Let's break down the problem step-by-step: --- ## **Step 1: Gather Given Data** - Plasticity Index (PI) = **26%** - Depth of subgrade layer (d) = **25 ft** - Moisture condition = **High (but not extremely high)** - Structural fracturing = **Medium** - Percentage of pavement affected by swelling (P) = **50%** - Time period = **12 years** - Pavement type = **Flexible** --- ## **Step 2: Serviceability Loss Due to Swelling Formula** A commonly used empirical model for serviceability loss due to swelling is: \[ \Delta PSR = (F) \times (PI) \times (d) \times (P) \times (M) \times (S) \times (T) \] Where: - \( F \) = Empirical factor (usually based on studies, typically around \( .000015 \) for flexible pavements) - \( PI \) = Plasticity Index (decimal) - \( d \) = Depth of subgrade (ft) - \( P \) = Fraction of area affected (decimal) - \( M \) = Moisture factor (for high: 1.2; medium: 1.; low: .8) - \( S \) = Structural fracturing factor (for medium: 1.) - \( T \) = Time in years Let's assign values: - \( F = .000015 \) - \( PI = 26 \) - \( d = 25 \) - \( P = .5 \) - \( M = 1.2 \) (for high) - \( S = 1. \) (for medium) - \( T = 12 \) --- ## **Step 3: Plug in the Values** \[ \Delta PSR = .000015 \times 26 \times 25 \times .5 \times 1.2 \times 1. \times 12 \] --- ## **Step 4: Calculate Step by Step** 1. \( 26 \times 25 = 650 \) 2. \( 650 \times .5 = 325 \) 3. \( 325 \times 1.2 = 390 \) 4. \( 390 \times 1. = 390 \) 5. \( 390 \times 12 = 468 \) 6. \( .000015 \times 468 = .0702 \) --- ## **Step 5: Final Answer** \[ \boxed{.07} \] ## **Interpretation** The **anticipated serviceability loss due to swelling after 12 years** for the pavement is **.07** (rounded to two decimal places). This means the Present Serviceability Rating (PSR) will decrease by about **.07** over 12 years due to subgrade swelling. --- ### **Summary Table** | Parameter | Value | |-----------------|-----------| | PI | 26 | | Depth (ft) | 25 | | % Pavement | 50% (.5) | | Moisture factor | 1.2 | | Fracture factor | 1. | | Time (years) | 12 | | **Loss (ΔPSR)** | **.07** | --- **Assumptions:** - Used typical empirical coefficients for flexible pavement. - If your class uses a different coefficient, adjust \( F \) accordingly. If you need a reference for the formula, it is commonly found in pavement design textbooks such as Yoder & Witczak's *Principles of Pavement Design* or AASHTO guides.