Tunnel Linings

Abstract

In this chapter, important research results for the development of a robust and damage-tolerant multimaterial tunnel lining are presented. This includes the production, design and optimization of fiber-reinforced hybrid segmental lining systems based on numerical models and experimental investigations under tunneling loads. In addition, novel tail void grouting materials are developed and optimized regarding their infiltration and hardening behavior while taking the interaction with the surrounding ground into account. In order to expand the applicability of mechanized tunneling regarding soils characterized by significant swelling potential due to water uptake by clay minerals, a deformable segmental lining system is presented. The risk of damage due to high localized loads is reduced by the integration of additional radial protective layers on the lining segments and a compressible annular gap grout, which protect the tunnel structure by undergoing high deformations after reaching a certain yielding load. However, the deformability of such support systems affects the distribution of the stresses around the tunnel which governs the magnitude and buildup of the swelling pressure in the soil. Therefore, the development of damage tolerant lining systems requires a material and structural design which ensures an optimal soil-structure interaction through a synergy of computational and experimental techniques.

5.1 Introduction

Segmental linings form the load-bearing ring of mechanically excavated tunnels. Most commonly, they are divided into 5 to 10 segments per ring and joined together with direct concrete to concrete contact at the longitudinal and ring joints. Typically, segmental linings are prefabricated in serial production with hundreds or thousands of repetitions [41]. As reinforced concrete (RC) members they are made of higher-strength normal concretes and detailed with classic reinforcing steel. Regarding the design, the regions of load introduction at the longitudinal joints and at the jacks, the coupling joints for the shear load transfer from ring to ring [64] as well as the ring cross-section itself with compressive axial force and moderate eccentricities [29, 36] are essentially relevant. In addition, high geometric accuracy must be met in contact at the longitudinal and ring joints. Otherwise damage due to cracking and spalling can occur.

Analytical methods, beam-spring, and continuum models have become established for the analysis of internal forces [102, 75]. They usually consider only the final state of the finished tunnel and take into account the interaction between soil and structure in a conservative and simple way. This interaction however, plays a crucial role [13]. It is already activated by the annular gap grouting in the construction state of the tunnel and is decisive especially in case of one-sided loading–e.g., from buildings–and swelling soils. Since a stiff ring can hardly compensate for axial deformation, only compensation with lateral displacement into the soil bedding remains possible.

The research findings outlined in the remainder introduce a holistic optimization of the coupled tunnel system comprising the concrete segments, the annular gap grouting and the surrounding bedding soil [26, 63]. It is both experimental and numerical in nature. The segments are formed with hybrid reinforcement consisting of reinforcing steel bars and steel fibers [88], whereas the regions of load application subject to partial area loading are locally reinforced. At that, different concretes and precast elements are combined employing ‘‘fresh to fresh’’ or ‘‘fresh to solid’’ casting technologies. The steel fibers are aligned in a tailored way in direction of the tensile stresses exploiting the wall effect of temporary formworks during casting. As a result, load-bearing capacities, especially for biaxial load transfer at the longitudinal joints [67, 74], increase significantly. In the sense of an optimization-aided design [23, 73], this results in segments with uniform stress utilization across the ring. The segments get significantly more material-efficient and adapt to the variable loading and stress conditions along a tunnel. In addition, the robustness against spalling, unintended deformations or offsets, variations in load scenarios, or non-uniform ground conditions greatly increases.

The grout for the annular gap is developed as a single or a multi-component material – depending on the demands of the individual soil-structure interaction and the process of grouting [7]. Experimentally this is performed from small scale up to real scale level in a novel prototype testing facility [70].

For swelling soils, a deformation-tolerant segmental lining is developed. It consists of two components. The first one is a compressible annular gap mortar that deforms nearly plastic at defined pressures [89]. The second component is a hybrid segment of high-strength concrete with reduced thickness which is supplemented by a soft and deformable mortar layer at the outside. Thus, two layers protect the sensitive lining. The development proceeds in interaction and captures the locally concentrated soil deformations [9], which are derived from numerical simulation accounting for specific swelling scenarios activated by the tunnel construction [103]. Experimental implementations are treated up to full scale proving true bearing capacities of the segments as well as the deformation capacities of the soft layers [60].

5.2 Steel Fiber Reinforced Concrete (SFRC) for Tunnel Linings

Experiments have been conducted to examine the behavior of fiber pullout, considering fiber shape, dimensions and inclination angles, among others. Also, the following sections introduce the experiments along with the corresponding numerical models that were set up to adequately describe steel fiber reinforced concrete for tunnel linings.

5.2.1 Fiber Pullout

For the analysis, design and optimization of robust and damage-tolerant tunnel linings made of SFRC, it is essential to first understand the mechanical behavior of the material at a structural level. As is well known, the mechanical behavior of SFRC is related to the bridging effect across a crack opening, which is governed by the total contribution of all the fibers intercepting the crack. Since the bridging efficiency of each individual fiber depends largely on the fiber-matrix bond characteristics, the pullout behavior of single steel fibers embedded in a concrete matrix was systematically investigated by means of laboratory tests and analytical models. The results provide the basis for the numerical modeling of the mechanical processes in SFRC (Sect. 5.2.3), and on the other hand, they are the basis for further specific investigations on the use of SFRC in lining segments, such as investigations on the behavior under partial-area loading (Sect. 5.2.2).

5.2.1.1 Experiments on Fiber Pullout

The bond mechanisms of steel fibers in a concrete matrix were experimentally investigated by performing pullout tests on single steel fibers. The test program included the variation of relevant parameters influencing the pullout behavior, such as fiber shape, fiber dimensions, fiber tensile strength, inclination angle and concrete strength. A detailed description of the properties of the investigated fibers and the concrete mixes used as the pullout matrix is given in [11, 77, 79].

For the pullout tests, cylindrical specimens with dimensions of 60 mm $\times$ 60 mm (Fig. 5.1, left) were prepared. The fibers were fixed and embedded in the concrete as described in [11, 77, 79]. The test setup used for the pullout tests is shown in Fig. 5.1, right.

Fig. 5.1

Concrete specimen with embedded steel fiber [11, 79] (left) and pullout test setup [11, 77] (right)

The pullout behavior was analyzed focusing on the correlation between the pullout force and the fiber displacement. Moreover, the failure mode of fibers and concrete matrix were taken into account. The key findings of the experimental investigations on the pullout behavior are presented in the following. Further results and more detailed analyses can be found in [11, 63, 77, 79]

5.2.1.2 Fiber Shape and Dimensions

In Fig. 5.2, left, the influence of fiber shape (straight, crimped, hook-ended and twin-coned) on the pullout response is illustrated for fibers with embedded length of $L_{f}=20$ mm and inclination angle of $\theta=0$° in a high-strength concrete matrix ($f_{c}=84$ MPa). As shown in Fig. 5.2, left, the pullout responses differed significantly from each other depending on the fiber shape. In the case of straight fibers, the bond between fiber and matrix is only provided by interfacial friction. As a result, complete debonding occurred at an almost very low pullout load, which was accompanied by a sudden drop of the load. In contrast, due to the mechanical anchorage of deformed fibers, the pullout loads continued to increase after debonding. Thus, deformed fibers provided significantly higher resistance against pullout than straight fibers. As a consequence of the strong anchorage in the high-strength concrete matrix, both crimped and twin-coned fibers failed in all tests by rupture shortly after the ultimate load was exceeded. However, to achieve ductile material behavior, fiber rupture at small pullout displacements should be avoided. Beneficial pullout behavior was observed for the hook-ended fibers. For these fibers, the hook tended to be progressively mobilized and straightened, which caused a decrease of the pullout force and a further increase of the pullout displacement.

Fig. 5.2

Influence of fiber shape [11, 77] (left) and fiber dimensions [79] (right) on pullout response

Figure 5.2, right, shows the effects of fiber dimensions by comparing hook-ended fibers with $l/d=60/0.75$ mm/mm and $l/d=35/0.55$ mm/mm under otherwise identical testing conditions. As can be seen, the load-displacement curves of the two fibers were similar and almost parallel to each other. However, the fibers with larger dimensions exhibited a considerably higher ultimate pullout load ($+77.0\%$). With increasing diameter and hook size, respectively, the bending stiffness of the fiber and the contact area between fiber and matrix increases, which led to an increase in the energy required for plastic deformation of the hook. However, by comparing the two fibers in terms of the ratio between ultimate pullout load and fiber strength, it was shown that the efficiency of the fibers differed only marginally (60.1% and 61.0%) [11].

For concretes with the same fiber content, the number of shorter/thinner fibers is several times higher than that for longer/thicker fibers. As a result, for smaller/thinner fibers, the number of fibers intercepting potential cracks is also comparatively higher. Therefore, although the results showed a higher pullout resistance for longer/thicker fibers compared to smaller/thinner fibers, poorer load-bearing behavior must not be expected for the latter in general. Rather, a positive synergy effect can be concluded from the results. In fiber cocktails the advantages of fibers with different dimensions (longer/thicker and shorter/thinner fibers) can be used in combination. In particular, such fiber cocktails could have a positive effect in edge areas of linings segments to prevent cracking and spalling.

5.2.1.3 Fiber and Concrete Strength

The influence of fiber strength ($f_{t}=1225$ MPa, 2600 MPa) and concrete strength ($f_{c}=44$ MPa, 84 MPa) are depicted in Fig. 5.3. For both concrete strengths, the pullout resistance up to the frictional sliding stage was almost twice as high for the high-strength fibers as for the normal-strength fibers. As expected, fibres embedded in high-strength concrete exhibited a comparatively higher pullout load than those in normal-strength concrete. However, this effect was much more pronounced for high-strength fibers.

Fig. 5.3

Influence of fiber and concrete strength on pullout response [79]

While the load-displacement curves of high-strength fibers differed significantly for both concretes tested, normal-strength fibers showed similar load-displacement behavior regardless of the concrete strength. The curves of high-strength fibers in normal-strength concrete dropped relatively slowly after the ultimate pullout load was reached. This indicates that the mechanical anchorage of the hook was only utilized to a low level due to the low concrete strength. Hence, the fiber efficiency for high-strength fibers tested in normal-strength concrete was comparatively low (42.7%). However, it should be noted that these fibers had a remarkably higher efficiency in high-strength concrete (61.6%), as shown in [79]. Therefore, to achieve an optimal fiber efficiency, the properties of steel fibers in terms of tensile strength should be adjusted to the concrete strength.

5.2.1.4 Fiber Inclination Angle

Results of pullout tests on normal-strength and high-strength hooked-end fibers embedded with different inclinations angles (0°, 15°, 30°, 45° and 60°) in high-strength concrete are shown in Fig. 5.4. As expected, the high-strength fibers exhibited significantly higher ultimate pullout loads for all inclination angles considered. In the case of normal-strength fibers pulled out without rupture, the ultimate pullout loads were very similar independent from the inclination angle (Fig. 5.4, left). In contrast, high-strength fibers showed a remarkable increase in the ultimate pullout load at inclination angles between 30° and 45° (Fig. 5.4, right).

Fig. 5.4

Influence of fiber inclination angle on pullout response: normal-strength fiber (left) and high-strength fiber (right) embedded in high-strength concrete [11, 77, 79]

In general, the pullout displacement at ultimate load increased and the slope of the pre-peak branch decreased as the inclination angle rose, particularly for inclination angles greater than 30°. With the increase of the inclination angle, also an increase of matrix crushing and spalling at the exit point of the fiber was observed. This phenomenon was more pronounced in the case of high-strength fibers. Furthermore, fiber rupture occurred more frequently as the inclination angle increased, especially for normal-strength fibers [79].

5.2.2 SFRC Under Partial-Area Loading

As mentioned in the introduction, lining segments are subjected to high concentrated loads (i.e. partial area loadings) during both, the construction and final stage. The resulting splitting tensile stresses are usually decisive for the design and must be resisted by appropriate reinforcement. Among other advantages, the use of steel fiber reinforcement can increase the load-bearing capacity and positively affect the fracture behavior under such concentrated loading situations. In order to realize an optimized use of SFRC especially for lining segments, the bearing and fracture behavior of SFRC under partial-area loading was systematically investigated through experimental and numerical approaches.

5.2.2.1 Experiments on SFRC Under Partial-Area Loading

To characterize the bearing and fracture behavior of SFRC under partial-area loading, a comprehensive experimental program containing the variation of both non-fiber-related and fiber-related parameters has been conducted. The non-fiber-related variables included concrete strength, specimen dimensions, area ratio and eccentricity of load. Furthermore, taking into account the two relevant concentrated loading scenarios that occur in the ring and longitudinal joints of lining segments, partial-area loadings with spatial (i.e. point loading) and plane load distribution (i.e. strip loading) were considered (Fig. 5.5, left). The fiber-related factors included fiber properties (e.g. strength, dimensions, geometry and aspect ratio), fiber content and orientation as well as the combination of different fiber types (i.e. fiber cocktails).

The base mix designs of the investigated concretes corresponded to those typically used for lining segments. Detailed information regarding the mix designs and the various fiber reinforcements as well as the production and preparation of specimens can be found in [62, 78].

Figure 5.5, right, shows the test-setup as an example for partial-area loading with spatial load distribution on a specimen with dimensions of 150 mm $\times$ 150 mm $\times$ 300 mm. Further details on the test-setup are given in [62, 78].

Fig. 5.5

Schematic illustration of partial-area loading with spatial and plane load distribution (left) and test setup for partial-area loading with spatial load distribution [62, 78] (right)

The effects of the various parameters on the load-bearing and fracture behavior were studied by analyzing the ultimate local compressive stress (i.e. ultimate bearing capacity), the stress-displacement response as well as the failure and cracking characteristics. In the following, some of the main results and key findings are presented. A more detailed description of the experiments and the results obtained is published in [10, 55, 62, 78, 8, 80].

5.2.2.2 Area Ratio and Fiber Addition

In Fig. 5.6, left, the effects of area ratio ($\delta$ = $A_{c1}/A_{c0}=2.25$, 4, 9 and 16) and fiber addition (60 kg/m${}^{3}$ of hooked-end steel fibers) on the local compressive stress versus longitudinal displacement response are presented for high-strength concrete ($f_{c}=84.5$ MPa) under centric partial-area loading with spatial load distribution. Note that in the following, local compressive stress $\sigma_{\text{pal}}$ is defined as the applied load divided through the loaded area $A_{c0}$. As can be seen from the stress-displacement curves, the addition of fibers to the plain concrete (PC) led to a strong increase in the ultimate local compressive stress $\sigma_{\text{pal,ult}}$ (up to 47%). In Fig. 5.6, right, the ratios between ultimate local compressive stress and concrete strength ($\sigma_{\text{pal,ult}}/f_{c}$) are depicted for each area ratio. Comparing PC and SFRC, it can be seen that the difference of these values is larger at higher area ratios. Therefore, is obvious that the influence of the fibers on the load-bearing capacity is more pronounced at higher area ratios.

