# Effect of Stirrup on Bond Strength Degradation in Concrete Cracked by Expansion Agent Filled Pipes
## 4 Evaluation of Bond Strength Degradation in Cracked Specim with Stirrup  
### Bond Degradation and Surface Crack Width Relationship  
In this paper, we used the residual bond strength ratio (\(R_{B}\)) to evaluate bond degradation. It is the ratio of the bond strength of a cracked specimen to that of the uncracked specimen. Assuming that the splitting surface is in the entire embedded bar length, Equation (3) [32] is used to calculate the bond strength of uncracked specimen without stirrup (S0):  
\[\tau_{b,max}=0.601\cdot\sigma_{t}\cdot\frac{r_{u}}{d_{b}}\cdot\cot\alpha \tag{3}\]  
Figure 11: Examples of bond stress versus slippage curves.  
Figure 12: Bond strength-induced crack width relationship: (**a**) Serie S0; (**b**) Serie S1; (**c**) Serie S2.  
where, \(\tau_{b,max}\): bond strength, \(\sigma_{t}\): tensile strength of concrete, \(r_{u}\): \(C+d_{b}/2\), \(d_{b}\): diameter of the rebar (19 mm), \(C\): the thickness of cover concrete, \(\alpha\): the angle between the longitudinal axis and splitting force (=30 degrees).  
On the other hand, the bond strength for uncracked specimens with stirrup (S1 and S2) is calculated by the Equations (4) and (5) reported in a previous study [33]:  
\[\tau_{b,max}=\sigma_{l}\cdot\cot\theta \tag{4}\]  
\[\sigma_{l}=\sqrt{0.018\cdot\frac{b\cdot p_{w}}{N\cdot db}\cdot\frac{h}{l_{w}} \cdot E_{St}\cdot\sigma_{B}} \tag{5}\]  
where, \(\tau_{b,max}\): bond strength, \(\sigma_{l}\): lateral confinement stress, \(\theta\): angle between the longitudinal axis and principal bond stress, \(d_{b}\): the diameter of the rebar (19 mm), \(b\): width of specimen, \(p_{w}\): stirrup ratio, \(h\): rib height of the rebar, \(N\): number of rebar (=1), \(l_{w}\): bond effective length of stirrup (9 times diameter of stirrup = 90 mm), \(E_{st}\): elastic modulus of stirrup, and \(\sigma_{\text{B}}\): compressive strength of concrete.  
The calculated bond strengths of uncracked concrete are 9.86 MPa, 10.92 MPa, and 11.78 MPa for S0, S1 and S2 specimens, respectively. Figure 13 displays the relationships between residual bond strength ratio and crack width. One aim of this study is to explore the relationship between bond degradation and induced crack width. To build this model, various studies have assessed the efficacy of exponential fitting to provide better correlation [15; 18; 34].  
The following equation form evaluates the degradation of the bond by regression analysis:  
\[R_{B}=1-a(1-(\text{e}^{b\cdot w_{cr}})) \tag{6}\]  
where, \(a\) and \(b\) are empirical coefficients, \(w_{cr}\) is the induced crack width in mm.  
Table 3 shows the result of regression analysis, and Figure 14 compares the three obtained formulas for S0, S1, and S2. Until a crack width of 0.15 mm, the stirrup does not much influence bond degradation due to induced crack. This fine crack was still able to transmit the load from the bar to the surrounding concrete. However, when the crack width is more than 0.15 mm, the stirrup can restrain the opening of the induced crack and limit the bond deterioration in S1 and S2. After cracking, the stirrups provide valuable confinement of the concrete. The opening of the crack was inhibited by stirrups, which can increase the residual tensile strength of concrete. The advantageous impact of stirrups on the bond strength degradation due to induced crack is well reflected.  
Figure 13: Residual bond strength ratio versus crack width: (**a**) series S0; (**b**) series S1; (**c**) series S2.