## 3 Results and Discussion
### Crack Propagation Life Analysis
#### 3.4.2 Critical Crack Size  
Critical crack size \(a_{c}\) is determined by the size \(a_{\mathit{of}}\) calculated from fracture toughness and the crack propagation range constrained by plate width or detail geometry. For the anchorage structure studied, the boundary of the web stiffeners determines the propagation range in which a crack would grow, as shown in Figure 16.  
Crack size \(a_{\mathit{of}}\) related to fracture toughness was calculated as follows:  
\[a_{\mathit{cf}}=K_{c}^{2}/\pi(Y\sigma_{\mathrm{max}})^{2} \tag{6}\]  
where shape function \(Y\) of the upper and lower crack tips is taken as a constant of 1.5; \(\sigma_{\mathrm{max}}\) is the maximum stress at the part in consideration (assuming stress ratio \(R=0\)).  
Fracture toughness \(K_{c}\), crack tip opening displacement \(\Delta_{\mathit{m}}\), fracture energy \(G_{c}\), and \(J\)-integral \(J_{c}\) are related as follows in LEFM [5]:  
\[K_{c}^{2}=E\sigma_{\mathrm{s}}\delta_{\mathit{m}}=EG_{c}=EJ_{c} \tag{7}\]  
Crack size \(a_{\mathit{of}}\) of the anchorage zone was calculated according to Equations (6) and (7) and \(\Delta_{\mathit{m}}\) in Table 2 and is listed in the fourth column of Table 3. The modulus of elasticity \(E\) is 2.05 \(\times\) 10\({}^{5}\) MPa, and yield strength \(\sigma_{\mathrm{s}}\) of Q345 qD is 353.2 MPa at room temperature (20 \({}^{\circ}\)C) and 384.9 MPa at low temperature (-60 \({}^{\circ}\)C) [17].  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Operating** & \(K_{c}\) & \(\sigma_{\mathrm{max}}\) **or** & \(\Delta\sigma\) & \(a_{\mathit{cf}}\) & \(a_{\mathit{c,upper}}\) & \(a_{\mathit{c,lower}}\) & \(\Delta K_{\mathit{th}}\) **(MPa\(\cdot\sqrt{\mathrm{m}}\))** & \(a_{\mathit{0}}\) \\
**Case** & **(MPa-\(\sqrt{\mathrm{m}}\))** & **(MPa)** & **(m)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline I & 291.31 & 40.21 & 7.429 & 238.1 & 408.8 & 3.11 & 0.85 & 0.85 \\ II & 291.31 & 47.76 & 5.266 & 238.1 & 408.8 & 3.11 & 0.60 & 0.60 \\ III & 173.84 & 47.76 & 1.875 & 238.1 & 408.8 & 3.89 & 0.94 & 0.94 \\ \hline \end{tabular}
\end{table}
Table 3: \(a_{\mathit{of}}\), \(a_{\mathit{c}}\), \(a_{\mathit{of}}\), and \(a_{\mathit{0}}\) of anchorage structure under different operating cases.  
As can be seen in Table 3 for operating cases I and II, when the temperature effect is taken into account for the cable force range, the stress range increases by 19%, and \(a_{\mathit{cf}}\) decreases by 29%. A comparison of operating cases II and III in Table 3 reveals that since the \(K_{c}\) of steel decreases significantly as the temperature drops, the \(a_{\mathit{cf}}\) at low temperature is only one-third of that at room temperature.  
Considering the effect of low temperature on both the load response and fatigue resistance, the \(a_{\mathit{cf}}\) at low temperature is only one-quarter of that at room temperature. The resistance of the anchorage structure to fracture instability at low temperatures is poorer than that in normal operating conditions. However, even for case III, the \(a_{\mathit{cf}}\) is as large as 1.875 m, indicating that the bridge steel has excellent toughness and that the risk of a brittle fracture is small.  
Considering the geometric limits of crack propagation, the critical crack size is \(a_{c}=238.1\) mm for the upper crack tip and \(a_{c}=408.8\) mm for the lower crack tip, as listed in the fifth and sixth columns of Table 3.