As expected, ultimate local compressive stress was also significantly enhanced with increasing area ratio for both PC and SFRC due to the confinement effect of the surrounding concrete. In the case of partial-area loading with spatial load distribution, a proportionality between the ultimate local compressive stress and the square root of the area ratio could clearly be ascertained [78]. Due to the incomplete confinement in the case of partial-area loading with plane load distribution, the increase in the ultimate local compressive stress was comparatively lower. In this case, a proportionality to the cubic root of the area ratio was found [62]. These correlations provided the basis for an analytical approach to calculate the bearable local stress of high-strength SFRC, which is presented in [55, 62].

Fig. 5.6

Influence of area ratio and fiber addition on the stress-displacement response (left) and on the stress ratio ($\sigma_{\text{pal,ult}}/f_{c}$) (right) for high-strength concrete under partial-area loading [78, 8]

Shortly after reaching the maximum stress, the stress-displacement curves of PCs were characterized by a sharp drop (Fig. 5.6, left), indicating a sudden failure of the specimens during the test. In contrast, SFRCs exhibited a gradual reduction in stresses with continuously increasing displacement, corresponding to a ductile fracture behavior. In the case of partial-area loading with plane load distribution, this effect was less pronounced due to the incomplete confinement [62].

Representative fracture patterns of high-strength PC and SFRC specimens are presented in Fig. 5.7, matching the respective stress-displacement curves shown in Fig. 5.6, left. In general, for both PC and SFRC, no visible cracking or spalling was observed until shortly before reaching the ultimate load. Soon afterwards, all PC specimens failed in a more or less explosive manner, in particular in the case of high-strength concrete, accompanied by the sharp drop of the load-displacement curves shown in Fig. 5.6, left. While for large area ratios single major splitting cracks developed and spread across the testing and lateral surfaces to the bottom (Fig. 5.7a), PC specimens tested with small area ratios completely collapsed (Fig. 5.7b). In contrast, all SFRC specimens retained their integrity until the end of testing and exhibited a multiple cracking pattern. At large area ratios, cracking in conjunction with minor concrete spalling occurred mainly on the upper half of SFRC specimens (Fig. 5.7c). With decreasing area ratio, the cracks and spalling tended to increasingly develop also in the lower half (Fig. 5.7d).

Fig. 5.7

Fracture patterns of high-strength PC (a, b) and SFRC (60 kg/m${}^{3}$ hooked-end macrofiber) (c, d) loaded with area-ratios 9 (a, c) and 2.25 (b, d) [78]

5.2.2.3 Fiber Type and Cocktails

As can already be assumed from the studies on the pullout behavior of steel fibers (Sect. 5.2.1), the properties of fibers also had a significant influence on the bearing and fracture behavior of SFRC under partial-area loading. Figure 5.8, left, shows the ultimate local compressive stress of high-strength SFRC produced with various types of steel fibers at a fiber content of 60 kg/m${}^{3}$ (spatial load distribution, $\delta=9$). Compared to PC, the increase in the ultimate local compressive stress was between 28% and 51%. As expected, the high-strength hook-ended macrofibers (Lh60) appeared to be most effective in enhancing the ultimate bearing capacity. However, it was found that SFRC with high-strength macrofibers had only an increment of 15% in its ultimate bearing capacity compared to SFRC produced with normal-strength macrofibers (L60), although the results of the pullout tests showed that the ultimate pullout load was almost twice as high for the high-strengths fibers (cf. Fig. 5.3).

For SFRC produced with fiber cocktails, a positive synergy effect could be observed for certain combinations of fiber types. For high-strength concrete, however, this effect was only clearly visible at relatively high fiber contents ($>$ 80–100 kg/m${}^{3}$) [78]. Investigations on different fiber cocktails in [55, 62, 78] showed that under such loading conditions the combination of hooked-end macrofibers and straight microfibers proved to be most advantageous. Even though the ultimate local compressive stresses of the two fiber cocktails compared in Fig. 5.8, right, were almost identical, it can be seen that SFRC with 50% macrofibers and 50% microfibers (SFRC L60+S60) was characterized by a much more ductile behavior in the post-cracking stage (compared to SFRC L40+M40+S40). A positive influence on the load-bearing and fracture behavior due to the reinforcement with fiber cocktails was particularly evident in the case of partial-area loading with eccentric load introduction [78].

Fig. 5.8

Influence of fiber type on ultimate local compressive stress of high-strength SFRC (left) and stress-displacement response of high-strength SFRC with different fiber cocktails under partial-area loading [78] (right)

5.2.2.4 Fiber Orientation

As is well known, fibers aligned to the acting direction of tensile stresses have the best crack-bridging capacity. In the case of partial-area loading, the resulting splitting tensile stresses perpendicular to the loading direction usually limit the bearable local compressive stress and dominate the failure mode [39, 78, 90]. Accordingly, it has to be expected that the fiber orientation has a significant influence on the bearing and fracture behavior of SFRC under partial-area loading. In order to study this influence, partial-area loading tests were performed on SFRC in which the fiber orientation was intentionally manipulated through varying the casting direction in the production of specimens in standing and lying formworks.

In Fig. 5.9, left, the fiber orientation (measured by means of an electromagnetic device described in [78]) with respect to the loading direction is depicted by percentage in the three spatial directions ($x$, $y$, $z$) for SFRC specimens (60 kg/m${}^{3}$ hook-ended macrofibers) cast in both standing and lying formworks (150 mm $\times$ 150 mm $\times$ 300 mm). For specimens produced in lying formworks, about 46% of the fibers were oriented towards the loading direction. Consequently, compared with specimens cast in standing formworks, fewer fibers were oriented in the two directions perpendicular to the loading direction, which correspond to the directions of principle tensile stresses (in the case of partial-area loading with spatial load distribution). As a results, the specimens produced in lying formworks exhibited considerably lower values of ultimate local compressive stress (stress drop up to 23%, Fig. 5.9, right). Furthermore, after reaching ulimate local compressive stress, the stress-displacement curves were characterized by a comparatively abrupt stress drop, indicating a less ductile fracture behavior. As can be seen in Fig. 5.9, right, this effect was clearly proven independent of the area ratio.

The effects of a preferred fiber orientation depending on the casting direction (standing/lying formwork) were even more significant for partial-area loading with plane load distribution, since in this case the principle tensile stresses act predominantly in only one transverse directions (perpendicular to the load direction) compared to partial-area loading with spatial load distribution [55, 62].

Fig. 5.9

Influence of the casting direction on the fiber orientation in SFRC specimens with respect to loading direction (left) and influence of the casting direction on the stress-displacement response of high-strength SFRC (right) [10, 78, 8]

5.2.3 Numerical Modeling of SFRC

An integral part for the design and optimization of damage-tolerant tunnel linings is the development of a virtual-lab to analyze fiber reinforced concrete structures. Through the interaction of experiment and simulation, the developed models are validated and together novel segment designs are developed. In order to analyze the post-cracking behavior of arbitrary combinations of fiber type, concrete strength and optional bar reinforcement, the numerical models have to capture the major physical behavior of SFRC and RC subjected to tunneling related loading.

5.2.3.1 Multi-Level SFRC Model

Steel fibers provide a residual strength after onset of cracking depending on the type, content and orientation of the fibers. Available guidelines (i.e. [44]) characterize the residual strength of SFRC based on bending tests and derive uniaxial stress-strain relationships for Ultimate Limit State (ULS) and Service Limit State (SLS) design. In the framework of the Collaborative Research Center SFB 837 Interaction modeling in mechanized tunneling, a multi-level SFRC model has been developed (see [100] for more details), which allows to directly assess the influence of the individual fiber type, content, and orientation on the structural response. In contrast to the guideline approaches, the explicit modeling of important SFRC design parameters allows for their consideration as material uncertainties in the model based design of SFRC structures.

As illustrated in Fig. 5.10, the proposed multi-level model consists of submodels related to three different scales involved in the numerical analyses of SFRC structures:

Fig. 5.10

Multi-level modeling of SFRC: Semi-analytical model for single fiber pullout considering various ‘‘key states’’ during elongation, plastification and concrete spalling of hooked-end fibers (Fiber Scale); Integration of the pullout response of all fibers crossing a representative crack and considering their orientation to calculate the traction-separation law (Crack Scale); Modeling of discrete cracks via interface elements inserted between the regular bulk elements and consideration of conventional reinforcement (Structural Scale)

• Fiber Scale: At the level of the individual fibers and the matrix, the pullout behavior of a single fiber is controlled by the interface conditions, the fiber shape and the fiber inclination with respect to a crack. A semi-analytical model predicting the pullout force-displacement relation of single fibers $F(w,\theta,\tilde{x})$, which depends on the position of the centroid $\tilde{x}$ and the inclination $\theta$ of the fiber with respect to the crack plane (Fig. 5.10, top center), has been developed in [99]. The model is capable of capturing the major mechanisms (straightening of the hooked-end, concrete spalling and fiber rupture) activated during the pullout of a single steel fiber embedded in a concrete matrix, accounting for different configurations of fiber type and strength, concrete strength, fiber inclination and embedment length (see Fig. 5.10, top).

• Crack Scale: At the level of an opening crack within the fiber-concrete composite, the fibers crossing the crack are activated and ensure a residual post-cracking strength depending on the fiber content and the fiber orientation. In the multi-level SFRC model, the post cracking response is approximated by a traction-separation law which is derived via the integration of the pullout force-displacement relations $F(w,\theta,\tilde{x})$ of all single fibers intercepting the crack and taking an anisotropic orientation of fibers into consideration [100].

• Structural Scale: At the structural level, the post-cracking behavior is captured by a discrete crack model based on cohesive interface elements [100, 50]. Between the regular finite elements (bulk elements), interface elements are inserted (Fig. 5.10, bottom), which allow a discrete mapping of cracks and provide direct information on the crack widths. The behavior of the interface elements is governed by the traction separation law derived on the crack scale with the multi-level SFRC model. In order to account for reinforcement bars if present, a mesh independent contact-based tying algorithm for the discrete representation of steel reinforcement was developed [24]. The rebars are modeled as truss elements and are coupled with the concrete matrix using a constraint condition between control points located on the rebar elements and their respective projection points within the solid elements in which they are embedded (see Fig. 5.10, bottom). The constraint condition includes the bond-slip mechanism as provided in [44] and the steel behavior itself is considered with an elastoplastic v. Mises yield surface with linear hardening.

5.2.3.2 Validation for Fiber Pullout

The laboratory tests published in [11, 77, 79] (see also Sect. 5.2.1) are also used for the validation of the semi-analytical fiber pullout model incorporated in the multi-level SFRC model (Sect. 5.2.3.1). Beside the aforementioned material parameters of the concrete and the fiber, the explicit fiber geometry (length $L_{f}=60$ mm, diameter $d_{f}=0.75$ mm and hooked-end geometry described by $l_{1}^{\text{h}}=1.8$ mm, $l_{2}^{\text{h}}=1.3$ mm, $\alpha^{\text{h}}=45$°, $\rho^{\text{h}}=1.4$ mm) and the fiber-concrete bond characteristics ($\tau_{\text{max}}=1.70$ MPa, $\tau_{0}=0.42$ MPa, $s_{\text{ref}}=0.38$ mm and the friction coefficient $\mu=0.3$) are used as model input. The model predictions for the different concrete and fiber strengths are compared to the experimental results in Fig. 5.11.

Fig. 5.11

Model validation for hooked-end fiber pullout without inclination (0°): Model results vs. experimental results for different combinations of concrete and fiber strength [101]

A remarkably good correlation is observed in all cases. In each diagram, the first nearly vertical ascending branch corresponds to the elastic state of the hooked-end. The further ascent of the curve leads to the pullout force peak which is reached when both of the arcs of the hook become fully plastic and the hooked-end starts to enter the straight part of the fiber channel. Afterwards the pullout force decreases, accompanied with the progressive straightening and sliding of the segments of the hooked-end. Finally, the whole hook is straightened and a residual pullout force remains corresponding to the frictional sliding in the fiber channel.

Fig. 5.12

Model validation for the pullout of hooked-end fibers embedded in high-strength concrete: Model results vs. experimental results of the load-displacement relation for different inclination angles [101]

In addition, the pullout tests on inclined fibers (Sect. 5.2.1.4) are used for the validation of the semi-analytical fiber pullout model (Sect. 5.2.3.1). All the model results and the respective test results are plotted in Fig. 5.12.

A good agreement between the model predictions and the experimental results can be observed. While the pullout response for fibers with an inclination of $\theta=15$° is similar to non-inclined fibers (see Fig. 5.11), the pullout characteristics change considerably when the inclination angle is increased from 30° to 60°. Particularly, in the case of 60°, instead of a steep ascending branch prior to the peak force, the pullout force continuously increases and reaches its maximum at a rather high level of pullout displacement due to matrix spalling.

5.2.3.3 Validation for Partial Area Loading

For the validation of the multi-level SFRC model, the partial area loading experiments with a plane load distribution and an area ratio of $\delta=A_{c1}/A_{c0}=4$ are used [62] (Sect. 5.2.2). All available experimentally obtained material parameters and the explicit fiber geometry (Fig. 5.13, left) are used as a direct model input. A detailed description of the model input can be found in [47]. With these parameters, first the single fiber pullout responses $F(w,\theta,\tilde{x})$ of arbitrary inclined fibers with homogeneous distributed embedment lengths (see Fig. 5.10, top) are calculated, and then the crack bridging behavior is predicted (see Fig. 5.10, middle). The derived traction separation laws are validated for mode I fracture by re-analyzing the experimental 4-point bending tests. A comparison between the experimentally and numerically obtained nominal stress-midspan-deflection curves are presented in Fig. 5.13, right.

Fig. 5.13

Measurement of Dramix 3D 80/60 hooked-end geometry (left); Strength characterization of SFRC with different fiber contents by 4-point bending tests based on the DAfStb guideline [82] (right) [47]

In general, a good agreement between the numerically obtained stress-deflection-curves and the experimental results is observed. The flexural strength, the post-cracking peak and the qualitative response is captured for all investigated fiber contents.

For the re-analysis of the strip loading tests, a plane stress model with an unstructured mesh and an element size between 4 mm up to 10 mm is used (see Fig. 5.14, left). To consider the influence of friction between the load plate and the specimen to the structural response, the load plate is explicitly modeled, and contact elements are inserted between the plate and the specimen. Due to the high uncertainty regarding the shear strength, different values of the model parameter $\beta$ controlling the shear strength are investigated in the re-analysis of the strip loading experiments. In Fig. 5.14, the numerically and experimentally obtained maximum load of the PC and the SFRC specimens are shown for different values of $\beta$ (blue).

Fig. 5.14

Numerical model used for the re-analysis of the strip loading experiments with the distribution of the principal tensile stresses in the elastic regime (left); Comparison of the experimental and predicted peak load with different values for the shear strength parameter $\beta$ (right) [47]

It can be observed, that the load bearing capacity for the plain concrete specimen (i.e. for zero fiber content) is mainly influenced by the tensile strength, whereas for the fiber reinforced specimens, the shear strength governs the failure load. Furthermore, depending on the fiber content, an increased shear strength ($\beta$ values between 5.0 and 6.0) is required to capture the experimentally obtained failure loads. Employing the empirical formulas for estimating the shear strength [47, 76] a reasonably good agreement between experimental results and numerical predictions can be observed (see red line in Fig. 5.14).

