Design of Rock Support in Hard and Layered Rock Masses Using a Hybrid Method: A Study Based on the Construction of the New Skarvberg Tunnel, Norway

Abstract

Hard rock formations with layered, anisotropic geological structures are rather common in Norway. The design of tunnel rock support under such ground conditions has been traditionally done with an empirical approach and assisted with engineering rock mass classification. In most cases, rock stability has been achieved with the use of fiber-reinforced sprayed concrete combined with rock bolts, sometimes supplemented with ribs of reinforced sprayed concrete (RRS) and bolt spiling. However, the discontinuous character and more complex ground behavior of layered rock can lead to design challenges as experienced during the construction of the new Skarvberg highway tunnel in Northern Norway. A significant portion of the tunnel exhibited consistent overbreak, delamination problems and a tunnel failure. In this article, the ground behavior and rock support performance at different locations of the tunnel are investigated to find a basis for design optimization of rock support under similar ground conditions. For this purpose, a thorough site investigation program was carried out, which formed the basis for further assessments and analyses with a hybrid design methodology and the numerical code UDEC (Itasca). As a result, the main challenges of classification systems to capture ground behavior and derive optimal support design in layered rocks were identified. As such, this study has investigated design optimization possibilities, which resulted in the development of, among others, a specific ground behavior classification for layered grounds and a new, optimized RRS-support concept for layered rock masses of very poor quality (e.g., Q ~ 0.1–1).

Highlights

• Classification systems have limitations to address ground behavior and support design in layered rock masses of poor quality.

• Optimal designs of tunnel rock support in layered rock masses of poor quality require a comprehensive characterization of the ground conditions.

• Hybrid design methodologies can address ground behavior and utilize this input to derive optimal rock support in layered rock masses.

• A new concept of reinforced ribs of sprayed concrete has been developed and performed satisfactorily in numerical models for layered rock masses.

• A hybrid method for the design of rock support in layered rock masses of poor quality can help to a significant reduction in rock support consumption.

1 Introduction

Layered rock masses of hard rock with anisotropic geological structure are rather common in nature. They typically exhibit persistent and weak bedding planes which represent a significant source of mechanical weakness in the rock mass. Upon tunnelling, the ground behavior of layered rock is therefore different and more complex than that of blocky rock masses without a dominant weak orientation. Such complexity in the ground behavior of layered rocks is the result of the interaction of several engineering geological parameters such as the joint spacing and persistence, bed thickness, the mechanical properties of joints, tunnel span, roof geometry and the in-situ rock stresses, as well as the characteristics of the rock reinforcement.

As a result of a more complex behavior in layered rock masses, the design of permanent rock support also turns more challenging. Some examples have been addressed by authors like Perras and Diederichs (2009) and Diederichs (2020) from case studies in Canada and Australia, respectively. Similar cases with tunnel stability problems and/or design challenges have been also found in Norwegian tunnels constructed during the last decades in layered rocks of poor quality. For example, in the case of the North Cape tunnel, unforeseen and significant roof stability problems over long tunnel sections resulted in high support consumption (Melby et al. 2002; Palmstrom, personal communication 2023). The recent construction of the new Skarvberg tunnel, investigated in this study, showed extensive sections with roof overbreak, delamination problems, and a tunnel failure, resulting in a significant increase of the required rock support (Bøgeberg and Skretting 2021; Gildestad and Bakkevold 2021).

Some of the mentioned challenges can be attributed to the limitations of current design methods such as empirical classification systems being unable to capture and describe the basic ground properties controlling the anisotropic ground behavior of layered rocks. As noted by authors like Palmstrom and Broch (2006), Palmstrom and Stille (2006), Marinos et al. (2005), Anagnostou and Pimentel (2009), Palmstrom (2009), Vibert and Vaskou (2011), Høien et al. (2019), and Terron-Almenara and Li (2023), the latter concerns well-established and widely used classification systems that are built upon the assumption of structural isotropy such as the Rock Mass Rating (RMR) of Beniawski (1973; 1989), the Q-system (Barton et al. 1974; NGI 2015), and the Geological Strength Index (GSI) of Hoek (1994). Alternatively, analytical solutions based on the Voussoir beam analogue of jointed rock roofs may be also used to evaluate the roof stability of an excavation as suggested in Sofianos (1996) and Diederichs and Kaiser (1999), among others. However, such an approach also presents limitations to predict roof stability due to the challenges in some of the mentioned analytical solutions to simulate tunnel shape and jointing, the effect of in-situ stresses and rock support, and the failure mechanisms of the beds (Oliveira and Pells 2014; Bakun-Mazor et al. 2009; Abousleiman et al. 2021, 2023). With the use of a numerical approach based on the Distinct Element Method (DEM), detailed analyses of tunnel stability in layered, discontinuous rock masses can be performed. However, numerical methods applied for the analysis of discontinuum ground also pose limitations, as noted in the work of Starfield and Cundall (1988), Oliveira and Pells (2014), and Walton and Sinha (2022). These limitations are mostly related to simplifications, uncertainty, selection of constitutive models, and representativity issues in relation to the input data and modelling.

Hence, there is a lack of specific design methods enabling the integration of different methodologies to assess ground behavior and adequate designs of tunnel rock support in anisotropic, layered ground. Although there is published work in which the role of ground behavior is studied as a direct input for design (i.e., ÖGG 2010; Stille and Palmstrom 2008; Marinos 2012), there is a lack of a unified design procedure specific for anisotropic, layered rocks. The latter constitutes a design gap that led previous research work (Terron-Almenara et al. 2023) to the development of a hybrid approach. This approach was built on the basis of a retrospective stability analysis of more than one hundred hard rock tunnel cases, including anisotropic rock masses, from Norway and abroad. The goal was to provide a practical design procedure in which adequate rock support could be assessed on the basis of an elaborated study performed from the combination of different design approaches and the evaluation of ground behavior.

In this article, the new Skarvberg tunnel is studied as an example to demonstrate the limitations of the current empirical methodology used in Norway (NPRA 2020, 2022a) for tunnel rock support design in layered rocks. Two main objectives were outlined for this study. The first aims to find a basis for developing improved design recommendations for the assessment of optimal and more cost-effective support designs in layered ground. The second objective consists of the practical implementation and testing of a hybrid approach (Terron-Almenara et al. 2023) with the purpose of documenting its use in layered and hard rock conditions. To fulfill these objectives, the performance of the current design methodologies was numerically studied in the layered ground conditions of the Skarvberg tunnel by comparing the tunnel stability with empirically based and hybrid-based designs. The latter required a detailed characterization of the ground conditions, involving laboratory testing, measurement of in-situ rock stresses and deformation monitoring, which formed the basis for a comprehensive numerical simulation in UDEC. As a result, this article presents a new design procedure for tunnel rock support in layered rock masses, based on the application of a hybrid methodology (Terron-Almenara et al. 2023) in the anisotropic rock masses of the new Skarvberg tunnel. In this context, a new classification for layered rock masses called Layered Ground Behavior Classification (LGBC) is also proposed. This is intended as a supplementary design tool to forecast ground behavior and support loading conditions in anisotropic ground. Finally, a new and optimized rock support concept consisting of modified ribs of sprayed concrete called Crown-RRS (C-RRS) is proposed for the reinforcement and support of horizontally layered rock masses of quality Q 0.1–1. The numerical analyses performed for the proposed C-RRS support concept showed tunnel stability and an associated potential for a reduction in rock support consumption. Hence, in addition to an improvement in tunnel stability, optimization with the use of hybrid designs can also pose a beneficial contribution to the environment.

2 The New Skarvberg Highway Tunnel

A major upgrade has been undertaken by the Norwegian Public Roads Administration (NPRA) during the last decades in the E69 highway to North Cape in the Finnmark county in Northern Norway. A part of this upgrade comprised the construction of the subsea North Cape tunnel and the Honningsvåg tunnel, completed in the late 1990s, and the recent construction of the new 3.5 km long Skarvberg tunnel (Fig. 1a). The new Skarvberg tunnel is D-shaped, 12.5 m wide, drill-and-blast excavated, and has a rock support design based on empirical rock mass classifications. Shotcrete and rock bolts, as well as RRS support arches are the main elements of the designed permanent rock support.

Fig. 1

The new Skarvberg tunnel. a Location along the E69 highway in Finnmark County, b Tunnel alignment plotted over an aerial photograph of the terrain surface, including identified weakness zones (represented as solid black lines with double cross-ticks) and rosette plots of joint sets, and c Geological profile (modified from NPRA 2017). The two test locations for stress measurements are marked with blue dots and the two study locations with red dots. (Color figure online)

The rock mass in the new Skarvberg tunnel was mainly characterized as hard rock with a distinct anisotropic, layered rock mass structure. The rocks belong to the Kalak formation (Ramberg et al. 2013), mainly composed of sequences of metasandstones and micaschists disposed subhorizontally and intersected by subhorizontal, 0.1–1 m thick intrusions of metagabbro. The rock mass structure is formed by three main joint sets as observed in the rosette diagrams assembled upon the joint registrations done during construction (Fig. 1b). The first and most dominant joint set corresponds to the bedding, which exhibits a high degree of persistence. The bedding has a main strike in the NNE-SSW direction and a dip of 10 degrees towards the NW. The two additional sets were mainly striking in the NE-SW and NW–SE directions, both with steep dip angles. The latter two joint sets therefore represent cross-joints intersecting the bedding planes at approximately normal angles.

From aerial photograph interpretation of the surface above the tunnel alignment, up to nine weakness zones were identified (Fig. 1b). From the geometrical correlation between the recognized structures on the surface and the encountered (mapped) weakness zones at the tunnel level during construction, it was also confirmed that the weakness zones were vertical to subvertical as illustrated in Fig. 1c (NPRA 2022b). In addition to the nine weakness zones intersecting the tunnel alignment, minor fracture zones were also encountered at the tunnel level. These had similar strike and dip, but a lower degree of jointing and shearing.

The construction of the new Skarvberg tunnel showed a significant deviation between the forecasted and the encountered distribution of rock mass classes and rock support. While a proportion of only 15% was forecasted for rock masses of “Very poor quality” or lower in the tender studies based on Q-logging, the actual proportion encountered during construction was as high as 51.6% (Table 1). This deviation was mostly related to the frequent occurrence of tunnel roof instabilities caused by the unfavorable combination of layered rock mass structure and insufficient horizontal confinement in long sections of the tunnel, and the evident limitations of empirical systems used in the tender design to forecast such behavior.

Table 1 Forecast versus actual distribution of rock mass classes for the new Skarvberg tunnel in relation to the NPRA classification of rock masses and rock support classes (NPRA 2022a)

Due to the mentioned stability problems and the relatively high proportion of rock masses of very poor quality, several tunnel sections along the tunnel were supported with RRS arches and bolt spiling. Despite the heavy rock support measures in tunnel sections with very poor rock mass quality, a failure of the tunnel roof took place at profile 3 + 162 during construction (Fig. 2). The failure occurred in a tunnel section with mapped rock mass quality Q 0.1 and supported with single (Si)-RRS 30/6 spaced c/c 2 m, radial bolts L 4 m c/c 2 m, and bolt spiles in the crown. As shown in Fig. 2, the failure occurred in the center part of the roof and extended about 9 m behind the tunnel face. The center part of the RRS arches in the flat tunnel roof is visibly buckled downwards, causing the development of tensional fractures in the shotcrete that mostly concentrate in the intrados part or tensional zone of the reinforced shotcrete members forming the RRS arches.

Fig. 2

Buckling failure of RRS-support in the new Skarvberg tunnel chainage 3 + 162. Courtesy of Skanska Norge AS

In addition to the tunnel roof failure shown in Fig. 2, frequent instabilities during the excavation of the tunnel were observed in the form of delamination, rock fall, and overbreak in the tunnel roof when the rock mass was exposed at the tunnel face immediately after each excavation advance. As a consequence of these instabilities, several additional site investigations were conducted. The findings were used by the contractor`s specialist consultant to derive an adjusted support design which was implemented for the remaining part of the project. The encountered rock conditions along the tunnel finally lead to an increased use of RRS-arches and rebar bolt spiles corresponding to a factor of 13.4 and 6.6 respectively, higher than the tender forecast (Gildestad and Bakkevold 2021). The tunnel section at 3 + 162 is further studied in this article, together with a monitored tunnel section at 2 + 227. Locations are shown in Fig. 1.

3 Design of Tunnel Support in Horizontally Layered Rock Masses of Poor Quality in Norway

The design of permanent rock support in Norwegian tunnels is based on an empirical approach. It combines the application of engineering rock mass classification and experience. The rock mass quality index Q of Barton et al. (1974) and the subsequent updates of Grimstad and Barton (1993), Grimstad et al. (2002), and NGI (2015) have normally been used for rock support design. Rock support design in Scandinavian hard rock tunnelling is similarly based on the utilization of the rock mass as a load-bearing structure. This is achieved by conserving -or improving- the self-bearing capacity of the rock mass with rock reinforcement measures. This is typically done with a combination of rock bolts and fiber-reinforced sprayed concrete that form part of the permanent rock support. In rock masses of poor quality that have partially lost their self-bearing capacity, load-bearing support is normally added by the installation of RRS arches. In the Q-system, different configurations -or designs- of RRS support are available by adjusting the longitudinal center spacing, thickness, the number of steel layers -Single (Si) or Double (D)- and the number of rebars in the RRS arches. This approach permits a wide design- and construction -flexibility to suit different loading conditions (NFF 2008).

The design of rock support in Norwegian tunnels must additionally comply with the national design guidelines for tunnel support in road tunnels as published by NPRA in Pedersen et al. (2010), and the subsequent revisions (NPRA 2020, 2022a). The NPRA design guidelines are basically a simplification of the Q-system, and bases the support design on Q-assessments to derive support classes and prescribed support classes or designs (Table 1). For rock masses of very poor quality (Q < 1), the support class IV is divided into two sub-classes IVa an IVb at Q 0.2. The two sub-classes mark the boundary between the recommendation of only rock reinforcement with rock bolts and sprayed concrete (for Q > 0.2), and load-bearing support in the form of RRS arches (for Q < 0.2).

