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In laboratory tests, the fatigue life of asphalt mix is frequently assessed by using a sinusoidal or haversine waveform at a particular frequency.
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In this article, we are going to discuss the ways to improve the whole procedure. The shape and frequency of the tensile response pulses at the base of the asphalt layer are influenced by the pavement structure and loading conditions.
In order to better simulate field conditions, two methods for predicting the loading frequency in laboratory asphalt fatigue tests are introduced in this paper. The longitudinal and transverse stress and strain response pulses for 5,000 (five thousand) four-layered pavement sections were calculated.
The effective length of the response pulses was established after the haversine function was fitted to them using the equal-energy pulse concept.
The effective length (i.e., frequency) of tensile stress and strain pulses in longitudinal and transverse directions were then predicted using the haversine waveform using two techniques: Multivariate Adaptive Regression Splines (MARS) and Artificial Neural Network (ANN) methods.
According to the data, both approaches (MARS and ANN) are very accurate at predicting the frequency of loading in HMA fatigue tests under controlled stress and strain modes. However, the ANN method is more accurate than the MARS method.
Furthermore, it is demonstrated that a straightforward equation enables the results of the current study to be extrapolated to sinusoidal waveforms.
One of the most frequent forms of failure for flexible pavements is fatigue cracking brought on by repeated traffic load. In laboratory tests, the fatigue life of asphalt mixes is frequently assessed by applying a dynamic load with a sinusoidal or haversine waveform at a particular frequency. Several variables, including vehicle speed, loading characteristics, environmental conditions, and pavement structure, affect the frequency of horizontal tensile stress and strain pulses in both the longitudinal and transverse directions.
To predict the duration of the vertical stress pulse at various asphalt layer depths, a number of relationships were put forth.
However, there is still a lack of understanding regarding the frequency of horizontal stress and strain pulses at the base of the asphalt layer.
Restrepo-Velez (2011) examined how various variables affected how long tensile strains persisted at the base of the asphalt layer. On the WAY-30 project's perpetual section AC 664, the pavement responses were measured.
Restrepo-Velez (2011) also came to the conclusion that the pulse durations were longer at lower values of speed and temperature, which is consistent with Hernandez's observation.
Furthermore, Restrepo-Velez (2011) compared the pavement responses that were actually observed with those that were forecast using the MEPDG method and the multilayer elastic analysis program JULEA.
Based on this comparison, the MEPDG procedure produced strain pulse durations that were about 80% longer than those actually measured in the field.
It is pointless to mention that some studies have demonstrated that the loading times derived from the longitudinal strain gauges are consistent with the approach advised by the MEPDG method.
According to the findings of these studies, there is disagreement regarding the MEPDG method's applicability for predicting loading times in a longitudinal direction.
This is due to the MEPDG method's initial development, which involved computing the loading time brought on by vertical stress rather than horizontal responses.
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Fakhri et al. (2013) proposed regression equations for calculating the duration of tensile stress and strain pulses at the bottom of the asphalt layer in both longitudinal and transverse directions based on the viscoelastic analysis of 112 flexible pavement sections.
Through the use of weighted nonlinear regression, proposed equations were developed based on the haversine waveform. With this regression technique, the fitted haversine waveform only depends on three variables: the speed of the moving wheel, the thickness of the asphalt layer, and the temperature of the asphalt layer.
The shape and frequency of the longitudinal and transverse response pulses at the base of the asphalt layer are influenced by a number of factors, including the thickness of the various layers, the ratio of resilient modulus for the two succeeding layers, and the contact radius of the tire.
Therefore, all of these factors should be taken into account to develop an all-encompassing method for forecasting the loading time (or frequency) of longitudinal and transverse response pulses.
We are aware of no general relationship or method that can be used to calculate the frequency of horizontal tensile stress and strain pulses at the base of asphalt layers.
Existing relationships are essentially limited to vertical stress durations, and a small number of equations that were created using measurements of horizontal strain pulses taken in the field cannot be applied in most other circumstances where the loading characteristics and pavement section are very dissimilar from the desired sections.
This study's goal is to present two approaches based on haversine and sinusoidal waveforms for forecasting the frequency of tensile stress and strain pulses at the base of asphalt layers in both longitudinal and transverse directions.
These techniques allow for a more realistic determination of the frequency of loading in HMA fatigue laboratory tests, such as four points bending beam and indirect tensile (IDT) fatigue tests, based on pavement design and loading characteristics.
The study's findings can be applied to more accurately simulate dynamic loading for asphalt mix fatigue tests in both stress control and strain control modes.
5000 flexible pavement sections were examined using layered elastic theory to create the dataset related to the normalized stress and strain pulses at the base of the asphalt layer (LET).
At various radial distances from the contact area's center, the longitudinal and transverse stress and strain values were calculated. A single wheel with a 700 kPa contact pressure was taken into consideration as the moving load.
Only the contact radius was taken into account as a variable because the contact pressure has no impact on the normalized shape and duration of response pulses.