5.2.3.4 Casting Simulation of SFRC Structural Members

A major source of uncertainty involved in the mechanical properties of SFRC is the orientation and distribution of the fibers in the concrete matrix, which is influenced by the casting process and the geometry of the mold. If scale bridging material models are employed for the simulation of SFRC structures (see Sect. 5.2.3.1), assumptions regarding the orientation and distribution of fibers are required to predict the crack bridging behavior of the SFRC in the hardened state.

In order to simulate the casting process of SFRC and taking the formwork geometry into account, a numerical model based on Smooth Particle Hydromechanics (SPH) was developed and implemented in the open-source Code DualSPHysics [16]. The fiber orientation during the casting process is described by the Folgar-Tucker model [1, 22] in combination with the weakly compressible SPH method and allows the determination of the spatially as well as temporally varying probability density function of the fiber orientation. The flowable cement matrix is modeled as a viscous, regularized Bingham-Papanastasiou fluid, where the influence of the fiber content on the flow behavior is taken into account by the micromechanical model of Ghanbari and Karihaloo [30]. Finally, the orientation tensors can be easily visualized by ellipsoidal surfaces, where the eigenvalues denote radii, and the eigenvectors the axes orientations. The resulting fiber orientation after casting of a tunnel lining segment is shown in Figure 5.15. A detailed description and validation of the model can be found in [32].

Fig. 5.15

SPH flow model to predict the fiber orientation resulting from the casting of a tunnel lining segment (left). Fiber orientation visualized by ellipsoids, where the orientation component $a_{11}$ shows the probability of fibers aligned towards the horizontal direction (right) [32]

5.3 Hybrid Segmental Tunnel Linings

The peripheral zones of lining segments (i.e. corners, edges, joints) are usually exposed to significantly higher forces than the interior regions due to various complex loading scenarios (e.g. concentrated load introduction, bending moments and shear force transmission) and are also particularly vulnerable to damages, such as concrete spalling. The risk of such damages is particularly high when lining segments are conventionally reinforced with steel bars, since such type of reinforcement cannot cover the edge areas due to the requirement of minimum concrete cover. A potential improvement of this problem could be achieved by the use of SFRC. Due to the small dimensions of steel fibers and their homogeneous distribution in the concrete, lining segments made of SFRC are more robust than conventionally reinforced lining segments, especially with regard to corner and edge spalling. However, the steel fiber content of conventional SFRC for lining segments usually ranges between 30 and 50 kg/m${}^{3}$. The reinforcement efficiency of SFRC with such low (subcritical) fiber content is limited, so that the reinforcement exclusively with steel fibers is not expedient in the case of high loadings. SFRC with a supercritical fiber content (approximately > 100 kg/m${}^{3}$), which is necessary for an adequate reinforcement efficiency as well as for achieving the intended crack distribution and crack width limitation, is no reasonable alternative for the use in lining segments, especially from an economic point of view.

A robust, damage-tolerant as well as economical efficient solution is provided by the concept of hybrid lining segments. The basic idea of the hybrid concept is to use high-performance steel fiber reinforced concretes (HPSFRC) with a high fiber content (> 100 kg/m${}^{3}$) – even in combination with conventional rebars – only locally in the highly stressed and vulnerable zones of the segment, while in less stressed interior regions a conventional lining concrete (LC) with reduced steel bar reinforcement or a moderate steel fiber content is used, respectively. In addition, the segment cross-section can be significantly reduced by an optimized hybrid design (Sect. 5.3.5). This leads to a smaller excavation cross-section and lower material consumption, resulting in both economic and ecological benefits.

The development of the hybrid concept was based on a multi-level approach, which included the tunnel lining system, the individual segments and details in terms of material and contact areas. For an optimization-based design, the developed numerical models and methods for the modeling of hybrid SFRC components at the structural level were used. On the one hand, the focus was on the design, taking into account aspects of robustness and an experimental validation of the methodology, and on the other hand, the development of adequate production methods and suitable joint and material concepts.

5.3.1 Production Methods

For the production of hybrid lining segments two main concepts were pursued, resulting in different production methods and casting techniques, respectively [58, 61]. The first concept (‘‘wet-on-solid’’) aims to use geometrically tailored built-in components made of HPSFRC, which are pre-fabricated in a separate production process. The hardened precast components are installed at the critical zones (i.e. corners, edges, joints) in the conventional segment formwork, followed by the placing of a wet mixed conventional lining concrete. In this way, the precast components are integrated into the lining segment ‘‘wet-on-solid’’ (Fig. 5.16, left). In the case of precast components with additional rebars, appropriate connection reinforcement must be provided. Apart from this, only minor modifications compared to the conventional production process are necessary. However, to ensure a sufficient bond between the precast components and the subsequently added lining concrete, appropriate measures are required (cf. Sect. 5.3.2.1).

Fig. 5.16

Schematic illustration of production methods for hybrid lining segments: ‘‘wet-on-solid’’ (left), ‘‘wet-on-wet’’ (right)

The second concept (‘‘wet-on-wet’’) is characterized by a simultaneous casting of the two concretes (HPSFRC/LC) in a standing segment formwork (Fig. 5.16, right). The concretes are initially separated from each other by a temporary interior formwork (i.e. moveable partition plates). During compaction, this interior formwork is continuously lifted up so that the two concretes get into contact ‘‘wet-on-wet’’, without getting mixed. This production method was first successfully tested on small-scale. It was shown that in a standing formwork, the HPSFRC can be precisely installed in horizontal but also in vertical thin layers together with an adjacent lining concrete (Fig. 5.17, left). Samples taken from such hybrid specimens consistently showed a monolithic bond between the two concretes.

In order to investigate this production method on the real scale, a standing segment formwork was used, which was designed and manufactured exclusively for this purpose by Herrenknecht Formwork. This formwork allows to cast rectangular segments with an external ring diameter of 7.80 m, a thickness of 0.40 m and a width/height (length of segment in longitudinal direction of the tunnel) of 1.00 m. The production of hybrid segments in this formwork (Fig. 5.17, right) has demonstrated that the ‘‘wet-on-wet’’ production method can be successfully realized on real scale.

For the ‘‘wet-on-wet’’ production of hybrid lining segments with additional rebars in the zones close to the longitudinal joints, alternative systems were also investigated for the use as an interior formwork, instead of monolithic steel plates. On the one hand, slotted steel plates were used, which could also be pulled during compaction, and on the other hand, steel mesh was used, which remained permanently in the segment. The latter solution, however, led to a significant weakening of the bond between the two concretes and is therefore not appropriate.

Moreover, it should be mentioned that in the case of the ‘‘wet-in-wet’’ production method additional production steps are required compared to the production in a conventional segment formwork. For example, in order to comply with the dimensional tolerances for lining segments and to install the circumferential sealing properly, mechanical finishing of the longitudinal joint is necessary, which is the unmolded surface in the standing segment formwork.

Fig. 5.17

Standing wooden formwork for the ‘‘wet-on-wet’’ production of hybrid specimens (top left), cutting surface through a hybrid specimen (bottom left) and standing segment formwork for the ‘‘wet-on-wet’’ production of hybrind lining segments in real scale (right)

5.3.2 Experimental Investigations on Hybrid SFRC Systems

As the following sections demonstrate, investigations on the bonding behavior between precast components and the added lining show that a so-called hybrid SFRC system can substantially improve the overall system behavior, in particular when considering partial-area loading.

5.3.2.1 Investigations on the Bonding Behavior (‘‘wet-on-solid’’)

The weak point of the ‘‘wet-on-solid’’ concept is the bond between the precast components and the subsequently added lining concrete. In order to ensure a durable tensile and shear resistant bond between the two concretes, the bonding surfaces of the precast components must have a certain macro and micro roughness. If necessary, also an additional mechanical connection with appropriate connection reinforcement may even be required. To investigate the pure bond behavior (i.e. without reinforcement) between the precast components and the lining concrete, experimental tests were carried out [59, 61]. To this end, plate-like precast components (50 cm $\times$ 30 cm $\times$ 5 cm) made of HPSFRC ($f_{c}=120$ MPa, 120 kg/m${}^{3}$ steel fibers) were first produced. By means of various measures [59, 61] (e.g. use of formliner with a specific geometrical structure, surface ‘‘washing’’ with high-pressure water jets), the roughness of the bonding surface of the precast components was varied in the range from tenths of a millimeter (micro roughness) to several centimeters (meso roughness). The hardened precast components were then positioned and fixed in a formwork (50 cm $\times$ 30 cm $\times$ 25 cm), followed by the placing of a wet mixed lining concrete ($f_{c}=70$ MPa, 50 kg/m${}^{3}$ steel fibers). The so produced hybrid specimens represent sub-segments of a hybrid lining segment. Drilling cores were taken from these hybrid specimens to investigate the bonding behavior under shear and tensile loading (Fig. 5.18, left, middle).

Fig. 5.18

Test setup for investigations on the bonding behavior under shear (left) and tensile loading (middle) [61]; structure of the bonding surface of precast components (right)

In the case of a bonding zone with a pyramid or frustum of pyramid shaped structure in combination with surface roughening trough ‘‘washing’’ with high-pressure water jets (Fig. 5.18, right), the ultimate bearable shear stress was enhanced by 40% to 50% compared to a flat bonding zone (Fig. 5.19, left). Specimens with a flat bonding zone showed an abrupt and brittle failure immediately after reaching the ultimate shear stress. The failure occurred trough a complete shear-off of the HPSFC layer along the flat bonding zone (Fig. 5.18, top right). In contrast, specimens with a specific structure of the bonding zone (pyramid, frustum of pyramid) clearly showed a more robust fracture behavior [59, 61]. In these cases, a significant load increase was observed after first cracks appeared. After reaching the ultimate shear stress, the stress-displacement curves were characterized by a sharp drop. However, a residual load-capacity was observed (Fig. 5.18, left) and no complete shear-off failure has occurred until the end of testing (Fig. 5.18, top right). Under tensile loading the fracture always occurred in the lining concrete (Fig. 5.18, bottom right), irrespective of the structure of the bonding zone [59, 61]. Therefore, a sufficient bond under tensile loading could be proven.

Fig. 5.19

Influence of the structure of the bonding zone on the stress-displacement response under shear loading (left); characteristic failure patterns of hybrid specimens under shear (top right) and tensile loading (bottom right) [61]

5.3.2.2 Hybrid SFRC Systems Under Partial-Area Loading

As the investigations on pure SFRC under partial-area loading (Sect. 5.2.2) clearly demonstrate, the use of steel fiber reinforcement can significantly increase the bearable local compressive stress and positively influence the fracture behavior. However, with regard to the stress distribution prevailing in concrete elements subjected to partial-area loading, it is known that critical tensile stresses develop predominantly in the load propagation zone (i.e. St. Venant disturbance zone) and vary as a function of the distance to the loaded area [39, 78, 90]. Therefore, following the basic idea of the hybrid concept, it is not necessarily appropriate to reinforce the entire concrete element with steel fibers, rather only those zones where (critical) tensile stresses are expected. To experimentally verify the hybrid concept on the material level, partial-area loading tests were performed analogue to Sect. 5.2.2, however, on hybrid specimens. Instead of a full range fiber reinforcement, the hybrid specimens were partially strengthened with steel fibers just in the St. Venant disturbance zone. To this end, the hybrid specimens were produced with both PC and SFRC using the ‘‘wet-on-wet’’ production method. In [56, 57, 58, 78, 81] the experimental investigations on hybrid SFRC elements under partail-area loading are presented in detail.

In Fig. 5.20, left, the stress-displacement response of PC specimens is compared to those of SFRC and hybrid specimens with L60 fiber reinforcement (60 kg/m${}^{3}$ hooked-end macrofibers) tested under partial-area loading (spatial load distribution, $\delta=9$). For hybrid specimens, the thickness of the fiber reinforcement layer was 50, 100 and 150 mm (total specimen height: 300 mm). Figure 5.20, right, shows the corresponding ultimate local compressive stresses. In addition, Fig. 5.20, right, also shows the ultimate compressive stresses for SFRC and hybrid specimens produced with L60+S60 fiber reinforcement (120 kg/m${}^{3}$ fiber cocktail).

Fig. 5.20

Influence of the reinforcement layer thickness on the stress-displacement response (left) and on the ultimate local compressive stresses (right) of hybrid specimens under partial-area loading [78]

For all hybrid specimens, a considerable increase in the ultimate bearing capacity was observed with increasing thickness of fiber reinforcement compared to PC specimen, especially for those reinforced with fiber cocktails (up to 41% for L60 and 76% for L60+S60, Fig. 5.20, right). In the case of fiber cocktail reinforcement, by incorporating an only 50 mm thick reinforcement layer (1/6 of the total specimen height), an increase of 30% in ultimate bearing capacity was recorded. With increasing thickness of fiber reinforcement, hybrid specimens exhibited not only an increase in the ultimate bearing capacity, but also showed a considerable improvement in their ductility (Fig. 5.20, left). However, hybrid specimens with a 150 mm thick layer of fiber reinforcement (1/2 of the specimen height) exhibited, besides an almost identical bearing capacity, also nearly the same ductile behavior as SFRC specimens. This clearly shows, that a full range fiber reinforcement is not necessary.

In the case of partial-area loading with plane load distribution (Fig. 5.21, left), it was also found that the installation of a relatively thin layer with fiber cocktail reinforcement (120 kg/m${}^{3}$ L60+S60) in the load propagation zone already leads to a significant improvement in the ultimate bearing capacity (Fig. 5.21, right). Analogous to specimens with a full range fiber reinforcement, the casting direction (standing/lying formwork) had a remarkable influence on the ultimate bearing capacity, in both plane and spatial cases of partial-area loading. In the plane case, hybrid specimens produced in standing formworks exhibited an up to 30% higher bearing capacity compared to hybrid specimens produced in lying formworks, depending on the thickness of the reinforcement layer (Fig. 5.21, right). Thus, it could be shown that the production in a standing formwork, as in the case of the ‘‘wet-on-wet’’ production method for hybrid lining segments, can have a positive effect with regard to the fiber orientation.

Fig. 5.21

Dimensions of PC and hybrid specimens tested under partial-area loading with plane load distribution (left); influence of the reinforcement layer thickness and casting direction on the ultimate local compressive stress of hybrid specimens (right) [58]

Although ductile fracture behavior was less pronounced for hybrid specimens under partial-area loading with plane load distribution due to the incomplete confinement, the high content of steel fibers in the reinforcement layer nevertheless caused a considerable slower crack propagation and prevented complete splitting, as in the case of unreinforced specimens (Fig. 5.22). Moreover, concrete spalling was prevented by the fibers and was only present in the unreinforced areas of the specimens. Above a certain reinforcement layer thickness (here: 20 cm), the fracture behavior could even be changed to the effect that no cracking or spalling occurred in the peripheral zones close to the loaded area.

Fig. 5.22

Fracture of PC (left) and hybrid specimens (middle, right) under partial-area loading with plane load distribution [58]

5.3.3 Reinforcement Layout by Topology Optimization

Topology optimization gives the opportunity to distribute a limited amount of material in a geometrical domain $\Omega$, so that the structural stiffness $K$ is maximized or the weight is minimized [71]. From a mathematical point of view, this is a minimization problem steered by the principles of elasticity. The design space is discretized with finite elements. Iteration and variation of the material distribution therein yields the optimal relation between the load $F$ and the displacements $u$ [34]. Among the three established methods of structural optimization – sizing, shape and topology optimization – topology optimization is the most suitable one for this purpose. In the past, it has been shown a valuable design tool especially for structures with complex stress states. In certain conditions employing distinct parameters, it automatically gives distributions of materials in structural elements comparable to strut-and-ties models (STM) [23].

