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Evaluation of crack width in FRC structures and application to tunnel linings

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Abstract

When steel bars are placed in a concrete structure, the evaluation of crack width and crack spacing is generally required in the serviceability stage. According to more or less aggressive conditions, crack width shall be limited in order to avoid, for instance, the corrosion of steel reinforcement. The presence of fibers in the concrete cast may help to achieve this goal, since fibers remarkably increase the bridging actions across a crack. However, new mechanical models are needed to evaluate these effects, which are generally neglected by classical approaches. Code requirements are based on semi-empirical formulae, in which the average structural performances are analyzed by referring to a single cross-section, instead of a wide portion of an R/FRC or RC element in bending. To evaluate crack patterns more accurately, a suitable block model is therefore introduced in this paper. With the new approach, the bridging effects of fibers, as well as the bond-slip mechanism between steel bars and FRC in tension, are taken into account. By means of such model, it is possible ble to predict at one time the values of crack width, crack spacing, and crack depth, and compare them to data obtained by bending tests on concrete beams. Moreover, to evaluate the possible crack patterns in R/FRC tunnel linings, the proposed block model has been extended to the serviceability stage of massive structures subjected to combined compressive and bending actions. This paper follows a previous work by the same authors (Chiaia et al. Mater Struct 40(6):593–694, 2007) and completes the design procedures for FRC cast-in-place tunnel linings.

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Acknowledgments

The authors wish to express their gratitude to the Italian Ministry of University and Research for financing this research work (PRIN 2006), and to Bekaert for its continuous support.

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Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, 10129, Italy

    Bernardino Chiaia, Alessandro P. Fantilli & Paolo Vallini

Authors
  1. Bernardino Chiaia
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  2. Alessandro P. Fantilli
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  3. Paolo Vallini
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Corresponding author

Correspondence to Alessandro P. Fantilli.

Appendix I: Notations

Appendix I: Notations

The following symbols are used in this paper:

  • A c = Concrete area in a RC or R/FRC cross-section

  • A c,eff = Effective concrete area in tension

  • A s = Area of steel reinforcing bars in tension

  • \(A^{\prime}_{\text{s}} \) = Area of steel reinforcing bars in compression

  • A s,min = Minimum amount of steel reinforcing bars in tension

  • B = Width of the beam cross-section

  • c = Concrete cover

  • d a = Maximum aggregate size

  • E c = Young’s modulus of concrete

  • E s = Young’s modulus of steel

  • f c = Cylindrical compressive strength of concrete or FRC

  • f ck = Characteristic value of f c

  • f ct = Tensile strength of concrete or FRC

  • f ctm,fl = Average flexural tensile strength of concrete or FRC

  • f R,1, f R,4 = Residual strengths obtained from a three point bending test on notched FRC beams [8]

  • f y = Yield strength of the longitudinal steel reinforcing bars

  • f yk = Characteristic value of f y

  • H = Height of the beam cross-section

  • h w = Depth of a primary crack

  • k 1, k 2, k 3, k 4, k 5 = Non dimensional coefficients in Eqs. (2, 3)

  • L/D = Fiber aspect ratio (L = length of the fiber; D = diameter of the fiber)

  • l tr = Length of the block between type 1 and type 2 cross-sections

  • M = Bending moment applied to a beam cross-section

  • M rd = design bending moment resistance of a beam cross-section

  • M sd = Design bending moment applied to a beam cross-section

  • N = Normal force applied to a beam cross-section

  • n = Number of segments contained within the domain l tr

  • N rd = Design normal force resistance of a beam cross-section

  • N sd = Design normal force applied to a beam cross-section

  • P = Loads applied to a beam in four point bending

  • p s = Perimeter of steel reinforcing bars in tension

  • s = Slip between steel and concrete

  • s 1 = Slip between steel and concrete at τmax

  • s r = Crack distance

  • s r,m = Average value of crack distance

  • s r,max = maximum value of crack distance

  • w = Crack width at level of reinforcement

  • w k = Characteristic value of w

  • w max = Maximum value of w

  • y = Vertical coordinate

  • z = Horizontal coordinate

  • α = Exponent in Eq. (7)

  • β = Coefficient relating the average crack spacing to the design value

  • χ i  = Coefficient of similarity of Eqs. (8a, b).

  • Δz = l tr/n = Length of the ith part of the domain l tr

  • ε c(y) = Strains in concrete or FRC

  • ε cm = Mean strain in concrete or FRC at level of reinforcement between cracks

  • ε c,crit = Strain in concrete or FRC at level of reinforcement in the type 2 cross-section (Fig. 2b)

  • ε s = Strain in steel reinforcing bars in tension

  • ε sm = Mean strain in steel reinforcing bars between cracks

  • \( \varepsilon ^{\prime }_{\text{s}}\) = Strain in steel reinforcing bars in compression

  • Φ = Bar diameter

  • ρ = A s/A c,eff = Effective reinforcement ratio

  • σ c(y) = Stress in concrete or FRC

  • σ s = Stress in steel reinforcing bars in tension

  • \( \sigma ^{\prime }_{{\text{s}}} \) = Stress in steel reinforcing bars in compression

  • τ = Bond stress between steel and concrete

  • τ f = Residual bond stress between steel and concrete

  • τ max = Maximum bond stress between steel and concrete

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Chiaia, B., Fantilli, A.P. & Vallini, P. Evaluation of crack width in FRC structures and application to tunnel linings. Mater Struct 42, 339–351 (2009). https://doi.org/10.1617/s11527-008-9385-7

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  • DOI: https://doi.org/10.1617/s11527-008-9385-7

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