In the mentioned NPRA guidelines, however, there are no specific design recommendations for rock support in anisotropic, horizontally layered rock masses. Despite the frequent occurrence of layered and hard rock formations encountered in Norwegian tunnels, there has been a lack of development or adjustment of the mentioned procedures to reflect the more complex nature of layered rocks in design recommendations. In compensation, experienced geologists apply some mapping adjustments and recommendations to derive representative values of Q-parameters that reflect best the expected ground response of layered rocks. For example, the joint roughness number (${J}_{\text{r}}$) and the joint alteration number (${J}_{\text{a}}$) of the most unfavorable joint set for the tunnel stability -i.e., the bedding planes-, are selected. The value of the stress reduction factor (SRF) is set to represent the impact of in-situ stresses in relation to the main joint directions. In addition, thorough evaluations of the rock quality designation (RQD) in the Q assessment are performed to differentiate natural joints representing mechanical bedding discontinuities from incipient, closed joints as suggested in NGI (2015). Similarly, measurement of RQD in horizontally bedded rock is normally performed in the vertical direction since it then permits a more representative measurement of the degree of jointing in the most critical orientation for stability. However, the presence of cross-joints also has an important effect on tunnel stability and needs to be accounted for in the RQD assessment. As such, additional methodologies can be included in the assessment of representative RQD values to account for both the bedding and the cross-joints. For example, with the study of the volumetric joint count (Jv), as suggested by Palmstrom (1982, 2000) and NGI (2015). Applying the respective empirical correlations from Jv towards RQD, enables a better description of the overall degree of jointing, thus using a better basis for further adjustment and evaluation of the RQD to account for the cross-joints in layered ground.

A frequent design solution to reinforce and support layered rock masses in Norway has been rock support with RRS arches combined with bolt spiling. The reinforcement effect and principles of this combined methodology are illustrated in Fig. 3. As such, radial rock bolts reinforce the laminated rock mass above the tunnel, which results in a thicker and stronger beam of reinforced rock beds (illustrated with dark brown color in Fig. 3). Rock bolts do similarly tie the load-bearing support (RRS) to competent and undetached ground above the tunnel roof, restricting therefore deflection and providing the needed ground-RRS-arch interaction. In this context, the main functionality of the RRS arches is to hold ground pressure and provide a resistance force against radial deformations, as illustrated by the thick black arrows in Fig. 3.

Fig. 3

Interaction between layered ground, ground reinforcement at the tunnel face with radial bolts (dark brown color) and ahead of the face with bolt spiles (grey color), and Si-RRS arches. a Longitudinal section. b Cross section (tunnel span ~ 10 m). (Color figure online)

Bolt spiling constitutes the pre-reinforcement of the rock mass with the installation of steel bolts ahead of the tunnel face and above the tunnel crown prior to an excavation pass in poor rock conditions (Fig. 3). The aim of such support methodology is to reinforce the ground ahead of the face to improve tunnel stability and minimize deformations in poor ground conditions. Among the wide variety of tunnel pre-reinforcement methods, bolt spiling in hard rock conditions falls within the category of an umbrella arch (UA) reinforcing system (Oke et al. 2014a). Among other authors, Peila (2013), Volkmann and Schubert (2006), FHA (2009), and Oke et al. (2014a, b; 2016) have published important contributions to the understanding of the mechanical behavior and interaction between UA and the ground. The design practice for bolt spiling in Norwegian hard rock tunnelling, however, is primarily based on previous experience and basic empirical design recommendations given in Holmøy and Aagaard (2002), NFF (2008), NGI (2015) and NPRA (2022a). These basically recommend the use of bolt spiling when rock mass quality Q < 0.6. The latter basically leads to designs where the actual (and necessary) interaction between the ground, the tunnel support and the bolt spiling remains unaccounted for. In other words, tunnel support design runs principally independent of the beneficial (and known) stabilizing effect of bolt spiling.

Such interaction of the bolt spiling with the tunnel support and the ground is illustrated in Fig. 3, and its stabilizing effect attributed to three main characteristics: (a) Spiling helps to keep the arched roof geometry, thus increasing stability, (b) Spiling reinforces the rock mass in both the longitudinal and radial tunnel directions, generating a protection vault (grey-shaded color in Fig. 3) that can reduce loosening and radial movements, and (c) Spiling contributes to a better load transfer and arching in the tunnel direction, thus leading to a more even distribution of ground load onto the support.

4 Ground Behavior of Horizontally Layered Rock Masses

The response or behavior of a rock mass upon tunnelling is controlled by several rock mass parameters where the geological structure plays a significant role (Stille and Palmstrom 2008). The analysis of the deformational behavior in tunnels has been traditionally linked to the Convergence Confinement Method (CCM), an analytical elastoplastic model for weak and isotropic rocks (Fenner 1938; AFTES 1978; Carranza-Torres and Fairhust 2000). However, in hard and layered rock, the persistent and often weak planes of bedding print a discontinuous character in the ground that invalidates the premises, therefore, the use, of the CCM. In the following, a summary of the main geometrical and engineering geological parameters controlling ground behavior of layered rock is presented in an attempt of formulating a new proposal for the classification of ground behavior in layered rocks.

4.1 Failure Mechanics of Jointed Roof Beds

The mechanics of tunnel roof beds have been studied at least since Fayol (1885) when the study of tunnel roof stability was mostly based on the classic beam theory for elastic and continuous beams. However, the presence of cross-joints in bedded rock formations reduces the allowable beam tensile strength and invalidates the continuum approach in favor of the jointed or Voussoir beam analogue approach developed by Evans (1941) and later updated by authors like Wright (1972), Sterling (1980), Stimpson and Ahmed (1992), Sofianos (1996), Diederichs and Kaiser (1999), Oliveira and Pells (2014), and Abousleiman et al. (2021). The essential principle of the voussoir concept is that jointed, discontinuous rock beams can develop moment resistance from the generation of a compression arch between the upper beam midspan and the lower abutment. Based on this approach, roof deflection and beam stress can be used to evaluate roof bed failure in relation to the intrinsic elastic properties of the rock, the geometry of the beds, and the span of the opening. Accordingly, four basic failure modes of voussoir roofs were categorized by Diederichs and Kaiser (1999) as illustrated in Fig. 4.

Fig. 4

Basic failure modes of voussoir rock beams. a Buckling or snap-through, b crushing, c shear, d diagonal tensile rupture. (modified after Diederichs and Kaiser 1999). (Color figure online)

As noted by Diederichs and Kaiser (1999) and Brady and Brown (2006), buckling (Fig. 4a) and crushing (Fig. 4b) in multi-layered rock masses would be expected upon a configuration with a high ratio of span to bed thickness, typically greater than 10, i.e., beds relatively thin compared to the excavation span. If the rock is hard, buckling failure (Fig. 4a) is produced when the moment capacity of a bed is exceeded by the moment loading acting on the bed. In cases where rock material is relatively weak, failure by crushing can be initiated at the upper midspan and at the lower support (hinges) by an excess of compressive stresses (Fig. 4b). Both buckling and crushing failures would be accompanied by interbed slip and delamination causing cracking of rock beds and roof deflection. In instances where the ratio span to bed thickness is low, however, the beds are considered thick and shear failure (Fig. 4c) can take place when the limiting shear resistance at the abutments is surpassed by the exerted shear loads. A fourth failure type is diagonal tensile rupture (Fig. 4d), which also can occur under low span-to-bed thickness ratios, provoking tensional breakage of the rock from the upper midspan to the lower abutments as a result of excessive overlying loads.

4.2 Additional Engineering Geological Parameters Controlling Ground Behavior of Layered Ground

Other parameters like the joint spacing and joint persistence have also a direct influence on the extent of the dome of loosened rock formed above the tunnel, hence on the ground loads as noted by Terzaghi (1946), Cunha (1987), Tsesarsky and Hatzor (2005), Bakun-Mazor et al. (2009), and Perras and Diederichs (2009). The in-situ rock stress in the tunnel is also a parameter that controls ground behavior since horizontal confinement usually contributes to roof arching and stability. However, in-situ rock stresses are seldom accounted for in the voussoir solutions as addressed by Carvalho and Carter (2020) and Diederichs (2020). Similarly, rock mass structure and changes in rock stiffness across the geological sequence can also contribute to stress reorientation and fluctuation in the stress magnitude (Amadei 1996; Zang and Stephansson 2010) which can lead to misinterpretation of the stress state. Other important parameters for roof stability analysis are roof geometry and the presence of roof reinforcement as noted by Lang and Bischoff (1982), Diederichs and Kaiser (1999), Peck et al. (2013), and Abousleiman et al. (2021). The mentioned authors conclude that roof stability is generally improved in arched roof geometries and in rock-reinforced roofs since the effective stiffness of the roof increases.

4.3 Proposed Classification of Ground Behavior in Layered Ground

On the basis of the complex interplay between the geometrical and engineering geological parameters involved in the ground behavior of layered rocks, and in view of the importance of describing failure mechanisms for rock support design, a classification of ground behavior for layered ground has been elaborated (Fig. 5). It combines the basic failure modes of jointed rock beams (Fig. 4) and the relevant engineering geological parameters discussed in Sect. 4.2. Such Layered Ground Behavior Classification (LGBC) is similarly based on a literature study of relevant research work as that presented in Sects. 4.1 and 4.2, together with rock engineering experience gained during the construction and the design of the tunnel support in the new Skarvberg tunnel.

Fig. 5

Chart for layered ground behavior classification (LGBC) of layered and hard rock masses into layered ground behavior type (LGBT). Based on D-shaped tunnels with arched roof and rounded walls. Bedding is assumed horizontal

The proposed LGBC classification aims to supplement a hybrid procedure for tunnel rock support design in layered and hard rock masses as proposed by Terron-Almenara et al. (2023) and explained in Sect. 6. The classification chart in Fig. 5 is intended as an aid or guideline to identify and characterize failure mechanisms and ground behavior in layered, hard rock masses. According to such characterization, the rock mass can be classified into one of the categories of Layered Ground Behavior Type (LGBT) in the chart. The main purpose of such classification is to better anticipate ground loading conditions (approximate distribution and size) to assist in the rock support design. Final, hybrid designs of permanent rock support should, however, be performed on the basis of an elaborated analysis process as described in Sect. 6.

According to Terron-Almenara et al. (2023), such a hybrid procedure should principally be used when rock mass quality is Q < 1 in anisotropic and hard rock masses. Defining a lower boundary, on the other hand, may be challenging as for when the utilization of LGBC is not recommended. The reason for this is that rock mass anisotropy is rather scale- and stress-dependent. As such, the relative size of the excavation span to the bed thickness along with the in-situ stress conditions should therefore be evaluated for every case when rock mass quality is very poor. Some authors like Barton (1998) and Brady and Brown (2006) suggest that rock masses of quality Q below 0.1 may behave more isotropically as a result of a denser jointing. The latter agrees well with both, the observations in the Skarvberg tunnel, and the LGBC. However, it is also advised that Q-values are not always representative of the rock mass anisotropy as the Q-system itself presents some limitations to capture such anisotropy (Palmstrom and Broch 2006). Therefore, several rock mechanical parameters and geomechanical systems should be employed when performing a classification of LGBT. The LGBC is similarly built on the basis of civil rock tunnelling experience where in-situ rock stresses were not especially high to create stability problems related to a brittle failure of rock and rock masses. The classification is therefore limited to ground conditions subject to low-moderate stresses, i.e., in-situ stresses not exceeding approximately 10 MPa.

As observed in Fig. 5, the LGBC is divided into three main vertical blocks, which correspond to the definition of “ground conditions”, “ground behavior”, and “support loading” (or design considerations). Each of the six different LGBT categories is therefore defined by a set of characteristics that can be found in the corresponding horizontal rows of each category in the chart. By selecting the representative characteristics of the layered ground conditions and navigating from left to right in the chart, both an LGBT category and a prognose or description of the ground behavior and the loading conditions with the associated support design considerations can be derived.

In the following, descriptions and guidelines are given for the practical application, the mechanics, and the premises of the LGBC based on the explanation of each of the vertical blocks:

• The level of horizontal confinement and stress anisotropy are relevant parameters controlling the stability of voussoir arches. Therefore, the six categories are grouped first into two main subcategories, “a” and “b”, which describe two different confinement conditions. As such, the subcategory “a” refers to “LOW and k₀ < 1” and defines rock masses subject to low in-situ stresses (maximum horizontal stress typically below 5 MPa) with a horizontal to vertical stress ratio k₀ < 1. In such situations, rock masses present challenges to the arch and often leads to a higher propensity for tunnel instabilities and a greater need of tunnel support. On the other hand, subcategory “b” or “MODERATE and k₀ > 1” represents rock masses where the maximum horizontal stress is in the order of 5–10 MPa, and a horizontal to vertical stress ratio (k₀) equal or above the unit. In such situations, the horizontal confinement contributes to rock arching. Hence, the rock mass can take a greater share of the ground loads.

• Stress measurements in civil tunnel projects are often performed once the tunnel excavation has started. In the absence of such measurements, best estimates can still be made based on an interpretation that combines the rock overburden, the topography, rock mass structure, and the geological/tectonic context of the site. This allows for the application of the LGBC in the early phases of tunnel projects. Once tunnel excavation starts, the latter interpretations can be updated with the observation of the engineering behavior of the rock at the face, and ultimately, be confirmed or updated with the measurement of in-situ stresses inside the tunnel.

• The categories I, II and III in the proposed classification represent rock masses with decreasing joint spacing (relative to tunnel span) and quality of joint surfaces, i.e., declining rock mass quality from category I to III. Categorization between the subcategories I, II and III is primarily done by studying the tunnel span to bed thickness ratio. The provided ratios “<25”, “25–50”, and “>50” for the respective categories I, II and III are based on literature values for flat roof openings (Diederichs and Kaiser 1999; Diederichs 2020) and adjusted on the basis of the experience gained from the construction of the new Skarvberg tunnel. Such adjustment of the threshold values given above is to reflect the supporting effect and the inherent reduction of roof beam span provided by the tunnel abutments in D-shaped, circular tunnel roofs, if compared to flat spanning beds.

• Further guidance is additionally given in a third and last sub-section called “rock mass conditions” within the block “ground conditions” in the LGTC chart. As such, typical values describing rock mass conditions with well-known geomechanical classifications and joint surface conditions are given as an aid for classification.

• In the second block of LGBC, an interpretation or prognose of the failure mechanisms that may arise upon excavation is performed in conjunction with Fig. 6. As such, a tunnel with thick roof beds and low horizontal confinement (category Ia) could result in shear failure of the immediate roof when the effective shear strength of the lateral boundaries in blocks at the abutments is surpassed by the own load of the loosen rock. This failure also assumes roof detachment along bedding planes and would have a limited vertical extent. In turn, an increase in the lateral confinement would stabilize the arch roof, and only unconfined blocks cutting the tunnel periphery may fall and produce overbreak (Fig. 6, Ib). In categories IIa and IIb, the joint spacing and the quality of joint surfaces decrease, which results in larger loosening zones. However, the extent of the loosening zone in category IIa is deemed larger than that of category IIb as a consequence of less arching in the less confined condition. Failure in category IIa can then take place by a partial or total detachment of the loosen dome through a combination of shear failure of the lateral block boundaries, gravitative failure, and progressive detachment at bedding planes. In turn, failure in category IIb could occur by block overbreak, possibly followed by tensile rupture or buckling depending on the bed thickness and rock properties. In laminated ground, roof beds close to the opening tend to fail by buckling whilst ground above it may fail by shear through discontinuities (Fig. 6, IIIa and IIIb). The loosened ground forms peaked domes of different sizes and geometry as observed for the low-confinement condition IIIa if compared to the more confined ground in category IIIb.

