Fibre-Reinforced and Hybrid-Reinforced Concrete: An Updated Bridged Crack Model with Softening Pull-Out
Bibliographic record
Abstract
The Bridged Crack Model is a fracture mechanics approach able to describe the crack propagation process in the critical cross-section of brittle-matrix reinforced members.The model was originally proposed to interpret the fracturing behaviour of steel-bar lightly reinforced concrete (RC) beams [1,2], and it has been recently updated to the case of fibre-reinforced concrete (FRC) [3] by introducing a softening pull-out constitutive law for the reinforcing fibres.Now, a further extension of the model to the case of hybrid-reinforced concrete (HRC) beams ─in which the reinforcing phase consists in a combination of continuous steel rebars and short discontinuous fibres─ is discussed.The Bridged Crack Model assumes the concrete matrix as a linear-elastic perfectly-brittle primary phase, its toughening contribution being defined by the fracture toughness, KIC.On the other hand, nonlinear constitutive laws are assumed to describe the toughening action of the reinforcing secondary phases, which are related to the yielding of steel rebars and to the pull-out of the short fibres.Under these assumptions, it is possible to evaluate the stress-block diagram for each crack depth and to describe the mechanical response in terms of fracturing moment vs localized rotation of the notched crosssection.Different post-cracking regimes can be predicted by the model, as a function of three scale-dependent dimensionless numbers: the bar-reinforcement brittleness number, NP, which is directly related to the steel-bar reinforcement percentage, ρ; the fibre-reinforcement brittleness number, NP,f, which is directly related to the fibre volume fraction, Vf; and the pullout brittleness number, Nw, which depends on the critical embedment length of the fibre-reinforcement, wc.A parametric analysis makes evident how these three dimensionless numbers allow to fully capture the different transitions in the postcracking regime, which can range from softening to hardening, including hyper-strength phenomena.The focus of the present work is on the minimum reinforcement condition, i.e., the combination of ρmin and Vf,min required to guarantee a stable post-peak response, which is defined by the critical values of the two reinforcement brittleness numbers, NP and NP,f.It is found that, at the critical conditions, NP and NP,f can be put in connection with a linear relationship, thus providing an effective tool to the minimum reinforcement design of HRC members, which is still lacking in the current structural design codes [4,5].The validity of the proposed approach is discussed on the basis of several numerical simulations and supported by experimental campaigns reported in the current scientific literature.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".