Let's go through the step-by-step solution to calculate the number of ESALs (Equivalent Single Ax Loads) for each vehicle based on the given traffic data. --- # Step 1: Understand the Data and Given Parameters - **Traffic Data:** - \( p_t = 2.5 \) (axle load factor for the traffic) - \( SN = 5 \) (number of years) - Number of Passes for each vehicle - **Vehicles and Number of Passes:** | Vehicle | Number of Passes | |------------------------------|------------------| | 18-kip single axle | 1,500,000 | | 32-kip single axle | 255,000 | | 32-kip tandem axle | 400,000 | | 40-kip single axle | 95,000 | | 40-kip tandem axle | 800,000 | | 40-kip tridem axle | 750,000 | | 48-kip tandem axle | 600,000 | | 66-kip tridem axle | 120,000 | --- # Step 2: Understand the ESAL Calculation Formula The general formula for calculating the **ESALs** for each vehicle is: \[ \text{ESAL} = \text{Number of Passes} \times \left( \frac{\text{Axle Load}}{p_t} \right)^4 \] *Note: The ratio is raised to the 4th power because the damage potential of a load is proportional to the 4th power of the axle load ratio.* --- # Step 3: Calculate the Load Ratio and ESALs for Each Vehicle **Step 3.1:** Calculate the load ratio: \[ \text{Load Ratio} = \frac{\text{Vehicle axle load}}{p_t} \] **Step 3.2:** Calculate the damage factor: \[ \text{Damage Factor} = \left( \frac{\text{Axle Load}}{p_t} \right)^4 \] --- # Step 4: Calculate the ESALs for Each Vehicle Let's compute these step-by-step: ### For 18-kip single axle: - Load ratio: \[ \frac{18}{2.5} = 7.2 \] - Damage factor: \[ 7.2^4 = (7.2)^4 \] Calculate: \[ 7.2^2 = 51.84 \\ 7.2^4 = (7.2^2)^2 = 51.84^2 \] \[ 51.84^2 = 2686.5 \] - ESALs: \[ 1,500,000 \times 2686.5 = 4,029,750,000 \] --- ### For 32-kip single axle: - Load ratio: \[ \frac{32}{2.5} = 12.8 \] - Damage factor: \[ 12.8^4 \] Calculate: \[ 12.8^2 = 163.84 \\ 12.8^4 = (12.8^2)^2 = 163.84^2 \] \[ 163.84^2 = 26843.4 \] - ESALs: \[ 255,000 \times 26843.4 \approx 6,841,491,000 \] --- ### For 32-kip tandem axle: - Load ratio: \[ \frac{32}{2.5} = 12.8 \] - Damage factor: \[ 12.8^4 = 26843.4 \] - ESALs: \[ 400,000 \times 26843.4 \approx 10,737,360,000 \] --- ### For 40-kip single axle: - Load ratio: \[ \frac{40}{2.5} = 16 \] - Damage factor: \[ 16^4 \] Calculate: \[ 16^2 = 256 \\ 16^4 = 256^2 = 65,536 \] - ESALs: \[ 95,000 \times 65,536 \approx 6,221,920,000 \] --- ### For 40-kip tandem axle: - Load ratio: \[ 16 \] - Damage factor: \[ 65,536 \] - ESALs: \[ 800,000 \times 65,536 \approx 52,428,800,000 \] --- ### For 40-kip tridem axle: - Load ratio: \[ 16 \] - Damage factor: \[ 65,536 \] - ESALs: \[ 750,000 \times 65,536 \approx 49,152,000,000 \] --- ### For 48-kip tandem axle: - Load ratio: \[ \frac{48}{2.5} = 19.2 \] - Damage factor: \[ 19.2^4 \] Calculate: \[ 19.2^2 = 368.64 \\ 19.2^4 = (368.64)^2 \] \[ 368.64^2 \approx 135,911.4 \] - ESALs: \[ 600,000 \times 135,911.4 \approx 81,546,840,000 \] --- ### For 66-kip tridem axle: - Load ratio: \[ \frac{66}{2.5} = 26.4 \] - Damage factor: \[ 26.4^4 \] Calculate: \[ 26.4^2 = 696.96 \\ 26.4^4 = (696.96)^2 \] \[ 696.96^2 \approx 485,752.4 \] - ESALs: \[ 120,000 \times 485,752.4 \approx 58,290,288,000 \] --- # **Final Step: Sum All ESALs** To get the total ESALs over the 5-year period, sum all individual ESALs: \[ \text{Total ESALs} = \sum \text{individual ESALs} \] \[ \approx 4.03 \text{ billion} + 6.84 \text{ billion} + 10.74 \text{ billion} + 6.22 \text{ billion} + 52.43 \text{ billion} + 49.15 \text{ billion} + 81.55 \text{ billion} + 58.29 \text{ billion} \] Adding: \[ \approx 4.03 + 6.84 + 10.74 + 6.22 + 52.43 + 49.15 + 81.55 + 58.29 = 268.21 \text{ billion ESALs} \] --- # **Summary:** | Vehicle | Approximate ESALs (over 5 years) | |----------------------------|----------------------------------| | 18-kip single axle | 4.03 billion | | 32-kip single axle | 6.84 billion | | 32-kip tandem axle | 10.74 billion | | 40-kip single axle | 6.22 billion | | 40-kip tandem axle | 52.43 billion | | 40-kip tridem axle | 49.15 billion | | 48-kip tandem axle | 81.55 billion | | 66-kip tridem axle | 58.29 billion | | **Total ESALs** | **~268.21 billion** | --- **Note:** These are approximate values for the purpose of pavement design calculations.