1 Introduction
2 Backgrounds
2.1 Bond and stress components
2.2 Failure Mode, Bond Strength, and Bond–Slip Behavior
2.3 Influencing Parameters
2.4 Existing Models
Researcher | Equation* |
---|---|
Orangun et al. (1977) | \(\tau_{{{\text{max}}}} = \left( {1.2 + \frac{3c}{{d_{{\text{b}}} }} + \frac{{50d_{{\text{b}}} }}{{L_{{\text{e}}} }} + \frac{{A_{{{\text{st1}}}} f_{{{\text{yt}}}} }}{{72,500S_{{{\text{st}}}} d_{{\text{b}}} }}} \right)0.083\sqrt {f^{\prime}_{{\text{c}}} }\) |
Xu (1990) | \(\tau_{{{\text{max}}}} = \left( {1.6 + \frac{0.7c}{{d_{{\text{b}}} }} + 20\rho_{{{\text{sv}}}} } \right)f_{{\text{t}}} = \left( {1.6 + \frac{0.7c}{{d_{{\text{b}}} }} + \frac{{20A_{{{\text{st1}}}} }}{{cS_{{{\text{st}}}} }}} \right)f_{{\text{t}}}\) |
Harajli et al. (1995) | \(\tau_{{{\text{max}}}} = 0.78\left( {\frac{{c + K_{{\text{t}}} }}{{d_{{\text{b}}} }}} \right)^{2/3} \sqrt {f^{\prime}_{{\text{c}}} } = 0.78\left[ {\frac{{c + \frac{{7A_{{{\text{st}}}} }}{{S_{E} n}}}}{{d_{{\text{b}}} }}} \right]^{2/3} \sqrt {f^{\prime}_{{\text{c}}} }\) |
Esfahani and Kianoush (2005) | \(\tau_{{{\text{max}}}} = \tau_{{\text{c}}} \frac{{1 + \frac{1}{M}}}{1.85 + 0.024\sqrt M }\left( {0.88 + \frac{{0.12c_{{{\text{med}}}} }}{c}} \right)\left( {1 + \frac{{0.015A_{{{\text{st1}}}} A_{{\text{s}}} }}{{cS_{{{\text{st}}}} }}} \right),\) \({\text{where}} \;M = \cos \,h\left( {0.0022L_{{\text{e}}} \sqrt {3\frac{{f^{\prime}_{{\text{c}}} }}{{d_{{\text{b}}} }}} } \right)\) and \(\tau_{{\text{c}}} = 2.7\frac{{\frac{c}{{d_{{\text{b}}} }} + 0.5}}{{\frac{c}{{d_{{\text{b}}} }} + 3.6}}\sqrt {f^{\prime}_{{\text{c}}} }\) |
Wu and Zhao (2013) | \(\tau_{{{\text{max}}}} = \frac{{2.5\sqrt {f^{\prime}_{{\text{c}}} } }}{{1 + 3.1e^{{ - 0.47(K_{{{\text{co}}}} + 33K_{{\text{t}}} )}} }}\), where \(K_{{{\text{co}}}} = \frac{c}{{d_{{\text{b}}} }}\) and \(K_{{\text{t}}} = \frac{{A_{{{\text{st}}}} }}{{nS_{{{\text{st}}}} d_{{\text{b}}} }}\) |
Researcher | Equation |
---|---|
Rehm (1961) | \(\tau = f_{{\text{c,cub}}} \left( {\varphi s^{\alpha } \pm \psi s} \right)\) |
Nilson (1968) | \(\tau = 998.4s - 58,400s^{2} + 852,200s^{3}\) |
Martin (1973) | \(\tau = \tau_{0} + as^{b}\) |
Mirza and Houde (1979) | \(\tau = 539.8s - 25,610s^{2} + 592,200s^{3} - 5,574,000s^{4}\) |
Eligehausen et al. (1982) | \(\tau = \tau_{max} \left( {\frac{s}{{s_{1} }}} \right)^{\alpha }\), where \(0 \le s \le s_{1}\) |
Comité Euro-International du Béton (1993) | \(\tau = \tau_{max} ,\) when \(s_{1} < s \le s_{2}\) \(\tau = \tau_{max} - \left( {\tau_{max} - \tau_{f} } \right)\frac{{s - s_{2} }}{{s_{3} - s_{2} }}\), when \(s_{2} < s \le s_{3}\) \(\tau = \tau_{{\text{f}}}\), when \(s_{3} < s\) |
Wu and Zhao (2013) | \(\tau = \frac{{\tau_{{{\text{max}}}} }}{{\left[ {e^{{ - B\,\,{\text{ln}}\left( {B/D} \right)/\left( {B - D} \right)}} - e^{{ - D\,\,{\text{ln}}\left( {B/D} \right)/\left( {B - D} \right)}} } \right]}}\left( {e^{Bs} - e^{Ds} } \right)\), where \(B = \frac{{0.0254 + K_{t} }}{{ - 0.0232 - 8.34K_{t} }}\), \(D = 3{\text{ln}}\left( {\frac{0.7315 + K}{{5.176 + 0.3333K}} - 0.13} \right) - 3.375\) and \(K = K_{{{\text{co}}}} + 7K_{t}\) |