• It should be noted that the categorization presented in Fig. 6 represents rock masses where cross-joints are truncated at almost every bedding plane, simulating an ideal masonry block structure. The extent, geometry, and location of the loosening zones over the tunnel excavation can therefore vary if persistent cross-joints were to cut both the bedding and the tunnel excavation.

• The above interpretations of typical, potential failures modes define the loading conditions that may be endorsed in the tunnel support. The last block called “Support loading” in the chart (Fig. 5) provides typical loading conditions for the six different LGBT. The given descriptions do not intend to give precise or numerical estimates of load but to provide useful parameters for further engineering or design, such as the approximate sizes and distributions of loads. Accordingly, loading in the categories Ia and Ib are principally of minor to moderate magnitudes as a result of blocks and slab instabilities which are limited in size. Shotcrete and rock bolt support would normally stabilize the roof in these conditions. Loading in categories IIa and IIb is expected to have a rather marked uneven (or asymmetric) geometry as instabilities are especially affecting to the roof. A priori, a higher load would be expected in IIa than in IIb. Support in the latter category with rock bolts and shotcrete may be sufficient whilst an upgrade to lean variants of load-bearing support (arches) would be likely needed in category IIa. The support design for categories IIIa and IIIb has to handle, in addition to the larger roof instabilities, the loading produced in the tunnel walls as a consequence of the fractured, unstable rock in the abutments and sidewalls. In that case, it should be expected that the support for both IIIa and IIIb is formed by heavier and full arches entirely covering the perimeter to the invert level.

Fig. 6

Qualitative categorization of ground behavior in hard, horizontally layered rock for unsupported, D-shaped tunnels. Approximate tunnel span is 10 m. Joint spacing and rock mass quality decrease from category I–III. (Color figure online)

5 Ground conditions at Profiles 3 + 162 and 2 + 227

The roof failure during the construction of the new Skarvberg tunnel motivated additional site investigations in the area of the tunnel failure, further engineering, and upgraded tunnel support designs to overcome the complicated stability problems. In the following, the investigation works carried out by NGI, NPRA and NTNU are presented (Bøgeberg and Skretting 2021). The site investigations focused on the ground conditions at the two study locations, 3 + 162 and 2 + 227.

5.1 Rock Mass Mapping

The mapping performed at the two study locations shows that the rock mass is predominantly formed by layered metasandstones and thin intercalations of metagabbro (Figs. 7 and 8). From the face mapping at 3 + 162, the rock mass was characterized by a well-marked and thin lamination, joint surfaces with poor conditions, unstable tunnel abutments with overbreak and flat roofs, and roof delamination. The assessment of RQD was primarily based on measurements in the vertical direction (across the lamination) to capture the effect of the thin lamination on the degree of jointing and block size. Cross-joints were also accounted for due to their importance to form rock blocks and create instabilities in layered rock masses. The latter was done through the study of Jv, as described in Sect. 3, which gave the necessary basis to adjust and derive a final RQD that was representative of the overall jointing degree at the face. This resulted in Jv in the order of 32 joints/m³ and RQD 10–20% at profile 3 + 162.

Fig. 7

Interpretation of the geological conditions at 3 + 162, based on face mapping and a 45° inclined core hole behind the face (3 + 180). Tunnel azimuth ca. 180°. (Color figure online)

Fig. 8

a Tunnel face at 2 + 227 (Photo: Bøgeberg and Skretting 2021) b Interpretation of the geological conditions based on face mapping. Tunnel azimuth ca. 180°. Legend for lithology and structures as in Fig. 7. (Color figure online)

In connection to the tunnel failure at 3 + 162, there was also a core hole drilled at 3 + 180. With a length of 18 m and an angle of 45° pointing towards 3 + 162, the core hole reached an area located about 12–13 m above the tunnel roof at profile 3 + 162. The rock mass interpretation at the immediate roof in profile 3 + 162 was therefore based on the geological mapping of the face whilst the interpretation of the ground conditions above the tunnel roof based on the observations done in the core hole. As such, both the lithological sequence and rock mass structure logged in a core hole at 3 + 180 were projected to 3 + 162 as observed in the logged layers 1 and 2 (Fig. 7) of the metagabbro. In general, the rock mass was characterized by a rock mass quality Q 0.1, RMR 30–35, and a GSI 25–30. A summary of the assessed component values and categories for each of the mentioned geomechanical classifications is presented in Table 2.

Table 2 Assessment of component values for geomechanical classifications Q, RMR and GSI at location 3 + 162

The mapping at the second study location at profile 2 + 227 (Fig. 8) shows slightly better rock mass conditions. The degree of jointing of the rock mass was lower than that of profile 3 + 162, and by employing the same approach to measure the degree of jointing in laminated rock as for chainage 3 + 162, the resulting Jv was 22 joints/m³ and RQD 35–45% for chainage 2 + 227. An improvement in the conditions of the joints was also mapped, presenting in general larger spacing, higher roughness of joint surfaces, and less presence of infillings. This resulted, as observed in Fig. 8, in a better self-bearing condition of the rock mass and less overbreak. The rock mass for this portion was characterized by rock mass quality Q 0.3–0.4, RMR 35–40, and GSI 30–35. A summary of the assessed component values and categories for each of the mentioned rock mass classifications is presented in Table 3.

Table 3 Assessment of component values for geomechanical classifications Q, RMR and GSI at location 2 + 227

5.2 Joint and Rock Properties

At the two study locations, three main joint sets were identified. The bedding S₀, and two sets of cross-joints, J₁ and J₂. Each joint set was characterized on-site with the assessment of the Joint Roughness Coefficient (JRC), Joint Compressive Strength (JCS), the roughness number in the Q-system (${J}_{\text{r}}$), and measurements of joint spacing (S) and persistence (P). As observed in the summary of the main geometrical and rock mechanical properties of the joints (Table 4), joint conditions at 3 + 162 were significantly worser than those at 2 + 227.

Table 4 Joint properties assessed for locations 3 + 162 and 2 + 227 in the Skarvberg tunnel (tunnel azimuth 180°)

Rock testing was performed in samples retrieved outside the failure zone (3 + 162), both from core holes and from samples collected at the face during construction. The strength and deformational properties of the intact rock were derived from testing of the uniaxial compressive strength ${(\sigma }_{\text{ci}})$, Point Load Testing (PLT), determination of Young’s modulus of intact rock $({E}_{\text{i}})$ and Poissons ratio $(\nu )$. From testing of the uniaxial compressive strength in rock samples retrieved from core holes close to the failure zone, the strength anisotropy was also studied in intact rock samples. In average, a relatively low (10%) strength anisotropy was obtained from applying the load in the core specimens parallel and perpendicular to the direction of the rock fabric. Because of the relatively low strength anisotropy and the laboratory scale of the samples the main source of anisotropy in the rock mass was interpreted to be related to the mentioned mechanical weakness introduced by the penetrative and weak bedding planes. Density of the rock material was also tested, resulting in 27 kN/m³ for the metasandstone and 31 kN/m³ for the metagabbro. A summary of the properties is presented in Table 5.

Table 5 Rock properties from laboratory testing of intact rock material from retrieved samples at different places along the Skarvberg tunnel during construction

5.3 In-situ Rock Stresses

Estimation of the in-situ stress state for underground works for a particular site should, in addition to stress measurement methods and the integration of results in numerical models, be based on an assessment of the broader geological and stress context of the region (Stephansson and Zang 2012). In that sense, the research work and studies performed by authors like Fejerskov (1996), Myrvang (1996), Fejerskov and Lindholm (2000), Simonsen (2018), and Heidbach et al. (2018) describe N–S trends for the orientation of the major horizontal stress in the regions of Finnmark in Northern Norway and in the Barents Sea close to the Norwegian coast (Fig. 9). Myrvang (2002), also addresses significant variations in the degree that these compressional stresses of tectonic origin are locked in rocks placed onto the Norwegian continent. These variations depend on the geological history of the bedrock. As such, bedrock has traditionally been subdivided into two main tectonic belts in Northern Norway, one of Precambrian age with crystalline rocks like gneiss and granites, and another of Cambrian-Silurian age representing fractured metasediments. While relatively large horizontal stresses have been reported in the former, rocks of the Cambrian-Silurian belt have been generally characterized by lower stresses (Myrvang 2001, 2002). The latter applies to the region where the Skarvberg tunnel is placed (Fig. 9) as the Kalak rock formation belongs to the Cambrian-Silurian belt.

Fig. 9

Orientation and relative magnitude of horizontal stresses in Northern Norway. Modified after Myrvang (2002) and the World Stress Map (WSM) of Heidbach et al. (2018). (Color figure online)

According to Amadei and Stephansson (1997), both the magnitude and the orientation of horizontal stresses can be influenced by the rock mass structure and the degree of jointing. If the general lineament patterns in the region of Finnmark are considered, authors like Gabrielsen et al (2002) and Bergø (2016) note that geological structures are mostly oriented NW–SE, NE–SW and N–S. The latter orientations agree in general with the mapped lineaments and the structural measurements performed at the tunnel site (Fig. 1b; Table 4). In that sense, some degree of rotation of the major horizontal stresses (oriented N-S) should be expected to take place and align with the mentioned structures. The mentioned stress rotation and influence of geological structure at a regional scale, need to be accounted for when evaluating the results of the in-situ stress measurements. A description of the stress measurements carried out on site and the interpretations that followed such study are given below.

The in-situ rock stress measurements were carried out at two test locations in the tunnel, 1 + 509 and 2 + 886 (Fig. 1), outside the failure zone, and well behind the face during the construction stage. The stress measurements were done by SINTEF (SINTEF 2020), utilizing the doorstopper method (Leeman 1964, 1971; Myrvang 1976). The method is based on the determination of the stress state from boreholes where overcoring of rock at the bottom of the hole produces a strain relief that can be registered by a strain cell glued to the bottom rock and then correlated to stress by knowing the elastic properties of the rock (Amadei and Stephansson 1997). To be able to find the stress state with two boreholes, the method assumes that at least one of the principal stresses is aligned with one of the two boreholes and that the rock is homogeneous, continuous, isotropic and presents elastic behavior (Sjöberg et al. 2003).

At each of the two test locations, two boreholes were drilled. Each hole was about 7–8 m long where 6 to 7 measurements were performed at different depths (Fig. 10). The logged rock mass quality in both boreholes was good, with the presence of 2–3 joint sets, and a lithology dominated by metasandstones and thin intercalations of metagabbro. The tunnel alignment is mainly oriented N–S, with a rather distant hillside to the fjord, and a relatively flat plateau over the tunnel giving a rock overburden of 270 m at both test locations (Fig. 1). Based on the observed roof stability problems during construction, and the presence of structural anisotropy produced by the subhorizontal bedding, it was anticipated that the major principal stress would be oriented in the vertical direction, thus representing the vertical stress $({\sigma }_{\text{v}})$. Accordingly, boreholes at the two test-locations were drilled in the center roof and upwards across the geological structure, and horizontally in the tunnel West wall (the wall not facing the fjord).

Fig. 10

(Reproduced from SINTEF 2020). (Color figure online)

Measurement of in-situ rock stresses with 2-dimmensional doorstopper in the New Skarvberg tunnel a Profile 1 + 509 b Profile 2 + 886. Tunnel is not to scale. Tunnel azimuth 180°.

Interpretation of stress measurements in hard and layered rock is, as noted by Borsetto et al. (1984), a complex task. Since the doorstopper method is based on both the elastic properties of rock and the measured strain, disturbance and fluctuations in the measured stresses can naturally occur if one or two of the mentioned parameters suffer variations. In this context, while the vertical stress ${\sigma }_{\text{v}}$ at both test-locations (6–7 MPa) seems to agree with the overburden, the magnitude of the maximum horizontal stress $({\sigma }_{\text{H}})$ measured in the vertical holes shows variations as distant as 1 to 10 MPa at location 1 + 509 and 0 to 6 MPa at 2 + 886, over a distance of a few meters near the excavation. The latter clearly reflects rock stiffness contrasts and different stress concentrations across the layers, which confirms the observations of Amadei (1996) in that lithology changes can affect and truncate the distribution of the in-situ stresses across the bedding. As such, the gravity-induced horizontal stress estimated from elastic theory results in 2.5 MPa and gives a natural gap with the measured magnitudes of ${\sigma }_{\text{H}}$ (0–10 MPa) that suggests the existence of either a tectonic component in the horizontal stress, interbed slippage, rock block movement, or a combination. In the light of this discussion, it seems that both the high values (8–10 MPa) and the low values (0–2 MPa) of ${\sigma }_{\text{H}}$ measured in both vertical holes are not representative of the virgin, in-situ rock stresses, but rather a measure of the induced horizontal confinement (or lack of confinement) across the rock beds. Based on this, a comparison of the magnitude and distribution of ${\sigma }_{\text{H}}$ in the two vertical holes suggests a representative value for ${\sigma }_{\text{H}}$ of 5 MPa, as it seems a common value at the two test-locations that also lies outside the mentioned stress disturbance of unconfined or stressed layers. Such value is also in agreement with parametric analyses done by Bøgeberg and Skretting (2021) to derive a best-fit value for ${\sigma }_{\text{H}}$ in the Skarvberg tunnel. In Norwegian rocks, this value is not especially high for the given tunnel depth, which is likely related to the mentioned effect of the rock mass structure.

As addressed by Amadei and Stephansson (1997), both the magnitude and the orientation of the horizontal stresses can exhibit variations due to the rock mass structure. In that sense, it seems that while the orientation of the tectonic component of the maximum horizontal stress in the region around the Norwegian and the Barents Sea close to Finnmark is mostly oriented N–S (Heidbach et al. 2018), rock mass structure and faulting at the scale of the tunnel project has introduced a rotation of ${\sigma }_{\text{H}}$ as derived from Fig. 10. As such, ${\sigma }_{\text{H}}$ is mostly oriented E-W and N30E at two different depths for location 1 + 509, which correlates well with the possible influence of two throughgoing geological faults of similar strike (Fig. 1) in the vicinity of the test location. At location 2 + 886, the presence of nearby faults striking NE-SW and N120E seems to also correlate with the measured orientation of ${\sigma }_{\text{H}}$ at different depths in the borehole.

In the same horizontal plane as for ${\sigma }_{\text{H}}$, the minor horizontal stress $({\sigma }_{\text{h}})$ is also contained. Its magnitude is rather low, ranging from positive 3.6 MPa to negative (tension) values (−2.4 MPa), which supports the above interpretation of relatively low horizontal stresses at the project location. Because of the dominant rock faulting striking approximately NE–SW, it is interpreted that ${\sigma }_{\text{H}}$ has likely been rotated to follow a similar NE–SW orientation, leaving then an approximate strike NW–SE for ${\sigma }_{\text{h}}.$ A summary of the measured and interpreted in-situ stresses is presented in Table 6.

