Fig. 3: Calculation of equilibrium between internal and external forces a cross section under typical tunneling conditions, there are two equations (refer to Fig. 3) N C T N M Cz T z M C T = − + = = − − + = 0 0 0 0 The internal lever arm z, as well as the height of the compression zone x and height of the tension zone y are calculated as follows z d a z d a C C T T = − = − 2 2 x c d y d + c t t + c t = for 0 and c t 0 = Because only two equations for the equilibrium are available, all but two variables must be known to compute a unique solution. However, at first there are four unknowns: the resulting thrust (C), the resulting tensile force (T), and their respective levers (zC, zT). All four unknowns are directly related to the existing strain condition. By selecting a specific strain condition, the moment capacity, as well as the normal force capacity, can be calculated and the result is unique. Theoretically, the reversed approach—selection of the external forces followed by the calculation of the corresponding strain condition—is possible. However, this solution is practically not achievable because, typically, an SSR of FRC is discontinuous and depends on numerous parameters. In addition, the solution often provides multiple equilibriums and is therefore not unique.9 As a result, an iterative process is necessary to solve the equations, which requires a lot of computation effort.14 LOAD-BEARING BEHAVIOR AND DESIGN OF FRC UNDER COMBINED M/N LOADING Moment-normal force interaction diagrams (MNID) are typically used during the design of tunnel linings (and columns under combined M/N loading in general) for steel bar reinforced linings as well as FRC. However, while generic MNID are available for bar-reinforced members, an SSR-specific MNID has to be developed for FRC. Generic MNIDs for FRC can be developed in a similar fashion to bar-reinforced members if the diagrams are dimensionless and all strength values are defined (that is, relative to the compressive strength fc.) The dimensionless factor n = N/(fc × b × d) can hereby be interpreted as the use toward the maximum thrust under pure compression. The SSR used for the following parameter studies is defined in Table 1 and represents Table 1: Stress-Strain Relationship used in the Parameter Study Stress Tension Compression ft3 ft2b ft2a ft1 fc1 fc2 N/mm2 0.5 1.0 4.0 4.0 –40.0 –40.0 % of fc 1.25 2.5 10 10 100 100 Strain et3 et2b et2a et1 ec1 ec2 ‰ 10.0 0.16 0.16 0.12 –2.0 –3.5 www.shotcrete.org Spring 2017 | Shotcrete 31

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