3 Materials and Methods
3.1 Materials
\({\varvec{\tau}} - {\varvec{s}}\) data (Pull-out specimens) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Researchers | S | \(N_{{\text{s}}}\) (EA) | N (EA) | \(f^{\prime}_{{\text{c}}}\)(MPa) | c (mm) | \(d_{{\text{b}}}\) (mm) | \(c/d_{{\text{b}}}\) | \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) | s (mm) | \({\varvec{\tau}}\) (MPa) |
Baena et al., (2009) | P | 2 | 14 | 27 | 92–94 | 12–16 | 5.8–7.8 | 0 | 0–25 | 0–13 |
Campione et al., (2004) | P | 4 | 28 | 34 | 36 | 12 | 3 | 0–0.9 | 0–12.1 | 0–16 |
Eligehausen et al., (1982) | P | 8 | 56 | 29–32 | 57–96 | 19–32 | 3 | 0–3.6 | 0–12.1 | 0–16 |
Engstrom et al., (1998) | P | 3 | 21 | 28–29 | 16 | 16 | 1 | 0–0.4 | 0–15 | 0–8 |
Harajli et al., (1995) | P | 2 | 14 | 22 | 60–80 | 20–25 | 3–3.2 | 0.9–1.1 | 0–18.1 | 0–13 |
Kim et al., (2015) | P | 2 | 14 | 29–33 | 75 | 16 | 4.7 | 0 | 0–4.6 | 0–25 |
Lee et al., (2012) | P | 5 | 35 | 26–42 | 45 | 13 | 3.4 | 0 | 0–10 | 0–18 |
Liu et al., (2020) | P | 5 | 35 | 38–82 | 90–94 | 12–20 | 4.5–7.8 | 0 | 0–12 | 0–32 |
Ma et al., (2017) | P | 3 | 21 | 25 | 30–66 | 18–22 | 1.4–3.7 | 0 | 0–1.5 | 0–8 |
Melo et al., (2015) | P | 1 | 7 | 16 | 94 | 12 | 7.8 | 0 | 0–10 | 0–13 |
Mo and Chan (1996) | P | 1 | 7 | 27 | 66–69 | 13 | 5.1–5.3 | 0 | 0–1.1 | 0–9 |
John Robert Prince and Singh (2013) | P | 6 | 42 | 20–37 | 38–94 | 12–25 | 1.5–7.8 | 0 | 0–15.3 | 0–18 |
Solyom and Balázs (2020) | P | 6 | 42 | 35 | 47–72 | 6–8 | 7.8–12 | 0 | 0–10 | 0–15 |
Song et al., (2015) | P | 48 | 336 | 8–53 | 15–67 | 14–16 | 1.1–4.2 | 0–0.3 | 0–16.1 | 0–25 |
Soroushian and Choi (1989) | P | 5 | 35 | 30–32 | 48–96 | 16–32 | 3 | 0.8–1.5 | 0–12.1 | 0–17 |
Soroushian et al., (1991) | P | 8 | 56 | 24–54 | 75–96 | 25–32 | 3 | 0–1 | 0–12.7 | 0–20 |
Sturm and Visintin (2019) | P | 3 | 21 | 150 | 20–75 | 16 | 1.3–4.7 | 0 | 0–2 | 0–48 |
Wu et al., (2021) | P | 132 | 924 | 5–48 | 64–69 | 12–22 | 2.9–5.8 | 0 | 0–14.2 | 0–22 |
Yalciner et al., (2012) | P | 6 | 42 | 23–51 | 15–45 | 14 | 1.1–3.2 | 0 | 0–11.9 | 0–27 |
Zhou et al., (2015) | P | 1 | 7 | 18 | 91 | 18 | 5.1 | 1.3 | 0–10 | 0–5 |
de Almeida Filho et al., (2008) | P | 4 | 28 | 32–50 | 45–72 | 10–16 | 4.5 | 0 | 0–6.3 | 0–22 |
Total (or min–max) | 255 | 1,785 | 5–150 | 15–96 | 6–32 | 1–5.8 | 0–3.6 | 0–25 | 0–48 | |
\({\varvec{\tau}} - {\varvec{s}}\) data (Beam specimens) | ||||||||||
Desnerck et al., (2010) | B | 5 | 35 | 64 | 38–59 | 12–40 | 1.4–3.7 | 0.6–1.6 | 0–2 | 0–28 |
Hou et al., (2020) | B | 3 | 21 | 39 | 30 | 16 | 2 | 0–0.4 | 0–2.2 | 0–13 |
Robert and David (1961) | B | 10 | 70 | 24–31 | 38–44 | 13–25 | 1.5–3.4 | 1 | 0–0.8 | 0–11 |
Petean et al., (2013) | B | 4 | 28 | 50–51 | 41–42 | 16–18 | 2.3–2.6 | 1.7–2 | 0–4 | 0–25 |
Seis and Beycioğlu (2017) | B | 3 | 21 | 38 | 44–46 | 8–12 | 3.7–5.8 | 1 | 0–0.7 | 0–8 |