Each time, the pavement structure was viewed as a four-layered system with linear elastic properties applied to each layer. Two succeeding layer interfaces were regarded as fully-bounded.
Using the layered elastic analysis program Non-PAS, which allows the analysis of pavement systems composed of up to ten linear or nonlinear elastic layers subject to up to ten circular contact loads, each section of pavement was examined.
The Non-PAS program's ability to precisely predict the pavement responses to single and multiple loading has been thoroughly verified using the Kenlayer program.
It was necessary to develop such a program due to the Kenlayer and ELSYM 5 and other pavement analysis programs' limitations on the number of response points.
The created code makes it easier to calculate responses (strains, deflections, and stresses) for an infinite number of radial direction points.
As long as the amount of tensile stress or tensile strain reduces to 1% of the maximum stress or strain, respectively, stress and strain values in the longitudinal and transverse directions were calculated at various radial distances with one-centimeter intervals.
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The energy injected into the system per loading cycle can explain the effect of the loading waveform on the fatigue life of asphalt mixes. The area occupied by a loaded waveform in stress (or strain) against a time coordinate system has been hypothesized to be inversely correlated with energy.
The haversine function, on the other hand, is a good representation of strain pulses in the longitudinal and transverse directions, according to the results of full-scale tests.
These observations led to the fitting of the haversine function to the analytical response pulse such that the area under the haversine function was equal to the analytical response pulse obtained by LET analysis for each record of the dataset (stress and stain pulses in both longitudinal and transverse directions at the bottom of asphalt layer for 5000 pavement sections).
For resolving issues of this nature, Friedman (1991) created the theory of Multivariate Adaptive Regression Splines (MARS).
Due to its classification as a nonparametric regression procedure that makes no assumptions about the type of functional relationship (such as a linear or logistic relationship) between the dependent and predictor variables, the MARS technique has grown particularly popular in the field of data mining.
Instead, even in circumstances where the relationship between the predictors and the dependent variables is non-monotone and challenging to approximate with parametric models, useful models (i.e., models that yield accurate predictions) can be derived.
The entire input variable space is divided into a number of sub-regions by MARS. For each sub-region, a different mathematical equation is defined. The core concept behind MARS is to approximate the model using a combination of linear truncated basis functions.
As a result, to account for interactions, the functions of MARS are either single spline functions or the sum of two or more truncated power functions.
The forward pass and the backward pass are the two stages through which models are built on MARS.
The initial stage. MARS starts with a model that only contains the intercept term and gradually adds basis functions in pairs. The basis function pair that offers the greatest reduction in sum-of-squares residual error is identified at each step.
The only difference between the two basic functions in the pair is that each function uses a different side of a mirrored hinge function.
A new hinge function is multiplied by each new basis function made up of an existing model term. A variable and a knot together define a hinge function.
Therefore, in order to add a new basis function, MARS must search through all possible combinations of already existing terms (parent terms), all variables (in order to choose one for the new basis function), and all possible values for each variable (for the knot of the new hinge function).
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The forward phase is carried out up until the maximum number of terms is reached or the change in residual error is smaller than the threshold. The user has previously specified these parameters.
Due to the nature of hinge functions, this search can be completed more quickly by using a heuristic method that limits the number of parent terms taken into account at each step, or by using a fast least-squares update technique.
The Reverse Pass. Typically, the forward pass results in an over fitted model. To create a model with better generalization capabilities, a backward deletion phase is therefore activated. To find the best sub-model, the model is pruned in this phase by removing the least effective sub-model.
From this study, the following conclusions can be made.
In comparison to the transverse direction, the haversine function fitted stress and strain pulses better in the longitudinal direction.
While a transverse response pulse typically causes the HMA layer to experience tensile stress or strain, a longitudinal response pulse typically consists of two compression zones and one tension zone. The strain pulses calculated by LET resemble those seen in full-scale pavement tests very closely.
Both in the longitudinal and transverse directions, there are significant correlations between the effective lengths of the stress and strain pulses.
The effective length for other responses can be easily calculated or measured using these relations and the effective length for one of the stresses or strain pulses.
When the effective length is large, the MARS approach predicts it with a high degree of accuracy (e.g., wide-base tires or pavement sections with a thick asphalt layer). As a result, when the effective length is small, a more precise method is required.
At the base of the asphalt layer, a feed-forward backpropagation neural network with a 7-15-4 architecture accurately predicts the effective length of tensile stress and strain pulses in both the longitudinal and transverse directions.
An Excel-based computing code was created using ANN to forecast the frequency of various responses at the asphalt layer's base.
The contact radius of the wheel, the thickness of the asphalt layer, the ratio of the resilient modulus of the asphalt layer to the base layer, and the ratio of the resilient modulus of the base layer to sub base layer are the most important variables that influence the effective length of tensile response pulses.
With an increase in these factors, both the longitudinal and transverse effective lengths of the tensile response pulses grow.
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