Table 6 Measurements and interpretation of in-situ rock stresses at test locations 1 + 509 and 2 + 886

To derive in-situ rock stresses for the UDEC calculations at the two study locations 3 + 162 and 2 + 227, further assessment has been needed. In that context, the stress boundary conditions in UDEC are specified by means of the individual components of the in-plane stress tensor (Itasca 2011), where ${\sigma }_{xx}$ represents the in-plane horizontal stress, ${\sigma }_{yy}$ the in-plane vertical stress, and ${\sigma }_{zz}$ the out-of-plane (in the tunnel direction) horizontal stress. As such, the vertical stress ${\sigma }_{yy}$ is derived from the overburden 200 m and 275 m at 3 + 162 and 2 + 227, respectively. Regarding the magnitude of ${\sigma }_{xx}$ and ${\sigma }_{zz}$, the angle (30°–60°) formed between the tunnel plane and the respective orientations of ${\sigma }_{\text{H}}$ and ${\sigma }_{\text{h}}$ results naturally in a reduction of the effective magnitude of the in-plane stresses. In addition to the necessary geometrical transformation to an in-plane stress state, the magnitudes summarized in Table 7 are also the result of an iterative process with back-calculations to calibrate the stress state based on measured ground deformations (Sect. 5.4).

Table 7 Interpreted stress tensor for UDEC calculations at 3 + 162 and 2 + 227. Tunnel azimuth 180°

5.4 Deformation Monitoring

As part of the construction process, convergence measurements were systematically taken behind the face, with the use of total stations and reflectors attached to the tunnel contour. However, convergence measurements were taken about a couple of tens of meters behind the face, which limited the anticipation of some sort of instability or trend that might have anticipated the tunnel failure at 3 + 162. The available deformation records at the failure area were then derived from the comparison of scanned tunnel profiles before and after the failure event. This gave a total deflection of the center roof of about 150–200 mm at profile 3 + 162.

Four multi-point borehole extensometers (MPBX) were also installed vertically in the center roof at four tunnel sections 2+245, 2 + 231, 2 + 227 and 2 + 186. These were 12 m long, with anchors placed at different depths, and a reading accuracy of ± 0.5 mm. The aim was to study the behavior of the layered rock over time and the extent of loosening zones. The measurements from extensometers were also utilized for the necessary calibration process in the numerical analyses of tunnel stability.

The measured vertical displacements at different anchor depths in extensometers 2 + 227 and 2 + 231 for a period of three months are illustrated in Fig. 11. The readings are referenced to the anchor point at the borehole collar. Considering that on average, a 4 m long excavation round was taken per day and the characteristics of the hard and anisotropic rock mass (of quality Q ~ 0.3–0.4), displacements appear to occur in connection to block movements, its associated dilation, and possible rock detachment. When comparing the two extensometers, the measured displacements are of small magnitude in general, with up to 3 mm of maximum vertical displacement in MPBX 2 + 231, and approximately 7 mm for MPBX 2 + 227 in anchors placed at 12 m depth above the roof. Some differences in the distribution of vertical displacements measured at the anchors placed at different depths were also observed. For example, anchoring points placed closer to the tunnel contour (depths 3, 4, 5 and 6 m) showed in general rather limited displacements (< 2 mm) if compared to anchor 12 m in MPBX 2 + 227. Such response may be interpreted either as rock mass movements mass occurring at any depth between 4 and 12 m, as the beneficial reinforcement effect of the 5 m long rock bolts stabilizing the arch roof, or a combination.

Fig. 11

(Reproduced from Bøgeberg and Skretting 2021). (Color figure online)

Displacements recorded by vertical roof borehole extensometers at chainages 2 + 227 and 2 + 231 close to the tunnel face. Readings in anchors are referenced to a collar. Rock mass quality Q 0.3–0.4 at both MPBX locations. Tunnel is not to scale.

6 Analysis Methodology

Four numerical models were run to study the ground behavior and the performance of different design approaches for tunnel rock support in layered rocks of the new Skarvberg tunnel. The four models replicate different support designs that are the result of a sequence of supplementary site investigations and studies done at different design stages in connection to the mentioned tunnel failure (Table 8). As such, the first numerical model (model A1) is an analysis of the tunnel failure at 3 + 162 with an empirically designed rock support. Model B1 represents tunnel sections 2+227, with rock mass quality Q 0.3–0.4 and a rock support design equally based on empirical classifications and aided by numerical analyses. Models A2 and B2 represent tunnel sections 3+162 and 2 + 227 respectively, but study the support performance and ground behavior considering that the design of the rock support had followed a hybrid design approach as that proposed by Terron-Almenara et al. (2023). The pairs of models A1–A2 and B1–B2 therefore cover an empirical and a hybrid design approach for both study sections 3+162 and 2 + 227. This has enabled a comparison of the results to determine the performance of the current empirical methodology used in Norway for layered hard rock masses. The results are presented in Sect. 8.

Table 8 Model description in relation to the design stage, design approach and performed investigations

A summary of the analysis methodology based on a hybrid approach (Terron-Almenara et al. 2023) is presented in Fig. 12. This is followed by a detailed description of each of the analysis steps that form such a procedure. It should be noted that to allow comparison between the empirically based rock support designs of models A1 and B1, and the hybrid-based designs proposed for models A2 and B2, the full sequence of analysis steps of a hybrid approach presented in Fig. 12 has been only applied to A2 and B2 models (Sects. 8.2 and 8.4, respectively). In turn, the analysis of ground behavior and support performance for models A1 and B1 has been based on UDEC numerical analysis.

• Step 1. The first step consists of the identification of ground behavior and assessment of potential failure mechanisms based on the evaluation of, among other rock mechanical parameters, the rock mass structure and competence, jointing conditions, tunnel geometry, and the in-situ stresses. The ground conditions are then categorized into a GBT according to the classification presented by Terron-Almenara et al. (2023) for the hybrid design of rock support in hard rock tunnelling conditions. In the case of the new Skarvberg tunnel, the investigated ground conditions presented in Sect. 5 clearly refer to hard and bedded rock masses indicating a case that falls within the category “Anisotropic rock mass”. Based on tunnel inspection, mapping, and an assessment of the rock mechanical conditions described above, an additional categorization of the behavior of layered rock masses can be done with the classification for Layered Ground Behavior Type (LGBT) proposed in Sect. 4.3. The Layered Ground Behavior Chart (LGBC) and the procedure explained in Sect. 4.3 is utilized to select the LGBT category that defines best the rock mass conditions to be analyzed. With the use of such classification, one can derive simple but useful estimates of failure mechanisms and loading conditions to be used in the design process.

• Step 2. Specific site investigations are recommended in such hybrid methodology for anisotropic rock masses when rock mass quality Q < 0.4 and/or GSI < 40. Accordingly, deformation monitoring, joint properties and jointing geometry, deformability and strength properties of rock, and rock stress measurements should be performed. Both the scope and the outcome of such investigations are presented in Sect. 5. The resulting set of rock mechanical properties is then utilized in the analyses of tunnel stability, especially in the ones carried out by analytical solutions in Step 4, and in the numerical calculations in the analysis Step 5.

• Step 3. Empirical and basic estimates of rock support design are derived in this step from using well-established classifications for hard rock tunnelling like the RMR, the Q-system, or the empirical classification proposed by NPRA (NPRA 2022a). There can naturally be some difference in the level and details of the recommendations from one to another classification. In this step, however, a comparison of the empirical designs based on the mentioned classification systems is performed with the aim to obtain a basic estimate of empirical rock support design concerning rock bolting, thickness of sprayed concrete, and the design of RRS arches.

• Step 4. Analytical solutions are employed for the assessment of tunnel stability and possible optimizations of empirical support designs obtained in Step 3. In this case, the authors have selected analytical solutions based on limit equilibrium analysis of voussoir roof beds like the ones proposed by Lang and Bischoff (1982), Diederichs and Kaiser (1999), and Abousleiman et al. (2021). The authors have chosen these three analytical solutions since their joint application can capture the actual ground-and-support behavior of the jointed roof beds. As such, the solution of Lang and Bischoff (1982) considers arched tunnel roof geometry and the effect of in-situ stresses for the assessment of roof stability in bolted tunnel roofs, but it obviates anisotropic rock mass structure. The solution of Diederichs and Kaiser (1999) can simulate the behavior and stability of anisotropic but single-layered roof beds based on the study of the maximum vertical deflection and the buckling limit of a voussoir roof bed. As such, the equilibrium analysis of the beam`s resistance to buckling is studied with Eq. 1 whilst deflection is calculated by using Eq. 2.

  $${\text{FS}}_{\text{buckling}}=\frac{{M}_{R-\text{beam}}}{{M}_{W-\text{beam}}}$$

  (1)
  $$\delta =\frac{{\upgamma D}^{4}}{{32ET}^{2}}$$

  (2)

  where (FS_buckling) is the factor of safety against buckling failure of the roof bed, (M_R-beam) is the moment resistance of the bolted roof beam, (M_W-beam) is the moment loading acting in the bolted roof beam, ($\delta )$ the maximum deflection of the reinforced rock beam, D the span of the excavation, (E) Young’s modulus of the rock, and (T) the thickness of the reinforced rock beam.

  The more recent solution of Abousleiman et al. (2021) is based on the solution of Diederichs and Kaiser (1999). Both solutions consider rock mass stiffness. However, while rock mass stiffness in Diederichs and Kaiser (1999) is based on a single-layer beam solution with sub-vertical joints, the solution of Abousleiman et al. (2021) considers bedding planes in a passively bolted multi-layer beam. The Abousleiman et al. (2021) solution therefore permits a more realistic simulation of the collapse mechanics of bolted voussoir roofs since the intrinsic reduction in rock mass deformation modulus produced by the layering is captured in the solution. The application of Eq. 2 in the Abousleiman et al. (2021) solution requires then the replacement of the Young’s modulus of the intact rock (${E}_{\text{i}})$ by an adjusted rock mass modulus (${E}_{\text{rmn}})$ that accounts for the number of rock beds within the bolted, reinforced voussoir beam.

  For the determination of tunnel stability based on analyses performed with the three analytical procedures mentioned above, a study that involves comparison and analysis of (FS_buckling) and ($\delta$) for the models A2 and B2 is done. Such a study should additionally consider the results obtained in the calculations of A1 and B1 models, together with the estimates of support needs performed in the preceding analysis steps (steps 1–3).

• Step 5. The numerical analyses in the models summarized in Table 8 are conducted with the numerical code UDEC v5.0 (Itasca 2011). It is a two-dimensional numerical program based on the distinct element method (DEM). Its essential feature is that the rock mechanical problem is treated as an assemblage of distinct, interacting, and deformable blocks by solving the equations of motion and contact forces between blocks (Cundall 1980). This permits the simulation of large displacements due to slip and/or opening of joints together with block deformation (Eberhardt 2023), which is a behavior that suits the observations at the two study locations, 3 + 162 and 2 + 227.

Fig. 12

Analysis methodology for the hybrid design of rock support in models A2 and B2 in layered rock, based on Terron-Almenara et al. (2023). Description of the analysis steps is numbered and presented below in the text

Each of the models was built with input parameters of the ground conditions as presented in Sect. 5. The four models A1, A2, B1 and B2 were calibrated before performing the analyses presented in Sect. 8. Such calibration process is described in Sect. 7. The calibrated models were then used to evaluate ground behavior in the form of rock displacements and induced rock stresses around the tunnel opening. The exerted loading conditions on the support systems in each of the models were then compared with the calculated resistance and capacity of such support systems. In that sense, since the mode of shotcrete failure in hard rock masses not subject to high stresses is primarily related to shear and bending loading (Kaiser and Tannant 1997, 2001; Hoek 2007), the analyses performed for Skarvberg tunnel have then focused on the stability analysis of these parameters. Accordingly, the analysis of the support loading and performance was based on the calculation of the factor of safety (FS) against failure of the support system as defined by Eqs. 3, 4 and 5.

$${\text{FS}}_{c-\text{bending}}=\frac{{M}_{\text{RC}}}{M}$$

(3)

$${\text{FS}}_{c-\text{shear}}=\frac{{V}_{\text{RC}}}{V}$$

(4)

$${\text{FS}}_{b}=\frac{{T}_{b}}{{Q}_{b}}$$

(5)

where (FS_c-bending) is the factor of safety of the sprayed concrete to fail in bending, (M_RC) is the moment of resistance of the shotcrete support system, (M) the acting bending moment on the shotcrete support, (FS_c-shear) the factor of safety of the concrete to fail in shear, (V_RC) the shear resistance of the shotcrete support, (V) the shear forces acting on the shotcrete support, (FS_b) the limit state at axial failure of the bolt, ${(T}_{\text{b}})$ the tensile capacity of the bolt, and ${(Q}_{\text{b}})$ the axial load acting on the bolt.

By further evaluation and comparison of the distribution and size of the support loading and FS in each of the pair models A1–A2 and B1–B2, it was possible to assess design optimization possibilities. Such comparison similarly aided to evaluate the performance of the current empirical methodology of tunnel rock support in Norway if compared to a hybrid methodology as proposed in Terron-Almenara et al. (2023). Several trials were then put in an iterative process for models A2 and B2 in UDEC where different designs were analyzed until satisfactory levels of FS and rock displacements were obtained. In both cases, A2 and B2, this resulted in design optimizations aligned in the direction of optimized bolt length and bolt density together with leaner RRS arch-support designs (Sects. 8.2 and 8.4).

7 Set up of the Numerical Models

The UDEC models are built in plane-strain conditions, with elasto-plastic Mohr–Coulomb material, and the non-linear Barton-Bandis joint model (Barton and Bandis 1990). Such a combination of material and joint models is intended to be representative of the studied rock mass conditions where rock displacements are not only related to shear translations through existing joints but also to buckling failure of rock when subject to loading. The models contain three explicit joint sets that divide the models into an assemblage of discrete rock blocks (Fig. 13). The individual rock blocks were similarly subdivided into a mesh of finite-difference elements to make the rock material deformable, where each element responds to the mentioned stress–strain failure criterions. To not compromise the accuracy of the results, the size of the mesh inside the rock blocks was established on the basis of the rock mass conditions, the dominant failure mechanisms, and the size of the excavation. As such, the maximum element length was set to not exceed a value of 1 m in the UDEC models. The latter is in line with modelling recommendations given by, among others, Lorig and Varona (2013) and Itasca (2011), to provide sufficient accuracy in numerical simulations with UDEC for underground excavations in hard and jointed rock.

