The design assumptions for the calculation of the sectional strength for FRC based on an SSR can be summarized as follows10 (ACI 318-14, Section 22.2,11 ACI Design Handbook, Section 7.412): 1. Equilibrium shall be satisfied at each section; 2. Strain in the cross section of the member shall be assumed directly proportional to the distance from the neutral axis (Bernoulli’s theorem). The cross section also remains plane during loading; 3. The stress-strain relationship for the FRC in compression is defined; thus, the stress for a given strain is known within defined limits; and 4. The stress-strain relationship of the FRC under tension is defined; thus, the stress for a given strain is known within defined limits. A comparison of the essential design assumptions for moment and axial strength at sections for reinforcing bar reinforced concrete design in ACI 318 shows that the first and second assumptions—equilibrium (ACI 318-14, Section 22.2.1.111) and Bernoulli (ACI 318-14, Section 22.2.1.211)— are adapted for FRC. However, by citing two additional design assumptions from ACI 318, two major differences between FRC and classical bar reinforced designs assumptions can also be highlighted. According to ACI 318-14, Section 22.2.2.211: “Tensile strength of concrete shall be neglected in flexural and axial strength calculations.” For sectional strength calculation of FRC, the tensile strength under uncracked as well as cracked conditions is used. This is one of the major differences between the modeling of FRC in comparison with unreinforced or barreinforced concrete. According to ACI 318-14, 22.2.1.211: “Strain in concrete and non-prestressed reinforcement shall be assumed proportional to the distance from the neutral axis.” This design assumption is based on the hypothesis of perfect bonding between steel and concrete. While barreinforced concrete is modeled as a composite of concrete and steel, where each component has its own material properties (refer to ACI 318-14, Section 22.2.2, for concrete and Section 22.2.3 for non-prestressed reinforcement11), FRC is assumed to be a macroscopically homogeneous and isotropic material.13 The material properties of a single fiber in the model becomes irrelevant. Therefore, the fibers and the concrete are modeled using a single SSR relationship and not two (that is, as for steel reinforcing bar and concrete). After the cracking of the FRC material under tension, the material properties in the model are based on strains rather than a discrete crack. In the model, the cracked material is also viewed as homogeneous and isotropic. Because this is in the area around the crack, it is obviously not the case. This circumstance is very important to realize and understand when evaluating the sectional strength of FRC using an SSR. During the evaluation of material testing data based on beam tests (and subsequently the design of the structure), it is assumed that the crack is “smeared” over a certain Fig. 1: Schematic post-peak response of fiber-reinforced concrete length into an “equivalent strain,” which is also referred to as the “integral approach.”10 Fibers influence the bearing behavior in multiple ways. However, three properties are most relevant for application in tunnels.13 They slightly increase the flexural tensile strength (1), which is mostly needed if improved properties under uncracked conditions are desired (that is, to design for serviceability.) However, for the case of ultimate bearing capacity of tunnel linings, the residual flexural tensile strength under cracked conditions (2) and the increase of the toughness (3) are the major benefits. The focus of this paper is on the performance improvements attributable to (2) and (3). The provision of a reliable and usable post-cracking tensile strength transforms the brittle failure mechanism of plain concrete into a ductile failure mode (refer to Fig. 1). This is a material property that provides major engineering and economic advantages, especially if used to facilitate system failure of a tunnel lining rather than a cross section failure at one presumably most-critical location. A concept for the design of a system failure will be presented later in this article. According to Dietrich,10 the load-bearing response of FRC under bending can be subdivided into three phases. The first “uncracked” phase is based on the behavior of the concrete matrix alone. The concrete matrix and fibers are assumed to be in “perfect bond” and the ratio of load supported by the concrete compared to the fibers is dependent on the moduli of elasticity of the materials. Due to the relatively small volume of fibers compared to concrete, the load-bearing share of the fibers is relatively small. Microcracks develop in the matrix during the second phase of load response. The development of cracks is hampered by the fibers and leads, according to Dietrich,10 to a more stable “strain softening” with a restricted expansion of cracks and less brittle material behavior. Phase two ends with crack widths of approximately 0.004 in. (0.1 mm).10 In the third phase, the concrete matrix no longer provides significant bearing capacity at the crack. The opening cracks are bridged by the fibers and the load transfer is effectively provided by the fibers alone. www.shotcrete.org Spring 2017 | Shotcrete 29

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