# Flexible Pavement Performance Probability Calculation ## Given Data - **Pavement Structure:** - Asphalt Wearing Surface: 4 in - Dense-Graded Crushed Stone Base: 6 in - Crushed Stone Subbase: 8 in - **All Drainage Coefficients (\(m\))**: 1. - **Design Load:** 18-kip single-axle, 129 per day - **Performance Indices:** - Initial Serviceability Index (\(PSI_{initial}\)): 4.5 - Terminal Serviceability Index (\(PSI_{terminal}\)): 2.5 - **Soil Resilient Modulus (\(M_R\)):** 12,000 lb/in² - **Overall Standard Deviation (\(S_o\)):** .40 - **Design Life:** 20 years ## Step 1: Calculate Total Number of Axle Load Repetitions \[ \text{Total repetitions} = 129 \text{ axles/day} \times 365 \text{ days/year} \times 20 \text{ years} \] \[ = 129 \times 730 = 9,417,000 \text{ axles} \] ## Step 2: AASHTO Flexible Pavement Structural Number (\(SN\)) AASHTO equation for performance: \[ \log_{10}(W_{18}) = Z_R S_o + 9.36 \log_{10}(SN + 1) - .20 + \log_{10} \left( \frac{\Delta PSI}{4.2 - 1.5} \right) / [.40 + \frac{1094}{(SN+1)^{5.19}}] + 2.32 \log_{10}(M_R) - 8.07 \] But for this problem, you want the **probability** that the terminal PSI is exceeded, given the standard deviation. ## Step 3: Probability Calculation The AASHTO method assumes the predicted serviceability loss is a random variable: - **Mean predicted loss:** \( PSI_{initial} - PSI_{terminal} \) - **Standard deviation:** \( S_o \) - **Probability that actual loss < predicted (i.e., actual PSI > terminal PSI):** This is found by calculating the Z-score for the design: \[ Z = \frac{(x - \mu)}{\sigma} \] Where: - \( x \): Actual (terminal) serviceability index (2.5) - \( \mu \): Mean predicted terminal serviceability (2.5) - \( \sigma \): Standard deviation (.40) But more correctly, the **probability that after 20 years, the PSI is above 2.5** is the probability that the actual loss is less than or equal to the predicted loss. Since the design is for PSI to reach 2.5 after 20 years (i.e., the mean \( \mu = 2.5 \)), and the standard deviation is .40, the Z-score is: \[ Z = \frac{2.5 - 2.5}{.40} = \] ## Step 4: Find Probability from Z-Score - For \( Z = \), using the standard normal distribution: \[ P(Z > ) = .5 \] ## Step 5: Interpretation - **Probability that PSI after 20 years is greater than 2.5:** **50%** ## Summary Table | Parameter | Value | |-----------------------------------|-----------------------| | Initial PSI | 4.5 | | Terminal PSI | 2.5 | | Standard Deviation (\(S_o\)) | .40 | | Design Life | 20 years | | Probability \(PSI > 2.5\) @ 20 yr| **50%** | --- ## Conclusion **The probability that the pavement will have a PSI greater than 2.5 after 20 years is 50%.** --- _This is because, by AASHTO design convention, the terminal serviceability index is selected at the mean expected value. With a normal distribution and zero deviation from the mean, there is a 50% chance the actual value will be above (better) or below (worse) the terminal PSI._