intact rocks, including unit weight (γ), uniaxial compressive strength (σci), point load strength index (IS(50)), and Young’s modulus (Ei). In addition to this, the rock quality designation (RQD) was defined from drillings and scan-line surveys using the methods proposed by Deere (1964) and Priest and Hudson (1976). The average RQD values and rock quantities for the geotechnical unit A1, unit A2, unit B1, unit B2, unit C and unit D were found to be 84% good, 60% fair, 62% fair, 88% good, 3% very poor and 73% fair, respectively. Table 1 demonstrates the results of laboratory test and RQD values.

Discontinuity readings were taken from outcrops to identify the main discontinuity sets. The discontinuity orientations were identified by using the Dips v7.0 (Rocscience Inc. 2016a) software according to stereographic projection (Fig. 5 and Table 2). Additionally, the identification of the discontinuities in geotechnical units such as orientation, aperture, spacing, roughness, persistence, weathering degree and infilling were obtained by examining the core samples and scan-line surveys by the help of the technique recommended by ISRM (2007). The quantitative descriptions of the discontinuities are shown in Table 3.

Fig. 5
figure 5

Stereographic projection of the discontinuity sets in geotechnical units

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Table 2 Orientations of the discontinuity sets
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Table 3 Quantitative description of the discontinuities in geotechnical units
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In the present study, the most commonly preferred classifications such as the RMR and Q system were employed to describe the ground throughout the tunnel route and to determine empirical lining design. The data acquired from field works, scan-line surveys, boreholes and laboratory studies were utilized in the classification systems.

The RMR classification system was suggested by Bieniawski (1974) and updated by Bieniawski (1989). This classification system utilizes the following five rock mass parameters: strength of rock material, RQD, spacing of discontinuities, condition of discontinuities and groundwater. In order to determine the ratings for each five parameters, the charts for the RQD, intact rock strength and discontinuity spacing and rating tables suggested by Bieniawski (1989) were used. The basic RMR values were determined by the addition of these ratings. In the next stage, adjustments for the discontinuity orientations considering the tunnel excavation direction were applied to the basic RMR ratings. Based on the both basic and adjusted RMR values, the character of ground throughout the Cankurtaran Tunnel route varies between “very poor” and “fair” (Table 4).

Table 4 RMR and Q classifications of the rock masses along the Cankurtaran Tunnel
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The Q classification system was proposed by Barton et al. (1974) and it is determined by the following Eq. (1).

$$ Q=\left(\frac{RQD}{J_n}\right)\left(\frac{J_r}{J_a}\right)\left(\frac{J_w}{SRF}\right) $$
(1)

where Jn is joint set number, Jr is joint roughness number, Ja is joint alteration number, Jw is joint-water reduction factor and SRF is stress reduction factor.

In terms of Q values, the characteristics of the rocks located throughout the tunnel alignment vary from “extremely poor” to “very poor” (Table 4).

To be on the safe side, the rock mass quality values of the fault zones F1-F2 and fault zone F3 were assumed as half of the rock mass quality values of the unit B2 and unit C, respectively. According to rock mass classifications, the results obtained from the Q system exhibited more conservative estimates than basic and adjusted RMR values owing to different input parameters. The main difference between the results of these systems was originated from the lack of a stress parameter in the RMR system.

The rock mass parameters are essential data for the numerical modeling. To define the rock mass constants (mb, s, a), the Hoek–Brown failure criterion was utilized. The RMR rating can be used to predict the Geological Strength Index (GSI) for RMR > 23. In this case, the modified Q value (Q´) should be used. Therefore, the GSI values of the geotechnical units and fault zones were obtained using Eqs. (2) and (3) suggested by Hoek et al. (1995).

$$ GSI=9\ln {Q}^{\hbox{'}}+44 $$
(2)
$$ {Q}^{\hbox{'}}=\left(\frac{RQD}{J_n}\right)\left(\frac{J_r}{J_a}\right) $$
(3)

Variation of the RQD, RMR, Q and GSI values along the tunnel route are depicted in Fig. 6.

