Assessing the Role of Longitudinal Variability of Vertical Track Stiffness in the Long - Term Deterioration

The longitudinal variation of support stiffness is believed to be a key driver for the differential settlement of the ballast. This paper presents an innovative approach to correlate these two variables. In order to overcome the lack of input data, a statistical model is applied and the spatial correlation of the input stiffness data is controlled to generate naturally occurring variations. A large set of simulations is run and the relationship between deterioration rate and support properties is then analysed using log-linear multiple regression models. The results show that it is not only its mean value that has an impact in the long term settlement behaviour, but it is also its longitudinal variability. Also, main results suggest that the speed effect might be more crucial in localized corrective maintenance needs than in preventive maintenance needs. The proposal of this research is to statistically generate significant input data sets for track stiffness out of a known finite set of measurement values and then use a vehicle-track interaction model to predict track force in the time domain and derive long-term settlement of the track. The novelty of this approach resides both in the control of the spatial correlations of the input stiffness data to generate naturally occurring variations and in the attempt at producing a correlation between track settlement and track stiffness. The focus on spatial correlations is in line with current research in statistical modelling of rail track irregularities (Andrade and Teixeira, 2015). In Section 2, the statistical approach used to create the new sets of track stiffness data which can appropriately reproduce the statistical spatial properties (i.e. auto-correlation) of the measured ones is presented. Section 3 describes the vertical model of vehicle/track interaction system and the iterative process used to evaluate the long-term track behaviour. The main results in terms of correlation between deterioration rate and support properties are presented in Section 4. The influence of vehicle speed is also discussed. Finally, in Section 5 conclusions are drawn and future works are discussed. 2 STATISTICAL REPRESENTATION OF THE TRACK STIFFNESS Two sets of sleeper support stiffness data have been analysed, whose main characteristics are reported in Table 1. The data was measured in the U.K. using the FWD equipment. Table 1. Main characteristics of the measured sites. SITE Number of measured sleepers Support stiffness mean value [kN/mm/sleeper end] Support stiffness SD [kN/mm/sleeper end] Minimum value kN/mm/sleeper end] Maximum value kN/mm/sleeper end] KS test pvalue A 155 84.6 14.4 (Var[Kz]=208) 44.4 143.8 0.32 B 80 110.4 16.2 (Var[Kz]=262) 59.8 157.9 0.90 The distribution curves are shown in Figure 2, assuming a normal distribution of the support stiffness. This hypothesis has been validated performing the Kolmogorov-Smirnov (K-S) goodness-of-fit test and checking that the p-value of each set of data (Table 1) is not less than the 10% significance level (Dodge, 2008). Figure 1. Distribution curves for the two sites considered. Such distribution function in theory allows generating any number of data sets with the correct mean value and distribution. However, it does not ensure that the correct spatial distribution of stiffness, i.e. the correct variation between one sleeper and the next, is achieved along the track. In order to reproduce the spatial properties of the measured data, the ARIMA modelling approach has been used in the present study (Cryer and Chan, 2008). In a general way, the ARIMA model distinguishes three components: a mean component ( ) and/or a weighted sum of neighbouring values and/or a weighted sum of neighbouring error values ( ). From a mathematical point of view, a time series is said to follow an Integrated Autoregressive Moving Average (ARIMA) model if the d difference is a stationary Autoregressive Moving Average (ARMA) process. If follows an ARMA (p, q) model, then is an ARIMA (p, d, q) process. For practical purposes, values for d are usually assumed to be equal to d=1 or at most d=2. For instance, for a stationary ARIMA (p, 0, q) model with d=0 and with a mean equal to : ⋯ ⋯ (1) Different model specifications can be compared based on the Akaike’s Information Criterion (AIC) (Akaike, 1973). This criterion conducts model selection based on the one with minimum value for the AIC: AIC log ∗ (2) Where L* is the maximum likelihood and k is the number of parameters (k=p+q+1). Table 2 provides the best ARIMA models with estimated values for the associated parameters, with the respective standard deviations in parenthesis. Table 2. ARIMA models with estimated values for each site analysed. Site Model φ1 φ2 φ3 φ4 φ5 θ1 σ AIC A ARIMA (5, 1, 0) -0.5683 (0.0809) -0.4721 (0.0905) -0.4753 (0.0908) -0.3511 (0.0898) -0.0438 (0.0838) 16