Skip to main content
SpringerLink
Log in

Dimensional analysis approach to study snap back-to-softening-to-ductile transitions in lightly reinforced quasi-brittle materials

  • Original Paper
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The analysis of lightly reinforced concrete beams taking into account the nonlinear behavior of concrete in tension is typically addressed by numerical approaches based on the Cohesive Crack Model. In the present paper, Dimensional Analysis is applied to this context, in order to obtain a synthetic description of the phenomenon by reducing the number of the governing parameters. It will be analytically demonstrated that the structural response is more complex than that obtained by the Bridged Crack Model and based on Linear Elastic Fracture Mechanics. In this case, in fact, two dimensionless parameters, N P and s, are responsible for the brittle-to-softening-to-ductile transitions in the mechanical response. The results of a parametric analysis and of an experimental campaign available in the literature are interpreted in this new light.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Baluch M, Azad A, Ashmawi W (1992) Fracture mechanics application to reinforced concrete members in flexure. In: Carpinteri A (eds) Applications of fracture mechanics to reinforced concrete. Elsevier Applied Science, London, pp 413–436

    Google Scholar 

  • Barenblatt GI, Botvina LR (1980) Incomplete self-similarity of fatigue in the linear range of fatigue crack growth. Fatigue Fract Eng Mater Struct 3: 193–202

    Article  Google Scholar 

  • Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge

    Google Scholar 

  • Bažant ZP (1989) Identification of strain-softening constitutive relation from uniaxial tests by series coupling model for localization. Cem Concr Res 19: 973–977

    Article  Google Scholar 

  • Bazant ZP, Beisel S (1994) Smeared-tip superposition method for cohesive fracture with rate effect and creep. Int J Fract 65: 277–290

    Google Scholar 

  • Bosco C, Carpinteri A (1992a) Softening and snap-through behavior of reinforced elements. J Eng Mech 118: 1564–1577

    Article  Google Scholar 

  • Bosco C, Carpinteri A (1992b) Fracture mechanics evaluation of minimum reinforcement in concrete structures. In: Carpinteri A (eds) Applications of fracture mechanics to reinforced Concrete. Elsevier Applied Science, London pp 347–377

    Google Scholar 

  • Bosco C, Carpinteri A (1995) Discontinuous constitutive response of brittle matrix fibrous composites. J Mech Phys Solids 43: 261–274

    Article  Google Scholar 

  • Bosco C, Carpinteri A, Debernardi PG (1990) Minimum reinforcement in high-strength concrete. J Struct Eng 116: 427–437

    Article  Google Scholar 

  • Brincker R, Henriksen MS, Christensen FA, Heshe G (1999) Size effects on the bending behaviour of reinforced concrete beams. In: Carpinteri A (eds) Minimum reinforcement in concrete members. Elsevier Science Ltd, Oxford, pp 127–180

    Google Scholar 

  • Buckingham E (1915) Model experiments and the form of empirical equations. ASME Trans 37: 263–296

    Google Scholar 

  • Carpinteri A (1980) Size effect in fracture toughness testing: a dimensional analysis approach. In: Sih GC, Mirabile M (eds) Analytical and experimental fracture mechanics. Sijthoff & Noordhoff, Alphen ann den Rijn, pp 785–797

    Google Scholar 

  • Carpinteri A (1981a) A fracture mechanics model for reinforced concrete collapse. In: Thurlimann B (ed) Advanced mechanics of reinforced concrete (Proc. IABSE Colloquium). Delft University Press, Delft, The Netherlands, pp 17–30

  • Carpinteri A (1981b) Static and energetic fracture parameters for rocks and concretes. Mater Struct 14: 151–162

    Google Scholar 

  • Carpinteri A (1982) Notch sensitivity in fracture testing of aggregative materials. Eng Fract Mech 16: 467–481

    Article  Google Scholar 

  • Carpinteri A (1984) Stability of fracturing process in RC beams. J Struct Eng 110: 544–558

    Article  Google Scholar 

  • Carpinteri A (1985) Interpretation of the Griffith instability as a bifurcation of the global equilibrium. In: Shah SP (eds) Application of fracture mechanics to cementitious composites. Martinus Nijhoff Publishers, Dordrecht, The Netherlands, pp 287–316

    Google Scholar 

  • Carpinteri A (1989a) Post-peak and post-bifurcation analysis of cohesive crack propagation. Eng Fract Mech 32: 265–278

    Article  Google Scholar 

  • Carpinteri A (1989b) Cusp catastrophe interpretation of fracture instability. J Mech Phys Solids 37: 567–582

    Article  Google Scholar 

  • Carpinteri A (1990) A catastrophe theory approach to fracture mechanics. Int J Fract 44: 57–69

    Article  Google Scholar 

  • Carpinteri A, Massabò R (1996) Bridged versus cohesive crack in the flexural behaviour of brittle-matrix composites. Int J Fract 81: 125–145

    Article  Google Scholar 

  • Carpinteri A, Colombo G, Ferrara G, Giuseppetti G (1989) Numerical simulation of concrete fracture through a bilinear softening stress-crack opening displacement law. In: Shah SP, Swartz SE (eds) Fracture of concrete and rock. Springer, New York, pp 131–141

    Chapter  Google Scholar 

  • Carpinteri A, Ferro G, Ventura G (2003) Size effects on flexural response of reinforced concrete elements with nonlinear matrix. Eng Fract Mech 70: 995–1013