Fig. 6
figure 6

Rock mass characterization along the Cankurtaran Tunnel

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Hoek–Brown constants were determined using Eqs. (4)–(6) proposed by Hoek et al. (2002).

$$ {m}_b={m}_i{e}^{\left(\frac{GSI-100}{28-14D}\right)} $$
(4)
$$ s={e}^{\left(\frac{GSI-100}{9-3D}\right)} $$
(5)
$$ a=\frac{1}{2}+\frac{1}{6}\left({e}^{- GSI/15}-{e}^{-20/3}\right) $$
(6)

where mi is constant of the rock material and D is disturbance factor.

Factor of disturbance (D) was taken into account to be 0.5 for poor quality blasting and zero for the mechanical excavation. The mi values of the rock materials were identified using the RocData v5.0 (Rocscience Inc. 2016b) software and were taken as 13, 7, 8, 8, 7 and 25 for unit A1, unit A2, unit B1, unit B2, unit C and unit D, respectively (Table 5). It was assumed to be 7, 7 and 8 for intact rocks in fault zones F1, F2 and F3, respectively.

Table 5 Calculated rock mass parameters of the geotechnical units and fault zones for numerical analyses
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The rock mass strength (σcm) and deformation modulus (Em) were obtained using Eqs. (7) and (8) proposed by Hoek et al. (2002) and Hoek and Diederichs (2006).

$$ {\sigma}_{cm}={\sigma}_{ci}.\frac{\left({m}_b+4s-a\left({m}_b-8s\right)\right){\left({m}_b/4+s\right)}^{a-1}}{2\left(1+a\right)\left(2+a\right)}\left(\mathrm{MPa}\right) $$
(7)
$$ {E}_m={E}_i\left(\frac{1-\left(D/2\right)}{1+{e}^{\left(60+15D- GSI\right)/11}}\right)\left(\mathrm{GPa}\right) $$
(8)

where σci is uniaxial compressive strength of intact rock in MPa and Ei is Young’s modulus in GPa.

Because it is hard to determine the σci and Ei values of intact rocks in fault zones due to sampling difficulties, the σcm and Em values were estimated using Eqs. (9) and (10) that are the basis for the RMR proposed by Aydan et al. (1997). These equations are more suitable to identify the elastic properties of weak rocks such as phyllite, mudstone, siltstone, salt, potash or weathered and/or sheared/faulted rocks at relatively shallow depths (<400 m).

$$ {\sigma}_{cm}=0.0016 RM{R}^{2.5}\left(\mathrm{MPa}\right) $$
(9)
$$ {E}_m=0.0097 RM{R}^{3.54}/1000\left(\mathrm{GPa}\right). $$
(10)

The post-peak attitude of a rock mass is an essential input in the numerical modeling of a tunnel, because it has an important influence on the excavation stability. Residual parameters of the rock mass are needed to design underground structures properly. Much investigation has been focused on the specification of peak parameters, and limited attempts have been made to predict the residual parameters of rocks. In this study, because no field testing was performed to investigate the post-peak behavior of geotechnical units upon excavation, it was estimated by the help of the method suggested by Cai et al. (2007). The residual GSI (GSIr) value and residual Hoek–Brown constants (mbr, sr, ar) were calculated using Eqs. (11)–(14).

$$ GS{I}_r= GSI{e}^{-0.0134 GSI} $$
(11)
$$ {m}_{br}={m}_i{e}^{\left(\frac{GSIr-100}{28}\right)} $$
(12)
$$ {s}_r={e}^{\left(\frac{GSIr-100}{9}\right)} $$
(13)
$$ {a}_r=\frac{1}{2}+\frac{1}{6}\left({e}^{- GSIr/15}-{e}^{-20/3}\right) $$
(14)

In order to determine the Poisson’s ratio (νm) of geotechnical units dynamically, geophysical studies were carried out using the seismic refraction method in six lines (Fig. 2). With the help of the seismic studies, primary-wave (Vp) and shear-wave (Vs) velocities were defined for each geotechnical unit cropped out along the tunnel alignment. Considering the elastic theory, the dynamic Poisson’s ratio was determined using the seismic velocity values (Eq. 15).

$$ {\nu}_m=\frac{{V_P}^2-2{V_S}^2}{2\left({V_P}^2-{V_S}^2\right)} $$
(15)

where Vp is primary-wave velocity in m/s and Vs is shear-wave velocity in m/s.

