tensile strain zero – all members are under full compression. The εt = 0.12‰ respectively εt = 0.16‰-lines represent the end of the elastic behavior (Phase I/II) respectively the beginning of macro-cracking (Phase II/III). The εt = 10.0‰ line represents equilibriums reaching the maximum tensile strain defined for this example. Following this line into the area between N = 0 kN and N = 500 kN (Fig. 5) shows the area of pure bending in which axial thrust has a very small influence. In this area, the bearing capacity in the elastic state (εt = 0.12‰) and the microcracked state (εt = 0.16‰) is larger than the load in the cracked phase (0.16‰ < εt ≤ 10.0‰). That means the ultimate load is less than the peak load reached around the elastic/uncracked state (also refer to Fig. 6, e/d = ∞, 2.0, 1, 0, and 0.75 and Fig. 7, N = 0, 250, 500 kN). This shows the typical strain-softening behavior of FRC in bending tests (refer to Fig. 1). The intersection of the εt = 0.16‰ – line and the εt = 10‰ – line in the MNID marks the point of quasi ideal elasto-plastic behavior, meaning the maximum load level reached under uncracked or microcracked conditions can be maintained, which is reached in this example at roughly N = 500 kN (Fig. 1, 5, and 7). The simulated examples, which are representative of behavior observed in tests, also show that the momentbearing capacity in the elastic, the microcracking, and the cracked phase are all increased under the influence of an increasing normal compressive force. While typical FRC simply supported bending tests show a strain-softening behavior, it can be observed that an increased normal force leads to a quasi elasto-plastic and a quasi strain-hardening effect. The term “quasi” is used because the bending behavior is a characteristic of the structural system; material properties do not actually change. For the same material, the bearing behavior changes with an increased normal force influence (Fig. 1, 5, 6, and 7). Figure 1 shows the difference between elastic-brittle, elasto-plastic, and strain-hardening behavior, and strain softening in a simplified manner. Strain-softening behavior is typical for pure bending; the moment-bearing capacity decreases after the peak load. An increased normal force influence leads to nearly elasto-plastic system performance (in our example for N ≈ 500 kN refer to Fig. 7 and 0.5 < e/d < 0.75 refer to Fig. 6), respectively) and under a further increased normal force influence, the behavior progresses to a quasi strain-hardening effect. In addition to a change in failure mode, represented by the shape of the curve, there is also an increase in the peak moment capacity. While in the example the maximum bearable moment at 10‰ tensile strain is around 20 kNm, every 100 kN additional normal force increases the moment-bearing capacity by roughly 10 kNm in this example. FRC SPECIFICATION BASED ON RANGE OF NORMAL FORCE What do these results mean for a tunnel lining design and FRC specification? The general desire from a structural perspective for a tunnel lining design under cracked conditions requires that the bearing capacity under cracked conditions shall be equal to or higher than the bearing capacity in the elastic state. Referring to Fig. 1, the behavior shall be at least “elasto-plastic” or display “strain hardening.” In the previous section, it was shown that these conditions are highly dependent on the amount of normal force in the system. However, current tunnel designs do not take the range of expected normal force into consideration when specifying the material properties of FRC. Therefore, a lot of structural potential of FRC remains underused. If elasto-plastic behavior or strain-hardening behavior is desired, material specifications should take the expected range of normal force, represented by the mean compressive stress in the lining, into consideration. The range of expected normal force, respectively the compressive strength in a lining, can be easily evaluated based on preliminary lining designs. As shown in the parameter study herein, a project-specific SSR could be specified that meets or exceeds the requirements. Subsequently, pure bending or tests under combined moment-normal force could be used to prove that the SSR requirements could be met. Rather than the absolute values for the residual strength, it is suggested to specify an SSR with strength values relative to the compressive strength of the material. INELASTIC STRUCTURAL ANALYSIS USING PLASTIC HINGES ACI 318 and other international codes provide several options for a structural analysis of reinforced concrete structures. In a typical tunnel design the forces of the lining are determined in a linear-elastic model. Representative pairs of moments and normal forces from this analysis are then transferred into a MNID to ensure that the load combinations can be born by the FRC lining. As discussed previously, the inclusion of fibers increases the moment-bearing capacity compared to unreinforced concrete when a section is subject to a large compressive normal force. However, typically even light steel bar reinforcement can do the same or even exceed the bearing capacity of FRC. Where, then, is the benefit of FRC in the structural design? The benefit of FRC lies in the added toughness of the material, which allows—under elastoplastic or strain-hardening conditions—to “hold” a moment in a lining even under severe deformation of the lining. However, these benefits are not used in a standard linear-elastic structural analysis. The structural and economic potential can be activated in an in-elastic structural analysis using, for example, a concept typically used for a simplified method for an in-elastic design of steel frames. Structurally a cracked FRC lining acts like a “plastic hinge,” which still transfers a moment while rotating. In a classical elastic analysis, the capacity of the plastic hinges could be used as follows for a quasi inelastic procedure: While increasing the load on a tunnel lining, the peak elastic moment will be reached at a specific point. The elastoplastic or strain-hardening behavior would allow for the introduction of a hinge at this location and altering the overall static (elastic) system of the lining. In a next step, the external www.shotcrete.org Spring 2017 | Shotcrete 33

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