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Chapter 4. Bond behavior of fibers in cementitious materials: a unifiedformulationPull-out ProcessAnalytical assumption described in section 4.3Initial behavior – point OElastic stage (E)s i s eElastic limititSoftening-Elastic stage (SE)s i > s el emb l slShort fibersPeak strength for short fibers – point B’l emb > l slLong fibersPeak strength for long fibers – point BSoftening stage (S)Debonding-SE (DSE)s i > s B’ s i > s upoint C’point CDebonding-Softening i stage (DS)snap-back responseUltimate slip – point DFailures i = s uFigure 4.4: Schematic representation of the overall debonding process.while the other extreme (C ′ of Figure 4.5) is given by the following coordinatess ′ C = s u (4.20)P ′ C = πα 2 −1 d f tan[α 2 l e ]([k E + k S ]s e − k S s u ). (4.21)78
4.3. Elasto-plastic joint/interface model with isotropic linear softeningP iP Bs B =s C' =s D =s us CP CΒCP B'Β'criticalanchorageP e long fibershort fiberP i,C'C'DΟ s 0 1 2 3 4 s5iFigure 4.5: Typical analytical curves of the applied load P i vs. debonding displacement s i incase of short and long anchorage condition.Long fibersIn case of “long” anchorage (l emb > l sl ), the bond-slip process following the previous SEstageis characterized by a debonded zone in the neighbors of the loaded-end, whoselength is identified as l u . The analytical solution of this stage, namely Debonding-Softening-Elastic (DSE), is detailed in Table 4.2 in terms of slips s[z] and global pulloutcurve P i − s i . The closed relations depend on the two parameters l e and l u whichdetermine the configuration states of the fiber-to-concrete bond. The values of thosetwo parameters are strictly connected by means of the following relationshipwith⎛⎞l u = l e − α −12 sec−1 ⎝ k E + k S d f tanh 2 [α 1 (l e − l emb )]⎠k E + k S − k (4.22)S s us e+ k 1[(k 1 = k 2 S τ 2 0 tanh 2 [α 1 (l e − l emb )] (s e − s u )(k E k S s u − τ 0 (2k E + k S )) +τ 2 0 tanh 2 [α 1 (l e − l emb )] )] 1/2 .Consequently, the debonding process in this stage can be followed by assuming thevalue l e and then evaluating l u through Eq. (4.22). Once the elastic contact vanishes(l e = l emb ), the DSE-stage terminates reaching its ultimate point C , which coordinatesis analytically obtained as follows(s C = 1 + k )E[1 + α 2 l u sin[α 2 (l emb − l u )] −cos[α 2 (l emb − l u )]] s e (4.23)k S79
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Dottorato di Ricercain Ingegneria d
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AcknowledgementsI would like to exp
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Abstractinclusion of fibers and the
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ContentsAcknowledgementsAbstractTab
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Contents8.6.1 Tensile tests . . . .
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List of Figures2.1 Fine and coarse
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List of Figures4.16 Pull-out simula
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List of Figures6.14 Crack paths of
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List of Tables2.1 Mix design per cu
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Chapter 1. Introduction1.1.1 Fiber
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Chapter 1. Introductionthe fiber ad
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Chapter 1. Introductionand mechanic
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Chapter 1. Introduction[Vrech and E
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Chapter 1. IntroductionDespite the
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Chapter 1. Introduction(Figure 1.4)
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Chapter 1. IntroductionFigure 1.8:
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Chapter 1. Introductionfor structur
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Chapter 1. Introductioncracking beh
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Chapter 1. Introduction• "c" if 0
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Chapter 1. Introductionsumed due to
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Chapter 1. Introductionaleatoric na
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Chapter 1. IntroductionFigure 1.13:
Page 54 and 55: Chapter 1. IntroductionElement Meth
Page 56 and 57: Chapter 2. Experimental characteriz
Page 70 and 71: Chapter 3. Zero-thickness interface
Page 93 and 94: 4 Bond behavior of fibers in cement
Page 95 and 96: 4.2. Bond behavior of fibers in con
Page 97 and 98: 4.3. Elasto-plastic joint/interface
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Page 107 and 108: 4.4. Fracture energy-based interfac
Page 109 and 110: 4.4. Fracture energy-based interfac
Page 111 and 112: 4.5. Numerical results and experime
Page 117 and 118: 4.6. Closing remarksMPa3.02.5Pi N2.
Page 119 and 120: 5 Model performance and numericalpr
Page 121 and 122: 5.1. Numerical analysesones mention
Page 123 and 124: 5.1. Numerical analysesshown in Fig
Page 125 and 126: 5.1. Numerical analysesconsidered,
Page 127 and 128: 5.1. Numerical analysestic behavior
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Page 131 and 132: 5.2. Cracking analysis of the propo
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Page 139: 5.3. Closing remarksconnecting node
Page 142 and 143: Chapter 6. Structural scale failure
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Chapter 6. Structural scale failure
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7 Cracked hinge numerical modelfor
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7.1. Basic assumptionsSection at no
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7.1. Basic assumptions1997, Stankow
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7.2. Bond-slip bridging of fibers o
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7.4. Numerical predictionsvertical
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7.5. Closing remarksthe experimenta
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Chapter 8. Elasto-plastic microplan
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Chapter 9. Conclusionsof first-crac
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Chapter 9. Conclusionsfibers in con
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Chapter 9. Conclusions9.6 Future re
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BibliographyN. Banthia and N. Nanda
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BibliographyM. Butler, V. Mechtcher
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BibliographyCUR. Bepaling van de Bu
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BibliographyEN-12390-3. Testing of
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BibliographyT. Guttema. Ein Beitrag
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BibliographyE. Kuhl, P. Steinmann,
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BibliographyO. Manzoli, J. Oliver,
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BibliographyK. Park, G. Paulino, an
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BibliographyI. Singh, B. Mishra, an
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BibliographyG. Wells and L. Sluys.
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