Because the seismic studies were not feasible in the fault zones due to the steep morphology, the Poisson’s ratio values of the fault zones were estimated using the empirical equation (Eq. 16) that is the basis for the rock mass strength suggested by Aydan et al. (1993). The given equation gives the best solution for weak, weathered and/or faulted rocks at shallow depths (<400 m).

$$ {\nu}_m=0.25\left(1+{e}^{-{\sigma}_{cm}/4}\right) $$
(16)

where σcm is rock mass strength in MPa.

The rock mass parameters of the geotechnical units and fault zones are presented in Table 5. These parameters were later used in the FEM analyses as input parameters.

The vertical stress (σv) acting at a point below the ground surface is due to the weight of overburden. The loading condition for vertical stress was calculated by:

$$ {\sigma}_v=\gamma H\left(\mathrm{MPa}\right) $$
(17)

where γ is rock mass unit weight in MN/m3 and H is overburden depth in meters.

The horizontal stresses (σh) acting on a rock element at a depth below the surface are harder to predict than the vertical stresses. The horizontal stress in fractured rock mass was obtained from the following Eq. (18) proposed by Sheorey et al. (2001). On the other hand, the horizontal stress in the fault zones was assumed to be equal to the vertical stress. The rock mass parameters considered in the analysis models are given in Table 5.

$$ {\sigma}_h=\frac{\nu_m}{1-{\nu}_m}{\sigma}_v+\frac{\beta {E}_mG}{1-{\nu}_m}\left(H+1000\right)(MPa) $$
(18)

where 훽= 8 × 10−6/oC (coefficient of linear thermal expansion), G = 0.024 °C/m (geothermal gradient), υm is the Poisson’s ratio of rock mass and Em is deformation modulus of rock mass in GPa.

Tunnel lining design

Bieniawski (1989) published a set of guidelines for the selection of support in tunnels in rock for which the RMR value has been defined. The support chart is applied only for a tunnel with a 10-m horseshoe-shaped span, excavated using drill-blast technique and in situ stress <25 MPa. However, in the Q system, there are no specific limitations such as span of underground openings and excavation shape. Therefore, because of restrictions of the RMR system, the empirical support systems (final lining) recommended by Q classification (Barton, 2002) were implemented for the Cankurtaran Tunnel. Based on the Q system values, four support categories were determined for 15 segments (Table 6). The planned construction properties, including round length, construction phase, support time and stand-up time, determined considering the adjusted RMR ratings are also given in Table 6.

Table 6 Empirical tunnel support categories (final lining) for the rock masses along the Cankurtaran Tunnel
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To clutch the process of tunnel lining design, it is important to investigate basic notions of how a rock mass around an opening deforms and how the supports check this deformation. Therefore, the convergence–confinement (CC) method was utilized to determine the interaction among the surrounding rock mass and lining support. This methodology, which satisfies the Hoek–Brown criterion, has been delineated by Carranza-Torres and Fairhurst (1999) for rock masses. The CC method authorizes the load charged on a lining applied behind the tunnel face to be predicted. In this method, uniform internal pressure pi and far-field stress σo are taken into account. The Pi and So are described as (Carranza-Torres and Fairhurst 2000);

$$ {P}_i=\frac{p_i}{m_b{\sigma}_{ci}}+\frac{s}{{m_b}^2}\left(\mathrm{MPa}\right) $$
(19)
$$ {S}_o=\frac{\sigma_o}{m_b{\sigma}_{ci}}+\frac{s}{{m_b}^2}\left(\mathrm{MPa}\right) $$
(20)

where mb and s are the Hoek–Brown constants, σci is the intact rock strength in MPa, pi is the support pressure (uniform internal) in MPa, σo is the far-field stress (σo = (σv + σh)/2) in MPa, Pi is scaled internal pressure in MPa and So is scaled far-field stress in MPa.