Topology optimization is commonly applied to continuous or truss structures. Then, the objective is to find the optimal distribution of material in terms of densities $\rho$ or cross-section areas $A$ among the elements $e$ that constitute the design space. In this context, reinforced concrete poses a special challenge as a composite of two materials–concrete and steel reinforcement–whose behavior is strongly nonlinear. In the past, especially the hybrid truss-continuum optimization (HTCO, e.g. [27, 4, 74, 92]) has provided good solutions for reinforced concrete. Different to the classical approach, the design space (geometrical domain) is filled with continuous volume ($\Omega_{c}$) and truss elements ($\Omega_{t}$) simultaneously. Compressive ($\sigma_{c}$) and tensile ($\sigma_{t}$) stresses result from the applied loads $F$ and steer the use of both materials (Fig. 5.23). That way, it helps designers to identify suitable STMs for complex cases. Thereby, HTCO provides an objective – non subjectively biased – computational solution that suggests rebars in direction of the principal tensile stresses. Just the task to choose suitable diameters remains to the designer. The constrained optimization problem is formulated in terms of the vector of densities $\boldsymbol{\rho}$ and cross-section areas $\mathbf{A}$ according to [27, 74]:

$$\begin{aligned}\min_{\boldsymbol{\rho},\mathbf{A}}\quad & c=\mathbf{F}^{\top}\mathbf{u}\end{aligned}$$

(5.1a)

$$\begin{aligned}\text{such that}\quad & \mathbf{K}(\boldsymbol{\rho},\mathbf{A},\boldsymbol{\sigma}_{\mathbf{t}},\boldsymbol{\sigma}_{\mathbf{c}})\mathbf{u}=\mathbf{F},\end{aligned}$$

(5.1b)

$$\begin{aligned} & \sum_{e_{c}\in\leavevmode\nobreak\ \Omega_{c}}\rho_{e}V_{e}+\sum_{e_{s}\in\leavevmode\nobreak\ \Omega_{s}}A_{e}L_{e}\leq\beta\leavevmode\nobreak\ V_{0},\end{aligned}$$

(5.1c)

$$\begin{aligned} & \rho_{\text{min}}\leq\rho_{e}\leq 1,\leavevmode\nobreak\ \forall\rho_{e}\in\Omega_{c}\,,\end{aligned}$$

(5.1d)

$$\begin{aligned} & A_{\text{min}}\leq A_{e}\leq A_{\text{max}},\leavevmode\nobreak\ \forall A_{e}\in\Omega_{t}\,.\end{aligned}$$

(5.1e)

Therein the total compliance $c$ is computed from the sum of the continuum and truss contributions. The total volume is restricted to a fraction $\beta$ of the initial volume $V_{0}$. Further constrains limit the design variables. Usually $\boldsymbol{\rho}$ takes values between 0 and 1 to represent void or full material at each element. Minima for $A_{e}$ and $\rho_{e}$ avoid numerical singularities when solving the central FEM equation (Eq. 5.1b) to obtain $\mathbf{u}$, whereas $A_{max}$ keeps the design feasible, limiting the diameter of rebars to those available on the market.

Fig. 5.23

Left: Mesh and material definition in hybrid truss-continuum optimization. Right: stress-strain relationship employed for concrete and steel. Modified with reference to [74]

This scheme was applied to an idealized longitudinal joint of a segmental lining of width $d$ and depth $2d$ – in order to fully capture the discontinuity region – subjected to single and multiple loads ($q_{e-}$, $q_{e0}$ and $q_{e+}$) in Fig. 5.24 [74]. In practice, the ring loads can be introduced at the longitudinal joint not only centered, but also eccentrically. As expected, tensile stresses below the load introduction appear and the algorithm places suitable ties. Directly below load introduction, up to a depth of $0.42d$, a region of small trusses is identified. This motivates the replacement of rebars by steel fibers in this region to cover the smeared tensile forces and simplify the design. Conventional reinforcement still takes over the larger tensile stresses below that region (Fig. 5.24, right). Such a design bears potential to save material and manufacturing time as discussed in Sect. 5.3.2.2.

Fig. 5.24

Left: Hybrid optimization of an isolated longitudinal joint subjected to multiple (centric and eccentric) loads. Rebars represented as black truss elements. Right: Design simplification after replacement of smeared trusses by steel fibers. Modified with reference to [74]

These results motivated the extension of the experimental campaign to hybrid SFRC systems. In the experiments not only SFRC was employed for the strengthening of the longitudinal joint, but also conventional rebars. Below load introduction, a layer of HPSFRC with a depth of $0.42d$ was cast wet-on-wet [58, 74] to the normal strength concrete of the specimen. To quantify the performance of the strengthened joints, geometrically equivalent specimens with and without conventional reinforcement were cast for testing. Also the impact of relocating the gasket to the center was investigated. In this configuration, even higher bending moments are bearable by the same thickness and the splitting forces are reduced. In total, this leads to less reinforcement [74].

The bearing capacities reached by all alternative configurations tested are summarized in Fig. 5.25. Longitudinal joints with hybrid reinforcements have up to 18% more bearing capacity than those reinforced with rebars only. The relocation of the gasket to the center provides an increase of 37% with respect to the classical location at the edge and bar reinforcement, whereas for hybrid reinforcement, an increase of 40% was registered [74].

Fig. 5.25

Bearing capacity of the tested specimens. Modified with reference to [74]

New reinforcement concepts derived from topology optimization proved to be more efficient in terms of bearing capacity of longitudinal joints. Strengthening the region below load introduction, and the relocation of the gasket to the center of the segment, leads to a considerable increase of the bearing capacity of longitudinal joints. That way, longitudinal joints are no longer decisive for the design and segments get a more homogeneous material utilization throughout [74].

5.3.4 Numerical Investigations of RC, SFRC and Hybrid Segmental Lining Designs

In order to investigate the performance of hybrid reinforced segmental linings on the tunnel scale and compare it to conventional RC segment designs, the structural responses of different reinforced segmental linings is analyzed using a detailed 2D model consisting of the grout layer and the individual tunnel lining segments, respectively (Fig. 5.26, left). For this purpose, a reference tunnel project is defined (overburden 20 m, sandy soil, lateral earth pressure coefficient of 0.5), where the loading acts on the lining (at steady state) is determined by the process-oriented tunnel simulation as described in Sect. 6.4.1.

The focus of the analyses is on the durability, quantified by means of the maximum crack opening exceeding the tolerated width of 0.2 mm. A RC segment with minimum reinforcement ($\varnothing$ 10–10 $+$ additional rebars in high stressed regions), a SFRC segment with 60 kg/m${}^{3}$ of 60 mm normal-strength hooked-end fibers (see Sect. 5.2) and a hybrid reinforced segment are analyzed (125 mm SFRC band and 200 mm long SFRC cap). The total steel mass of the SFRC and RC segment is similar ($\approx$260 kg/segment), whereas the steel amount in the hybrid segment is reduced by 65% (92 kg/segment). The simulations are carried out using a C40/50 concrete with a design tensile strength of 2.5 N/mm${}^{2}$. For the designs containing SFRC an isotropic fiber orientation is assumed. A comparison of fiber and conventionally reinforced segments and different safety concepts in non-linear FE analysis can be found in [45].

Fig. 5.26

Left: Dimension of the ring and the longitudinal joint contact and maximum principal stresses in the segmental lining model and indication of potential failure mechanisms.; Right: Characteristic normal force and bending moment distribution around the ring from a linear elastic analysis

The resulting maximum principal stresses and stress resultants are shown in Fig. 5.26. Under the characteristic loading, no large cracks occur in the plain concrete segmental lining. However, two potential failure modes are indicated: At the crown segment, the lowest normal force ($-1375$ kN/m) in conjunction with the highest bending moment (329 kNm/m) can be observed, potentially resulting in bending cracks. Around the longitudinal joint at 78°, the largest tensile stresses occur due to a normal force of $-1943$ kN/m and a bending moment of $-217$ kNm/m. This could lead to a potential chipping of the segment corner (Fig. 5.26, left). Based on these observations, two loading cases are created for design purposes. For the design of the crown segment subjected to bending, the loads are modified such that the resulting bending moment is increased by a factor of 1.5 and the normal force is reduced by a factor of 0.5. For the design of the area around the longitudinal joint, where chipping could occur, the normal force and bending moment are increased by a factor of 1.5 to increase the stress transmission and maintain the joint rotation.

5.3.4.1 Load Case: Bending

The responses of all lining designs, or rather, reinforcement schemes, to the bending load case are shown in Fig. 5.27. Only the bending response at the crown segment is shown, as this is the area in which maximum cracking is observed. Further, only the crack width is shown. The lining system does not fail and continues to function as a load bearing structure. Therefore, the only limiting factor controlling the choice of any of these scenarios is the predicted crack width, which may not exceed the project dependent requirements for the SLS (around 0.2 mm).

Fig. 5.27

Crack patterns at the crown segment for a plain concrete (PC), a conventional reinforced (RC), a steel fiber reinforced (SFRC) and a hybrid reinforced (Hybrid) segment design (magnification of 300$\times$) [25]

As expected the plain concrete lining ring (PC), exhibits the largest bending cracks, with the dominant crack occurring at the inner edge of the tunnel lining at the crown, which is where the maximum bending moment is predicted. What is of note, is that the PC case exhibits multiple cracking behavior (due to the compressive normal force), rather than a single monolithic crack which leads to segment failure. The RC segment exhibits a similar response to that of the PC segment, with the primary difference being that the crack pattern is much more regular. Rather than predicting a crack size that more or less corresponds to the moment applied at a point, as is the case in the PC segment, the cracks that are observed in the RC segment are more or less all of similar size, indicating that the reinforcement bar effectively normalizes the cracking response along the length of the rebar. The maximum cack width predicted by the RC segment is 0.158 mm, which is within a tolerable range of below 0.2 mm as prescribed by the serviceability limits. Interestingly, the SFRC segments are the most effective in reducing crack width along the length of the segment. Both reinforcement schemes lead to cracks that are significantly below the serviceability limit and therefore represent valid reinforcement designs. In the case of the hybrid reinforcement scheme, the bending performance of a full SFRC segment is only slightly better (0.112 mm vs. 0.142 mm). This equals an increase in crack width of approximately 25% relative to a total decrease of SFRC of 75%.

5.3.4.2 Load Case: Chipping at the Longitudinal Joint

The responses of all lining designs, or rather, reinforcement schemes, subjected to the chipping load case are shown in Fig. 5.28. The maximum chipping cracks occur at the longitudinal joint located 78° away from the tunnel crown, as measured clockwise from the top of the ring.

Fig. 5.28

Crack patterns at the longitudinal joint located at 78° for a plain concrete (PC), a conventional reinforced (RC), a steel fiber reinforced (SFRC) and a hybrid reinforced (Hybrid) segment design (magnification of 100x) [25]

Generally, the trend in predicted maximum crack width is similar to that as of the bending variants as shown in Fig. 5.27 and discussed above. However, in the case of chipping, the SFRC proves to be significantly more effective in reducing crack widths. Significantly, the minimum reinforcement provided against splitting stresses in the RC design is not sufficient to inhibit the formation of cracks in exceedance of the tolerance. This indicates that SFRC is much better suited to resist chipping cracks than RC. While the performance of the hybrid reinforcement scheme is better than the conventional RC design and below the serviceability limit, it’s effectiveness can be increased by using a higher SFRC cap length. The splitting crack is initiated outside the SFRC layer and therefore the SFRC cap can only limit further crack development. In order to estimate an optimal SFRC cap length, optimization algorithms can be applied (see Sect. 5.3.5).

5.3.5 Optimization Based Design for Hybrid SFRC Linings

A robust segment design in tunneling projects, going along with the reduction of the risk of damage, also needs to satisfy the economic feasibility. As shown in the previous sections, SFRC segments provide a sufficient load bearing capacity and an increased serviceability performance compared to conventional RC segments. If a proper hybrid reinforcement is chosen, a similar performance to a SFRC reinforcement scheme can be achieved while the steel content is further reduced. In this section, hybrid SFRC segment designs are derived by using optimization algorithms in conjunction with complex numerical models and by taking material uncertainties into account [68]. Due to the high computational effort of solving optimization tasks under uncertainty, the structural models are replaced by Artificial Neural Networks (for more details i.e. [47]).

5.3.5.1 Robust Optimization Earth Pressure Loadings

In Sect. 5.3.2.2 as well as in Sect. 5.3.4 it was shown that hybrid segments are capable to prevent critical cracking around the longitudinal joint and to simultaneously reduce the amount of steel. Here, a robust optimization [6, 98], with the design parameters summarized in Fig. 5.29 (top) and using the numerical model described in Sect. 5.3.4, is performed to determine a robust hybrid segment design with a minimal SFRC material usage (see also [43, 46]). In addition, conventional optimization is performed to minimize the SFRC cap width and the required fiber content. Besides the savings on material costs, the minimization of the fiber amount reduces the likelihood of fiber clumping and therefore contributes to a more consistent performance. A maximum tolerable crack width of 0.2 mm is employed as constraint. In case of the robust optimization, also the variance of the maximum crack width due to uncertain material properties is included as an objective function. The tensile strength of the C40/50 concrete is modeled as Gaussian distribution ($\mu=3.5$ N/mm${}^{2}$, $\sigma=1.0$ N/mm${}^{2}$) and also the fiber orientation is considered as interval representing favorable to unfavorable orientations. The optimization task is solved by the particle swarm optimization [37].

Fig. 5.29

Optimized design vs. robust optimization of longitudinal joints. Specification of the optimization tasks and resulting width of the SFRC cap and fiber content (top). Comparison between an optimized and robust optimized segment design in terms of installation as well as material variance (bottom) [43]

The optimized design leads to a 280 mm long SFRC cap with 25 kg/m${}^{3}$ of fibers, while the robust optimized design leads to a 420 mm SFRC cap with 45 kg/m${}^{3}$ fibers. In order to evaluate the vulnerability of both designs, their performance in case of an improper segment installation, resulting in a misalignment $\Updelta$r between adjoining segments (Fig. 5.29, bottom left), and their reliability regarding variances in the concrete strength as well as in the fiber orientation (Fig. 5.29, bottom right) are investigated. In terms of installation variances, both designs are damage tolerant with respect to small misalignment, but only the robust optimized design prevents the development of critical cracks at larger misalignment. In addition, the robust optimized design provides a consistent performance in the full range of possible material properties. Considering the material variance, unfavorable realizations of concrete strength and fiber orientation, results in a probability of 27% that the maximum tolerable crack width of 0.2 mm is exceeded when using the optimized design.