Fig. 13

Set-up of the UDEC model for numerical calculations in tunnel profile 3 + 162. (Color figure online)

Boundary conditions were set first by the creation of an external geometry or box of 100 × 100 m that contains the geological model and the tunnel in its center. The external boundaries of the box were also restrained. That is, constrained to zero-horizontal movement in the vertical walls of the external box, and zero-vertical movement in the horizontal (top and bottom) sides of the box, to prevent the model to rotate in the space. To model in-situ stresses, both the in-plane and the out-of-plane in-situ stresses (Table 7) were also applied. The modelled tunnel geometry was based on a D-shaped, 12.5 m span tunnel, and updated to match the actual excavation profiles obtained from tunnel surveying (Fig. 13). The lithological alternance was similarly included in the UDEC models as mapped at the face (Figs. 7 and 8). The rock properties of the metasandstone and metagabbro as defined in Sect. 5.2 are used in the models and represent deformable material.

Jointing in the models was simulated to replicate the actual rock mass structure. However, some simplifications were included since the presence of joint contacts in UDEC modelling normally occupy a relatively large computational memory. A typically adopted approach to reduce the number of joints and computational time whilst keeping accuracy is by simplifying the joint geometry. This was done by following one of the approaches recommended by Chen et al. (2001), where a larger joint spacing is simulated outside the area of interest (far from the tunnel excavation) whilst maintaining a joint geometry of similar characteristics in the region close to the tunnel (ca. 1 × tunnel diameter). Although such simplification has an impact on the size of rock blocks and rock mass deformability, the deformational properties of the modelled rock were adjusted, through the mentioned numerical calibration process, to reflect the deformational character of the in-situ rock mass.

The four models A1, A2, B1 and B2 were calibrated before performing the analyses presented in Sect. 8. Tunnel deformations registered with deformation monitoring as described in Sect. 5.4 were utilized as a basis for the necessary back-calculations in the calibration process. In such a process, the rock and joint properties obtained from field and lab investigations were adjusted through an iterative process until consistent outputs (results) with the measured behavior were obtained. The calibrated properties of the joints and the rock that were finally used in the UDEC analyses are presented in Tables 9 and 10, respectively. In that sense, the bulk modulus (K) and shear modulus (G) of rock were also derived to describe material behavior of the rock blocks in the UDEC models.

Table 9 Calibrated joint properties used for the UDEC calculations of tunnel stability at profiles 3 + 162 and 2 + 227

Table 10 Calibrated properties of rock material used for UDEC calculations of tunnel stability at profiles 3 + 162 and 2 + 227

Calibration of numerical models performed on the basis of deformation monitoring in hard rock tunnels is a widely used procedure in rock engineering. As pointed out by Sakurai (2017), it provides a rather useful posteriori rock engineering indicator reflecting the true response of a set of rock mass properties to tunnel excavation. However, the same author along with Walton and Sinha (2022) advises that the use of such a back-analysis process normally involves limitations and assumptions. For the case of hard and layered rock of very poor quality like in the two studied tunnel sections, 3 + 162 and 2 + 227, the main assumptions and considerations for the modelling and calibration processes were:

• The utilized constitutive models for joint and rock material are assumed to capture the behavior of layered hard rock together with the behavior of distinct joints subject to relatively low levels of horizontal stress confinement.

• The properties of rock and joints derived from field and laboratory investigations (Sects. 5.1 and 5.2) are representative of the ground conditions at tunnel section  3 + 162 and 2 + 227. That is, the amount of rock mechanical data from the site investigations at each of the locations was sufficient to define representative models containing a sufficient level of detail for the analysis purpose.

• There is no evidence of time-dependent or creep deformations in the monitored rock masses during the monitoring period. In that context, the deformations measured over a period of three months (Sect. 5.4) represent the final, stabilized rock mass after tunnel excavation.

• The displacements registered from deformation monitoring in the four MPBX (at section ca. 2 + 227) and from several tunnel scans of the tunnel section 3 + 162 were assumed representative of the ground behavior at the corresponding study locations. The measurements were similarly assumed reliable.

• The UDEC calculations are performed in computational steps or stages which provide a representative simulation of the sequential installation of support. Such simulation in the UDEC models similarly permits differentiation between deformations occurring ahead and behind the face. The calculated deformations behind the face were then compared to observed (measured) ground behavior behind the face.

• The material properties of the rock support elements are not subject to calibration since they represent fixed, known design values given by the design itself.

• The calibration process is principally based on the comparison between the observed (measured) displacements and the calculated results (outputs) for ground behavior. In that context, it is assumed that rock displacement is the parameter that defines best the behavior of layered and hard rock subject to relatively low stresses. However, various combinations of input parameters used in the back-analyses can also yield a similar response. To limit reliability issues associated to such non-uniqueness in the back-analysis and calibration processes, all the available rock-mechanical information was utilized to constrain the model as much as possible, as suggested in Sjöberg (2020). On the basis of the numerical results (outputs), and the amount of data and model parameters considered in the calibration process of this study, it is assumed that the calibration process resulted in sufficiently reliable models.

As noted by Walton and Sinha (2022), however, there can always be remaining possibilities for reliability improvement since every parameter added to the back-analyses intrinsically adds a source of uncertainty. This is also in line with Elmo and Stead (2021). The two authors conclude that besides the attempts of minimizing uncertainty and subjectivity by increasing knowledge (of the ground conditions), a certain level of uncertainty and subjectivity can still remain due to the limitations of the methodologies (i.e., rock testing, in-situ testing, rock mass classification and mapping, numerical method) involved in the study. In that sense, we believe that the involvement and integration of different design tools and approaches as suggested in a hybrid approach by Terron-Almenara et al. (2023) have contributed to the identification of possible deviations and to reduce the level of uncertainty.

Before excavation in the models, the models were run in elastic conditions to equilibrium. Then, the full face excavation was computed, and the support installed sequentially behind the face. Considering the mechanical characteristics of the rock mass and that the tunnel roof was pre-reinforced with bolt spiles, it was considered that about 30–50% of the total elastic tunnel deformation takes place ahead of the face (Hoek et al. 1997) and before the support installation. To allow such simulation of rock mass relaxation and support loading with respect to a sequential support installation in a two-dimensional program, UDEC permits the splitting of the computing analyses in numerical steps or intervals. Accordingly, the span of each construction stage was modelled by intervals defined by a determined number of computing cycles representative of the relaxation occurred prior to that stage. Once the computing of the final installation of the permanent support was complete and cycled, the models were then run to solve for equilibrium.

Bolt spiling ahead of the face was also modelled at the two studied locations, from the 10 to 16 o’clock tunnel position. To simulate bolt spiling in 2-dimmensional numerical analyses, a zone of ca. 0.5 m thickness in the immediate tunnel roof with improved ground properties was modelled as suggested by Hoek (2001). Based on the cross-sectional area of Ø32 mm spiles, ca. Ø55 mm holes, and a center spacing between spiles of 0.3–0.5 m, the properties of the spiling-reinforced ground were derived (Table 11).

Table 11 Weighted material properties of spiling-reinforced ground

With regards to the modelling of RRS support, UDEC permits the construction of composite beam support that may simulate RRS in the tunnel direction. As such, the modelling of RRS-support requires the definition of normalized beam sections to derive equivalent beam properties of the RRS-support. This is a process that weights the amount of steel, the dimensions of the RRS arches, and the c/c distance of arches to obtain equivalent beam sections that fit in the unitary (1 m) out-of-plane slice thickness of UDEC. The equivalent section approach to model ribs or arches in geotechnical software has been used in similar studies by authors like Donovan et al. (1984), Carranza-Torres and Diederichs (2009), Chryssanthakis (2015) and Høien and Nilsen (2018), and proven satisfactorily. The necessary UDEC properties of the rock support material and the interface ground-support and support-support are summarized in Table 12. The characterization of the properties was based on both, rock support data sheets obtained from the rock support suppliers, and from a literature search within similar case analyses such as Chryssanthakis et al. (1997), Malmgren (2005), Itasca (2011), and Bahrani and Hadjigeorgiou (2017).

Table 12 UDEC rock support material and interface properties for models A1, A2, B1, and B2

8 Analysis Results of Ground Behavior and Support Performance

8.1 Model A1 at the Tunnel Failure Zone Profile 3 + 162

Model A1 (Fig. 15) represents the tunnel failure occurred at 3 + 162, characterized by a buckling failure of the roof support and a vertical deflection of about 150–200 mm into the opening. With the application of the NPRA guidelines for tunnel support design (NPRA 2020) and a mapped rock mass quality Q 0.1, rock support class IVb was empirically derived. Accordingly, it was installed a layer of Sfr 150 mm, systematic rock bolting c/c 2–4 m long in the roof and 3 m in walls-, and Si-RRS 30/6 spaced c/c 2 m, followed by an excavation round length of 4 m. The asymmetric geometry of the excavation profile (Figs. 2, 14, and 15) should also be noted. This is a consequence of the dominant overbreak following the main discontinuity sets.

Fig. 14

UDEC model of an unsupported case for the new Skarvberg tunnel 3 + 162. a Plot of total vertical displacements and b Plot of stress contours for major principal stresses. Horizontal and vertical model scale in (m × 10). (Color figure online)

Fig. 15

UDEC model A1 with empirical support at profile 3 + 162. a Ground behavior of the supported and failed tunnel roof, with plot of total vertical displacements, bending moments on support and bolt axial loading, and b Plot of stress contours for major principal stresses. Horizontal and vertical model scale in (m × 10). (Color figure online)

Before modelling the tunnel support, a computation of model A1 in UDEC was run first in a separate and unsupported model to investigate ground behavior. In such analysis, the calculations were run to equilibrium according to the default criterion in UDEC. The displacements reached equilibrium were then interpreted to be indicative of a scenario likely to exhibit failure in reality.

As observed in Fig. 14a, the spatial distribution and magnitude of the vertical displacements clearly indicate that failure takes place predominantly through the subvertical joints, and to a less extent through the subhorizontal bedding above the tunnel roof. From the magnitude and distribution of compressive stresses plotted in Fig. 14b, it is similarly interpreted that a domed zone of loosened rock is formed and reaches about 5–6 m above the tunnel roof.

Detachment and rotation of rock blocks are also observed at the immediate roof slab and towards both abutments (Fig. 14a), which together with the calculated magnitude of vertical displacements suggests the presence of buckling failure of the immediate roof beds. This combination of buckling failure with detachments at bedding planes and rock translations by shear through subvertical joints cutting the tunnel abutments would place this tunnel case in Category IIIa of LGBT (Fig. 6). Such behavior would basically result in uneven loads acting at both the center roof and at the intersection of subvertical joints with the excavation.

As for the unsupported model (Fig. 14), the distribution of rock displacements in the supported model (Fig. 15a) also follows the geological structure. Although the vertical rock displacements are rather restricted by the tunnel support if compared to the unsupported model (Fig. 14a), they are of sufficient magnitude (ca. 140 mm in center roof) to generate buckling and failure of the RRS-support. From the distribution and magnitude of displacements and principal stresses (Fig. 15), the loosened/failure zone is observed to span about 4–5 m vertically above the tunnel roof. This is a significant reduction of rock loosening if compared to the unsupported model (Fig. 14) and confirms the beneficial effect of rock reinforcement to stabilize layered ground.

In analyzing the magnitude and distribution of axial forces on the RRS-support in model A1, these were mostly concentrated in the tunnel roof and the right abutment. In addition, a rather uneven distribution of shear forces was observed in the roof and tunnel walls, suggesting shear loading by differential movements of rock blocks over the entire tunnel profile. Regarding bending moments on the RRS support, the plot shows an uneven distribution that mainly affects the tunnel roof (Fig. 15a), reaching up to 125 kNm. Assuming load-sharing between RRS arches in the tunnel direction, the moment resistance of the installed RRS-support design is analytically calculated to about 88 kNm after Holmgren (1992). Given that the moment resistance of the RRS support is substantially exceeded, and in the provision that a flat RRS geometry in the roof has a reduced capacity to build moment resistance, the modelled behavior of the failure (Fig. 15a) matches fairly well with the observed failure in bending of the support occurred at 3 + 162 (Fig. 2). In applying Eq. 3, the latter yields an FS against moment failure of the RRS-support FS_c-bending ~ 0.7.

The failure of the roof is, however, not only explained by the sole failure of the load-bearing support but also by the rock bolting design. As discussed above, the decompressed/loosen rock above the tunnel roof reaches about 4–5 m whilst roof bolts are 4 m long. Hence, the roof bolts’ ability to reinforce the ground is limited, since some of the bolts do not reach stable, undetached ground. This is observed in Fig. 15a by the two center roof bolts not taking as much load as the nearby bolts at both abutments since the former partially moves with the ground. In the light of bolt spacing, the center distance of bolts seems also in disagreement with the expected ground loads. Considering that the installed bolts had an ultimate load capacity close to 200 kN, roof bolts towards the two abutments were very close (190 kN) to reach failure. Bolts in the tunnel sidewalls, however, were loaded less than one-third of their capacity. This suggests the need for longer bolts and a closer spacing for the roof bolts to obtain a more effective load-sharing of the ground loads. The latter would have similarly contributed to the formation of a thicker beam or artificial pressure arch of reinforced rock to take a greater load share, hence preventing overloading of the RRS arches and bolts.

Bolt spiling is observed to have contributed to stability in spite of the flat tunnel roof shape. This was confirmed in the calculation step between the full excavated tunnel and the first layer of sprayed concrete where rock displacements and rock loosening in the roof were kept relatively low due to the confinement provided by the spile-reinforced roof. Such stabilizing effect provided by the bolt spiling has been further analyzed in a separate study presented in Appendix A. The study involves UDEC numerical analysis of ground behavior and support loading of the A1 and B1 models without bolt spiling to enable comparison and an evaluation of the actual contribution of such pre-reinforcement to tunnel stability.

8.2 Model A2 with Hybrid Support Design at Profile 3 + 162

Model A2 (Fig. 16) represents an idealized case where hybrid rock support had been designed with a hybrid approach (Terron-Almenara et al. 2023) at 3 + 162. This is done to allow comparison against model A1 (which has a support design based on the empirical method, only). The rock mass conditions, the tunnel geometry, and in-situ rock stresses are kept identical as for model A1. In the following, a step-by-step procedure is followed according to the analysis methodology (and the step structure) presented in Sect. 6 for hybrid support design in layered rock masses, to present the mechanics of such hybrid methodology when implemented in practice. A summary of the results is also presented in Table 13.

