Deterministic and Statistical Size Effect in Plain Concrete

  • Jacek Tejchman
  • Jerzy Bobiński
Part of the Springer Series in Geomechanics and Geoengineering book series (SSGG)


The numerical FE investigations of a deterministic and stochastic size effect in concrete beams of a similar geometry under three point bending were performed within an elasto-plasticity with a non-local softening. The FE analyses were carried out with four different sizes of notched and unnotched beams. Deterministic calculations were performed with a uniform distribution of the tensile strength. In turn, in stochastic calculations, the tensile strength took the form of random correlated spatial fields described by a truncated Gaussian distribution. In order to reduce the number of stochastic realizations without losing the calculation accuracy, Latin hypercube sampling was applied. The numerical outcomes were compared with the size effect law by Bažant and by Carpinteri.


Concrete Beam Localize Zone Latin Hypercube Sampling Plain Concrete Nominal Strength 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ABAQUS, Theory Manual, Version 5.8, Hibbit. Karlsson & Sorensen Inc. (1998)Google Scholar
  2. Bažant, Z.P.: Size effect in blunt fracture: concrete, rock, metal. Journal of Engineering Mechanics ASCE 110(4), 518–535 (1984)CrossRefGoogle Scholar
  3. Bažant, Z.P., Lin, K.L.: Random creep and shrinkage in structures sampling. Journal of Structural Engineering ASCE 111(5), 1113–1134 (1985)CrossRefGoogle Scholar
  4. Bažant, Z.P., Chen, E.P.: Scaling of structural failure. Applied Mechanics Reviews 50(10), 593–627 (1997)CrossRefGoogle Scholar
  5. Bažant, Z., Planas, J.: Fracture and size effect in concrete and other quasi-brittle materials. CRC Press LLC (1998)Google Scholar
  6. Bažant, Z., Novak, D.: Proposal for standard test of modulus of rupture of concrete with its size dependence. ACI Materials Journal 98(1), 79–87 (2001)Google Scholar
  7. Bažant, Z.: Probability distribution of energetic-statistical size effect in quasibrittle fracture. Probabilistic Engineering Mechanics 19(4), 307–319 (2004)MathSciNetCrossRefGoogle Scholar
  8. Bažant, Z.P., Yavari, A.: Is the cause of size effect on structural strength fractal or energetic-statistical? Engineering Fracture Mechanics 72(1), 1–31 (2005)CrossRefGoogle Scholar
  9. Bažant, Z., Vorechovsky, M., Novak, D.: Asymptotic prediction of energetic-statistical size effect from deterministic finite-element solutions. Journal of Engineering Mechanics ASCE 133(2), 153–162 (2007a)CrossRefGoogle Scholar
  10. Bažant, Z., Pang, S.-D., Vorechovsky, M., Novak, D.: Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model. International Journal for Numerical Methods in Engineering 71(11), 1297–1320 (2007b)CrossRefGoogle Scholar
  11. Bažant, Z., Yavari, A.: Response to A. Carpinteri, B. Chiaia, P. Cornetti and S. Puzzi’s comments on “Is the cause of size effect on structural strength fractal or energetic-statistical”. Engineering Fracture Mechanics 74(17), 2897–2910 (2007c)CrossRefGoogle Scholar
  12. Bielewicz, E., Górski, J.: Shell with random geometric imperfections. Simulation-based approach. International Journal of Non-linear Mechanics 37(4-5), 777–784 (2002)zbMATHCrossRefGoogle Scholar
  13. Bobiński, J., Tejchman, J., Górski, J.: Notched concrete beams under bending – calculations of size effects within stochastic elasto-plasticity with non-local softening. Archives of Mechanics 61(3-4), 1–25 (2009)Google Scholar
  14. Carmeliet, J., Hens, H.: Probabilistic nonlocal damage model for continua with random field properties. Journal of Engineering Mechanics ASCE 120(10), 2013–2027 (1994)CrossRefGoogle Scholar
