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2017SpringShotcreteEMag

SECTION DESIGN OF FRC USING STRESS-STRAIN RELATIONSHIP The three phases of crack development are also reflected in SSRs found in different codes and guidelines. Studies by Nitschke9,15 have shown that by using all three phases in computer simulations, test results obtained using beam tests can be simulated very accurately. Typically, all SSRs in codes and guidelines incorporate Phase I (elastic) and Phase III (macrocrack) behavior. However, because the distinction between uncracked and microcracking in Phase II is not clearly defined, Phase I and Phase II are oftentimes lumped together or Phase II is completely neglected.9 It is important to note that for the modeling of ultimate loadbearing capacity in the macrocracked phase, a detailed evaluation of the microcracking Phase II is irrelevant. However, it might be significant for serviceability design. A generic SSR and nomenclature of the variables used throughout this paper is shown in Fig. 2. The tension side is represented by the three sections discussed previously. The compression side uses a classical parabolic constant shape. Nitschke9 has conducted numerous simulations of beam test results under pure bending as well as combined M/N loading. The three load-bearing phases observed during the experimental studies could also be reflected with the simulation of the load-bearing behavior, using the SSR as follows. In general, it is possible to identify “typical” SSRs based on typical load-deflection curves from either tests or the simulation of results. By adhering to certain boundary conditions, it is almost possible to look at each of the three phases separately.9 The pure elastic (uncracked) behavior is related to the first part of the stress-strain relationship and conforms to the principles of elastic bending. The flexural strength ft1 results from the maximum elastic moment divided by the section modulus. The range of the related strain εt1 is very limited and can either be measured during the test or— based on the used SSR—be calculated using the original modulus of elasticity. Alternatively, and if the major focus of the interest is the bearing capacity under cracked conditions, a generic value between 0.1‰ ≤ εt1 ≤ 0.15‰ (100 to 150 microstrain) will yield sufficiently accurate results because the overall influence of the elastic section on the bearing capacity under cracked conditions is diminished.9 The interim section of microcracking is reflected by the second section of the SSR on the tension side. In general, two different types of curves are used between εt1 and εt2: 1) a plateau; or 2) a linearly decreasing curve (trapezoid). By using a plateau, the stress in the second section is constant (ft2a = ft1) (refer to Fig. 2). In general, the plateau creates loaddeformation curves with a distinct maximum and a “hard” decline of the moment-bearing capacity in pure bending conditions. On the other hand, a declining curve in the second section (ft2a ≥ ft2b) (refer to Fig. 2) “softens” this area of the moment-deflection curve.9 More complex curves can be used in the second section; however, the two selected types may encompass many other cases. As parameter studies have shown,9 the overall influence of the second section of the SSR controls the shape of a specific area of the simulation of the bearing capacity but has only a small influence on the overall bearing capacity. It was also shown that more important than the value of the stress ft2b is the specific strain εt2, which controls the shape of a moment-deflection curve in this area.9 However, by far the biggest influence on the load-bearing behavior under cracked conditions is the third section of the SSR. The tensile stress under cracked conditions is typically referred to as the “residual strength.” Under consideration of the conducted beam tests with a maximum deflection of 0.14 in. (3.5 mm), SSRs up to a strain of εt3 = 25‰ were investigated. The load-bearing capacity of a cross section based on the SSR is calculated by finding the equilibrium between internal and external forces. Only a discussion of the basic principle is covered in this paper. A complete solution for the calculation of the inner forces resulting from a specific strain scenario is provided by Nitschke.9 For the calculation of equilibrium between internal and external forces acting on Fig. 2: Generic stress-strain relationship for fiber-reinforced concrete 30 Shotcrete | Spring 2017 www.shotcrete.org


2017SpringShotcreteEMag
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