The scaled critical internal pressure Picr representing the elastic limit and the actual critical internal pressure picr are defined as (Carranza-Torres and Fairhurst 2000);

$$ {P_i}^{cr}=\frac{1}{16}{\left[1-\sqrt{1+16{S}_o}\right]}^2\left(\mathrm{MPa}\right) $$
(21)
$$ {p_i}^{cr}=\left[{P_i}^{cr}-\frac{s}{{m_b}^2}\right]{m}_b{\sigma}_{ci}\left(\mathrm{MPa}\right) $$
(22)

If the pi is less than the picr, failure is expected to occur. On the other hand, if the pi is greater than this picr, no failure will occur.

In this study, the CC analyses were carried out in two stages. In the first stage, the pi was supposed to be 0 for the unsupported state. The essential rock characteristics were replaced due to the rock units and they were utilized as data. The computed values of each segment are presented in Table 7. When pi and picr values are compared for unsupported tunnel segments, apart from segments 2 and 15, the stability problem is expected, and the deformation type is plastic.

Table 7 Results of the convergence–confinement analyses for the Cankurtaran Tunnel
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In the second stage, the psmax (maximum pressure) and the Ks (elastic stiffness) of the proposed lining supports were defined for each segment. In calculations, the methods suggested by Hoek (2007) and Carranza-Torres and Engen (2017) were used. According to the acquired results, the psmax values were higher than the picr values for segments 1, 2, 3, 10, 12, 14, and 15. However, the psmax values were smaller than the picr values for other segments. As a result, considering the CC analysis, the empirical lining design was not satisfactory for these segments.

Numerical analyses have been progressively used today compared to the past in tunnel designs to check the accuracy of results acquired from empirical methods owing to the quick developments in computer systems. Because numerical techniques consider the geometry of tunnels, field stresses and elastic/strength parameters, they are valuable design instruments in tunneling. The finite element method (FEM) has become one of the most chosen methods in underground applications by many researchers (i.e. Ozsan and Karpuz 2001; Javadi and Snee 2002; Kockar and Akgun 2003; Ozsan and Basarir 2003; Park 2004; Sari and Pasamehmetoglu 2004; Basarir et al. 2005; Basarir 2006; Genis et al. 2007; Gurocak et al. 2007; Sopaci and Akgun 2008; Kaya et al. 2011; Aydin et al. 2014; Kanik et al. 2015; Yalcin et al. 2015; Kaya and Sayin 2017; Lin et al. 2017; Ozdogan et al. 2018; Kanik and Gurocak 2018).

In the present study, to define the displacements and plastic zones around the opening and to check the validity of the suggested tunnel lining design given in Table 6, the 2D FEM software RS2 v9.0 and 3D FEM software RS3 v1.0 (Rocscience Inc. 2016c, 2017) were used in the numerical analyses. Unlike the RS3, RS2 uses a plane strain analysis where two principal in situ stresses are in the plane of the excavation, and the third principal stress is out of plane. In this model, the 3D stress tensor is divided into three orthogonal stresses which are aligned with the 2D model of excavation (2 in plane and 1 out of plane), and they decompose. An automatic mesh around the excavation was formed, and considering the elasto-plastic analysis, deformations and stresses were calculated using these software programs. A very simple model was used to analyze the tunnel stability and to identify the concept of rock support interaction. The six-noded triangular elements were applied to optimize the meshes, and sensitive zoning was used around the excavation. The boundary of excavation was generated based on its height and span in three steps as (i) top heading, (ii) bench and (iii) invert for the “unit C” and fault zone F3 (segments 12 and 13), and (i) top heading and (ii) bench for the other segments (Fig. 7). The external boundary of design was applied as ten times the excavation width. The Hoek–Brown failure criterion was applied to define the yielded elements and plastic zones in close vicinity to the excavation. In general, deformation of the rock mass reaches its maximum rate between one and two circular tunnel radii behind the face (Hoek 2007). However, noncircular tunnel cross sections are widespread in application. Therefore, based on the method proposed by Curran et al. (2003) for noncircular cross sections, the slice thickness was taken as 10 m in the 3D models.