• Step 1. With the proposed classification of layered ground behavior type (LGBT) for anisotropic, layered rock masses in Sect. 4.3 of this article, the definition of the stress situation should be done first. The stress measurements (Sect. 5.3) show a situation with relatively low horizontal stresses (~ 3 MPa) and k₀ < 1 (i.e., ${\sigma }_{xx}$/${\sigma }_{yy}$= 3/5.4 = 0.55). This places LGBT in the subcategory “a” in the LGBC chart (Fig. 5). Following the chart, further classification is then performed by the definition of the relative tunnel span to the bed thickness. For the given tunnel span and an average bed thickness of 0.1 m, the subcategory III is obtained (Table 13). The latter places the ground conditions at section 3 + 162 in LGBT IIIa. Such classification is further substantiated by the agreement existing between the mapped rock mass conditions Q 0.1, RQD ~ 15% at section 3 + 162, and the typical rock mass conditions that define category IIIa in the LGBC chart (Fig. 5).

• In using such classification in the context of a hybrid rock support design, category IIIa indicates the formation of a relatively deep and wide dome of loosened rock that may extend beyond the length of reinforcement rock bolts. Associated shear failure along joints and buckling of rock slabs at the roof may also take place as illustrated in Fig. 6. Such response can generate significant and uneven loading in the roof, which in turn would anticipate the need for tighter and longer rock bolting designs together with full-section and relatively heavy load-bearing support systems (arches).

• Step 2. A comprehensive site investigation program was performed to characterize the ground conditions and the ground behavior as described in Sect. 5. The results have provided valuable and necessary input for the analyses done from Step 3 to 5.

• Step 3. For the tunnel section 3 + 162 with rock mass quality Q 0.1 and a theoretical span of 12.5 m, the Q-system recommends an empirical support consisting of Sfr 150 mm, D-RRS 45/6 + 2 c/c 3 m, and rock bolts spaced c/c 1.3 m of length 4 m in the roof and walls (NGI 2015). A similar exercise done with the empirical design guidelines of NPRA (NPRA 2022a) yields a design composed of Sfr 150 mm, Si-RRS 30/6 c/c 2 m, and 4 m long rock bolts spaced c/c 2 m (as for model A1). In using the RMR, a RMR about 30–35 recommends systematic rock bolts 4–5 m long spaced c/c 1–1.5 m, a layer of 100–150 mm of wire mesh reinforced sprayed concrete, and light ribs spaced c/c 1.5 m where required. By looking to the support loading and the roof failure at model A1 (Fig. 15) together with the above empirical recommendations from three different rock mass classifications, optimization in this case should go in the direction of heavier support and an optimized rock bolting design. In that context, it is only the Q-system that gives a more specific recommendation with D-RRS 45/6 + 2 ribs; both the RMR and Q systems agree that a shorter bolt spacing (c/c 1–1.5 m) is needed; but only the RMR suggests bolts longer than 4 m.

• By unifying the recommendations given by these three classification systems with preliminary interpretations of ground loads and depth of loosened rock from Step 1, and on the basis of the numerical results obtained in model A1, the following empirical design recommendations are derived: (a) an upgrade of the load-bearing support in the direction of heavier (i.e., RRS design between I and II in NGI 2015) and arches with closer spacing (i.e., < 2 m), and (b) 5 m long bolts in patterns ca. c/c 1–1.5 m for the tunnel roof (Table 13).

• Step 4. The empirically recommended design of 5 m long rock bolts with spacing c/c 1.5 m from Step 3 has been utilized in the analytical analyses in Step 4 (Table 13). To derive a surcharge load, the interpreted depth of the loosened rock dome from Step 1 (category IIIa) was used (Table 13). As such, a 4 m thick layer of broken rock above the bolt-reinforced layer was employed in the analytical analyses.

• The application of the Lang and Bischoff (1982) solution for jointed rock gives an axial loading of the roof bolts Q_b 79 kN for bolt spacing c/c 1.5 m. Using Eq. 5, this results in an FS against bolt axial failure FS_b~2.5 (Table 13), which may be interpreted as an optimal bolt spacing and tunnel stability in relation to the ground loading.

• With the solution of Diederichs and Kaiser (1999), the calculated moment resistance of the 5 m thick reinforced roof beam M_R-beam (~ 1765 kNm) is slightly higher than the exerted bending moment (M_W-beam) acting on it, which yields an FS against buckling FS_buckling ~ 1 (Table 13). While this may be interpreted as a situation in equilibrium, it could also indicate that the load-bearing support (i.e., RRS) is taking a lower load share in compensation for a higher moment loading of the reinforced rock mass. By calculating the vertical deflection of the bolted roof beam using Eq. 2, we obtain 58 mm. The limited size of it is in agreement with the mentioned equilibrium. However, its relatively small magnitude could similarly be interpreted as a sign of apparent stability, disregarding then the potential need for heavier support solutions with load-bearing support.

• In using the solution of Abousleiman et al. (2021) with a multi-layered beam approach in Eq. 2, the calculated deflection of the center roof is 267 mm. Upon comparison of the deflection results from both solutions, Abousleiman et al. (2021) and Diederichs and Kaiser (1999), it seems that the former solution gives a more realistic result since it displays a behavior that is in a closer agreement with both the actual tunnel roof deformation (measured 150–200 mm at 3 + 162) and with the needs of load-bearing support. In turn, the equilibrium moment and the limited size of the deflection calculated after Diederichs and Kaiser (1999) could be misinterpreted as a nearly stable rock mass, thus yielding an optimistic assessment of tunnel support design.

• In the same context, a comparison of the results between the solutions of Lang and Bischoff (1982) and Abousleiman et al. (2021) would raise the question of whether the latter yields conservative results provided that the analytical solution does not consider arched roof geometry nor the stabilizing effect of horizontal stresses. Remarkably, we observe the opposite for the analyzed case at tunnel section 3 + 162. The Abousleiman et al. (2021) give a result that is closer to reality whilst Lang and Bischoff (1982) suggest stability. The latter optimism in the Lang and Bischoff (1982) solution may be explained, among other reasons, by the intrinsic stabilizing effect provided by the structural isotropy assumed in the solution and its associated improvement of block interlocking.

• In unifying the results obtained under Step 4 (Table 13), and taking into consideration the limitations of the employed solutions, some conclusions regarding the performance of the proposed rock bolting design for the hybrid design of model A2 can be drawn: (a) it appears that the proposed roof bolting design consisting of 5 m long bolts c/c 1.5 m can effectively contribute to arching and rock stabilization, and (b) the results also indicate the need for load bearing support able to take up deformations and ground loads not taken by the reinforced voussoir beams of case A2.

• Step 5. From the interpretation of the recommendations in the previous analysis steps, it is possible to build a preliminary, hybrid support design and test its performance with numerical calculations. After an iterative process where loading behavior and ground behavior are analyzed for different support configurations, a final, optimized design is derived. Such design is now put in model A2 (Fig. 16) and consists of Sfr 150 mm, rock bolting c/c 1.5 m, 5 m long in roof and 4 m in walls, and Si-RRS 35/6 c/c 1.5 m. The numerical results of model A2 (Fig. 16a) show that for the proposed hybrid design, the deformation at the center roof is reduced from ca 150 mm in model A1 to 47 mm in model A2. In addition, it is observed a significant reduction of the moment loading (51 kNm) on the RRS-support, in compensation for a larger load share in the bolts. The denser bolt pattern is observed to have similarly reduced the maximum axial load of roof bolts as anticipated in model A1. The 5 m long bolts in the roof are also loaded along their entire length, and an increase in the load share of the 4 m long wall bolts is also observed.

Fig. 16

UDEC model A2 with hybrid rock support design at profile 3 + 162. a Ground behavior and support loading, with plot of total vertical displacements, bending moments on support and bolt axial loading, and b Plot of stress contours for major principal stresses. Horizontal and vertical model scale in (m × 10). (Color figure online)

Table 13 Overview of results and interpretations using a hybrid approach as presented in Sect. 6 for model A2

When comparing the results for the stress situation between model A1 simulating the failed empirical support and model A2 simulating hybrid design, it is clearly observed an improvement in the latter. Although the results for model A2 also show unconfined rock above the tunnel roof (Fig. 16b), the mentioned improvement is shown in the form of (a) a generalized increase of the compressive stresses in the jointed voussoirs, and (b) a reduction of the area affected by unconfined, loosened rock. In other words, the optimized reinforcement is seen to have contributed positively to rock mass stabilization from the improvement of rock arching and the associated restriction of vertical movements.

8.3 Model B1 with Empirical Support at Profile 2 + 227

Model B1 (Fig. 17) shows the ground behavior and support loading of a monitored tunnel section with rock support design based on empirical classification at profile 2 + 227. The tunnel profile presents less overbreak and a more symmetric geometry than that of 3 + 162 due to the better rock mass quality (Q 0.3–0.4). The rock support class IVb consisted of Sfr 150 mm, systematic rock bolting c/c 1.5–5 m long in roof and 4 m in walls-, and Si-RRS 30/6 spaced c/c 1.5 m. Excavation round length was reduced to 3 m.

Fig. 17

UDEC model B1 with empirical support at profile 2 + 227. a Ground behavior and support loading, with plot of total vertical displacements, bending moments on support and bolt axial loading, and b Plot of stress contours for major principal stresses. Horizontal and vertical model scale in (m × 10). (Color figure online)

The calculated vertical deflection in the center roof was rather small (7–8 mm). Movements were also seen to be of limited magnitude in the sidewalls. The latter is interpreted to be a consequence of the improved rock arching in model B1 if compared to A1. The reduced extent of the dome of loosened rock for model B1 is observed to extend about 3–4 m vertically above the tunnel roof (Fig. 17a).

Both axial and shear forces in the load-bearing support are concentrated in the roof and in both abutments. Also, peaks in shear forces seem to be mostly linked to differential rock displacements at joints intersecting the tunnel roof periphery, which together with the ratio roof span to bed thickness, would place this study case into Category IIa of LGBT (Fig. 6). The higher magnitude of the shear force calculated in model B1 (187 kN) in relation to model A1 may be explained by the higher capacity of the heavier RRS-support in model B1 to take up loading if compared to model A1.

The maximum bending moment on the installed RRS-support is 59 kNm and is located at the right tunnel abutment (Fig. 17a). Provided a moment capacity of the RRS-support calculated analytically to 125 kN, this results in an FS against bending FS_c-bending = 2.1. Rock bolts in the roof are loaded to about 30–50 kN whilst bolts in the sidewalls present even lower axial loads, leading to a FS_b ~ 4. There is one exception that locally affects one bolt at the stressed right abutment, loaded to 159 kN. If one also looks at the bolt loading over the bolt length, it can be observed that bolts present marginal to no loading beyond the length of 4 m in the roof and beyond 3 m in the walls. The latter suggests room for improvement in the bolt length and spacing. However, with the provision that specific blocks in the tunnel profile should need local reinforcement. When looking at the analysis of principal stresses in Fig. 17b, a substantial increase of rock arching above the roof can be observed, when compared to the stress situation in model A1 (Fig. 15b). The latter implies that not only the rock support design influences rock arching in layered rock, but also the rock mass conditions as explained in Sect. 4.2.

8.4 Model B2 with Hybrid Support Design Proposing New C-RRS Support at Profile 2 + 227

Model B2 (Fig. 18) represents the analysis of ground behavior and the performance of a hybrid support design consisting of a crown reinforcement with modified RRS-support at tunnel section 2 + 227 and designed with a hybrid approach (Terron-Almenara et al. 2023). This is done to allow comparison against model B1 (which has a support design based on the empirical method, only). In model B2, the rock mass conditions, the tunnel geometry, and in-situ rock stresses are kept identical as for model B1. As for the study of hybrid support design in model A2 (Sect. 8.2), a worked case example showing a step-by-step hybrid design procedure for model B2 is also presented in the following. A summary is also presented in Table 14.

• Step 1. Similar to tunnel profile 3 + 162, the stress measurements (Sect. 5.3) also show a situation with relatively low horizontal stresses (~ 5 MPa) and a condition of k₀ < 1 (i.e., ${\sigma }_{xx}$/${\sigma }_{yy}$= 5/7.5 = 0.66). This places LGBT in the subcategory “a” or “LOW stress k₀ < 1” in the LGBC chart (Fig. 5). Further, the ratio of tunnel span to bed thickness gives a value of ca. 42, which places the ground conditions at 2 + 227 into category LGBT IIa (Table 14). Such classification is further substantiated by the agreement existing between the mapped rock mass conditions Q 0.3–0.4 and RQD 35–40% at 2 + 227 with the typical rock mass conditions that define category IIa in the LGBC chart (Fig. 5).

• By using the output of such classification for design purposes, category IIa basically suggests that ground behavior is primarily characterized by gravity failure of rock blocks located in unconfined rock portions of the tunnel roof. The latter can naturally lead to anisotropic loads affecting the stability of the tunnel roof. As such, it may be anticipated that the distribution of rock reinforcement and support measures may also follow the distribution or the location of such loads, i.e., an anisotropic distribution of the reinforcement systems around the tunnel, including simplified, leaner versions of load-bearing support.

• Step 2. A comprehensive site investigation program was performed to characterize the ground conditions and the ground behavior as described in Sect. 5. The results have provided valuable and necessary input for the analyses done from Step 3 to 5.

• Step 3. For the tunnel section 2 + 227 with rock mass quality Q 0.3–0.4 and a theoretical span of 12.5 m, the Q-system recommends an empirical support consisting of Sfr 120 mm, Si-RRS 30/6 c/c 4 m, and rock bolts spaced c/c 1.5 m of length 4 m in the roof and 3.5 m in the walls (NGI 2015). Using the empirical guidelines of NPRA (NPRA 2022a) the support design would be composed of only reinforced sprayed concrete 150 mm combined with systematic rock bolting c/c 1.5 m, length 4 m. With the application of RMR, a value about 35–40 recommends a support design consisting of systematic rock bolts 4–5 m long spaced c/c 1–1.5 m, a layer of 100–150 mm of wire mesh reinforced sprayed concrete, and light ribs spaced c/c 1.5 m where required.

• By unifying the recommendations given by these three classification systems with preliminary interpretations of failure mechanisms and loading from Step 1, combined with the numerical results obtained in model B1, a set of design considerations can be derived. For example, the buckling failure of Si-RRS 30/6 spaced 1.5 m in model B1 has an FS 2.1 (Table 15). This gives little room for optimization in the direction of increasing the center distance of the ribs as suggested in the Q-system (c/c 4 m ribs), or by omitting RRS-support as recommended in NPRA. The bolt spacing c/c 1.5 m of model B1, however, is seen to present some optimization possibilities as derived from a generalized FS 4 for bolt tensile failure (Table 15). Despite the limited correlation observed in the recommendations given in the three empirical systems, support recommendations can be still extracted on the basis of a combined assessment that considers both the anticipated ground behavior in Step 1 (Table 14) and the empirically based recommendations described above. As such, there seems to be agreement that 4 m long roof bolts -possibly shorter ones in the sidewalls- with a center spacing about c/c 1.5–1.7 m may be used at 2 + 227. It similarly seems that a leaner design of RRS arches could be applied on the basis of the calculated estimated conditions in the previous analysis steps (Table 14).

