Modeling of Load-Bearing Behavior of Fiber-Reinforced Concrete Tunnel Linings By Axel G. Nitschke The use of fiber-reinforced concrete (FRC)—with steel or macro-synthetic fibers—has technical and economic advantages that primarily stem from the fact that fibers transform the post-cracking behavior from a brittle failure mode typical of unreinforced concrete into an elasto-plastic behavior. Numerous codes and guidelines provide qualitative or quantitative design approaches.1-6 Modeling of the load-bearing behavior based on a stressstrain relationship (SSR) for tunneling applications is commonly used. This article discusses the modeling process and some typical results of a parameter study. It also identifies the weakness of the current concept and suggests a path to more fully use the structural and economic potential of FRC. The concept discussed herein is theoretical in nature and applicable for both steel and synthetic FRC. To limit the scope of this article, the discussion is focused on the load-bearing capacity under cracked conditions, which is typical for shotcrete initial linings. Therefore, design concepts that do not use the toughness potential of FRC (that is, by limiting it to uncracked conditions) are not discussed herein. Different international codes and guidelines for FRC provide testing procedures based on simply supported beam tests that are used to define an SSR by basically amending the known trapezoidal or parabolic SSR for concrete on the compression side with assumptions for an SSR on the tension side. The latter is the primary subject of this article. For this discussion, it is irrelevant which type of macrofibers— steel or synthetic—is used because the SSR models a homogeneous, composite material behavior and not discrete fibers. In general, the SSR design approach follows the concept to adapt existing concrete design concepts and procedures and simply extend the SSR on the tension side to account for the effect of the properties of the composite material. This article is focused solely on combined moment thrust or moment normal force (M/N) loading of tunnel linings in which bearing capacity relies on a tunnel arch. This is typical for soft ground tunnel linings and rock tunnels with soft-ground-like behavior. Nonetheless, the ideas and concepts can also be adapted in typical rock tunneling applications. However, they are not useful in tunnels with no arching effect, which is typical for tunnels with relatively thin linings or with an irregular shape. For these types of tunnels (that is, typical initial linings in classical rock tunneling), qualitative and empirical design concepts (for example, Barton chart7,8) are available but are not discussed in this paper. The use of an SSR is typically evaluated on the basis of beam test data. Under elastic (uncracked) conditions, the beam theory and the classical mechanics for materials apply. However, after the initial cracking of the FRC, the material is no longer homogeneous and the theoretical conditions for beam theory no longer apply. The bearing behavior of FRC in beam tests in a cracked state are better described using a stress-crack width relationship rather than a stress-strain relationship. It is important to understand that for the aforementioned reason, an SSR cannot be measured directly in standard FRC beam tests. The codes and guidelines are therefore describing testing procedures that measure external forces and deformations, which are then transformed into stresses and “equivalent” strains via an equivalence model, which implies several assumptions. Research by Nitschke9 has discussed flaws in some of these models by back-calculation of tests using the SSRs. It was shown that these flaws can be significant under loading conditions of combined moment and thrust, typical for tunneling. The same work also provided modified models to provide more useable and accurate procedures.9 STRUCTURAL BASICS OF FRC DESIGN The biggest difference between the sectional strength of unreinforced or steel bar reinforced concrete and FRC is that the concrete in unreinforced or bar-reinforced concrete has (theoretically) no bearing capacity in tension. In the modeling of conventionally reinforced concrete sections, all tension is supported by the reinforcing bar. Because the location of the reinforcing bar is known, the location of the resulting tensile force is also known, and this simplifies the calculation of the equilibrium compared to FRC sections. The computation of axial equilibrium in FRC sections is much more challenging because the location of the resulting tensile force is an unknown during the computation and moves if the external load and the distribution of the strain over the cross section changes. 28 Shotcrete | Spring 2017 www.shotcrete.org

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