Fig. 4: Moment normal force interaction diagram (MNID) Fig. 5: MNID—enlarged section relevant for tunneling typical values for FRC (that is, an initial tunnel lining.) For the nomenclature and shape, refer to the SSR in Fig. 2. The thickness of the lining was assumed to be 10 in. (0.25 m) and a 3.2 ft (1.0 m) wide tunnel lining section was assumed. Figure 4 shows the complete MNID for the SSR presently used, while Fig. 5 shows an enlarged section of the same MNID. Focus on investigations are typically lines with a constant normal force, parallel to the x-axis, or lines with a constant “e/d ratio,” which are inclined and intersecting the origin of the MNID. The dimensionless e/d ratio is defined as the eccentricity e over beam height d, with e = M/N. Figure 6 shows the results of a parameter study with varying e/d ratios; Fig. 7 shows a parameter study for varying normal forces. It is important to highlight that all figures are basically dif ferent ways of displaying the bearing capacity of the same material, defined in Table 1. The results represented—for example, along the e/d = 0.5 line in Fig. 5—are the same as shown in Fig. 6 for the identical case. The results represented in Fig. 5 on a line with a constant normal force, parallel to the x-axis—that is, N = 1000 kN—is the same as shown in Fig. 7 Fig. 6: Moment-strain diagram, parameter study e/d ratio Fig. 7: Moment-strain diagram, parameter study normal force for the same thrust. The bearing behavior under pure bending (N = 0; e/d = ∞) is represented on the x-axis in the MNID. The example shown could also be transferred into a generic dimensionless MNID by using the following equations for the normal force and the moment n N f b d m M = × × 2 f b d c c = × × The residual strength ft1 on the tension side can be expressed as a percentage of the compressive strength (refer to Table 1). The dimensionless MNID would be valid for all cases where the ratios between the tensile strength and the compressive strength are kept the same. The different lines in the MNID show cross section equilibriums for specific constant tension strains. A good rule of thumb is that tunnel linings are typically using between 5 and 30% of the compression capacity of a member.9 So, for example, in the MNID in Fig. 4, a typical use in a soft ground tunnel would be between 0.5 and 3 MN and only the lower one-third of the diagram would be relevant for the design; Fig. 5 shows this area of the MNID enlarged. Tunnel linings in this part of an MNID generally fail under tension by reaching the maximum allowable tensile strain εt3. The different lines in the MNIDs represent lines of specific strains (Fig. 4 and 5). Left of the line marked with εt = 0‰ – 32 Shotcrete | Spring 2017 www.shotcrete.org

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