5.3.5.2 Reliability Based Optimization of Thrust Jack Forces

Beside the earth as well as the water pressure loads acting on the tunnel structure, the thrust jack forces during TBM advancement are often decisive for the chosen segment thickness. Therefore, the same reference tunnel project as described in Sect. 5.3.4 is used as an example, where the maximum thrust jack force of the tunnel boring machine $F_{\text{design}}$ is specified to 5600 kN. Here, a hybrid fiber reinforced segment consisting of two SFRC layers (A and B) is designed, whose required thickness $d$ and the fiber content $c_{F}$ as well as the width $\Updelta L$ of the strengthening SFRC layer B (see Fig. 5.30) is determined by solving an optimization task under consideration of uncertainties (for more details see [47]). Through the consideration of material parameter uncertainties (fiber orientation $\lambda_{F}$ as interval, concrete tensile strength $f_{t}$ as log-normal distribution) as well as construction tolerances resulting in an eccentric placement of the thrust jacks $\Updelta$r and considering accepted failure probabilities as constraints in the optimization procedure, a cost-effective as well as durable hybrid fiber reinforcement segmental lining design can be derived. In order to design the lining segment subjected to thrust jack forces, two plane FE models, labeled as Model I (plane strain) and Model II (plane stress), are used analogous to the design models in engineering practice (Fig. 5.30). In accordance with the design models for the bursting stresses due to partial area loading, Model I is decisive for the load bearing capacity of a segment subjected to thrust jack loading. The main objective of Model II is to examine, if the serviceability requirements in terms of critical crack widths are fulfilled. Therefore, a plane stress model of the segmental lining is employed to check, if the derived design can prevent the occurrence of critical cracks between the two thrust jack pads.

Fig. 5.30

Design variables (red), the jack eccentricity and material uncertainties as uncertain a priori parameter (green) considered in two structural models used for the optimization task to generate a robust hybrid lining design. The potential cracks due to the thrust jack loading are indicated in the illustration of the segment at the bottom left [47]

The inner part of the segment is reinforced by 25 kg/m${}^{3}$ of Dramix 3D 65/60 hooked-end steel fibers (minimum fiber content according to [82]), whereas a SFRC cap with a higher content of fibers $c_{F}$ and a width $\Updelta L$ (both to be optimized) is placed at the outer part (see Fig. 5.30). The post-cracking behavior depending on the fiber type, content and the orientation is calculated by the multi-level SFRC model described in Sect. 5.2.3.1, which was systematically validated for the Dramix 3D 65/60 hooked-end steel fibers in [33].

Fig. 5.31

Constraint limit states for the characteristic case and the realizations of a favorable as well as an unfavorable fiber orientation [47]

Figure 5.31 illustrates the constraint limit state from the reliability based optimization in the design space expanded by the fiber content, the segment thickness and the width of the layer cap. The constraint limit state divides the design space into a feasible and an unfeasible region. As can be seen in Fig. 5.31, the segment thickness $d$ and the fiber content $c_{f}$ are dominating the shape of the constraint limit state for each case, whereas the cap width $\Updelta L$ has only a minor influence. With the minimum fiber content of 25 kg/m${}^{3}$, a segment thickness of $d\approx 460$ mm (characteristic case) will still lead to a feasible design. An increase of the fiber content up to 40 kg/m${}^{3}$ enables a reduction of the segment thickness to $d\approx 400$ mm for the characteristic case, but by further increasing the fiber content, only small reductions of the segment thickness can be achieved. The width of the SFRC cap $\Updelta L$ is the least sensitive design variable. If a certain width is provided (tensile stresses occur within the outer SFRC layer), only small improvements with increasing cap width $\Updelta L$ can be obtained (see characteristic case at higher fiber contents $c_{F}$ in Fig. 5.31). A reduction of the segment thickness up to $d\approx 375$ mm can be achieved. Furthermore, the characteristic case is investigated for an unfavorable fiber orientation ($\lambda_{F}=0.3$) and for a more favorable fiber orientation ($\lambda_{F}=0.45$) according to [86]. It can be seen in Fig. 5.31, the fiber orientation has a high impact on the performance of the constraint limit state, which can either result in a further reduction of the segment thickness $d$ by $\approx$ 17 mm or to an increase by $\approx$ 13 mm for the characteristic case.

For this reason, the first set of weights adopted in the multi-objective optimization problem prioritize the reduction of the thickness over the minimization of the remaining design variables and results in a 402 mm thick segment with a 200 mm wide SFRC cap containing 41 kg/m${}^{3}$ of fibers for the characteristic case. When compared to the RC reference segment design, the steel content is reduced by 59% (106 vs. 260 kg/segment) and the concrete volume is reduced by 20% (3.61 vs. 4.53 m${}^{3}$/segment).

5.4 Full-Scale Tester for Segmental Linings Subjected to Conditions in Real Tunnels

5.4.1 Experimental investigation

The knowledge gained on isolated longitudinal joints (Sects. 5.2.2 and 5.3.3 had to be transferred and extended to large-scale experiments on full lining segments. To this end, a testing rig capable to introduce up to 5 MN in horizontal and vertical direction was developed (Fig. 5.32). It has the advantage that the resources of the testing facility can be concentrated on a single segment. Nevertheless, it preserves the boundary conditions in real tunnels [53, 9]. The vertical load $V$ is introduced by means of a vertical steel frame that distributes it on eight contact points of the segment extrados. The transition between steel profile and curved specimen is realized with elastomeric bearings which, simultaneously, considerably reduce the transmission of shear stresses. Thus, the load introduction is radial, properly reflecting surrounding ground and water pressure conditions. The presence of adjacent segments is simulated by steel wedges that copy the geometry of the longitudinal joints, where the compressive load of the ring is introduced, following the static scheme of an arc. The load introduction mimics the conditions in real tunnels, i.e., the effects of partial area loading are taken into account simultaneously by compression with variable eccentricity. An experiment on a conventional segment showed that for an eccentricity $e\approx 0.14\cdot h$ (being $h$ the segmental width) the longitudinal joint becomes decisive, showing brittle failure. Further details can be taken from [53].

Fig. 5.32

Test setup (left) and load concept (right). Modified with reference to [9]

The experimental campaign was then extended to two segments employing hybrid reinforcement concepts, i.e., casting normal concrete at the central region (reinforced with conventional rebars), and high performance steel fiber reinforced concrete at the longitudinal joints, which constituted a strengthening of that region. Both were manufactured employing the methods described in Sect. 5.3.1. Strengthening the longitudinal joints led to an improvement in the capacity of 25.5% and 42.1% in the two specimens, respectively (Fig. 5.33b, c). The difference observed between both results is directly related to the compressive strength of the concretes cast in the central region of the segments, which the experiments showed to be the decisive factor for failure. In this case, however, a ductile failure due to compression with low eccentricity in the central region was found, which enables thickness reduction, and thus concrete and excavation savings.

Fig. 5.33

Bearing capacities of conventional and hybrid linings. NC: normal concrete. HPSFRC: high-performance steel fiber reinforced concrete

The use of high-performance materials was even exploited to a further extent. Next, the complete body of two segments with hollows were cast that increased the material savings by 20% with respect to a conventional design. The geometry of the hollows was set to create an arc effect, given that the strut above is subjected to local bending due to external loading. A highly deformable mortar was cast on top to allow large plastic deformations (see Sect. 5.6.2) and better distribute the radial loads over the bearing part of the segment. In the segment (d), the load was introduced at the longitudinal joint in a conventional way, which corresponds to an outer gasket location. In another segment (e), the gasket was moved to the middle as learned from the simplified isolated specimens, investigated in Sect. 5.3.2.2. The eccentricity was increased by 10 mm to match the load level at failure with the capacity of the testing device. All bearing capacities are compared in Fig. 5.33.

5.4.2 Validation and Numerical Exploration of Novel Segment Designs

In order to explore arbitrary combinations of reinforcements schemes in a virtual-lab, the numerical simulation platform described in Sect. 5.2.3 is used to replicate the presented real-scale segment test device.

After the model was validated for SFRC structures in Sect. 5.2.3, the predictive capability for conventional RC segments is checked by re-analysis of the experimental investigated reference segment. The material properties are determined on the basis of the experimentally obtained compressive strength ($f_{cm}=42.8$ MPa) using [44] and [18]. The tensile strength of the concrete is calculated to be $f_{ctm}=3.7$ MPa and the modulus of elasticity to $E_{cm}=29$ 658.7 MPa, assuming a mode I fracture energy of $G_{F,I}=0.1$ Nmm. The material behavior of the B500B rebars is assumed to be perfectly plastic (yield strength $f_{y}=500$ MPa, $E_{s}=200$ 000 MPa) and the bond-slip relationship from [44] is used. The plane FE model, representing a single layer of reinforcement, and an comparison between the numerically obtained crack pattern and the DIC measurement are presented in Fig. 5.34.

Fig. 5.34

FE model (top) and comparison of the DIC measurement with Aramis (GOM GmbH) and the damage predicted by the numerical model (down). Modified with reference to [52]

A good agreement between the experimental and numerical predicted load bearing capacity (2519.8 vs. 2503.5 kN) as well as the crack development (Fig. 5.34, bottom) can be observed.

After the validation of the numerical model, different novel segment designs (from pure SFRC over combinations of SFRC and RC to segments with hollows) are investigated. For the SFRC, a post-cracking behavior is assumed, which was derived within the framework of a systematic validation campaign [33] for concrete reinforced with 57 kg/m${}^{3}$ Dramix 3D 65/60 steel fibers. The load bearing capacity of the investigated variants and the crack patterns for the segmental designs containing hollows are shown in Fig. 5.35.

Fig. 5.35

Comparison of the maximum vertical forces of the investigated segmental designs (left) and the numerical crack patterns of the segments with hollows for two reinforcement schemes (right). Modified with reference to [52]

A pure SFRC segment can carry a similar load as the experimentally investigated conventionally reinforced segment, whereas a combination of bar and fiber reinforcement (SFRC/RC) increases the load bearing capacity by 25.6% (Fig. 5.35, left). If two hollows (479 x 65/125 mm) are placed, the concrete volume per segment can be reduced by 23.8%. When the reinforcement layout is adopted from the experiment of the reference segment, critical cracking occurs in the corner areas of the hollows (Fig. 5.35, top right) and results in a by 65.9% reduced load bearing capacity. By adding steel fibers, even a slightly increased load bearing capacity can be achieved compared to the reference segment tested experimentally. Adopting the reinforcement layout by placing additional reinforcements around the hollows, the performance of conventional segments can be improved (see Fig. 5.33).

5.5 Annular Gap Grouts in Mechanized Tunneling

During tunnel driving a gap remains between the segment lining and the surrounding ground with a thickness of about 10 to 20 cm, the so-called annular gap. This gap must be backfilled with a suitable grouting material [7]. The state of the art is to inject the grouting material under pressure in order to completely backfill the annular gap and to stabilize the tunnel lining soon after mounting of the segment rings. Simultaneously, settlements of the ground surface shall be minimized. The main material-specific requirements on annular gap grouting materials are optimal workability and sedimentation stability until grouting as well as, immediate strength development soon after grouting. Thereby, the grouting materials have to be defined considering the respective (hydro-)geological and constructional boundary conditions. While one-component grouts are usually used in permeable soils, two-component grouts are applied in cohesive, impermeable soils. The hardening or rather stiffening of one-component grouts is primarily achieved through dewatering under pressure into the surrounding soil and thus consolidating the grout. In contrast, the strength development of two-component grouts is based on a chemical reaction usually between cement and accelerator.

5.5.1 One-Component Grouts – Experimental Investigations on the Material Characteristics

One-component grouts are mixed before being pumped to form a final annular gap grout and are used in both unconsolidated and solid rock. The annular gap grouting is usually carried out by pilaster strips in the shield tail (Fig. 5.36).

Fig. 5.36

Schematic illustration of the annular gap grouting with one-component grout

According to the current state of the art, one-component grouts can be classified according to their binder content [17, 84]:

• Active grouts (Portland cement (OPC) $>$ 200 kg/m${}^{3}$)

• Semi-inert or reduced-active grouts (50 kg/m${}^{3}$ $<$ OPC $<$ 200 kg/m${}^{3}$)

• inert or inactive grouts (OPC $<$ 50 kg/m${}^{3}$)

The focus of the laboratory tests under varying material- and construction-specific parameters was on the decisive properties of a one-component grout namely workability/consistency, sedimentation stability, dewatering behavior and stiffness development. Through variations of the constituents and composition of the grout mixes the dewatering and thus the consolidation behavior of the annular gap grout can be influenced significantly. This affects the further dewatering process of the grout. Table 5.1 shows example mix designs of one-component grouts, which were already used in practice. The grout designation contains information the grout type and cement content, for example, ‘‘SI-120’’ is a semi-inert grout with a cement content of 120 kg/m${}^{3}$.

Tab. 5.1 Reference formulations of one-component grouts (kg/m${}^{3}$)

These grout mix designs served as base mixes for the material-specific variations of binder composition (type, ratio) and aggregate (type, grading curve). Besides the original binder components (cement and fly ash) additives such as fine limestone powder, quartz powder and granulated blast-furnace slag were also used. A detailed list of the various compositions can be found in [94].

5.5.1.1 Workability and Sedimentation Stability

In the laboratory tests sufficient sedimentation stability was given for all reference mixes throughout testing. Furthermore, the investigated reference mixes exhibited a sufficient flowability (as a characteristic value for the workability) at a target spread flow diameter of 15 $\pm$ 5 cm after 6 to 8 h [84] after mixing. A correlation between the granulometry of the fine material and the consistency could be determined. A high specific surface area ($S_{m,b}$ – area of grains per unit weight) and thus an increased fineness of the initial materials results in a reduced flowability of the grout. This effect could be identified both for the binder fineness and for grading curves with smaller grain size fractions and was probably due to the 20% higher water demand of the fines. Also the surface texture and the grain shape have an effect on the consistency. While angular, rough surfaces impeded the optimum sliding of the solids in the grout, smooth, round grain particles positively influenced the flowability.

Furthermore, the binder type had a decisive influence on the consistency of the grout. While active and semi-inert grout compositions stiffened faster due to the initial hydration reactions of cement, inactive or cement-free compositions showed a significantly higher flowability over a longer period of time (> 8 h after mixing).