    Article  Google Scholar 

  • Carpinteri A, Ruiz Carmona J, Ventura G (2007a) Propagation of flexural and shear cracks through reinforced concrete beams by the bridged crack model. Mag Concr Res 59: 743– 756

    Article  Google Scholar 

  • Carpinteri A, Corrado M, Paggi M, Mancini G (2007b) Cohesive versus overlapping crack model for a size effect analysis of RC elements in bending. In: Carpinteri A, Gambarova P, Ferro G, Plizzari G (eds) Fracture mechanics of concrete and concrete structures. Taylor & Francis, Leiden, The Netherlands, pp 655–663

    Google Scholar 

  • Carpinteri A, Corrado M, Paggi M, Mancini G (2009) New model for the analysis of size-scale effects on the ductility of reinforced concrete elements in bending. J Eng Mech 135: 221–229

    Article  Google Scholar 

  • Ciavarella M, Paggi M, Carpinteri A (2008) One, no one, and one hundred thousand crack propagation laws: a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth. J Mech Phys Solids 56: 3416–3432

    Article  Google Scholar 

  • Comité Euro-International du Béton (1993) CEB-FIP Model Code 1990. Thomas Telford Ltd, Lausanne, Bulletin No. 213/214

  • Gustafsson PJ (1985) Fracture mechanics studies of non-yielding materials like concrete. Report TVBM-1007, Div Bldg Mater Lund Inst Tech, Sweden

  • Gustafsson PJ, Hillerborg A (1988) Sensitivity in shear strength of longitudinally reinforced concrete beams to fracture energy of concrete. J ACI 85: 286–294

    Google Scholar 

  • Hawkins NM, Hjorsetet K (1992) Minimum reinforcement requirements for concrete flexural members. In: Carpinteri A (ed) Applications of fracture mechanics to reinforced concrete. Elsevier Applied Science, London, pp 379–412

    Google Scholar 

  • Hillerborg A (1990) Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams. Eng Fract Mech 35: 233–240

    Article  Google Scholar 

  • Hillerborg A, Modeer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6: 773–782

    Article  Google Scholar 

  • Irwin GR (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24: 361–364

    Google Scholar 

  • Jansen DC, Shah SP (1997) Effect of length on compressive strain softening of concrete. J Eng Mech 123: 25–35

    Article  Google Scholar 

  • Jenq YS, Shah SP (1989) Shear resistance of reinforced concrete beams—a fracture mechanics approach. In: Li VC, Bazant ZP (eds) Fracture Mechanics: Application to Concrete (Special Report ACI SP-118). American Concrete Institute, Detroit, pp 327–358

    Google Scholar 

  • Kotsovos MD (1983) Effect of testing technique on the post-ultimate behaviour of concrete in compressione. Mater Struct 16: 3–12

    Google Scholar 

  • Markeset G, Hillerborg A (1995) Softening of concrete in compression—localization and size effects. Cem Concr Res 25: 702–708

    Article  CAS  Google Scholar 

  • Palmquist SM, Jansen DC (2001) Postpeak strain-stress relationship for concrete in compression. ACI Mater J 98: 213–219

    Google Scholar 

  • Phatak DR, Deshpande NP (2005) Prediction of 28  days compressive strength of 53-grade cements using dimensional analysis. J Mater Civil Eng 17: 733–735

    Article  CAS  Google Scholar 

  • Phatak DR, Dhonde HB (2003) Dimensional analysis of reinforced concrete beams subjected to pure torsion. J Struct Eng 129: 1559–1563

    Article  Google Scholar 

  • Planas J, Elices M (1992) Asymptotic analysis of a cohesive crack: 1. Theoretical background. Int J Fract 55: 153–177

    Article  Google Scholar 

  • Ruiz G, Elices M, Planas J (1999) Size effects and bond-slip dependence of lightly reinforced concrete beams. In: Carpinteri A (eds) Minimum reinforcement in concrete members. Elsevier Science Ltd, Oxford, pp 127–180

    Google Scholar 

  • Ruiz G (2001) Propagation of a cohesive crack crossing a reinforcement layer. Int J Fract 111: 265–282

    Article  Google Scholar 

  • So KO, Karihaloo BL (1993) Shear capacity of longitudinally reinforced beams—a fracture mechanics approach. J ACI 90: 591–600

    Google Scholar 

  • Suzuki M, Akiyama M, Matsuzaki H, Dang TH (2006) Concentric loading test of RC columns with normal- and high-strength materials and averaged stress-strain model for confined concrete considering compressive fracture energy. In: Proc. 2nd fib Congress, Naples, Italy

  • van Mier JGM (1984) Strain-softening of Concrete under Multiaxial Loading Conditions. PhD Thesis, Eindhoven University of Technology, The Netherlands

Download references

Author information

Authors and Affiliations

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

    M. Corrado, E. Cadamuro & A. Carpinteri

Authors
  1. M. Corrado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Cadamuro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Carpinteri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Cadamuro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Corrado, M., Cadamuro, E. & Carpinteri, A. Dimensional analysis approach to study snap back-to-softening-to-ductile transitions in lightly reinforced quasi-brittle materials. Int J Fract 172, 53–63 (2011). https://doi.org/10.1007/s10704-011-9646-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-011-9646-2

Keywords

Search

Navigation