• Step 4. The empirical design recommendation of 4 m long rock bolts with radial spacing c/c 1.7 m and longitudinal spacing c/c 1.5 m from Step 3 (Table 14) has been utilized for the analytical analyses in Step 4. To derive a surcharge load, the interpreted depth of the loosened rock dome from Step 1 (category IIa) was used. As such, a 2 m thick layer of surcharge rock load above the bolt-reinforced layer was employed in the analytical analyses.

• In applying the solution of Lang and Bischoff (1982), a bolt loading in the roof Q_b 77 kN is obtained, which results in an apparent stable situation as FS_b = 2.6. With the use of the Diederichs and Kaiser (1999) solution, one obtains a moment resistance M_R-beam ~ 1766 kNm and a moment loading M_W-beam of 1472 kNm, which yields an FS against buckling FS_buckling ~ 1.2. The latter may indicate a stable situation since the voussoir is apparently able to build sufficient moment resistance and stabilize the endorsed moment loading. If vertical deflection of the 4 m thick bolted roof beam using Eq. 2 is performed, it is obtained 68 mm with the Diederichs and Kaiser (1999) solution. As for model A2, the limited size of such deflection agrees with the mentioned stable conditions from the moment analysis. At the same time, the latter could similarly be misinterpreted as a stable situation, thus disregarding the need for other forms of rock support.

• Determination of roof deflection using the Abousleiman et al. (2021) solution, however, gives an additional and valuable perspective to the analytical analyses in Step 4. From using the mentioned solution in Eq. 2, a deflection of 141 mm is anticipated for the tunnel roof at 2 + 227. The latter poses therefore questions on the representativity and optimism around the former two solutions utilized here. As such, whilst the results from both the Lang and Bischoff (1982) and the Diederichs and Kaiser (1999) solutions suggest a nearly stable situation, the results in Step 3 are recommending rock reinforcement and load-bearing support. In that sense, it seems that the Abousleiman et al. (2021) solution gives a less optimistic response as the level of calculated deflection appears to be more in line with the recommended empirical support in Step 3 (Table 14).

• In compiling the results obtained under Step 4, and taking into consideration the limitations of the employed solutions, the following design considerations may be made: (a) the proposed bolting design seems a satisfactory solution design to reinforce the voussoir roof beams and contribute to rock stabilization, (b) there are still ground loads that are not taken by the reinforced roof beams, and (c) there is a need for load bearing support that is mostly concentrated in the tunnel roof and abutments.

• Step 5. From the interpretations and recommendations gathered in the previous steps, it may be possible to derive a preliminary, hybrid support design and test its performance with numerical calculations. Such hybrid design has been additionally adjusted in an iterative process where several combinations of support configurations were numerically tested in UDEC until arriving at an optimized support design. Such design is now put in model B2 (Fig. 18) and consists of Sfr 150 mm, rock bolting with radial spacing c/c 1.7 and longitudinal spacing c/c 1.5 m, bolt length 4 m in roof and 3 m in walls, and optimized RRS-support.

• For the optimization of RRS support, the numerical results from model B1 and the output from the analyses performed through Steps 1 to 4 in this section were examined. In general, it was observed that rock instabilities, deformations, and support loading are mostly concentrated in the tunnel roof and abutments, which confirms the anisotropic behavior of the rock mass and suggests the need for anisotropic ground support. In the case of 2 + 227, such anisotropic ground support is formed by RRS arches covering the location where the larger load demands are allocated, at the crown and abutments. That is, reinforcement with RRS arches ending at the tunnel mid-walls under the abutments, as opposed to the ordinary, full profile RRS arches with foots extended down to the invert level. This hybrid concept leaves then the lower mid walls of the tunnel (ca. 3 m high from the invert) supported with only Sfr 150 mm and rock bolts whilst roof and abutments are reinforced with hybrid, modified Si-RRS 30/6, nominated Crown (C)-Si-RRS 30/6.

Fig. 18

UDEC model B2 with hybrid rock support design at profile 2 + 227. a Ground behavior and support loading with plot of total vertical displacements, bending moments on support and bolt axial loading, and b Plot of stress contours for major principal stresses. Horizontal and vertical model scale in (m × 10). (Color figure online)

Table 14 Overview of results and interpretations using a hybrid approach as presented in Sect. 6 for model B2

Table 15 Comparison of ground behavior and support performance for models A1, A2, B1, and B2. Values in between brackets represent the FS for the general bolt-loading condition in the tunnel roof

The numerical analysis of such support configuration in model B2 with C-Si-RRS 30/6 (Fig. 18a) yields a slight increase in the vertical roof displacement (up to 11.6 mm) if compared to model B1 (8 mm), yet within safe margins. Although the maximum bolt loading in model B2 (110 kN) is reduced by about 30% of the maximum load reached in model B1, the calculations for model B2 show more loading in the bolts (in the range of 30–90 kN) and more bolts taking up loads, especially in the roof and abutments (Fig. 18a). The larger load share in bolts explains, therefore, the lower load taken by the C-Si-RRS-support in model B2. As such, shear and moment loading in the C-Si-RRS 30/6 c/c 1.5 m of model B2 are reduced by about half and one-third, respectively, compared to model B1. This results in a slightly higher FS for bending (FS_c-bending) and shear failure (FS_c-shear) of RRS-support and an optimized design of bolts as indicated by an FS_b = 2 for tensile failure (Table 15).

From the numerical results presented above, the proposed hybrid support with modified Crown-RRS arches yields satisfactory results and can be deemed as a valid design for the given ground conditions. Such improvement in the support loading and load sharing is similarly a reflection of the improved rock arching in model B2 (Fig. 18b) if compared to model B1. In this case, the increase of compressional stresses in the voussoir rock beams (and arching) above the tunnel roof is not as evident as for the studied cases A1 and A2. However, of sufficient magnitude to allow for the mentioned modifications and optimization of the rock support.

The hybrid support design used in the calculations of model B2 (Fig. 18) with modified arches is illustrated in Fig. 19. This concept with crown reinforcement based on modified Si-RRS-arches extending only over the roof and abutments is deemed to support the overlying loads that result from unconfined rock and/or blocks moving through subvertical joints above the tunnel roof. Such a concept with rock support to address the anisotropic rock mass is similarly based on two premises. The first one demands a mechanical interaction between the modified arch support (C-RRS) and the ground through an effective load sharing (and transfer) that is enabled by the radial rock bolts. The second one imposes the requirement of rock stability in the lower sidewalls not supported by the RRS arches. This requirement is related to the need for ensuring an overall structural stability of the support system where the rock mass at the lower sidewalls can withstand the induced vertical loads from the excavation. In this sense, horizontal to subhorizontal bedding with nearly vertical cross-joints and hard rock is seen as a rather beneficial jointing geometry as tabular rock bocks are basically subject to vertical compression behind the walls. In turn, rock masses with inclined bedding and/or cross-joints dipping into the excavation may give a different response in terms of tunnel stability and likely invalidate the use of this new proposed C-RRS-support.

Fig. 19

New Crown (C)-Si-RRS 30/6 arches in a D-shaped tunnel, proposed for hard and horizontally layered rock masses of quality Q 0.3–0.4 and subject to low-moderate horizontal confinement. (Color figure online)

8.5 Summary of Results

The results for model A1 (Fig. 15) clearly indicate that the failure of the tunnel support occurs due to flexural failure of the RRS support. As noted in Sect. 8.1, the unfavorable combination of a wide excavation span combined with a flat roof and too short bolt lengths, too large center distance of bolts, and insufficient horizontal confinement did not contribute to rock mass arching. This resulted in an excess of ground load not taken by the rock mass nor by the roof bolts, and then transferred onto the RRS. By using a hybrid approach as in model A2, proposing an improved support design has been possible, in which the rock support elements are dimensioned so that they become loaded to a more optimal level. Hence, an improved load sharing in the rock support can be realized. Obviously, model A2 requires a generally heavier support design than that in A1. In turn, it provides stability whilst limiting oversupporting.

In terms of the shear loading of rock bolts in the four UDEC models, the calculations showed that the shear load was, in general, rather low, mainly in the range of 5–8 kN for models A1, A2, B1 and B2, with one maximum finding of 18 kN (in model A1).

When comparing models B1 and B2, the results shown in Figs. 17 and 18 (Sects. 8.3 and 8.4, respectively) suggest an improvement of the rock arching in the latter, which naturally results in a slight reduction of the support loading. The latter improvement in the loading conditions of the hybrid support in the B2 model can be visualized from the comparison of loading values of the support system between models B1 and B2 in Table 15. Besides the limited magnitude in the load reduction of the support systems in the hybrid B2 model, the improved loading condition has still allowed for an optimization of the rock support design from the use of modified RRS arches. A summary of the results is presented in Table 15.

9 Final Discussions and Recommendations

The study of ground behavior and support performance in the new Skarvberg tunnel has enabled the evaluation of rock mass conditions influencing the design of rock support in layered rock masses of poor quality. Such an assessment was found very useful to demonstrate the limitations of the current empirical design practice used in Norway. As such, the application of a hybrid design methodology allowed for design optimization as so indicated in the numerical results for models A2 and B2.

As shown in this article, layered rock mass conditions require a more comprehensive characterization of the ground and the combination of different analysis methods. The proposed methodology to classify layered ground (Sect. 4.3) together with a hybrid methodology as that presented in Sect. 6 were utilized to predict ground behavior, support performance, and optimization possibilities for the rock masses of poor quality in the new Skarvberg tunnel project. With a rational combination of the mentioned approaches and methods, including the voussoir beam models, the authors were able to predict the ground behavior of layered rock and derive more optimal and cost-effective support designs.

It should be noted that the main goal of this hybrid methodology has been to provide a complementary tool for ground characterization of layered rocks and the assessment of permanent rock support design. Practitioners should, therefore, be aware that further elaborated design work may be needed to derive final, detailed designs. As observed, a hybrid approach still contains an empirical essence since part of the rock mass characterization and support assessments are based on the application of several rock mass classifications. The latter allowed for a better characterization of rock masses and provided a wider predictive range of suitable support designs. Similarly, the use of the Voussoir beam model provides valuable input to the study of the ground behavior of bedded tunnel roofs. Although a detailed study with such an analytical approach would require more specific information on, among other parameters, the joint spacing, rock stiffness, and the level of horizontal confinement, simple analyses using the Voussoir beam model can still help to predict roof stability and the needs of support on the basis of roof -deflection and -loading. In light of this discussion, it is obvious that the type and amount of rock mechanical information needed for more elaborated analyses with such a hybrid procedure may theoretically place the use of such hybrid procedure more towards detail design- and/or the construction stages of a tunnel project. However, the structure of this procedure is conceived in such a manner that it could be used at any stage of a tunnel construction project by implementing updates in the analysis steps forming the procedure (Sect. 6) as more rock mechanical information is gained along the project execution.

Although such hybrid methodology has been proven rather effective to predict ground behavior and support design in layered rocks, some additional remarks should be given to practitioners intending to use this approach in other projects. The use of this approach requires that practitioners are familiar with a hybrid methodology (see also Terron-Almenara et al. 2023) and its premises, or at least, with the separate analysis methods that constitute this approach. It must be similarly emphasized that the use of this methodology should be only used in underground rock conditions that are comparable to those studied in this article.

9.1 Validity of Results

A substantial part of the study is based on numerical analyses done with UDEC where displacements and stresses in the rock are studied for the evaluation of ground behavior, whilst support loading and FS used as a measure of support performance. In view of this, the utilized failure criterions of rock and joints in the models and the used analytical solutions for roof bed stability and support failure are well-known approaches for such analyses in such ground conditions. In the same context, it is assumed that the used rock mechanical input in the models is representative of the ground conditions and that the detail level (or the amount of simplification) in the models complies with that of the study purpose. In light of this discussion, it should be argumented that simplification is normally accepted in rock mechanics modelling as long as behavior is captured realistically (Starfield and Cundall 1988; Walton and Sinha 2022).

As mentioned in Sect. 7, it should be additionally noted that representativity or uncertainty are believed to have been improved in relation to the thorough program of site investigations, deformation monitoring, and back-analyses. The combination of all these approaches provided a sound basis for the calculations as suggested in Sakurai (2017). Involvement of different investigation and design approaches as suggested in a hybrid approach of Terron-Almenara et al. (2023) has therefore minimized possible deviations and uncertainties in the calculations and results. At the same time, the conclusions derived from such a study (Sect. 9.3) take into consideration the limitations and the possible remaining uncertainties in the study.

9.2 Significance of the Findings

Rock reinforcement in the form of radial rock bolts was seen in the calculations to provide a significant stabilization of the layered roof beds. In general, a larger load share was taken by the roof bolts in the models with better rock mass qualities (Q 0.3–0.4). Rock masses with a declining rock mass quality (Q 0.1), however, presented a slight reduction in roof bolt loading, in exchange for an increased load share in the RRS-support. In more detail, the results in the four models showed that rock bolting design with an optimal spacing and bolt length can provide a solid reinforcement and stabilization effect as the improved characteristics of the reinforced rock mass increase the ability of rock arching. The latter led to a significant load reduction to the RRS arches, which then allowed for an optimization of the load-bearing support. This similarly permitted the employment of hybrid support solutions that involve optimization of the full profile RRS-designs as observed in model B2. The Crown (C)-Si-RRS installed from the mid-wall over to the tunnel roof and down to the opposite mid-wall is an example of that solution. Alternative crown reinforcement methods may, in this context, be also suitable provided that the geometrical and mechanical properties of the employed solution suit with that of C-Si-RRS. Further studies may be required to arrive at a detailed solution, but some proposals for such alternative solutions may also consider crown reinforcements with weldmesh-reinforced sprayed concrete or crowns with fiber-reinforced sprayed concrete of high energy absorption.