  15. Carpinteri, A.: Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. International Journal of Solids and Structures 25(4), 407–429 (1989)CrossRefGoogle Scholar
  16. Carpinteri, A., Chiaia, B., Ferro, G.: Multifractal scaling law: an extensive application to nominal strength size effect of concrete structures. In: Mihashi, M., Okamura, H., Bazant, Z.P. (eds.) Size Effect of Concrete Structures, vol. 173, p. 185. E&FN Spon (1994)Google Scholar
  17. Carpinteri, A., Chiaia, B., Ferro, G.: Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Materials and Structures (RILEM) 28(180), 311–317 (1995)CrossRefGoogle Scholar
  18. Carpinteri, A., Chiaia, B., Cornetti, P., Puzzi, S.: Comments on “Is the cause of size effect on structural strength fractal or energetic-statistical”. Engineering Fracture Mechanics 74(14), 2892–2896 (2007)CrossRefGoogle Scholar
  19. Chen, J., Yuan, H., Kalkhof, D.: A nonlocal damage model for elastoplastic materials based on gradient plasticity theory. Report Nr.01-13. Paul Scherrer Institute, pp. 1–130 (2001)Google Scholar
  20. Cusatis, G., Bažant, Z.: Size effect on compression fracture of concrete with or without V-notches: a numerical meso-mechanical study. In: Meschke, G., de Borst, R., Mang, H., Bicanic, N. (eds.) Computational Modelling of Concrete Structures, pp. 71–83. Taylor and Francis Group, London (2006)Google Scholar
  21. Florian, A.: An efficient sampling scheme: Updated latin hypercube sampling. Probabilistic Engineering Mechanics 7(2), 123–130 (1992)MathSciNetCrossRefGoogle Scholar
  22. Frantziskonis, G.N.: Stochastic modeling of hetereogeneous materials – a process for the analysis and evaluation of alternative formulations. Mechanics of Materials 27(3), 165–175 (1998)CrossRefGoogle Scholar
  23. Górski, J.: Non-linear models of structures with random geometric and material imperfections simulation-based approach, Habilitation. Gdansk University of Technology (2006)Google Scholar
  24. Gutierrez, M.A., de Borst, R.: Energy dissipation, internal length scale and localization patterning – a probabilistic approach. In: Idelsohn, S., Onate, E., Dvorkin, E. (eds.) Computational Mechanics, pp. 1–9. CIMNE, Barcelona (1998)Google Scholar
  25. Gutierrez, M.A.: Size sensitivity for the reliability index in stochastic finite element analysis of damage. International Journal of Fracture 137(1-4), 109–120 (2006)zbMATHCrossRefGoogle Scholar
  26. Hordijk, D.A.: Local approach to fatigue of concrete. PhD thesis. Delft University of Technology (1991)Google Scholar
  27. Hughes, T.J.R., Winget, J.: Finite Rotation Effects in Numerical Integration of Rate Constitutive Equations Arising in Large Deformation Analysis. International Journal for Numerical Methods in Engineering 15(12), 1862–1867 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Huntington, D.E., Lyrintzis, C.S.: Improvements to and limitations of Latin hypercube sampling. Probabilistic Engineering Mechanics 13(4), 245–253 (1998)CrossRefGoogle Scholar
  29. Hurtado, J.E., Barbat, A.H.: Monte Carlo techniques in computational stochastic mechanics. Archives of Computational Method in Engineering 5(1), 3–30 (1998)MathSciNetCrossRefGoogle Scholar
  30. Knabe, W., Przewłócki, J., Różyński, G.: Spatial averages for linear elements for two-parameter random field. Probabilistic Engineering Mechanics 13(3), 147–167 (1998)CrossRefGoogle Scholar
  31. Koide, H., Akita, H., Tomon, M.: Size effect on flexural resistance on different length of concrete beams. In: Mihashi, H., Rokugp, K. (eds.) Fracture Mechanics of Concrete, pp. 2121–2130 (1998)Google Scholar
