# Optimization of Bituminous Road Surfacing Rehabilitations Based on Optimization of Road Asset Value
## 4 Residual Service Life and Pavement Performance Model
### Residual Service Life Calculation  
Equation (10) is used for the calculation of the residual life expectancy of existing pavements:  
\[\sum_{i=1}^{n}Q_{i}\times\frac{\delta_{r,i}}{S_{N}*R_{i,l}}\leq 1 \tag{10}\]  
where:  
\(Q_{i}\): temperature condition coefficient during the period \"i\" (0.2 winter, 0.3 summer, 0.5 spring and autumn),  
\(\delta_{r,i}\): radial stress at the lower edge of the surfacing layer, which arises in the period \"i\" when loaded by the design axle [MPa],  
\(R_{i,l}\): calculated value of the flexural tensile strength of the material under consideration for the conditions in period \"i\" [MPa] normatively prescribed for new pavements or experimentally measured on samples from existing pavements,  
\(n\): number of standard axle load cycles,  
\(S_{N}\): fatigue coefficient. To calculate the fatigue coefficient, it is necessary to derive the actual modulus of elasticity and strength in the pavement surfacing layer. These moduli are estimated during diagnostics of the pavements' load-bearing capacity utilizing a falling weight deflectometer [9]. FWD induces and measures a flexible reaction of the pavement produced by an impact on the pavement surface. This reaction can be graphically represented by a deflection bowl. The shape of this bowl shows the deflection ordinates measured by sensors attached to the pavement at varying distances from the impact point. The actual modulus of elasticity of the individual pavement layers is calculated by a process called back-calculation. The back-calculation gives elastic modulus estimates of the pavement layer by calculation in the layered elastic half-space model [48; 49; 50; 51]. Subsequently, based on the modulus of elasticity and pavement layer thickness, the stresses in the individual layers of the road structure are calculated. The residual life expectancy is calculated following Equation (11). The reliability of this calculation is dependent on the reliability at which the fatigue coefficients a and b are derived. These fatigue coefficients are derived from experimental measurements of fatigue characteristics, which expresses pavement resilience against repeated loading. The test measures pavement resilience against repeated loading. This is done by repeated bending of a pavement surfacing layer test sample. Fatigue tests are carried out following European standards [52]. The results of the fatigue test are plotted in the form of a Wohler diagram shown in Figure 3.  
\[log\ \varepsilon 0_{j}=a_{j}+b\times logN \tag{11}\]  
where:  
\(\varepsilon 0\)): maximum amplitude ordinate of proportional deformation during the test conditions at the beginning of the test,  
\(a\), \(b\): fatigue parameters measured during the fatigue tests is the stress lines coefficient in the range of N,  
\(N\): the number of load repetitions. Following the calculation of \(\varepsilon\)j, the calculation of maximum design axle load repetitions that the pavement can withstand can be determined from Equation (12) [53]:  
\[DAL=10^{6}\times\left(\frac{\gamma\times\varepsilon_{6}}{\varepsilon_{j}} \right)^{B} \tag{12}\]  
where:  
_DAL_: number of design standard axle loads,  
\(\varepsilon_{6}\): average deformation derived from fatigue curve after \(10^{6}\) loading cycles in microstrain [\(\mu\)m/m],\(\varepsilon_{j}\): calculated relative deformation at the bottom of the bituminous bound sub-layer in the pavement construction (based on a multilayer system in homogenous half-space, calculation model with input values presented by [PERSON] [54]),  
\(\gamma\): fatigue test reliability factor--(in our case 1.6 in the line with [55]). \(B\): fatigue characteristics--falling gradient of the fatigue line, B = \(-\)1/b.