Since 1928 the ACI design code has required a minimum distributed mesh reinforcement of about 0.18% of the gross cross-section — the 'shrinkage and temperature' reinforcement — to keep lightly reinforced concrete members ductile and free of widely spaced localized cracks, but the ratio has been purely empirical. Rasoolinejad, Dönmez and Bažant (2020) gave the first quasibrittle-fracture-mechanics justification of this ratio: finite-element simulations with the microplane model M7 and the crack band model on notched three-point-bend beams indicate that, when the tensile force at reinforcement yielding exceeds the tensile force at concrete cracking, the energetic size effect is suppressed and plastic limit analysis becomes applicable. The justification is, however, incomplete: the paper concludes that the 0.18% minimum is "only approximate" and that the required ratio depends nonnegligibly on steel yield limit, concrete strength and fracture properties, yet no precise, design-usable formula exists beyond the proposed form ρmin⁡=fte/fy\rho_{\min} = f_{te}/f_y with a size-dependent effective tensile strength ftef_{te}; the size-effect-suppression prediction has never been checked by size-effect experiments on mesh-reinforced members; and the analysis covers only notched bend beams with perfect bond, with the shrinkage and thermal actions that motivate the code rule treated as context rather than modeled. Completing this justification — a precise minimum-ratio formula, experimental anchoring of the suppression prediction, and the paper's recommended extension to more complex configurations such as slabs, shear walls, two-sided reinforced members and boundary beams — remains open.

Background

Concrete is a quasibrittle material: it fails through the growth of a fracture process zone — a damage band of a finite length comparable to aggregate size — rather than a sharp plastic crack tip, so the nominal strength (failure load divided by a characteristic cross-sectional area) of geometrically similar structures decreases with increasing structure size. This energetic size effect is captured by Bažant's size effect law for blunt (bluntly notched) fracture, derived by asymptotic matching of plastic limit analysis (zero size effect, strength criterion) and linear elastic fracture mechanics (the classical d−1/2d^{-1/2} scaling) (Bažant, 1984, 'Size effect in blunt fracture: concrete, rock, metal'). In numerical work the process zone is regularized either by the crack band model, which spreads the fracture energy over a band of fixed width so that mesh refinement does not bias the energy dissipation, or by the microplane model M7, a constitutive model in which the material response is defined on planes of all orientations rather than in tensorial invariants.

A separate, 92-year-old empirical thread: the ACI 318 building code requires a minimum area of distributed reinforcement — placed orthogonally in walls and slabs and called shrinkage and temperature (S&T) reinforcement, currently about 0.18% of the gross concrete cross-section — so that restrained shrinkage and temperature changes do not produce a few wide localized cracks but many fine ones, and the member remains ductile rather than failing brittly upon first cracking. The ratio was fixed empirically; no theory related it to material properties.

Rasoolinejad, Dönmez and Bažant (2020) connected these threads. Working on notched three-point-bend beams, they adopted the premise that distributed (mesh) reinforcement produces widely spaced localized cracks if and only if the structure exhibits a size effect, and showed by finite-element analysis with the crack band model and microplane model M7 that the size effect is suppressed when, for the same cross-section area, the tensile force at yielding of the reinforcement exceeds the tensile force at cracking of the plain concrete. Their calculations approximately reproduce the code's 0.18% and indicate that fracture mechanics can and should be used to check ductility and size-effect implications in slabs, shear walls, two-sided reinforced members and boundary beams — configurations beyond their analyzed scope. They also note that the solution of the Appendix III crack-width calculation "appears to be too complex to offer good insight".

An adjacent, older line treats the same physical situation differently: the Carpinteri school's bridged-crack and brittleness-number framework, in which the minimum reinforcement of a reinforced concrete cross-section is set by requiring stable flexural crack propagation, expressed through a dimensionless brittleness number combining the reinforcement ratio and yield strength with member size and the concrete fracture toughness (Bosco & Carpinteri, 2018; recently extended to fibre- and hybrid-reinforced beams via the Updated Bridged Crack Model — a refined bridged-crack formulation in which the closing tractions of both bars and bridging fibres act across the crack — by Rubino, Accornero & Carpinteri, 2025). In that framework the required minimum ratio is size-dependent. This adjacent framework is not discussed in the 2020 paper — whose own Eq. (7) ratio ρmin⁡=fte/fy\rho_{\min} = f_{te}/f_y is itself size-dependent through the effective tensile strength ftef_{te}.