5.5.1.2 Dewatering Behavior by Means of a Modified Filter Press Test

In order to investigate the dewatering behavior of one-component grouts more realistically than previously possible, a specially modified filter press (cf. [94]) was developed (Fig. 5.37, left). Thereby, typical steering parameters (grouting pressure p = 2.5 $\pm$ 0.25 bar, driving speed/dewatering duration t = 30 min) of a tunnel driving with segment lining (segment length of 2.00 m) were applied on grout samples with a layer thickness of about 17 cm. The total amounts of filtrate water $f_{30}$ of the reference mixes with different binder combinations showed significant differences in the results (Fig. 5.37, right). With a linear regression analysis, a correlation between the total amount of filtrate water and the specific surface of the binder could be calculated, shown as trend lines at a mortar age $t_{M}$ of 0 h and 6 h, respectively. Only with increasing age ($t_{M}=6$ h) a downward trend of the total amount of filtrate water $f_{30}$ could be observed (Fig. 5.37, right, grey highlighted area). Apparently, the progressive hydration of cement led to a decrease of the amount of filtrate water. This effect could be clearly seen at mortars, containing ordinary Portland cement (OPC) and fly ash (FA). With an equal substitute amount of FA ($S_{m,b}=6.130$ to 8.910 cm${}^{2}$/g) with OPC ($S_{m,b}=6.070$ cm${}^{2}$/g) the amount of filtrate water was reduced by approx. 25%. Whereas, cement-free grouts ($S_{m,b}=5.580$ cm${}^{2}$/g) with identical initial water content exhibited an approx. 48% higher amount of filtrate water, which is attributed to the smaller specific surface ($S_{m,b}=5.580$ cm${}^{2}$/g) and thus smaller water demand of FA. A substitution of FA with limestone powder (LSP) and quartz powder (QP) demonstrated that the total amounts of filtrate water $f_{30}$ decreased significantly through a high amount of the respective fine (LSP: 75% by mass, $f_{30}=4.2$% by mass, QP: 20% by mass, $f_{30}=4.7$% by mass). Whereas, a high substitute amount of FA with ground granulated blast-furnace slag (GGBS: 45% by mass) resulted in approx. 22% more filtrate water $f_{30}$. What can be deduced from this is that beside the specific surface of the binder $S_{m,b}$ also the fineness of the respective fines contributes to the dewatering potential of the mortar [93].

Fig. 5.37

Complete test setup of the ‘‘modified filter press’’ (left) [97]. Total amount of filtrate water $f_{30}$ of grout mixes depending on the specific surface of the binder $S_{m,b}$ at the age $t_{M}$ of 0 h and 6 h (grey highlighted area) (right) [93]

Through dewatering under pressurization the grout consolidates, leading to an internal compaction of the grain system through a re-distribution of the fine particles. On the basis of the particle size distribution and the dry density, it could be demonstrated that fine and ultra-fine particles (< 0/0.25 mm) are rearranged from the pressurized side to the dewatering side. In the present case (Fig. 5.38, left, exemplary shown for a semi-inert grout), the coarse grain content (> 4 mm) was about 20% by volume before pressurization. At the end of the dewatering test the dewatered grout sample exhibited a grain content about 23% by volume on the pressurized side and only 16% by volume on the opposite side (towards the water release). This is reflected through the dry densities of the grout sample. Higher densities were determined at the dewatering side than at the pressurized side (Fig. 5.38, right). Accordingly, a more intense compaction occurred, resulting not only from the re-distribution of the fine particles but also from the increased water release at the pressurized side.

Fig. 5.38

Grain size distribution of individual layers of a dewatered one-component-grout (semi-inert grout ‘‘SI-60’’) (left). Dry density of individual layers (right) [95]

5.5.1.3 Shear Strength Development

In order to measure the development of shear strength at different ages and depths of the dewatered grout compositions, shear blades were already installed before pressurization into the fresh grout [94]. All investigated grout compositions consistently achieved the required minimum shear strength of $\tau\geq$ 2.0 kN/m${}^{2}$ after 30 min [84]. Overall, a positive effect on the shear strength was observed with an increase in binder fines, use of coarse-grained aggregate, higher drainage pressures, and at greater depths of placement. Significantly higher shear strengths were obtained when initial components with a high specific surface area were used and inert materials were substituted by finer cement particles. In addition to the granulometry effect and the associated physical filler effect of the fines, this was a result of the progressive hydration development of the cement. The development of shear strength correlated with the amount of filtrate water of the respective grout compositions. In the pressurized zones less consolidation could be observed due to the relatively soft consistency of the grout. In contrast, in deeper layers (towards the water release), an approximately complete dewatering and as a result of the continuous increase in the packing density of this layer, the highest shear strength values were determined (Fig. 5.39, right). Accordingly, it can be stated that due to the re-distribution of fines, the transportation routes for further dewatering successively decrease (Fig. 5.39, left).

Fig. 5.39

Development of shear strength at dewatered specimens of semi-inert mixes ‘‘SI-60’’ depending on the grout depth (left). Formation of different dewatered layers (dewatering gradient) (right) [96]

5.5.2 Two-Component Grouts – Experimental Investigations on the Material Characteristics

As an alternative to one-component grouts, two-component grouts have steadily become more relevant in practice. Such systems are indispensable in soils with very low permeability. In two-component grouts, an activating component B is added to the actual grout (component A) immediately before or during grouting in the annular gap. This causes an instantaneous stiffening/consolidation of the grout system. Therefor both physical effects (internal dewatering, e.g. via superabsorbent polymers, cf. Sect. 5.5.4) and chemical reactions (e.g. alkaline activation by water glasses) can be considered.

Currently, two-component grouts are formulated on a purely empirical basis. Component A usually consists of cement, bentonite, possibly pozzolanic/inert additives and a high water content (approx. 70% to 80% by volume) [65, 87]. To achieve sufficient workability retarders can be added. Accelerators, usually alkali silicates (water glass), are used as the activating component B. The performance of practice-established grouts was determined on a reference grout (designation: M0). The focus was on investigating characteristic properties of two-component grouts such as workability/consistency, sedimentation stability and strength development (gel phase, early strength and final strength). In addition to the reference grout, used in practice minima and maxima of every source material are listed in Table 5.2.

Tab. 5.2 Reference formulation and practical orientated minima and maxima of the corresponding source materials in kg/m${}^{3}$

Various modifications especially with regard to type and quantity (cement and bentonite content, ratio of cement to accelerator, retarder) were investigated, based on grouts used in practice. A detailed description of the respective grout mix designs (also cementless mixes) as well as an overview of typical requirements and reference values for two-component grouts can be found in [69].

5.5.2.1 Workability an Sedimentation Stability

The required processing time depends on the logistical conditions of the respective tunnel construction project. In this context the workability of component A is commonly determined by the Marsh time [83] as its use in several completed projects shows [69]. Sufficient workability and, consequently, pumpability of the two-component grout is considered at a Marsh time between 30 and 45 sec for 72 h [51]. The workability of the reference grout M0 was within the target range of the Marsh time from 30 to 45 sec up to approx. 60 h after production. Subsequently, a marked increase of the Marsh time can be observed with an excess of the target range of approx. 10%. Furthermore, with a higher content of granulated blast-furnace slag in the total binder, a reduced Marsh time was observed within the first 24 h. Thereby an increased flowability of the grout was a result of the cement substitution. After 48 h, the Marsh times of the considered grouts were approximately at the same level, which can be explained by the beginning hydration of the binder.

With regard to the sedimentation stability of the reference grout, the amount of bleeding water was determined to be below the limit value of 3% [51] after 3 h. The guideline value of 4% after 24 h [51], however, was exceeded by approx. 1.5 M.-%. With increasing time up to 48 h after mixing, the amount of bleeding water did not increase significantly.

5.5.2.2 Strength Development

The consideration of the strength development of a two-component grout can be divided into three phases:

• Gel phase up to 0.5 h,

• early strength from 0.5 up to 24 h,

• final strength.

The essential phase for the immediate bedding of the installed segments is the transition of the gel phase to early strength. Assuming an advance rate of 50 mm/min, a segment width of 1.5 m and a construction time of 30 min, the annular gap is grouted within 30 min by simultaneous grouting. After the grouting, the drift presses are released from the last and now freshly inserted segment ring. At this point, the lifting forces are decisive and the grouting material needs to be solidified so far that the segment ring remains stable in its position.

A previously developed test method was used to determine the gel time. For this purpose, the activation of the two-component grout takes place in a vertically arranged mixing device (Fig. 5.40, left). Components A and B are both filled separately into sealed cells. The cells are opened by valves and the components collide/homogenize in the static mixer arranged below. The two-component grouts, which is thus completely mixed, enters a funnel with a defined geometry and discharge opening (diameter: 6 mm). The time from the collision of the components to the break-off point of the volume flow is measured.

Fig. 5.40

Test method for determining the gel time of two-component grouts and visualizing of the mixing process of mortar (component A: green) and activator (component B: red) (left). Schematic illustration of the dependence between gel time and early strength with variable activator content (right) [69]

The gel times of the reference grout M0 ranged from 6–15 sec at an accelerator content of 70 to 120 kg/m${}^{3}$ and thus are within the frame of reference of 5–30 sec known from literature [51, 72]. Furthermore, a dependence between the accelerator content and the gel time was found. In this framework, contrary to expectations, an increase in the accelerator content lead to an increase in gel time. While a gel time of about 7.5 sec was determined with an accelerator content of 80 kg/m${}^{3}$ (6.1% by volume), the gel time with an accelerator content of 120 kg/m${}^{3}$ (9.8% by volume) was about 12 sec. (Fig. 5.41, left).

Fig. 5.41

Gel times alkaline activated two-component grout containing granulated blast-furnace slag with variable content of Portland cement (left) and cementless compositions with calcium hydroxide as solidification medium (right) [69]

In addition, the gel time and early strength interact directly with each other and can be controlled by the accelerator content if the binder content is kept steady. A high accelerator content is required for rapid development of the early strength. In contrast, the gel time is prolonged with an increased accelerator content. This reciprocal relationship is shown in Fig. 5.40, right.

The extent to which alkaline activation supports cement substitution and thus the use of latent-hydraulic and/or pozzolanic materials was further investigated. Through coordination of accelerator and binder, such additives can be stimulated and form their strengthening structures accordingly. Compared to the reference grout, the addition of granulated blast-furnace slag depending on an appropriate sodium/potassium water glass dosage reduced the cement content by up to 170 kg/m${}^{3}$ (approx. 68%) (Fig. 5.41, left). In addition, sodium-water glass leads to a faster strength development in cement, whereas potassium-water glass supports a faster strength development if blast-furnace slag is used.

Besides, the gel time for a cementless composition could be determined. The grout was modified by adding a calcium hydroxide (Ca(OH)${}^{2}$) suspension as a solidification medium together with blast-furnace slag. As seen in Fig. 5.41, right, the test results prove that the requirements for the gel time could be achieved with at a calcium hydroxide content of approx. 18 kg/m${}^{3}$ and an accelerator content of 90 kg/m${}^{3}$ (6.8% by volume). The actual long-term strength was subsequently due to the reaction of the blast-furnace slag. The material-technological decoupling of the gel and strength phases consequently allows the employment of different binder compositions of latent-hydraulic and/or pozzolanic materials.

The development of the shear strengths is the decisive characteristic for the stabilization of the segments. A shear vane (according to [49]) was used to determine the shear strength over a period of 25 to 80 min after preparation. The shear strength requirements (2 kN/m${}^{2}$ after 30 min) were achieved with an accelerator content of 90 kg/m${}^{3}$ (Fig. 5.42, left). Similarly, a rapid development of the compressive strength also occurred (Fig. 5.42, right). After only 2 h, the grout with an activator content of 90 kg/m${}^{3}$ had a compressive strength of approx. 0.3 MPa. The usual requirement of 0.5 MPa of compressive strength after 24 h [85] is thus already achieved immediately after grouting.

Fig. 5.42

Development of the shear strengths (left) and the compressive strength (right) of the reference grout M0 with different accelerator contents [69]

5.5.3 Test Setup for the Simulation of the Annular Gap Grouting

A special test device was developed to simulate annular gap grouting under realistic conditions. The relevant control variables were included in the design: Grouting volume, advance rate, grouting pressure and the grouting nozzles. With the test setup, the distribution of the material in the annular gap, and the homogeneity of the grout can be recorded during the grouting process.

5.5.3.1 Constructive Design

The grouting device can be separated into six assemblies: bucket (1, soil), slide (2, simulated tunnel boring machine or its shield tail), the annular gap (3), the nozzle (4, mixing of component A and B), the hydraulic system including the abutment (5) and the cover (6). The internal dimension of the simulated annular gap are 1500 mm $\times$ 600 mm $\times$ 150 mm. In Fig. 5.43 is a schematic overall view (left) and a photograph of the experimental setup (right). Further detailed information can be found in [70].

Fig. 5.43

Overall view of the test setup for annular gap grouts (left). Recording of the experimental setup (right) [70]

5.5.3.2 One-Component Grout

To verify the pressure distribution in the annular gap during a grouting test, the comparison between the set pressure (approx. 0.5 bar) at the grout pump and the measured pressure with the ceramic sensors (nominal pressure range: 0–4 bar) was investigated. The results of the pressure measurement are presented in Fig. 5.44.

Fig. 5.44

Pressure measurement in the annular gap in the plane of the mortar opening during a grouting process of a one-component grout (grouting pressure: 0.5 bar, advance rate: 60 mm/min) [69]

The set pressure of approx. 0.5 bar was reached within 90 sec after filling the first section. This value was kept constant during the entire test. Based on the individual curves of the pressure sensors (Fig. 5.44, right), it can be determined that the pressure and the measured fluctuations of the respective curves developed in parallel. This indicates constant pressure distribution in the annular gap over the height and width. Thus, it is ensured that there is an equal pressure at every point of the grouted annular gap and homogeneous material is obtained for subsequent tests.

During groutability verification of a semi inert one-component grout (cf. [69]), grouting pressures of about 0.5, 1.0 and 1.5 bar were applied. In order to evaluate the flow behavior, the grout was pigmented (Fig. 5.45). The color gradient reveals a ring-shaped distribution of the grout. In addition, the homogeneous distribution of the one-component grout confirmed the applicability of the material.

Fig. 5.45

Pressed one-component grout (left). Flow behavior of the grout (right) [70]

5.5.3.3 Two-Component Grout

In the course of the grouting tests, the two-component grout M0 (cf. Table 5.2) was also investigated with regard to the flow properties after activation and subsequent strength development. The tests were realized without pressure and an advance rate of 90 mm/min. The filling process or the expansion of the activated grout was observed directly upon entry into the annulus (Fig. 5.46).

Fig. 5.46

Filling of the annular gap with a two-component grout commonly used in practice (reference grout M0, activator content: 90 kg/m${}^{3}$, gel time: approx. 9 sec.) [69]

The grouted material was demolded after approx. 30 min. (dimensionally stable at every point) and homogeneous or evenly gelled (Fig. 5.47, top). On the exposed activated material, the shear strength development was determined with a shear vane according to [49] (Fig. 5.47, bottom left/center). The results of the shear strength tests are shown in Fig. 5.47 (bottom right) and compared with the shear strength development of the same grout produced separately in the laboratory, by hand.

Fig. 5.47

Demolded reference grout M0 (Age of the sample: 30 min., activator: 90 kg/m${}^{3}$, gel time: 9 sec.) (top). Procedure for determining the shear strengths (bottom, left/center). Comparison of the shear strengths of the reference grout M0 with the results on laboratory test specimens (bottom, right) [69]

The results reveal a higher shear strength of the grouted material compared to the laboratory specimens, especially up to 40 min after activation, which may be due to a more intensive mixing of the two components in the mixing unit. From a period of 80 min, the shear strengths were at approximately the same level, with the laboratory specimens showing a larger increase in strength.

5.5.4 Multiscale Simulation for the Strength Development of Two-Component Grouts

In order to provide reliable stiffness development for two-component grouts, physically acting superabsorbent polymers (SAP) could be considered as component B. The polymers bind an extensive amount of fluid within a very short time and thus contribute to the stiffening of the grout. Therefore, experimental tests with regards to the water absorption of SAPs under conditions close to the final application have been carried out in accordance to [48]. In parallel, a thermodynamically consistent model based on the Theory of Porous Media (TPM) was developed to account for the internal mass transfer between the pore fluid and the SAP particles in a two-component grout mix [66]. Finally, the parameters of the experimental results were used to generate the constitutive law for the mass transfer in the numerical model. In [66], the derivation for calibration of the multiphase multi-species model, using defined properties, balance equations, and the specific model description, is presented.