  32. Le Bellego, C., Dube, J.F., Pijaudier-Cabot, G., Gerard, B.: Calibration of nonlocal damage model from size effect tests. European Journal of Mechanics A/Solids 22(1), 33–46 (2003)zbMATHCrossRefGoogle Scholar
  33. McKay, M.D., Conover, W.J., Beckman, R.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  34. Pijaudier-Cabot, G., Haidar, K., Loukili, A., Omar, M.: Ageing and durability of concrete structures. In: Darve, F., Vardoulakis, I. (eds.) Degradation and Instabilities in Geomaterials. Springer, Heidelberg (2004)Google Scholar
  35. Skarżynski, L., Syroka, E., Tejchman, J.: Measurements and calculations of the width of the fracture process zones on the surface of notched concrete beams. Strain 47(s1), e319–e322 (2011)Google Scholar
  36. Skuza, M., Tejchman, J.: Modeling of a deterministic size effect in concrete elements. Inżynieria i Budownictwo 11, 601–605 (2007) (in Polish)Google Scholar
  37. Syroka, E., Górski, J., Tejchman, J.: Unnotched concrete beams under bending – calculations of size effects within stochastic elasto-plasticity with non-local softening. Internal Report, University of Gdańsk (2011)Google Scholar
  38. Tejchman, J., Górski, J.: Computations of size effects in granular bodies within micro-polar hypoplasticity during plane strain compression. International Journal of Solids and Structures 45(6), 1546–1569 (2007)CrossRefGoogle Scholar
  39. Tejchman, J., Górski, J.: Deterministic and statistical size effect during shearing of granular layer within a micro-polar hypoplasticity. International Journal for Numerical and Analytical Methods in Geomechanics 32(1), 81–107 (2008)CrossRefGoogle Scholar
  40. Vanmarcke, E.-H.: Random Fields: Analysis and Synthesis. MIT Press, Cambridge (1983)zbMATHGoogle Scholar
  41. van Mier, J., van Vliet, M.: Influence of microstructure of concrete on size/scale effects in tensile fracture. Engineering Fracture Mechanics 70(16), 2281–2306 (2003)CrossRefGoogle Scholar
  42. van Vliet, M.R.A.: Size effect in tensile fracture of concrete and rock. PhD thesis. University of Delft (2000)Google Scholar
  43. Vorechovsky, M.: Stochastic fracture mechanics and size effect. PhD Thesis. Brno University of Technology (2004)Google Scholar
  44. Vorechovsky, M.: Interplay of size effects in concrete specimens under tension studied via computational stochastic fracture mechanics. International Journal of Solids and Structures 44(9), 2715–2731 (2007)zbMATHCrossRefGoogle Scholar
  45. Walraven, J., Lehwalter, N.: Size effects in short beams loaded in shear. ACI Structural Journal 91(5), 585–593 (1994)Google Scholar
  46. Walukiewicz, H., Bielewicz, E., Górski, J.: Simulation of nonhomogeneous random fields for structural applications. Computers and Structures 64(1-4), 491–498 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Weibull, W.: A statistical theory of the strength of materials. Journal of Applied Mechanics 18(9), 293–297 (1951)zbMATHGoogle Scholar
  48. Wittmann, F.H., Mihashi, H., Nomura, N.: Size effect on fracture energy of concrete. Engineering Fracture Mechanics 33(1-3), 107–115 (1990)CrossRefGoogle Scholar
  49. Yang, Z., Xu, X.F.: A heterogeneous cohesive model for quasi-brittle materials considering spatially varying random fracture properties. Computer Methods in Applied Mechanics and Engineering 197(45-48), 4027–4039 (2008)zbMATHCrossRefGoogle Scholar
  50. Yu, Q.: Size effect and design safety in concrete structures under shear. PhD Thesis. Northwestern University (2007)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdansk-WrzeszczPoland
  2. 2.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdansk-WrzeszczPoland

Personalised recommendations