Reliability, brittleness, covert understrength factors, and fringe formulas in concrete design codes

Zdeněk P. Bažant*, Qiang Yu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


The paper analyzes the reliability consequences of the fact that the current design codes for concrete structure contain covert (or hidden) understrength (or capacity reduction) factors. This prevents distinguishing between different combinations of separate risks due to the statistical scatter of material properties, the error of the design formula, and the degree of brittleness of failure mode, and also makes any prediction of structural reliability (or survival probability) impossible. The covert formula error factor is implied by the fact that the design formula was calibrated to pass not through the mean but through the fringe (or periphery, margin) of the supporting experimental data. The covert material randomness factor is the ratio of the reduced concrete strength required for design to the mean of the strength tests. As a remedy, the covert understrength factor of design formula should be made overt, its coefficient of variation (based on the supporting test data) should be specified, and the type of probability distribution (e.g., Gaussian or Weibull) indicated (which then also implies the probability cutoff). Alternatively, the code could give the mean formula, specify its coefficient of variation and type of distribution, and either prescribe the probability cutoff or overtly declare the understrength factor. The mean of strength tests required for quality control should be figured out from the required design strength on the basis of a specified probability cutoff and the coefficient of variation of these tests. Furthermore, it is proposed that the currently used empirical understrength factor, which accounts mainly for the risks of structural brittleness (or lack of ductility), should be based on the expected maximum kinetic energy that could be imparted to the structure. The reliability integral taking into account the randomness of both the load and structural resistance is generalized for the case of multiple (statistically independent) understrength factors. Finally, it is pointed out that the currently assumed proportionality of the tensile and shear strengths to the square root of compressive strength of concrete is realistic only for the mean, but grossly underestimates the scatter of tensile and shear strengths.

Original languageEnglish (US)
Pages (from-to)3-12
Number of pages10
JournalJournal of Structural Engineering
Issue number1
StatePublished - 2006


  • Brittleness
  • Codes
  • Concrete structure
  • Design
  • Reliability

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction
  • Materials Science(all)
  • Mechanics of Materials
  • Mechanical Engineering


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