A THEORETICAL APPROACH FOR THE RELIABILITY ANALYSIS OF DESIGN OF FLEXIBLE PAVEMENTS
A theoretical model to calculate the reliability of pavement design by CBR method has been developed. The technique adopted is the statistical analysis of two different models viz. the traffic prediction model and the performance prediction model. In the analysis, the design variables are assumed to be probabilistic and normally distributed. By knowing statistical variation in the design inputs one can estimate the reliability of design by CBR method for flexible pavements.
Improving the performance of highway pavements has been the main concern of engineers for many years. A major area of interest is the development of highway pavements that require minimum maintenance during the in-service life. Improved construction practices, refinement in design procedures and scrutiny in material selection are some factors that contribute to the performance of pavements. Most of our National Highways are subjected to large traffic volumes, which when combined with the effect of age, environment and material variability cause deterioration and pre-mature failure of pavement structures. Estimation of pavement performance, within acceptable levels, is necessary in selecting the best design for a given project. Most design factors that affect pavement performance have some degree of variation, which may come from dispersion of their values and errors associated with estimating these factors.
Variation or uncertainty in design has been accounted for by the use of safety factors or arbitrary decision based on experience. However, the use of safety factors without considering the uncertainty of design may result in pre-mature failures. In order to assess the effects of uncertainty, probabilistic concepts need to be applied in the design procedure. The concept of reliability could be used to quantify the uncertainty associated with the design procedure. Reliability is a measure of the probability that a pavement will provide satisfactory service to the user throughout its design service life. Performance is understood primarily as a pavement’s ability to support a particular load without showing excessive deflection or cracking.
The CBR method of pavement design is still widely practiced in India. Generally, there are large variations in the inputs like initial traffic in terms of commercial vehicles per day and the CBR value of the subgrade from the actual design inputs. The current thickness design method doesn’t take into consideration the effect of variation from the actual design inputs. In this paper an attempt has been made to incorporate the reliability aspect into pavement design procedure. With a predetermined / known variation in the inputs, the proposed theoretical approach can determine the reliability of the design method.
2. THEORETICAL BACKGROUND
Lemer and Moavenzadeh (1971) developed one of the first models dealing with reliability of pavements. pointed out that the factors affecting the degree of variation in pavement system parameters have a significant effect on system reliability. Lemer attempted to apply the Monte Carlo Simulation method to a complex pavement design method for the Federal Highway Administration of U.S.A.
Another model developed is to apply the concepts of reliability to the systems analysis of rigid pavements for the Texas Highway Department by Kher (1970), He expressed the design output as an optimal pavement design, the design which can be expected to perform at the 95 percent reliability level with respect to the flexural strength of concrete and modulus of subgrade reaction. Another way to express is that 95% probability exists when a premature failure will not occur because of unexpected low strength in the slab or subgrade.
The reliability model developed by Darter and Hudson (1973), considers two major factors associated with the loss of serviceability or failure of a pavement: traffic and environment effects. Darter pointed out that two basic parameters associated with the prediction of the life of a pavement are the 8.16 tonne equivalent single axle load repetitions (ESAL) withstood by the pavement before the serviceability reaches terminal nt , and the number of 8.16 tonne ESAL applied to the pavement, Nt. Darter defined reliability, R, mathematically as follows
R = P [N, > Nr] ,
Reliability was then estimated by considering log Nt and log N7- to be normally distributed. VESYS model (Kenis 1977) uses a procedure similar to that developed by Darter. In this model the reliability is calculated in terms of the present serviceability index as follows:
p f = present serviceability index at time t; and p r is terminal serviceability index, generally set at 2.5 for AASHTO design.
3. RELIABILITY PREDICTION
Reliability prediction of the pavement design is important because of the uncertainties involved in it resulting from the variations in design inputs. Variations in the value of the annual growth rate of vehicles and the number of commercial vehicles per day considered for the pavement design are to be accounted by their statistical measurements of variations. The methodology can be adopted to find out the reliability of a pavement design with the known variations in the design inputs. It can also be used to suggest a limiting value of variation in the design inputs to get a desired reliability level. Different reliability levels of design can be suggested for National Highways, State Highways, Major District Roads etc depending upon the probable accuracy of data collection and subsequent variations in the design inputs.
4. THEORETICAL APPROACH
The reliability is found out from the combined error resulting from two models
4.1 Traffic Prediction Model
The computation of design traffic in terms of the cumulative number of standard axles, as suggested by IRC: 37-2001, is as follows:
where N= Cumulative number of standard axles to be catered
The above equation is analysed to incorporate the variation in the input parameters by taking the variance of the input variables. The variance of a random variable x is defined as the expected value of the square of the deviation from its expectation:
V[x] =E[(x-E[x]2] —— eqn. 2
Where V[x] is the variance of x and E[x] is the expected value of x
When the variable extends over several orders of magnitude, such as traffic volume, it is necessary to present it in log scale and assume its distribution as log Normal. In such cases the variance of a Log distribution is determined by the equation
The above equation (7) can be used to calculate the variance of Log [N].
4.2 PERFORMANCE PREDICTION MODEL
The performance of the pavement is defined by a relationship between the CBR value of the subgrade and the cumulative number of standard axles the pavement is expected to undergo. A performance prediction model has been suggested (S.K.Bagui, 2003) between the Equivalent Standard Axle Load and pavement thickness. It was done to put the relationship into a mathematical form so that statistical analysis can be done to find out the variance of CBR of the subgrade.
The total pavement thickness is proportional to the number of equivalent standard axles in msa for a given CBR value and the equation is in the form of
T = A + B Log N —— eqn. 8
For a given CBR value, two equations are proposed. First equation is applicable upto 10 msa repetitions and second equation from 10 to 150 msa. The proposed equations for different CBR and msa are tabulated below in Table 1
The cumulative equivalent standard axles have been computed as per IRC: 3 7-2001. The pavement was designed according to IRC and for the same thickness, reliability of pavement design was found out by substituting the same input variables in the AASHTO equations.
For the same number of commercial vehicles and different reliability levels the varying pavement thickness is found out from the AASHTO equation. The approach was to find out the reliability of pavement design for different levels of traffic intensities for the same type of soil (medium strength , soil CBR of 7%). The reliability charts developed are shown in Figure No: 1 & 2
The relationship between cumulative equivalent standard axles and the total pavement thickness is as given below
D = 72.336 Ln(CESA)+410.21, R2 =0.985 —— 95% reliability
Higher reliability is preferred for high volume and high density traffic corridors
The authors are thankful to Prof. P.K.Sikdar, Director, Central Road Research Institute, New Delhi, for his kind permission to publish this paper. The authors are also thankful to Shri. Satander Kumar, scientist for his valuable suggestions.