Yerlici and Ozturan (2000) | BE | 4 | 28 | 71–93 | 15 | 12–16 | 0.9–1.3 | 0 | 0–0.3 | 0–14 |
de Almeida Filho et al., (2008) | B | 2 | 14 | 30 | 42–45 | 10–16 | 2.6–4.5 | 0.6–1.6 | 0–4.2 | 0–12 |
Total (or min–max) | 31 | 217 | 24–93 | 15–59 | 8–40 | 0.9–5.8 | 0–2 | 0–4.2 | 0–28 | |
\(\tau_{\max }\) data only (Beam specimens) | ||||||||||
Lin and Zhao (2016) | B | 6 | 6 | 30 | 40 | 20 | 2.0 | 0.8–1.6 | – | 8–9 |
Kemp and Wilhelm (1979) | BE | 6 | 6 | 29–30 | 25–76 | 16 | 1.6–4.8 | 0–0.8 | – | 4–7 |
Yerlici and Ozturan (2000) | BE | 32 | 32 | 64–93 | 15–30 | 12–26 | 0.6–1.9 | 0–0.9 | – | 9–18 |
Total (or min–max) | 44 | 44 | 29–93 | 15–76 | 12–26 | 0.6–4.8 | 0–1.6 | – | 4–18 |
3.2 Methods
3.2.1 Random Forest
3.2.2 K-Means Clustering
3.2.3 Genetic Programming
4 Results and Discussion
4.1 Parametric Study
4.1.1 Feature Importance
4.1.2 Correlation Analysis
Pull-out | ||||||
---|---|---|---|---|---|---|
Cluster | \(f^{\prime}_{{\text{c}}}\)(MPa) | c (mm) | \(d_{{\text{b}}}\) (mm) | \(c/d_{{\text{b}}}\) | \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) | \(\tau_{{\text{p,max}}}\)(MPa) |
1 | 26 | 68 | 23 | 2.9 | 0.3 | 12 |
2 | 26 | 29 | 14 | 2.1 | 0.1 | 12 |
3 | 116 | 70 | 16 | 4.4 | 0.0 | 36 |
4 | 26 | 70 | 14 | 5.0 | 0.0 | 14 |
Beam | ||||||
---|---|---|---|---|---|---|
Cluster | \(f^{\prime}_{{\text{c}}}\)(MPa) | c (mm) | \(d_{{\text{b}}}\) (mm) | \(c/d_{{\text{b}}}\) | \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) | \(\tau_{{\text{b,max}}}\)(MPa) |
1 | 79 | 17 | 16 | 1.0 | 0.1 | 13 |
2 | 31 | 40 | 18 | 2.3 | 0.9 | 7 |
3 | 58 | 45 | 22 | 2.0 | 1.6 | 23 |
4.2 Derivation of Mapping Function
4.2.1 Evolutionary Parameter
Parameter | \({\varvec{\tau}}_{{\text{p}}} - {\varvec{s}}\) relationship (Bond slip of pull-out data set) | \({\varvec{\tau}}_{{\text{p}}} - {\varvec{\tau}}_{{\text{b}}}\) relationship (Mapping function) | |
---|---|---|---|
Instruction | \(f^{\prime}_{{\text{c}}}\), \(c\), \(d_{{\text{b}}}\), \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\), \(s\), \(+\), \(-\), \(\times\),\(\div\) | \(f^{\prime}_{{\text{c}}}\), \(c\), \(d_{{\text{b}}}\), \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\), \(\tau_{{\text{p}}}\), \(+\), \(-\), \(\times\),\(\div\) | |
Initialization | Method | Ramped half and half | |
Depth | 1 ~ 4 | ||
Ratio | 50% | ||
Population size | 1000 | 100,000 | |
Selection | Method | Tournament selection | |
Size | 7 | ||
Crossover rate | Probability | 90% | |
Max. size | 40 | 30 | |
Mutation rate | Probability | 10% | |
Max. size | 40 | 30 | |
Depth of mutation trees | 0 ~ 2 | ||
Generation | 250 | 50 |
4.2.2 Bond–Slip and Bond Strength Model for Pull-Out Specimens
4.2.3 Mapping Function Between Pull-Out and Beam Bond–Slip Data
5 Verification
5.1 Accuracy of Proposed Models
Fractile | γ |
---|---|
0.05 | 0.65 |
0.08 | 0.70 |
0.10 | 0.73 |
0.15 | 0.79 |
0.20 | 0.84 |
0.25 | 0.88 |
0.30 | 0.91 |
0.35 | 0.94 |
0.40 | 0.98 |
0.45 | 1.01 |
0.50 | 1.04 |