Fig. 7
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Tunnel excavation sections for segments (a) 1–11 and 14–15, and (b) 12–13

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All rock masses contain joints. Representation of these joints in numerical models differs based on the model type (discontinuum or continuum models). The fundamental significance in discontinuum models is the acting of discontinuity behavior. However, joints in continuum models are shown implicitly, with the intention that the behavior of the continuum design is substantially equivalent to the real jointed rock mass being represented. Continuum models assume material is continuous throughout the body. Joints are treated as special cases by introducing interfaces among continuum bodies (Wyllie and Mah 2004). Because the discontinuity spacing values of the geotechnical units varied between close and very close (Table 3), the rock masses surrounding the excavation are well-suited for use in the continuum model. Therefore, in numerical analyses, continuum models were used to represent rock mass behavior in the 15 segments. In this study, numerical modeling consisted of three stages.

In the first and second stages, deformations (Figs. 9, 10, 11, 12 and 13) and plastic zones (Figs. 14, 15, 16, 17 and 18) that occurred around the openings were examined using top heading followed by excavating the entire tunnel (bench/invert). In the final stage, the effectiveness of the lining support (i.e., shotcrete, rock bolting and steel set) was analyzed.

According to the 2D models for unsupported cases, the maximum value of total displacement for all segments ranges between 0.23 and 17.71 cm (Table 8, Fig. 8). On the other hand, the maximum total displacements are very small, and range between 0.19 and 3.42 cm when considering the 3D analyses (Table 8, Fig. 8). It was seen that higher deformations occurred in the 2D models of segments 5, 8 and 12 compared to the 3D models (Fig. 8). Moreover, there is a discrepancy among the convergence system of the 2D and 3D analysis results for segment 4. An exaggerated displacement has developed at the external walls considering the 2D model of segment 4. However, no wall deformations were detected in the 3D models of segment 4 (Fig. 10). This is a quirky finding and an unforeseen outcome of the 2D model since small deformations at the tunnel walls possibly occur under low field stresses. In general, the deformation concentrations in the 2D designs are larger than those of the 3D design. The justification in these variations may be that in the 2D design, the tunnel opening upright to the plane is infinitive. In addition to this, in the analogous direction, the excavation round is merely 10 m in the 3D designs. Figure 8 shows the maximum total displacement (Ut), normalized vertical displacement (Ux/span) and normalized horizontal displacement (Uy/tunnel height) as a function of vertical distance. Each of the 15 plots corresponds to a different value of displacement. The plots clearly showed that an increase in field stress leads to higher displacement in segments 5, 8 and 12.

Table 8 Results of the 2D and 3D FEM analyses for the Cankurtaran Tunnel
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Fig. 8
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Graphs showing the maximum total displacement (Ut), vertical displacement (Ux)/span and horizontal displacement (Uy)/tunnel height variations along the tunnel line for unsupported and supported cases according to 2D and 3D numerical analyses

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Strain is described as the ratio of tunnel closure to span. Hoek and Marinos (2000) proposed that for rock masses with strains less than 1%, few matters in terms of stability can possibly occur. On the other hand, minor squeezing problems are expected with strain values between 1 and 2.5%. In the present study, according to the 2D models, the strain values of 15 segments ranged between 0.05 and 2.22% (Table 8). However, the strain values obtained from 3D models varied between 0.03 and 0.57%, as shown in Table 8. According to 2D models, minor squeezing problems are expected in segments 5, 6 and 8 due to the great field stresses. However, considering the 3D models, few stability problems are expected in all segments. Thus, the lining application (rock bolts and shotcrete; sometimes steel ribs) is recommended for the security of rock masses whose quality ranges between extremely poor and very poor along the Cankurtaran Tunnel route.

