## 5 Study on Fracture Development
### Crack Width  
According to the GB50010 [1], the maximum crack width \(w_{\rm max}\) of flexural members combined depends on the load standard. And considering the long-term effect can be calculated according to the following formula:  
\[w_{\rm max}=\tau_{\rm t}\tau_{\rm s}w_{\rm m} \tag{16}\]  
where \(w_{\rm m}\) is the average crack width, \(\tau_{\rm s}\) is the expansion coefficient of short-term crack width, and \(\tau_{\rm f}\) is the expansion coefficient considering the long-term effect which is 1.5.  
The short-term maximum crack width \(w_{\rm max}\) can be calculated by the following formula:  
\[w_{\rm max}=\alpha_{\rm c}\tau_{\rm s}\psi\frac{\sigma_{\rm s}}{E_{\rm s}}l_{ \rm cr} \tag{17}\]  
where \(\alpha_{\rm c}\) is the influence coefficient of the concrete elongation between cracks on the crack width, taken as 0.77. \(\psi\) is the strain non-uniformity coefficient of longitudinal tensile reinforcement between cracks, which is calculated by Equation (18).  
\[\psi=1.1-0.65\frac{f_{\rm tk}}{\rho_{\rm t}\sigma_{\rm s}} \tag{18}\]  
When \(\psi<0.2\), take \(\psi=0.2\). When \(\psi>1.0\), take \(\psi=1.0\).  
In order to determine the expansion coefficient of the short-term crack width in the negative moment zone, the maximum crack width of the four main cracks closest to the column end is divided by the average crack width. The following 80 data are obtained, as shown in Table 4.  
In order to determine the value of \(\tau_{\rm s}\) with a 95% guarantee, 80 data collected from the test were analyzed mathematically, and the statistical histogram of crack width was plotted as shown in Figure 19. At the same time, K-S analysis was performed, \(p\) value = 0.2 \(>\) 0.05, and the data obeyed the normal distribution N (1.0, 0.25). The Quantile-Quantile diagram of the expansion coefficient of short-term crack width is shown in Figure 20. According to the statistical results, the value \(\tau_{\rm s}\) with a 95% guarantee rate is: \(\tau_{\rm s}=1.0+1.645\times 0.25=1.41\).  
Under the action of concentrated load \(P\) at the end of the beam, the change of crack width with the tensile stress \(\sigma_{\rm s}\) (along the length direction of the beam) of the steel bar is shown in Figure 21. The dispersion of the width of the frame beam is large. When \(P\geq 0.4P_{\rm u}\), the average crack width at the beam end is close to the maximum crack width \(w_{\rm smax}\) obtained from Equation (17). The average crack width in most other regions is much larger than the maximum crack width obtained from Equation (17). Although the average crack width outside the beam end is still greater than the maximum crack width obtained by Equation (17), the difference between the two gradually decreases as the load and the tensile stress of the steel bar gradually increase. For the frame beam, the maximum crack width at the beam end plays a controlling role. After amplifying the short-term maximum crack width obtained by Equation (17) by 1.25 times, it is in good agreement with the test results of the crack width at the beam end.  
Figure 21: \(Cont.\)  
Figure 20: Quantile plot of expansion factor statistics for short-term crack widths.  
When the load \(P=0.8P_{\rm u}\), the distribution of crack width along the length direction of the beam of RC specimen KJ-2 and specimen KJ-E1, and KJ-E2 with the ECC protective layer, is shown in Figure 22. The crack width of the ECC protective layer is significantly smaller than that of the RC specimen.