Problem Statement

Complete the fracture-mechanics justification of the empirical minimum mesh (shrinkage-and-temperature) reinforcement requirement in concrete design codes, as posed by Rasoolinejad, Dönmez & Bažant (2020). The source establishes the suppression criterion — the energetic size effect, and hence crack localization, is avoided only if, for the same cross-section area, the tensile strength of the yielding reinforcement exceeds the tensile strength of the concrete — and concludes that the code's 0.18% minimum is "only approximate". The common objective is to replace that empirical ratio with a precise, validated fracture-mechanics determination of the minimum ratio suppressing the energetic size effect and hence crack localization. The open targets indicated by the source are: (1) a simple, design-usable formula for the minimum mesh reinforcement ratio ρmin⁡\rho_{\min}, making explicit its dependence on the steel yield limit, concrete strength, concrete fracture properties (e.g., fracture energy GfG_f), structure size DD, and the loads applied in addition to shrinkage and temperature; (2) closing the acknowledged validation gap that the size-effect-suppression prediction has not been checked by size-effect tests on mesh-reinforced members ("no reports are found in the literature"; such tests "would be costly, and funding seems unobtainable at present") — the prediction is checked against size-effect observations, from new tests on mesh-reinforced members or from existing experimental size-effect data containing at least one lightly reinforced series at or above the minimum ratio ρmin⁡=fte/fy\rho_{\min} = f_{te}/f_y, and is supported when the measured size-effect trends show no significant size effect where the suppression criterion holds and a significant size effect where it does not; and (3) extending the fracture-mechanics ductility and size-effect check, as the source recommends, beyond notched three-point-bend beams to at least one more complex configuration — chosen from slabs (with a stated boundary condition), shear walls, two-sided reinforced members and boundary beams. All three targets are required components of the task.

The source analysis idealizes the steel–concrete interface as perfectly bonded (bar slip is argued to be important only for local behavior, not for the global size effect) and treats the shrinkage and thermal actions that motivate the code rule as context rather than modeling them directly; the answer may adopt these same idealizations.

References

  1. Rasoolinejad, M., Dönmez, A. A., & Bažant, Z. P. (2020). Fracture and size effect suppression by mesh reinforcement of concrete and justification of empirical shrinkage and temperature reinforcement in design codes. Journal of Engineering Mechanics, 146(10), 04020120. DOI: 10.1061/(asce)em.1943-7889.0001850. https://doi.org/10.1061/(asce)em.1943-7889.0001850
  2. Bažant, Z. P. (1984). Size effect in blunt fracture: concrete, rock, metal. Journal of Engineering Mechanics, 110(4), 518–535. DOI: 10.1061/(ASCE)0733-9399(1984)110:4(518). https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(518)
  3. Bosco, C., & Carpinteri, A. (2018). Fracture mechanics evaluation of minimum reinforcement in concrete structures. In Applications of fracture mechanics to reinforced concrete (pp. 347–378). CRC Press. DOI: 10.1201/9781482296624-14. https://doi.org/10.1201/9781482296624-14
  4. Augusthus-Nelson, L., Lees, J. M., & Weekes, L. (2022). Size effects in unreinforced and lightly reinforced concrete beams failing in flexure. Engineering Fracture Mechanics, 259, 107987. DOI: 10.1016/j.engfracmech.2021.107987. https://doi.org/10.1016/j.engfracmech.2021.107987
  5. Rubino, A., Accornero, F., & Carpinteri, A. (2025). Fracture mechanics approach to minimum reinforcement design of fibre-reinforced and hybrid-reinforced concrete beams. International Journal of Damage Mechanics, 34(6), 900–919. DOI: 10.1177/10567895241245865. https://doi.org/10.1177/10567895241245865
  6. Tang, P., Amin, A., Gilbert, R. I., & Kaufmann, W. (2024). Shrinkage cracking in restrained FRC members containing conventional reinforcement. Structural Concrete, 25(2), 1006–1017. DOI: 10.1002/suco.202300269. https://doi.org/10.1002/suco.202300269
  7. Ziaolhagh, N. (2023). Shrinkage and temperature reinforcement in concrete liquid-containing structures [Thesis, Toronto Metropolitan University]. DOI: 10.32920/ryerson.14655837. https://doi.org/10.32920/ryerson.14655837