On the other hand, it should be remembered that RS2 and RS3 are small-strain software programs and, thus, they cannot provide the very large strains. Therefore, it is more appropriate to take into account the size of the failure area rather than the deformation size. It can be seen from Figs. 14, 15, 16, 17 and 18 that the extent of the plastic zones demonstrates that there would be a serious matter in terms of stability, especially in segments 4–13, if they are not supported. When Figs. 15 and 16 and Table 8 are checked, the most problematical sections along the tunnel route are segments 4, 5, 6 and 8 driven in the units A1, B1, B2 and A1, respectively. A larger plastic zone and maximum total displacement were developed in these segments. However, the extents of plastic zones in the fault zones (F1, F2 and F3) are nearly half of these segments and are 6.29, 6.14 and 6.43 m, respectively (Table 8). In general, the maximum plastic zone concentrations occurred at the tunnel roof due to the raised field stresses. Considering the 2D designs, the dimensions of the plastic zone for segments varied between 1.22 and 14.21 m. On the other hand, since there is no a ruler option in RS3 v1.0, the failed areas in the 3D modeling could not be measured. In contrast, when the plastic zones occurred around the opening, given in Figs. 14, 15, 16, 17 and 18, are compared, it can be understood that the dimension of the failure areas in the 3D designs are partially larger than that of the 2D models, particularly in segments 5, 12, 13 and 15. This result is primarily owing to the lost arching activity in the 2D designs. It is believed that in numerical modelling, these distinctions are reasoned by a finite discretization.

In the last step, the efficiency of the suggested lining support was examined utilizing the same 2D and 3D analysis models. The lining designs were the same as those suggested in Table 6, and their properties implemented in FEM models are given in Table 9. Variances in the deformations and dimensions of the plastic zones following the support installations were analyzed, and outcomes were cross-checked with the unsupported states. The load split option in RS2 v9.0 software authorizes the user to “split” the field stress-induced load, between any stages of the model, rather than applying the entire field stress load in the first stage. Therefore, the load splitting option is used to liken the retarded support installation. However, RS3 v1.0 software does not have a load split option. Therefore, in order to compare the 2D and 3D analyses results realistically, it was implicitly assumed that the lining is installed instantly after excavation, and that no deformation occurs before the support installation.

Table 9 Characteristics of the support units used in the numerical analyses (TGDH 2013)
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The maximum total displacements for all supported segments range between 0.22 and 13.30 cm, and 0.10 and 1.84 cm with respect to 2D and 3D analyses, respectively. The size of the displacements was indistinctly decreased after support installation (Fig. 8, Table 8). It was concluded that no crown and wall displacements occurred in the 2D models of segments 6–10 compared to the 3D models (Figs. 10, 11 and 12). However, in the 2D models of segments 1, 4, 12 and 15, an overdone deformation condensation demonstrating a failure formed at the external walls. In contrast to the 2D FEM models, the reasonable deformations in the 3D analysis results of all segments occurred inside the reinforced area, and these results verify the stability of the tunnel (Figs. 9, 10, 11, 12 and 13).

Fig. 9
figure 9

2D and 3D numerical analyses showing the maximum total displacement developed in segments 1, 2 and 3 for the unsupported and supported cases

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Fig. 10
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2D and 3D numerical analyses showing the maximum total displacement developed in segments 4, 5 and 6 for the unsupported and supported cases

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Fig. 11
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2D and 3D numerical analyses showing the maximum total displacement developed in segments 7, 8 and 9 for the unsupported and supported cases

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Fig. 12
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2D and 3D numerical analyses showing the maximum total displacement developed in segments 10, 11 and 12 for the unsupported and supported cases

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Fig. 13
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2D and 3D numerical analyses showing the maximum total displacement developed in segments 13, 14 and 15 for the unsupported and supported cases

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Furthermore, checking against an unsupported state, the size of the plastic zone has been decreased remarkably by implementation of the rock bolts and shotcrete for all tunnel segments (Figs. 14, 15, 16, 17 and 18). Following the application of lining, the size of the plastic zone determined in the 2D designs of all segments reduced from 1.22–14.21 m to 0.00–6.61 m. However, as it may be observed from the 3D models shown in Figs. 14, 15, 16, 17 and 18, the plastic zones hardly ever formed. The empirical lining applications have been efficient in decreasing the plastic zone around the most problematic segments 7, 9 and 13 driven in fault zones (Figs. 16 and 18). The extremely poor rock mass quality of segments 12 and 13 was supported by the heaviest lining, including steel ribs. According to 3D analysis results of segments 4, 6 and 8, there are still plastic zones at the crown level, as in the 2D model (Figs. 15 and 16). Considering the 3D models, it was seen that the failure areas are in the zone of the assembled bolts for all tunnel parts except these segments. Therefore, it was suggested to use rock bolts longer than 4 m at the level of the crown in segments 4, 6 and 8.

