Let's solve the problem step by step: --- ## **Given Data** - **Diameter of tank, D \) = 10 m** (circular mat foundation) - **Depth of fill thickness, \( H_{fill} = 8 m \)** - **Modulus of elasticity of soil, \( E = 7,000 \) kPa** - **Maximum allowable settlement, \( s_{max} = 5 \) cm \( = .05 \) m** - **Water is the majority of weight** (so neglect tank self-weight) - **Find:** Maximum water height \( H_w \) --- ## **Step 1: Calculate Pressure from Water** The pressure exerted by water at the base: \[ q = \gamma_{water} \cdot H_w \] Where: - \( \gamma_{water} = 9.81 \) kN/m³ (unit weight of water) - \( H_w \) = height of water (unknown) --- ## **Step 2: Settlement of Flexible Mat on Cohesionless Soil** For a **circular flexible mat** foundation, the settlement is: \[ s = \frac{q \cdot B \cdot (1 - \nu^2)}{E} \] Where: - \( q \) = pressure from water (kPa) - \( B \) = diameter of mat (10 m) - \( \nu \) = Poisson's ratio (for sandy soil, typically .3) - \( E \) = modulus of elasticity (7,000 kPa) Set \( s = s_{max} = .05 \) m. --- ## **Step 3: Substitute and Solve for \( H_w \)** \[ .05 = \frac{q \cdot 10 \cdot (1 - .3^2)}{7,000} \] \[ 1 - .3^2 = .91 \] \[ .05 = \frac{q \cdot 10 \cdot .91}{7,000} \] \[ .05 = \frac{9.1q}{7,000} \] \[ .05 \times 7,000 = 9.1q \] \[ 350 = 9.1q \] \[ q = \frac{350}{9.1} \approx 38.46 \text{ kPa} \] --- ## **Step 4: Relate \( q \) to \( H_w \)** \[ q = \gamma_{water} \cdot H_w \] \[ H_w = \frac{q}{\gamma_{water}} = \frac{38.46}{9.81} \approx 3.92 \text{ m} \] --- ## **Step 5: Compare to Choices** The calculated height (**3.92 m**) is not among the answer choices (8, 9, 10, 11 m), suggesting a possible error in the formula or approach. Let's check if the **modulus of elasticity method** is applicable here: For **total settlement** of a mat on a finite layer over bedrock: \[ s = \frac{q \cdot H_{fill}}{E} \] Using this formula: \[ .05 = \frac{q \cdot 8}{7,000} \] \[ q = \frac{.05 \times 7,000}{8} = \frac{350}{8} = 43.75 \text{ kPa} \] \[ H_w = \frac{43.75}{9.81} \approx 4.46 ~\text{m} \] Still not matching the options. --- ### **Re-examine the Problem** It's possible the intent is to **ignore settlement distribution and use the full thickness** (since mat diameter = 10 m, fill thickness = 8 m). But based on the modulus and thickness, the **correct formula for settlement under a flexible mat is:** \[ s = \frac{q \cdot H}{E} \] So, the calculation above is correct. The answer is about **4.5 m**. --- ## **Final Answer** **The closest answer choice is not listed.** **If required to pick the closest from the provided options, 8 m is the lowest and closest, but the real answer is about 4.5 m.** --- ## **Summary Table** | Step | Formula | Result | |------|---------|--------| | 1 | \( s = \frac{q \cdot H_{fill}}{E} \) | \( .05 = \frac{q \cdot 8}{7,000} \) | | 2 | Solve for \( q \) | \( q = 43.75 \) kPa | | 3 | \( H_w = \frac{q}{9.81} \) | \( H_w \approx 4.46 \) m | --- **Closest answer: 8 m (but actual answer ≈ 4.5 m).**