Based on the water absorption capacity of SAP for different environmental solutions (tap water, different NaCl concentration), a drastic increase in water absorption with higher NaCl concentration could be observed. In the case of using tap water, the potential of water uptake continued to increase by up to 60% (Fig. 5.48, left). Accordingly, a particular salt concentration was assigned to the tap water in the numerical model. The comparison of the simulated model with the results from the experimental investigations is shown in Fig. 5.48, right, [66].

Fig. 5.48

Experimental water absorption tests of SAP with different environmental solutions (left). Fitted model parameter to experimental data (right) [66]

A resulting simplified volume balance was used to describe the absorption kinetics [66],

$$\displaystyle{\partial_{t}(\chi)}=\frac{n^{B}}{n^{B}_{0}}=\frac{1}{\tau_{W}}\left(\frac{a_{1}}{\chi}-a_{2}\chi\right).$$

(5.2)

The function $\chi={n^{B}}/{n^{B}_{0}}$ relates the absorbed water and the initial volume of the SAPs, while the partial derivative $\partial_{t}(\chi)$ represents the time. The first term in the function accounts for the addition of water due to a difference in salt concentration, while the second term simulates the development of a restoring force of the polymer network during swelling. Similar approaches can also be found in literature [12]. In this model, $a_{1}$ and $a_{2}$ are associated with the charges attached to the polymer chains and the elasticity of the network, respectively. Accordingly, $\tau_{W}$ scales the driving force and can be related to the friction between the polymer chains. Applying a least-squares fit to the parameters in Eq. 5.2, results to absorption curves that correspond to the experimental studies (Fig. 5.48, right) [66].

5.6 Deformation Tolerant Tunnel Lining

The stress redistribution and dilation due to the excavation in clay shale results in a so called Excavation Damaged Zone (EDZ) in the nearfield of tunnel. Taking into account the inhomogeneous structure of clay shales that consists of inherent cracks, joints and discontinuities, development of EDZ results in a reduction of shear strength while the increases in hydraulic conductivity forms a preferential flow path at the vicinity of the tunnel. In case of tunneling in shales with the potential of swelling, the water transport through the EDZ results in swelling of the clay rock that build up significant pressure at the tunnel support system. Apparently, development of a deformation tolerant lining can avoid extreme pressure evolution at the tunnel structure, however, the deformation of the tunnel support may lead to further development of the EDZ. In this context, proper prediction of the tunnel behavior requires adequate knowledge about the (a) swelling behavior of the host rock, (b) size and shape of the EDZ, and (c) load-deformation behavior of the tunnel support. Figure 5.49 indicates the coupled interactions between swelling pressure evolution and the deformability of the tunnel support system.

Fig. 5.49

The concept of deformation tolerant tunnel lining in swelling clay rocks which includes deformable cementitious layer and deformable annular gap grout around the lining segments. Typical mechanical behavior of the compressible material is characterized by 3 stages: elastic, plateau and densification [9]

5.6.1 Swelling Behavior of Clay Shales Under Tunneling Relevant Boundary Conditions

The swellable clay shales exhibit a significant volume or pressure increase upon hydration respectively in free or volume constant conditions, where the innercrystalline and osmotic swelling are the main mechanisms involved. The negative charge of smectite layers is compensated by interlayer cations which hydrate when getting in contact with water. This hydration is called innercrystalline swelling and results in swelling pressures larger than common overburden pressures in tunneling practice [20, 40, 42] which take place in the geological life of the formation. Therefore, innercrystalline swelling does not play a significant role in tunneling whereas the osmotic swelling occurs when the unloaded swellable clay shale adsorbs additional water. In this mechanism, the ions in the pore water form a diffuse double layer at the surface of negatively charged clay particles. The repulsion of such diffuse double layers is called osmotic swelling [40] that plays a role in tunneling practice. Dilation due to tunneling in the EDZ decreases the degree of saturation around the tunnel and initiates a suction-induced osmotic flux in radial direction towards the tunnel whilst, the higher hydraulic conductivity of the EDZ can trigger fluid water flow along the tunnel. This can result in swelling of clayey soils, if the three-sheet layered silicate smectite is present.

Fig. 5.50

Tunneling relevant boundary conditions for swelling test

To study the swelling behavior of a shale, it is important to consider the appropriate boundary conditions of the application in the test set-up. In the context of mechanized tunneling, these are explicitly the saturation modi and volumetric deformations due to the tunnel support reaction. As shown in Fig. 5.50, depending on the tunnel support condition, various volumetric deformation conditions can be adopted in the laboratory testing. For instance, free swelling, controlled-volume swelling and volume constant swelling tests represent unsupported tunnels as well as tunnels with flexible and rigid support structure, respectively. In terms of hydraulic boundary conditions, the advective flux in preferential flow paths in EDZ along the tunnel can be represented by hydration of sample with water fluid (i.e. single step hydration) and the diffusive flux in radial direction towards the tunnel due to a dilation induced suction increase can be reproduced by vapor saturation (i.e. multi step hydration). For this purpose, a novel multi-purpose swelling pressure device for tunneling relevant boundary conditions was designed and constructed (Fig. 5.51) in which free as well as controlled and constant volume tests are possible to be carried out. Water saturation from bottom inlets (single step) or by vapor saturation with a peristaltic pump, gas washing bottle and the cell’s top and bottom inlets/outlets (multi step) are enabled. The gas washing bottle is filled with an oversaturated salt solution which controls the relative humidity of the vapor and can be combined with a temperature to a suction via the Kelvin equation,

$$\displaystyle\psi_{\text{tot}}=-\frac{RT}{v_{w0}\omega_{v}}\ln(RH),$$

(5.3)

where $\psi_{\text{tot}}$ is the total suction, $R$ is the universal gas constant, $T$ is the absolute temperature, $v_{w0}$ is the specific volume of water, $\omega_{v}$ is the molecular mass of water vapour and $RH$ is the relative humidity.

For clay shales, also the sample preparation plays a significant role, since the undisturbed structure and fabric of the material has a substantial impact on the mechanical, hydraulic and hydro-mechanical behavior. Additionally, clay shales are usually strongly anisotropic due to their geological formation process (e.g., the approximately 170 million years old marine sediment Opalinus clay shale (OPA) [19]). Samples have to be prepared carefully regarding their bedding plane orientation (Fig. 5.52b). Considering the significant inhomogeneities like microfossiles, pyrite or quartz lenses in the intact samples that often complicate sample preparation and test results interpretation [21], it may be reasonable to destroy the fabric and structure to generate reconstituted powder samples for certain applications. In this frame a series of swelling tests on reconstituted shale samples have been carried out as well.

Fig. 5.51

Newly developed swelling pressure cell for tunneling relevant boundary conditions [9]

Fig. 5.52

Sample preparation on a lathe (a) considering bedding plane orientation (b) [103]

In the present study, the hydro-mechanical behavior of clay shales was investigated on OPA shale, since the New Belchen Tunnel (NBT) in Northern Switzerland was partly driven through this clay shale and the project was considered as the reference project. Samples were obtained from the construction site and total pressure sensors around the tunnel lining installed for long-term in-situ measurements. Results of swelling pressure tests showed a pronounced influence of anisotropy. The swelling pressure perpendicular to bedding planes (1.2–2 MPa) was 5 to 20 times larger than parallel to bedding planes (0.1–0.25 MPa), average swelling pressure curves of intact OPA in both directions are presented in Fig. 5.53a). Such a strong anisotropy is important to consider for the dimensioning of the tunnel lining, since the load on the individual lining segments can vary significantly regarding the bedding plane orientation of the surrounding soil which may vary along the tunnel. Discrepancies in test results with the same orientation exist due to naturally varying clay contents or inhomogeneities like pyrite and quartz lenses. The rate of saturation, and thus the increase in swelling pressure, depends on the hydraulic conductivity of the material. Considering volumetric deformations, minor strains of the material (less than 1%) resulted in approximately 50% reduction in swelling pressure. Due to structure and texture of the material the volumetric swelling of the material is significantly limited. Volumetric deformations can be used to upgrade the decreasing dry density of the material and derive a dry density-swelling pressure curve, which usually shows a logarithmic increase of swelling pressure with increasing dry density. For bentonites it is common to conduct swelling pressure tests with different initial dry densities to determine this unique correlation for its swelling potential. However, deriving this correlation at an intact clay shale due to stepwise strain release mimicking unloading and dilation of the material due to mechanized tunneling is a novelty.

In tunneling practice the newly developed deformation tolerant tunnel lining will allow volumetric deformations of the surrounding swellable clay shale in a controlled manner and therefore reduces the acting loads on the lining system. First results of sandwich tests, with samples composed of 2 cm OPA and 1 cm compressible grout, confirmed the effectiveness of this design. Test results of OPA powder with an initial dry density of 2.0 g/cm${}^{3}$ and OPA powder plus grout are presented in Fig. 5.53a) and show a reduction of 80% in swelling pressure. A compacted sample in the sample ring is shown in Fig. 5.53b) and the setup of a sandwich test in Fig. 5.53c).

Fig. 5.53

Experimental results of tests on intact OPA and powder with and without grout (a), sample ring (b) and test setup of the sandwich test (c)

5.6.1.1 Numerical Assessment of the Excavation Damaged Zones in Tunneling

Concerning the coupled interaction between the size and shape of the EDZ and swelling process in shale, assessment of the time dependent development of EDZ is very crucial. In the literature, different approaches have been proposed to model the fractures such as discrete element method, damage models or simulation of shearband localization using the finite element method. To enable simulation of not only the fracture development but also the swelling process, the approach involving finite element modeling was adopted. In this frame, the shear band localization approach in conjunction with a regularization technique can be successfully applied to model the EDZ without mesh dependency issues. In order to overcome the mesh dependence of strain localization in classical finite elements, it is essential to adopt an advanced model to properly address the post peak behavior as well as the localization aspect. Among different models, the second gradient model proposed by [14] is used as a regularization method where the continuum model kinematics includes macro- and micro-kinematics.

Fig. 5.54

Details of numerical modeling of the EDZ shape

To study the influence of tunneling parameters on the EDZ shape and size in partially saturated Opalinus shale, a 2-dimensional multi-phase model in conjunction with a the second gradient localization model was developed. The model is based on the model presented in [5]. To simulate the stress evolutions due to tunneling in the 2D model, a stress-release method was applied (Fig. 5.54). By this approach, stress reduction in front of the TBM as well as overcut and shield conicity (A-B), pressure increase due to grouting (B-D) and pressure drop upon TBM passing (D-E) are considered.

Fig. 5.55

Variation of the shape of EDZ by the TBM advance speed: 10% faster (left), reference case as Fig. 5.54 (middle), 10% slower (right)

Figure 5.55 shows that the EDZ develops less during faster tunnel excavation. This can be explained by quicker support of the semi-unsupported excavation zone in case of faster TBM advance. In this condition, less time for EDZ development is available before the clay rock gets supported by the tail grouting pressure.

5.6.1.2 Swelling Laws

For tunnel excavation in swelling clay rock, modeling of the swelling process in conjunction with an appropriate swelling law is essential. In the literature, various types of simplified analytical models like [31, 38, 54] and fully coupled swelling laws like the one of Wittke or the Barcelona Basic Model (BBM) [2] have been proposed. The analytical models generally provide a relationship between the swelling pressure and the swelling-induced volume increase on the basis of Huder and Amberg swelling tests. Referring to the complexity of coupled interactions due to mechanized tunneling, simplified analytical models are not sufficient. Therefore, Wittke developed a coupled semi-logarithmic swelling law according to [31]. Since the model parameters are calibrated only on the basis of Huder Amberg swelling tests, the model cannot be used to model the volume constant (e.g. rigid tunnel lining) boundary conditions.

The Barcelona Basic Model [2] is an extension of the modified Cam-Clay model that considers the stress-strain behavior of clays in partially saturated conditions. According to [28], the BBM is unable to reproduce large swelling strains, but only small reversible swelling in the elastic zone. Therefore, it is not suitable for so-called expansive soils where large volumetric strains upon saturation is expected. Therefore, [28, 3] extended the BBM with micro-structure effects to model large swelling strains; however, this approach is particularly relevant for double-porosity clay shales (e.g. micro- and macro-pores). Therefore, to study the swelling behavior of single-pore OPA shale, swelling tests have been carried out using the device shown in Fig. 5.51. In these experiments, each test consisted of multitude of volume constant and deformation release steps to consider the impact of tunnel deformations in the swelling behavior. Figure 5.56 illustrates graphically a mechanized-tunneling relevant swelling law that represents the correlation of swelling pressure, strain and time determined in the swelling pressure cells for tunneling relevant boundary conditions. A first peak in swelling pressure is reached for volume constant boundary conditions with a specific initial dry density. Each deformation release resulted in a drop of swelling pressure with a subsequent increase, where the water is absorbed in newly available pore space (i.e. desaturation upon unloading). As seen, an equilibrium swelling pressure can be assigned to any strain and additionally time by projecting the swelling pressure to the swelling pressure over time plane. In this frame, not only the maximum swelling pressure but also the trend of the swelling pressure evolution in time can be defined for each strain level using this swelling law.

Fig. 5.56

Exemplary correlation between swelling pressure, strain and time

5.6.2 Deformable Cementitious Materials

To realize a deformation compensating tunnel lining, compressible cementitious materials were developed, which can be arranged as annular gap grout or as an additional compressible layer at the exterior of the lining (Fig. 5.49). The functional principle of the materials is the same in both cases. By using highly porous additives (e.g. lightweight aggregates, expanded polystyrene) and highly air-entraining admixtures (foaming agents), the cementitious materials are characterized by a very high porosity. As an example, Fig. 5.57, left, shows the structure of two developed materials produced with a highly porous cement matrix (air void content of approx. 30% by volume) and different additives (expanded glass/expanded polystyrene). The high porosity of the materials enables a high degree of volume compressibility and thus a high plastic deformation capacity even for complete transverse strain constraint, as is more or less the case on the exterior of the tunnel lining.

The material behavior is characterized by a stress-deformation behavior shown qualitatively in Fig. 5.57, right. Under axial compression, a linear elastic behavior is initially present (stage I). This corresponds to the typical stress-deformation behavior of mortar and concrete, whereby the slope of the curve in this stage is indicating the initial material stiffness. When a certain stress ($\sigma_{\text{pl}}$) is reached, which is in the range of the compressive strength of the material, the material behavior changes from linear elastic to plastic (stage II). Optimally, in this stage a more or less constant stress level is maintained over a large range of deformation. With increasing plastic deformation, the volume compression gradually leads to a densification of the material (stage III). This stage is characterized by a huge increase of stresses. Total deformation capacity of the material is ultimately limited by this densification process.