Fig. 14
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2D and 3D numerical analyses showing the plastic zones developed in segments 1, 2 and 3 for the unsupported and supported cases

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Fig. 15
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2D and 3D numerical analyses showing the plastic zones developed in segments 4, 5 and 6 for the unsupported and supported cases

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Fig. 16
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2D and 3D numerical analyses showing the plastic zones developed in segments 7, 8 and 9 for the unsupported and supported cases

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Fig. 17
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2D and 3D numerical analyses showing the plastic zones developed in segments 10, 11 and 12 for the unsupported and supported cases

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Fig. 18
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2D and 3D numerical analyses showing the plastic zones developed in segments 13, 14 and 15 for the unsupported and supported case

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The 2D and 3D numerical analyses have become the most chosen ones in underground applications by many researchers. Dhawan et al. (2002) revealed that the 2D elasto-plastic tunnel analysis underestimates the deformations. On the other hand, it was determined that the 3D elasto-plastic analysis yields results which compare reasonably well with the in situ measurements. Ucer (2006) noted that the deformation analyses results of the 3D analyses in tunneling are in good agreement with the site data compared to the 2D analyses. Trinh et al. (2010) indicated that the 3D model is a much more powerful tool than 2D model in stress and strain estimations. Xu et al. (2015) noted that the precision of the 3D design is larger than that of the 2D prediction models. Furthermore, it was indicated that the 3D estimation models are required for absolute prediction owing to their higher accuracy and applicability in a wider range of complex practical problems. Kaya and Sayin (2017) reported that the 3D FEM model presents the best remedy in lining pattern compared to 2D models. It is clearly seen that the findings of the present study exactly coincide with the results of research mentioned above. According to results of the 2D models under great in situ rock stresses, the over-estimated displacements and plastic zones were obtained compared to the 3D models. Therefore, the utilization of 3D modeling in design of the tunneling seems to be safe.

Consequently, the outcomes of the CC method in all matters depending on the deformational behavior and performance of the empirical lining were not in good agreement with the related outputs of the FEM-based 2D and 3D models. However, as checked against to the 2D method, the 3D model may be very advantageous for tunnel engineers to define the safety in a more accurate way.

Conclusions

In the present study, the stability evaluation and the lining design of the Cankurtaran Tunnel, which is built in Artvin City (Turkey), was researched. Considering the data compiled from field and laboratory investigations, the quality of the tunnel ground segments were defined by the help of the RMR and Q systems as ranging from extremely poor to fair. After the application of the rock mass classifications, the four empirical lining design classes and the essential engineering geological characteristic properties forming the tunnel ground were obtained.

The applicability of the empirically determined support element was firstly controlled by the help of the convergence-confinement (CC) technique. Interaction between rock masses and tunnel support was also investigated analytically. According to the acquired results, the empirical lining design was only satisfactory for segments 1, 2, 3, 10, 12, 14 and 15. Except for these segments, it was concluded that the proposed lining designs are not successful to prevent the stability problem owing to the great field stresses.

In addition to the CC analysis, the FEM-based 2D and 3D software programs were utilized to check the efficiency of the empirical lining, and to numerically define the plastic zones and convergences that occurred around the ground surrounding the tunnel. Following application of the lining support, the size of the plastic zones around the opening was ordinarily decreased. Thus, it was determined that the empirical lining suggestions given for the Cankurtaran Tunnel project were generally acceptable. Support systems have been accomplished in suppressing the thickness of the plastic zone occurring around the tunnel excavation. However, it was determined that there are still plastic zones at the level of the tunnel roof in segments 4, 6 and 8. Application of rock bolts longer than 4 m was suggested for these zones. The results of the numerical analyses indicated that the most unsuitable deformation and the load scenario on lining elements are within admissible limits, and that the empirical support designs are considered to be reliable enough to support the excavation. Finally, it was concluded that the 3D FEM modelling presents the best answer in lining design compared to the CC method and 2D FEM analysis.

As an important addition of future studies on this topic, it is recommended that the in situ monitoring data should be taken throughout the construction stage of the excavation for adjustment of the FEM designs and for checking the accuracy of the suggested lining support.