Designing against size effect on shear strength of reinforced concrete beams without stirrups: I. formulation

Zdeněk P. Bažant*, Qiang Yu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


The shear failure of reinforced concrete beams is a very complex fracture phenomenon for which a purely mathematical approach is not possible at present. However, detailed modeling of the fracture mechanism is not necessary for establishing the general form of the size effect. The first part of this paper shows that the general approximate mathematical form of the size effect law to be calibrated by experimental data can be deduced from two facts: (1) the failure is caused by cohesive (or quasibrittle) fracture propagation; and (2) the maximum load is attained only after large fracture growth (rather than at fracture initiation). Simple dimensional analysis yields the asymptotic properties of size effect, which are characterized by: (1) a constant beam shear strength v c (i.e., absence of size effect) for sufficiently small beam depths; and (2) the linear elastic fracture mechanics size effect v c ∼ d -1/2 for very large beam depths d. Together with the recently established small- and large-size second-order asymptotic properties of the cohesive (or fictitious) crack model, this suffices to unambiguously support a size effect formula of the general approximate form v c=v 0(1+dld 0) -1/2112 (where v 0, d 0 are constants), which was proposed in 1984 for shear failure of beams on the basis of less general and less fundamental arguments. Verification and calibration are left for Part II of this paper which follows.

Original languageEnglish (US)
Pages (from-to)1877-1885
Number of pages9
JournalJournal of Structural Engineering
Issue number12
StatePublished - Dec 2005


  • Concrete
  • Concrete beams
  • Design
  • Fracture
  • Reinforced
  • Safety
  • Shear failure
  • Shear strength
  • Size effect

ASJC Scopus subject areas

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

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