With regard to the design of compressible materials, the stress-deformation behavior must be individually adjusted to the project-specific conditions. On the one hand, in order to avoid overloading of the segmental lining, the primary target value for the material design is a critical stress $\sigma_{\text{crit}}$ (Fig. 5.57, right), which must not be exceeded as a result of the densification process until a defined deformation is reached. On the other hand, to avoid early plastic deformation due to lower scheduled loadings (e.g. grouting pressure, water pressure), sufficient minimum strength also has to be ensured.

Fig. 5.57

Left: Compressible cementitious materials produced with a highly porous cement matrix and expanded glass (a) and expanded polystyrene (b); right: Qualitative illustration of the stress-deformation behavior of compressible cementitious materials under axial compression with transverse strain constraint [60]

In an extensive test program, compression tests were carried out to experimentally investigate the developed materials in terms of their stress-deformation behavior. Through variations of material composition (e.g. type of additives, air content, volume fractions, water-cement ratio) and test-related parameters (e.g. with/without lateral confinement, specimen geometry and dimensions, loaded area), essential influencing factors on the compressive behavior were identified, which can be used as control parameters for the design and optimization of the materials with regard to project-specific conditions.

Figure 5.58, left, schematically shows the compression tests performed on cylindrical specimens ($h_{0}=300$ mm, $D=150$ mm) with lateral confinement (i.e. complete transverse strain constraint) in a fully-enclosed, rigid specimen holder. As an example, Fig. 5.58, left, shows the structure of the material before and after compression.

In Fig. 5.58, right, the stress-deformation behavior of selected materials is presented, which have been compressed over the full cross-sectional area ($D_{s}=148$ mm), indicating the influence of the water-cement ratio ($w/c=0.7$, 1.4) and different additives (expanded glass, expanded polystyrene). Both variations clearly showed an influence on the stress $\sigma_{\text{pl}}$, which defines the transition from the linear-elastic to the plastic deformation stage. A lower water-cement ratio led to a higher matrix strength and thus to a higher value of $\sigma_{\text{pl}}$. The multiple higher strength of expanded glass in comparison to expanded polystyrene also led to an increase in $\sigma_{\text{pl}}$. The slope of the curves in the plastic stage was mainly influenced by the properties of the used additives. As can be seen, the stress-deformation curves of the materials with expanded glass were significantly steeper than those of the materials with expanded polystyrene in this stage. In contrast, the matrix strength had a comparatively small influence on the slope of the curves in the plastic deformation stage.

Fig. 5.58

Schematic illustration of a compression test and structure of the material before/after compression (left); influence of water-cement ratio and different additives on the stress-compression behavior (right) [9]

Under realistic conditions, it has to be assumed that deformations due to swelling pressures do not occur uniformly over the entire circumference of the tunnel, but rather locally. Against this background, also compression tests were carried out in which the load or deformation, respectively, was applied over a partial area (Fig. 5.59, left). As can be seen in Fig. 5.59, right, on the one hand, an increase in $\sigma_{\text{pl}}$ was observed with decreasing of the loaded area. This increase can be mainly attributed to the shear resistance of the material, which was activated by the penetration of the loading punch into the specimen [60]. On the other hand, the effect of the densification process leading to a steeper increase in the stress-deformation curve, occurred with increasing loaded area at smaller deformations.

Fig. 5.59

Section images of compressed specimens with different loaded areas (left) and influence of the loaded area on the stress-compression behavior (right) [60]

In addition to the fundamental investigations into compressive behavior, specific process and manufacturing aspects were also considered in the material design. For example, as the annular gap grout is pressed into the resulting gap between the outside of the segment and the surrounding soil, supplementary requirements have to be considered: sufficient flowability, pumpability, complete filling of the gap, none precompression of the highly porous additives due to the grouting pressure.

While the fundamental deformation behavior is the same for the annular gap grout and the compression layer, the formulation of the materials differ in order to satisfy the respective general requirements. The annular gap grout contains ingredients such as bentonite (to ensure sedimentation stability), retarders (to ensure a sufficiently long flowability) and a considerably higher water content, thus leads to a reduced strength of the mortar in contrast to the compressible layer.

In order to simulate the grouting and to take into account all the above needs, the test setup developed for evaluating the process qualification of annular gap grouts (Sect. 5.5.3) was modified and adapted to the compressive material (Fig. 5.60). For this purpose, an extended prototype of the static mixer was first developed, which could be considered for field-oriented projects. The static mixer contains alternating spiral-shaped sections, each of the elements being at a 90° angle to the following element and providing the required mixing effects by the radial mixing process forming, the rotational circulation as well as by the flow section. In enhancement, a magnetic inductive flow meter could be introduced into the circuit for the grouting test in order to record the volume of the grouted material and compare it with the calculationally necessary material requirement. This provides a direct indication of whether compression of the material is already occurring during the grouting process as a result of excessively high grouting pressures. Premature compression of the material could lead to a decrease in the subsequent compression potential. Based on the reference mix M0 (cf. Table 5.2), compressible two-component grouts (M0.mod) with varying porous additives (e.g. high air void ratio, micro glass bubbles, expanded polystyrene and vermiculite), were developed and subjected to a grouting test with set pressure of approx. 1.0 bar. Figure 5.60 includes the modified test setup with adapted static mixer and flow meter as well as a cut surface of a grouted two-component mortar with expanded polystyrene as porous additive.

Fig. 5.60

Design of the modified test set-up with adapted static mixer (1) and flow meter (2) for investigating the process qualification of compressible annular gap grouts (left). Detailed view of a cut surface on a two-component grout with expanded polystyrene as compressible additive (set pressure of approx. 1.0 bar) (right)

As seen in Fig. 5.61, left, the flow measurement during a grouting test demonstrated, with a relatively constant flow rate of approx. 8 l/min, that the calculated volume of 135 l was reached within the scheduled time of approx. 17 minutes at an advance rate of 90 mm/min. Thus, it can be ensured that no additional material was grouted as a result of an early compression of the grout. Furthermore, no significant decrease in deformation potential was observed for the grout. Figure 5.61, right, compares the stress-deformation behavior (of different mortars with various compressible additives) of the laboratory specimens prepared by hand and the samples of the grouted material taken from the test setup. Relative to the manually prepared reference specimens, the grouted mortar specimens showed a low slope in the linear-elastic range. Accordingly, the laboratory specimens indicated a minor increase in stiffness.

Fig. 5.61

Volume flow measurement during a grouting test (left). Comparison of the stress-compression behavior of manually produced reference specimens to grouted two-component mortar taken from the test setup (with varying porous additives) (right)

The knowledge gained in compressible cementitious materials was applied in form of a compressible layer on top of the lining segments tested in large scale, introduced in Sect. 5.4.1. As an example, Fig. 5.62 shows the stress-displacement response in radial direction of a lining segment with compressible layer in one of the experiments. The plastic behavior allowed for high deformations (> 100 mm for a compressible layer thickness of 175 mm) at a load level considerably lower than the ultimate load attained by the lining segment. As a result the lining segment was initially subjected to low utilization ratios.

Fig. 5.62

Compressible cementitious layer before (left) and after testing (middle); stress-displacement response in radial direction of a tested lining segment with compressible layer (right)

5.6.3 Model Based Design of Deformable Materials

The choice of a suitable deformable cementitious layer with the required deformation characteristics for certain geological conditions can be supported by performing computational simulations. The computational model should account for the material properties of the individual composite components, the geometry of the constituents, their volume and spatial distribution, in order to replicate the three stages characterizing overall behavior of compressible composites (an elastic stage, a plateau and a densification stage).

To this end, two different strategies for numerical description of highly compressible cementitious composites have been developed and analyzed: 1) a numerical strategy using the Discrete Element Method and 2) a multiscale approach based on continuum micromechanics.

5.6.3.1 Discrete Element Method

Simulations using the Discrete Element Method (DEM) can be used to model the underlying physical mechanisms such as a pore collapse occurring on the microscale of cementitious composites subject to compression.

In DEM models, the material is described as an assembly of discrete particles which interact and exert forces on each other. The dynamics of these DEM particles is governed by the second law of Newton. Within the DEM framework, cementitious materials are modeled as packing of discrete particles linked together by cohesive frictional forces. The induced forces are transmitted via a network of contacts between these particles. The current DEM formulation is based on the soft-sphere approach, where a contact is characterized by an interaction between overlapping spherical particles. The elastic bond between two discrete particles is characterized by a normal and tangential stiffness moduli. The interaction of two discrete particles in normal direction under tension is governed by a damage softening law, which enables one to replicate the behavior of cementitious materials in tension. The normal compression regime is characterized by an elasto-plastic constitutive law. Shear behavior is defined by a modified Mohr-Coulomb frictional law. Within this DEM framework, a highly porous cementitious composite material has been generated (Fig. 5.63, left) [89].

Fig. 5.63

Left: Comparison of the cut surface of the experimental sample with the generated numerical DEM model. Middle: Comparison of simulation results with experimentally obtained behavior in confined compression test. The experimental curve (green dotted line) has been used for calibration of the modeled behavior (red line). The behavior of another mix (yellow dotted line) has been used to validate the model prediction (blue line). Right: Damage distribution in a numerical specimen at two stages of compressive loading [89]

The DEM model has been calibrated for a cementitious mix of certain volume fraction of soft inclusions and validated for mixes of different inclusion volume fractions (Fig. 5.63, middle) [89]. According to the computational model, diffuse cracking occurs initially within the whole specimen arbitrarily. Damage initiation occurs at the top part of the sample where a compression band, characterized by compaction, is formed as well. Consequently a material compaction gradient can be clearly observed in Fig. 5.63, right. Simulation shows that most pores collapse ‘‘layer by layer’’. When all contacts of a discrete particle are ‘‘broken’’, it starts dynamically interacting with the other particles by means of newly created interaction (‘‘collision’’ of particles). The particles thus rearrange and fill the voids. This mechanism manifests itself at the specimen scale as a plateau behavior of the stress-strain curve also observed in the experiments. After all voids have experienced collapse, a densification process, characterized by a regain in stiffness, starts, which agrees with the laboratory observations.

DEM modeling enables one to capture the main physical mechanisms associated with compaction processes occurring in the cemenitious composite specimens. However, the composite microstructure of the investigated cementitious compositions is characterized by inclusions (pores, air voids, and EPS beads) at multiple scales. Thus, it requires a tremendous amount of very small discrete particles to resolve the topology of the small inclusions. Therefore, the prediction of the stiffness for materials characterized by inclusions at multiple scales is difficult. This issue can be solved by applying a multiscale approach based on continuum micromechanics.

5.6.4 Multiscale Model

Continuum micromechanics (as opposed to the numerical strategy using DEM) can resolve the complete range of inclusions and is computationally cheaper. A multiscale voxel model based on continuum micromechanics has been developed, where an individual voxel representing a particular volume of the material can compact (i.e. undergo irreversible deformation due to a pore collapse). The multiscale appoach includes three observation scales: the scale of the specimen (i.e. the dm scale), the scale of an individual voxel (i.e. the cm scale) and an analytical RVE (Representative Volume Element) scale (i.e. the mm scale) shown in Fig. 5.64, left.

Fig. 5.64

Left: The continuum micromechanics based multiscale strategy with three observation scales: the scale of specimen (i.e. the dm scale), the scale of an individual voxel (i.e. the cm scale) and an analytical Representative Volume Element scale (i.e. the mm scale). Right: Comparison of the model predictions with the experimental stress-deformation behavior of a set of compressible cementitious materials studied in Sect. 5.6.2

Each voxel is characterized at the lower scale by an RVE of cement paste matrix with embedded inclusions of zero stiffness. The RVE defines the strength and stiffness of the voxel. The voxel compacts when the strength of the corresponding RVE is reached. The strength is characterized by a damage initiation in the cement paste around the inclusion. Hexahedral finite elements are used to model the voxels, whereas the analytical framework of continuum micromechanics is used for description of the RVE and the processes occuring in it.

The RVE strength is defined by the stress field $\boldsymbol{\sigma}_{m}(\mathbf{x})$ in the cement paste matrix in the vicinity of the inclusion, which can be determined by localization. As soon as at any point $\mathbf{x}$ around the inclusion, the stress reaches the tensile strength of the cement paste $f_{t}$, damage is assumed to initiate and the voxel compacts. The elasticity parameters of the cement matrix and the microscopic tensile strength of the cement paste $f_{t}$ are assumed to be equal for all voxels. The inclusion volume fraction differs from voxel to voxel. It is obtained from a normal distribution among all voxels (Fig. 5.64, left).

Given the volume fraction of the soft inclusions and the properties of the cement paste matrix (determined within the experimental program), the multiscale approach enables one to predict the stiffness and the transitional stress $\sigma_{\text{pl}}$ of the cementitious composite material. Figure 5.64, right, provides comparison of the model predictions with the experimental stress-deformation behavior of a set of compressible cementitious materials studied in Sect. 5.6.2.

5.6.5 FE Simulations on Structural Scale

The choice of the deformable material for the protective layer around the tunnel lining is analyzed on the structural scale. Figure 5.65 presents the tunnel geometry, the boundary conditions together with the loading scenario and the mechanical properties of the Opalinus Clay [15] used in the simulation. Earth pressure is applied first and followed by the swelling pressure of 2 MPa (see Sect. 5.6.1) uniformly distributed on the tunnel.

Fig. 5.65

Left: Tunnel geometry, boundary conditions and the mechanical properties of Opalinus Clay. Scenario of swelling pressure uniformly distributed on the tunnel lining is presented. Right: Evolution of normal force in concrete lining and hydrostatic stress in compressible layer. The values are evaluated at the tunnel crown

The figure presents the normal force in the concrete lining at the crown. The black curve denotes the reference case without compressible layer around the tunnel. The green, red and blue curves denote the cases with compressible layer. The effect of three different compressible materials around the tunnel is analyzed. The materials have different stiffness and the transitional stress $\sigma_{\text{pl}}$ (see Sect. 5.6.2). It can be observed that the use of the compressible layer can reduce the normal force. The strength of the compressible material governs when the compressible layer yields, which leads to a reduction of the normal force in the tunnel lining. The yielding of the compressible material can be observed for the mix B (red line) in Fig. 5.65, right bottom, which depicts the hydrostatic stress in the compressible layer at the crown. The corresponding change in rate of the growth of normal force in the concrete lining at the crown can be observed in Fig. 5.65, right top. The choice of a too soft compressible material with low $\sigma_{\text{pl}}$ may not provide a sufficient tunnel support due to low normal forces in concrete lining (blue line in Fig. 5.65). In contrast, too stiff compressible material with high $\sigma_{\text{pl}}$ may not reach the yielding point which leads to high normal forces in the concrete lining (green dashed line in Fig. 5.65).

Figure 5.66 presents the second loading scenario where the swelling pressure is locally distributed on the tunnel lining. As compared to the case of the uniformly distributed swelling pressure, the locally applied load induces lower normal forces. That can be observed in the plot of normal force evolution at the invert of concrete lining (see Fig. 5.66, top right). The locally applied pressure induces, however, high bending moment in the concrete lining at the tunnel invert (see Fig. 5.66, bottom right). Simulations show that this bending moment can be reduced by using the compressible layer.

Fig. 5.66

Left: Tunnel geometry and boundary conditions. Scenario of a swelling pressure locally distributed on the tunnel lining is presented. Right: Evolution of normal force and bending moment in concrete lining at the tunnel invert