## 3 Results and Discussion
### Stiffness Modulus  
One of the primary objectives of modifying an asphalt binder is to reduce its susceptibility to putting by increasing its stiffness at high temperatures. Additionally, modifying the  
Figure 8: Penetration index results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Factors** & **DOF** & **Sums of Squares** & **Variance** & **F-Value** & **Pure Sum** & **Contribution** & \(p\)**-Value** \\ \hline Cecabase & 3 & 0.321 & 0.107 & 136.708 & 0.319 & 77.082\% & 0.0010 \\ Temperature & 3 & 0.024 & 0.008 & 10.371 & 0.022 & 5.848\% & 0.0431 \\ Time & 3 & 0.027 & 0.009 & 11.382 & 0.024 & 6.418\% & 0.0380 \\ Speed & 3 & 0.042 & 0.014 & 17.893 & 0.040 & 10.089\% & 0.0203 \\ \hline Error & 3 & 0.002 & 0.001 & & & 0.564\% & \\ \hline Total & 15 & 0.416 & & & & 100.000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: ANOVA table for softening point after RTFOT. binder aims to enable it to alleviate cryogenic stresses in winter by decreasing its stiffness at low temperatures. In other words, the purpose of asphalt modification is to widen the temperature range of the binder, enhance its ability to endure heavy traffic, and withstand extreme weather conditions [63]. The stiffness of asphalt is the main factor causing cracking at low service temperatures [64]. [PERSON] [65] established the concept of stiffness modulus, which defines the ratio of constant stress to the resulting strain at a given time. Based on tests on different bitumen samples, a nomograph was developed to determine asphalt binder stiffness without publishing the empirical equations used in its construction. The nomograph considers penetration, softening point, loading time, and temperature. Several researchers attempted to formulate a mathematical equation for the stiffness modulus of bitumen, S(t), based on P\({}_{25}\), T\({}_{SP}\), and [PERSON] et al. [66] proposed the equation given below to calculate the stiffness modulus. \[\text{S(t)}=1.157\times 10^{-7}\text{t}^{-0.368}\text{e}^{-\text{PT}}\text{(T}_{ \text{SP}}-\text{T})^{5} \tag{5}\]  
The t value indicates the duration of loading, and the T value represents the temperature of loading. Equation (5) is applicable within the loading duration of 0.01 \(<\) t \(<\) 0.1, penetration index range of \(-\)1 \(<\) PI \(<\) 1, and temperature range 10 \({}^{\circ}\)C \(<\) (T\({}_{\text{SP}}-\text{T})<\) 70 \({}^{\circ}\)C.  
The BitProps program was developed as a convenient tool for determining binder stiffness. It utilized the digitized version of [PERSON]'s nomograph, facilitating the calculations [67]. [PERSON] et al. [68] stated that the stiffness modulus found from nomographs showed the same pattern as the modulus found from the dynamic shear rheometer (DSR) test. Using an adjustment factor, they stated that the nomograph results were identical to the DSR test results. They concluded that the stiffness modulus found from nomographs can be used as a parameter to determine how much the additive influences the binder. [PERSON] et al. [69] noted an excellent agreement between the stiffness values of asphalt binders calculated from experimental tests and those estimated using the Van der Poel nomograph. They concluded that the nomograph remains a valid and accurate method for estimating binder stiffness, making it highly useful for practical applications. This study evaluated the stiffness properties using BitProps for the Van der Poel nomograph (Figure 9) and also using Equation (5). The loading duration was 0.05 s while loading temperatures ranged from \(-\)10 to 40 \({}^{\circ}\)C. The stiffness moduli at \(-\)10 \({}^{\circ}\)C calculated with Equation (5) were an average 13% less than the moduli found from the nomograph. The stiffness modulus of modified bitumen decreased at \(-\)10 \({}^{\circ}\)C with the second level of Cecabase RT BIO10, but it increased at the third and fourth levels compared to virgin bitumen. At 40 \({}^{\circ}\)C, with all levels of Cecabase RT BIO10, the stiffness modulus decreased slightly compared to virgin bitumen.