Energetic-statistical size effect in quasibrittle failure at crack initiation

Zdenëk P. Bazant, Drahomfr Novâk

Research output: Contribution to journalArticlepeer-review

85 Scopus citations


The size effect on the nominal strength of quasibrittle structures failing at crack initiation, and particularly on the modulus of rupture of plain concrete beams, is analyzed. First, an improved deterministic formula is derived from the energy release due to a boundary layer of cracking (initiating fracture process zone) whose thickness is not negligible compared to beam depth. To fit the test data, a rapidly converging iterative nonlinear optimization algorithm is developed. The formula is shown to give an excellent agreement with the existing test data on the size effect on the modulus of rupture of plain concrete beams. The data range, however, is much too limited; it does not cover the extreme sizes encountered in arch dams, foundations, and retaining walls. Therefore, it becomes necessary to extrapolate on the basis of a theory. For extreme sizes, the Weibull type statistical effect of random material strength must be incorporated into the theory. Based on structural analysis with the recently developed statistical nonlocal model, a generalized energetic-statistical size effect formula is developed. The formula represents asymptotic matching between the deterministic-energetic formula, which is approachedfor small sizes, and the power law size effect of the classical Weibull theory, which is approachedfor large sizes. In the limit of infinite Weibull modulus, the deterministic-energetic formula is recovered. Data fitting with the new formula reveals that, for concrete and mortar, the Weibull modulus m 24 rather than 12, the value widely accepted so far. This means that, for extreme sizes, the nominal strength (modulus of rupture) decreases,for two-dimensional similarity, as the -I/12 power of the structure size, and for three-dimensional similarity, as the -1/8 power (whereas the -1 / 4 power has been assumed thus far). The coefficient of variation characterizing the scatter of many test results for one shape and one size is shown not to give the correct value of Weibull modulus because the energetic size effect inevitably intervenes. The results imply that the size effect at fracture initiation must have been a significant contributing factor in many disasters (for example, those of Malpasset Dam, Saint Francis Dam, and Schoharie Creek Bridge.)

Original languageEnglish (US)
Pages (from-to)381-392
Number of pages12
JournalACI Structural Journal
Issue number3
StatePublished - Dec 1 2000

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction


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