# Long-Term Monitored Road Degradation Functions as a Tool to Increase Quality of Pavement Design
## 2 Research Methods
### Theoretical Approach
The starting point for determining road degradation is based on the relationship between the independent variables and the modeled parameter based on a general deterministic model [38; 39]:  
\[PCS_{t}=f(P_{0},ESAL_{st},H_{e},M_{R},C,W,I) \tag{2}\]  
where \(PCS_{t}\) is the road condition in t year, \(P_{0}\) is the initial road condition in the first year, \(ESAL_{st}\) is the equivalent load of one axle on the road in t age, \(H_{e}\) is the total equivalent structure thickness, \(M_{R}\)--is the modulus of elasticity of the subsoil, \(W\)--is the climatic or the environmental impacts, C--is the structural conditions of roads and I is interaction effects [40]. Most pavement management systems require degradation models designed from rut, IRI and pavement bearing capacity parameters [40; 41; 42; 43]. The rut parameter is one of the most important parameters in terms of influencing operational capability. The development of rut in degradation models is mainly represented by an increase in the depth of the wheel path. When creating degradation functions/models, it is appropriate to monitor only one wheel path, whose prediction of development reaches the limit value earlier than the other [13]. The processing of functions was based on the initial degradation models described in [39] used in Slovakia, Austria, Netherlands, Ireland and also in the software HDM-4. Expressing the IRI parameter through degradation models is a more demanding process than with the rut parameter. This is mainly due to the need of ensuring the same boundary conditions for long-term repeated measurements. The course of IRI in degradation models is represented by its own parameter: IRI (International Roughness Index) [13]. The starting models were the findings presented in [14; 44]. Due to the volume of measured data from long-term repeated measurements, deterministic (empirical) degradation models were created. In the creation of degradation functions, polynomial regression analysis was used, which prescribes a general regression relation in the form of a polynomial with an independent variable. Experience has shown that, for most of the modeled parameters, it is sufficient to consider polynomials of at most fourth degree. \[D(x)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4} \tag{3}\]  
where \(D(x)\) is a modeled degradation function, \(x\) is an independent variable (time or traffic load) and \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(a_{4}\) are regression constants. To mathematically model the degradation function, it was necessary to create a relationship between the relative values of the monitored parameters. When processing the amount of data, local extremes were eliminated, which could have occurred due to non-compliance with the wheel path of the measuring device, non-compliance with measurement procedures, extraordinary climatic impact, etc. The data prepared were thus further processed using statistical methods. Extrapolation methods were used to monitor further developments. The road condition can be expressed using the function as follows:  
\[Q(t)=q\times f \tag{4}\]  
where \(q\) are the transport conditions and climatic conditions and \(f\) is the time. After modification, this relationship can be defined as follows:  
\[q=\frac{Q(t)}{f(t)} \tag{5}\]  
The known time value T represents the limit value of the monitored parameter. Then, the mathematical ratio t/T determines the dimensionless monitoring of changes in the state of individual parameters, depending on the ratio between real time and time, which represents the end of life of a particular parameter. In the same way, it is possible to define the dependence on the traffic load n/N, which depends on the knowledge of the number of passes of the traffic load. Parameter N represents the total (limit) number of design axles for the whole life of the road and parameter n expresses the number of axles up to the moment of evaluation. The mathematical relationship t/T is called the time index and the relationship n/N is called the load index.