In looking to design optimization, the results obtained for models A2 and B2 with the application of a hybrid methodology (Terron-Almenara et al. 2023) combined with UDEC calculations show that such procedure can contribute to reduced amounts of support whilst tunnel stability is not compromised. The use of a hybrid approach demands more knowledge of the rock mass conditions and ground response, which naturally results in more site investigations and elaborated design work than otherwise in the current empirical approach used in Scandinavian hard rock tunnelling. This may appear, a priori, more expensive and time consuming. However, it would be quickly compensated, as extra costs, possible delays, and safety concerns related to tunnel instabilities can be significantly reduced. As shown in Table 15, and as derived from a comparison of the results for models A1-A2 and B1-B2, one can observe design optimization by just evaluating the load distribution to the different rock support elements, the FS against failure, and the resulting ground behavior. But to add further context in the light of optimization, rock support consumption has been calculated and compared in relation to the designs presented in models B1 and B2 (Fig. 20). Accordingly, the support consumption for the empirically based support designs currently practiced in Norway (and represented by model B1), is compared to a hybrid design (represented by model B2) in rock masses of quality Q 0.3–0.4. The calculations are based on a 12.5 m span tunnel and considering that about 175 m (5%) of the new Skarvberg tunnel length falls within rock mass quality Q 0.3–0.4 (NPRA 2022a). As observed, there is a significant reduction (ca. 30%) of sprayed concrete and steel reinforcement from the mere use of modified RRS arches. In addition, there is an overall optimization in the use of bolts of different lengths where optimized combinations of 4 m and 3 m bolts in the hybrid designs provide a notable reduction in rock support consumption when compared to empirical recommendation with 5 m and 4 m bolts. If the calculated amounts are put in an economic perspective in terms of 2022 prices for tunnel construction in Norway, a save of about 0.3 Million EUR may have been achieved by using a hybrid approach in just these 175 m of tunnel. Note that the equivalent construction time saved and the intrinsic save of carbon footprint were not considered in this analysis.

Fig. 20

Comparison of rock support consumption based on rock support design done with the current empirical approach used in Norway and a hybrid approach in a 12.5 m span tunnel, with hard and layered rock, and rock mass quality Q 0.3–0.4. (Color figure online)

9.3 Recommendations

In this article, analyses of tunnel stability in hard and layered rock masses of poor quality have served to both study the limitations of the current design methodologies and to develop a new proposal for hybrid support design in layered rocks. Based on the results and interpretations in this article, two sets of recommendations can be made.

The first one is of a more general character and covers best practices for design in connection with the mentioned limitations in the current empirical methodologies:

• (1) In any tunnel project through hard, layered rock masses with an anisotropic rock mass structure where jointed, voussoir rock beams are formed in the tunnel, the study of failure mechanisms and ground behavior should be an essential part of the design work.

• (2) In layered rock masses of very poor quality (i.e., Q < 1), special emphasis should be put on the characterization of the properties of rock, rock joints, and the in-situ rock stresses, to allow for a representative set of input data for stability analyses and design optimization.

• (3) Rock loosening and overbreak should be avoided with the use of bolt spiling to avoid flat and unstable tunnel roofs that demand higher amounts of rock support. The stabilizing effect of bolt spiling on the overall tunnel stability and its contribution to rock support should be studied further.

• (4) Rock bolting support in layered ground should be specifically investigated and included in empirical classification systems.

• (5) Rock mass classification systems for rock support design in layered rock masses should not be used to derive a final design of permanent support, but as a guideline for basic recommendations of support.

The second set gives new recommendations for the direct application of a hybrid approach in similar projects and for further research work:

• (6) In horizontally layered rock masses, a classification of layered ground behavior LGBC as that presented in Sect. 4.3 should be used as an aid for design since it allows the prediction or anticipation of failure mechanisms and loading conditions in the tunnel.

• (7) A hybrid approach as the one presented in this article, with the integration of different analysis methodologies, ground characterization, and numerical modelling should be used to derive optimal rock support design in layered rock masses of very poor quality (Q < 1) and subject to low-moderate stresses.

• (8) In horizontally layered rock masses of quality Q ~ 0.3–0.4, hybrid support designs in the form of modified Crown-arches as presented in Fig. 19 should be evaluated as a valid support solution.

• (9) Besides the intrinsic increase of engineering work, the associated (and considerable) reduction in rock support consumption observed in the results should motivate the use of hybrid design procedures as the one presented in this article for the evaluation of optimal support in layered rock.

• (10) Further case records of tunnels in hard and layered rock should be added and studied as a basis for potential improvements in hybrid methodologies as proposed in Terron-Almenara et al. (2023) and in the proposed new C-RRS-support concept.

• (11) As in any tunnel-case-based study, the possible projection of the obtained results and the proposed design optimizations into other projects needs to be studied separately and carefully based on the actual ground conditions. Rather than direct extrapolation, the proposed recommendations and designs could therefore be used in other projects as a design guideline or as a benchmark for future research. The addition of further case records of hybrid rock support design in layered ground conditions will naturally result in more knowledge and improvement.

10 Conclusions

The performance of the current empirical approach used in Norway for the design of rock support has been investigated in layered rock masses of poor quality (Q < 1) and subject to low horizontal confinement of the new Skarvberg tunnel. The results show that rock mass classifications such as the NPRA (NPRA 2020, 2022a) and the Q-system (NGI 2015) exhibit limitations in characterizing ground behavior and assess optimal designs of rock support in layered ground. With the application of a hybrid approach combined with numerical calculations performed with UDEC, it was possible to establish a better and wider description of the failure mechanisms dominating ground behavior and ground loads. This study demonstrated that a significant optimization of the support design would be possible, as shown in the studied models A2 and B2.

The presented results show the importance of conserving or improving the self-bearing capacity of the layered and jointed rock or voussoirs beams to arch and take part in the ground load. There is, in essence, a complex interaction of geometrical and rock mass properties that control the formation of such rock arching, or, in the absence of rock arching, the resulting domes of loosened rock above the tunnel roof. In addition to the assistance provided by a newly proposed classification for layered ground behavior types (LGBC), the performance of a comprehensive site investigation campaign was found to be important. Such a program mainly included the characterization of rock and joints, deformation monitoring, followed by the measurement and correct interpretation of in-situ rock stresses.

The function of rock reinforcement ahead of the face with bolt spiling together with rock bolts and shotcrete is important, as these measures were seen to actively contribute to ground stabilization and tunnel stability. Rock reinforcement of beds above the tunnel roof creates reinforced rock beams or voussoirs with an increased ability to arch, enabling a greater load share. Ground load taken by the rock mass itself and/or by the bolts is then only transferred to the RRS support to a limited extent, hence allowing for design optimization. In this context, the results showed a significant reduction of the load imposed on the load-bearing support systems in rock mass qualities Q 0.3–0.4. This permitted an optimization of the ordinary full-profile Si-RRS-arches into a support concept or proposal to reinforce only the crown as shown in Fig. 19. The satisfactory behavior and stability observed in the numerical results for C-RRS-support should enable the use of this or equivalent designs in tunnel projects with ground conditions comparable to those studied in model B2. With this in mind, further addition of case studies in hard, layered rocks will contribute to updates, improvement, and expand the use of the modified arches to fit with other rock mass conditions.

Appendix A: A 2D numerical analysis on the contribution of bolt spiling to stabilizing the rock mass at chainages 3 + 162 and 2 + 227 in the new Skarvberg tunnel

Bolt spiling is a pre-support reinforcement methodology that falls within the category of Umbrella Arch (UA) systems for ground stabilization ahead of the tunnel face (Oke et al. 2014a, b). As described in Sect. 3, bolt spiling consists of the installation of longitudinal steel bolts above and around the tunnel crown prior to an excavation pass, with the purpose of improving tunnel stability and reducing excavation-induced deformations. Such reinforcement method is a well-known procedure for weak ground and/or shallow tunnels where the spiling design and tunnel stability are often studied on the basis of numerical and analytical solutions. In that sense, the contributions of authors like Carrieri et al. (1991), Hoek (1999), Peila (2013), Volkmann and Schubert (2006, 2009), Song et al. (2013), and Oke et al. (2014a, b; 2016) has resulted in an increased understanding of the interaction mechanics between UA systems and the ground.

Besides the natural development of better design criterions achieved by the mentioned authors, the design of bolt spiling in hard and layered rock mass conditions as shown in Fig. 3 is normally based on empirical approaches and previous experience. This is the case of the new Skarvberg tunnel, where bolt spiling was applied in relatively long tunnel sections under the practical criterion of minimizing overbreak to let the rock mass conserving its inherent load-bearing capacity. In line with the observations of Volkmann and Schubert (2006), such an empirical approach often leads to uneconomical designs as the system behavior of this support is usually not addressed. In consequence, a limited or incomplete basis for the design of the permanent tunnel support may be also employed.

The authors of this article have therefore performed an additional study with the aim of investigating the stabilizing effect of bolt spiling in the laminated and hard rock mass conditions of the new Skarvberg tunnel at the failure tunnel sections 3 + 162 and at 2 + 227. Accordingly, numerical analyses with UDEC have been conducted by using the original rock mechanical models A1 and B1 employed in Sect. 8, but without bolt spiling. With such a study, the contribution of the bolt spiling to the global tunnel stability has been assessed from an evaluation of ground behavior and support loading with 2D analyses. The numerical results are presented in Fig. 

Fig. 21

UDEC models without bolt spiling showing results for total vertical displacements, bending moments on support and bolt axial loading in a Model A1* for 3 + 162 and b Model B1* for 2 + 227. Horizontal and vertical model scale in (m × 10). (Color figure online)

21 and to differentiate the results from models with and without bolt spiling, the latter are named with the superscript (*). The main observations regarding ground behavior are:

• In comparing the spiling-reinforced models A1 and B1 (Sect. 8) to models without spiling-reinforcement (A1* and B1*), the latter models are observed to present a generalized increase in the magnitude of the calculated total displacements (Fig. 21). It is also observed that whilst such increase of displacements represents about 33% in the A1* model, the more competent rock mass in model B1* is subject to a lower increment (up to 20%). Such increase of displacements in A1* and B1* is a measure of the stabilizing effect of different bolt spiling designs (i.e., c/c 0.3 m and c/c 0.5 m for the respective models A1 and B1) and is in line with previous investigations undertaken by Volkmann and Schubert (2006) and Oke et al. (2014b) for tunnels in weak rock and/or shallow conditions.

• In line with the previous observations, there are also indications of the different stabilizing effect of bolt spiling in relation to the rock mass conditions. As such, displacements calculated in model B1* (rock mass quality Q 0.3–0.4) are mostly located at loosened and single blocks cutting the excavation contour in the left spring line and at the roof. On the other hand, the more fractured and less competent rock mass in model A1* (Q 0.1) presents more difficulties to arch which derives into a larger mass of loosened rock blocks and significant displacements above the entire roof (Fig. 21a). Such difference in the model response suggests that bolt spiling in the more competent rock mass of model B1* (Fig. 21b) would have the primary function of holding rock blocks near the excavation contour in place to conserve the inherent bearing capacity of the rock mass. On the other hand, it is interpreted that bolt spiling in a rock mass of poorer quality (Q 0.1) as that of A1* would have the primary function of reinforcing the entire rock mass above the tunnel roof to reduce loosening of the ground and radial movements.

• The comparison of A1 and B1 against A1* and B1* models have also shown that it is not only the magnitude of displacements that is influenced by the absence of spile reinforcement but also the distribution of displacements. As such, it is observed that the absence of bolt spiling in models A1* and B1* has led to the development of displacements further down below the springlines and the tunnel walls. As also observed in Fig. 21, such an increase in displacements has expanded in the roof vertically into the rock mass, if compared to the spiling-reinforced models A1 and B1.

As derived from the observations given above, it seems that bolt spiling in hard rock has a direct stabilizing effect in terms of ground behavior and displacements. The latter naturally implies that the loading condition of the tunnel support system is influenced by bolt spiling. For this reason, the interaction of the tunnel support system with the bolt spiling should normally be studied jointly when conducting the design of reinforcement and permanent support measures in rock tunnels of poor quality. To evaluate such interaction between tunnel support and bolt spiling and to address the influence of bolt spiling on the loading of the tunnel support system, the loading condition of models A1, A1*, B1, and B1* has been compared (Table 

Table 16 Comparison of model response for roof displacements and support loading in models A1, A1*, B1, and B1*

16). The following observations regarding the contribution of bolt spiling to support loading have been derived:

• If compared to the spile-reinforced models A1 and B1, the overall loading condition of the tunnel support has increased in models A1* and B1*. This is as expected, provided the larger displacements calculated for the non-reinforced models as shown in Fig. 21. At the same time, such loading increase is observed to be of higher magnitude in the A1* model than in the B1* model, which again is a direct consequence of the lower magnitude of displacements calculated in the latter.

• If compared to model A1, the larger load endorsement due to the absence of bolt spiling in model A1* is observed in the form of further loading of roof bolts and bending moments of the RRS support. As such, the calculated bolt loading in the roof of model A1* presents a generalized increase with center bolts loaded above 200 kN, and bolts towards the tunnel abutments reaching up to 404 kN. That represents an increase of up to 100% with respect to the results for the roof bolts in model A1. While the increase of shear forces of the shotcrete support in A1* is of a limited magnitude, a significant development of the bending moment load (of more than 100%) is endorsed in the roof part. The latter comparisons suggest that bolt spiling in model A1 obviously introduces a significant (and necessary) stabilizing effect that otherwise would result in further excesses of loads and failure in the support system. The above evaluation of the stabilizing contribution of bolt spiling should be also seen in the light that insufficient bolt length and bolt spacing were used at 3 + 162, as described in Sect. 8.5.

• In model B1*, it is observed a total vertical displacement of the center roof of 9 mm and a single block in the left tunnel abutment with a total displacement of 10.4 mm. In relation to the limited increase of displacements if compared to the B1 model, the calculated response in the tunnel support of B1* is primarily related to a slight increase of the shear forces acting at the intersection of the distinct blocks with the shotcrete support. On the other hand, the increase of bending moment load is rather limited, which responds to the limited increase in calculated displacements. Regarding bolt loading in model B1*, the general loading condition for the radial bolts around the excavation is that of a very limited, almost negligible, increase. In turn, the maximum bolt loading is reduced, apparently as a consequence of the shotcrete support taking up a larger share of the loading.

It should be finally noted that the study presented in this Appendix was performed with 2D numerical analyses in which the spiling-reinforcement is simulated as a homogenous ring of rock with improved mechanical properties at the roof as described in Sect. 7. As noted by authors like Volkmann and Schubert (2007) and Trinh et al. (2007), the 2D simplification of a true 3D mechanical problem poses some limitations when performing the simulations of the tunnel response. In the same line, Oke et al. (2014b) discuss the intrinsic limitations of the Hoek (2001) approach to simulate spiling as a homogeneous, equivalent ring of reinforced rock. The authors conclude that such a type of modelling cannot capture important mechanical components for the analysis of spiling behavior, such as the local arching between the structural elements or spiles, and the necessary transfer of stresses in the tunnel direction.

Although more advanced and separate 3D analyses would be likely needed for detailed analyses of the stress–strain response of single bolt spiles, the performed 2D study has provided an additional and valuable understanding of the reinforcement effect of spiling in hard and layered rock conditions. The latter is actually in line with another set of findings presented by Oke et al. (2014b) in which the authors remark on the ability of the 2D numerical approaches to capture the expected reduction of crown convergence when other support systems come into play. With this in mind, the presented results show that there is indeed a beneficial reinforcement effect of spiles when utilized in layered ground. In this study, such stabilization has been numerically measured in the form of a generalized reduction of the vertical roof deflection and the ground support loading.