# Durability Investigation of Fiber-Reinforced Functionally Graded Concretes in Cold Regions
## 2 Test Methods and Proportioning Model
### Freezing Resistance Test  
The fast freezing method was selected for the freezing test, considering the engineering environment. Each category had 3 test specimens, and the prisms' size was 100 mm \(\times\) 100 mm \(\times\) 400 mm. The standard freeze-thaw cycle was adopted, with a starting temperature of 20 \({}^{\circ}\)C, falling to \(-\)20 \({}^{\circ}\)C at a rate of 10 \({}^{\circ}\)C/h over 4 h, maintaining a constant temperature for 3 h, and then rising to 20 \({}^{\circ}\)C at a rate of 10 \({}^{\circ}\)C/h over 4 h, before maintaining a constant temperature for 1 h. One freeze-thaw cycle required 12 h. The fundamental transverse frequency of the specimen was measured every 25 freeze-thaw cycles. The arithmetic mean of the fundamental transverse frequencies of each category is the measured value \(f_{n}\), where \(f_{0}\) is the initial value of the fundamental transverse frequency \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Concrete** & **Seepage** & **Impermeability** & **Seepage** & **Impermeability** & **Seepage** & **Impermeability** \\
**Strength Grade** & **Pressure (MPa)** & **Grade** & **Pressure (MPa)** & **Grade** & **Pressure (MPa)** & **Grade** \\ \hline C35 & \(1.0\) & \#1 & & & \#2 & & \#3 & W12 \\  & & & W9 & \(1.2\) & & W11 & \(1.3\) & \#6 & \\ C40 & \(1.1\) & \#4 & & & & & & \\  & & & W10 & \(1.4\) & & W13 & \(1.3\) & \#12 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Concrete water penetration resistance test results. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cement Type** & **56 d Diffusion Coefficient of CI-/\(\times\)10\({}^{-12}\) m\({}^{2}\)/s** \\ \hline C35 & \#1 & \#2 & \#3 \\  & 4.2 & \(2.5\) & \(2.6\) \\ \hline C40 & \#4 & \#5 & \#6 \\  & 3.9 & \(2.7\) & \(2.8\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Concrete resistance to chloride ion penetration test results. of the specimen before the freeze-thaw test, and the relative dynamic modulus of elasticity of the specimens after n freeze-thaw cycles [30] was calculated by Formula (3):  
\[P_{i}=\frac{f_{n}^{2}}{f_{0}^{2}} \tag{3}\]  
The test can be terminated when one of the following three conditions occurs:  
1. Considering the design requirements of the concrete freezing grade, the number of freeze-thaw cycles of specimens from #1 to #3 reaches 300 times, and the number of freeze-thaw cycles of specimens from #4 to #6 reaches 400 times;
2. The dynamic modulus of elasticity of the specimen falls below 60%;
3. The average weight loss rate of the specimens reaches 5%. Each group's average mass loss rate is determined from the arithmetic mean of the three specimens' mass loss test values. The test results are shown in Table 6 and Figure 8.  
Figure 8 shows the relationship between the frozen elastic modulus of concrete and the number of freeze-thaw cycles. The loss of dynamic elastic modulus was observed by changing the number of freeze-thaw cycles to judge the frost resistance of concretes of various strength levels. It can be seen that the concrete specimens of each ratio met the requirements of frost resistance. However, the dynamic elastic modulus loss of #2,