4. A soi-layer is fully saturated with a unit weight of $$ 20 \mathrm{kN} / \mathrm{m}^{3} $$ and a water table located 1 m below the surface. Calculate the total stress ( $$ \sigma $$ ), pore water pressure (u), and effective stress ( $$ \sigma^{\prime} $$ ) at depths of: a) 1 m b) 3 m c) 5 m

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To determine the **total stress (σ)**, **pore water pressure (u)**, and **effective stress (σ′)** at various depths within the soil layer, we'll follow a systematic approach based on soil mechanics principles.

### **Given Data:**
- **Unit Weight of Fully Saturated Soil (γ):** 20 kN/m³
- **Water Table Depth:** 1 m below the surface
- **Unit Weight of Water (γ₍w₎):** Typically, γ₍w₎ = 10 kN/m³

### **Key Concepts:**

1. **Total Stress (σ):**
   - **Definition:** The total stress at a depth $$ z $$ is the weight of the soil (and water, if present) above that point.
   - **Formula:** 
     $$
     \sigma = \gamma \times z
     $$
   
2. **Pore Water Pressure (u):**
   - **Definition:** The pressure exerted by water within the pores of the soil.
   - **Formula:** 
     $$
     u = 
     \begin{cases}
       0, & \text{if } z \leq \text{Water Table Depth} \
       \gamma_w \times (z - \text{Water Table Depth}), & \text{if } z > \text{Water Table Depth}
     \end{cases}
     $$
   
3. **Effective Stress (σ′):**
   - **Definition:** The stress carried by the soil skeleton, crucial for assessing soil stability.
   - **Formula:** 
     $$
     \sigma' = \sigma - u
     $$

### **Calculations for Each Depth:**

#### **a) Depth = 1 m**

- **Total Stress (σ):**
  $$
  \sigma = 20 \, \text{kN/m}^3 \times 1 \, \text{m} = 20 \, \text{kPa}
  $$
  
- **Pore Water Pressure (u):**
  - Since $$ z = $$ Water Table Depth, $$ u = 0 \, \text{kPa} $$
  
- **Effective Stress (σ′):**
  $$
  \sigma' = 20 \, \text{kPa} - 0 \, \text{kPa} = 20 \, \text{kPa}
  $$

#### **b) Depth = 3 m**

- **Total Stress (σ):**
  $$
  \sigma = 20 \, \text{kN/m}^3 \times 3 \, \text{m} = 60 \, \text{kPa}
  $$
  
- **Pore Water Pressure (u):**
  $$
  u = 10 \, \text{kN/m}^3 \times (3 \, \text{m} - 1 \, \text{m}) = 20 \, \text{kPa}
  $$
  
- **Effective Stress (σ′):**
  $$
  \sigma' = 60 \, \text{kPa} - 20 \, \text{kPa} = 40 \, \text{kPa}
  $$

#### **c) Depth = 5 m**

- **Total Stress (σ):**
  $$
  \sigma = 20 \, \text{kN/m}^3 \times 5 \, \text{m} = 100 \, \text{kPa}
  $$
  
- **Pore Water Pressure (u):**
  $$
  u = 10 \, \text{kN/m}^3 \times (5 \, \text{m} - 1 \, \text{m}) = 40 \, \text{kPa}
  $$
  
- **Effective Stress (σ′):**
  $$
  \sigma' = 100 \, \text{kPa} - 40 \, \text{kPa} = 60 \, \text{kPa}
  $$

### **Summary of Results:**

| **Depth (m)** | **Total Stress σ (kPa)** | **Pore Water Pressure u (kPa)** | **Effective Stress σ′ (kPa)** |
|---------------|--------------------------|----------------------------------|-------------------------------|
| 1             | 20                       | 0                                | 20                            |
| 3             | 60                       | 20                               | 40                            |
| 5             | 100                      | 40                               | 60                            |

### **Conclusion:**
At each specified depth, the total stress increases linearly with depth, while the pore water pressure starts to contribute only below the water table (1 m). Consequently, the effective stress, which is crucial for understanding soil behavior under load, increases with depth but at a slower rate due to the influence of pore water pressure.
At 1 m depth: - Total Stress (σ): 20 kPa - Pore Water Pressure (u): 0 kPa - Effective Stress (σ′): 20 kPa At 3 m depth: - Total Stress (σ): 60 kPa - Pore Water Pressure (u): 20 kPa - Effective Stress (σ′): 40 kPa At 5 m depth: - Total Stress (σ): 100 kPa - Pore Water Pressure (u): 40 kPa - Effective Stress (σ′): 60 kPa

A soi-layer is fully saturated with a unit weight of and a water table
located 1 m below the surface.
Calculate the total stress ( ), pore water pressure (u), and effective stress ( ) at
depths of:
a) 1 m
b) 3 m
c) 5 m

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A soi-layer is fully saturated with a unit weight of and a water table located 1 m below the surface. Calculate the total stress ( ), pore water pressure (u), and effective stress ( ) at depths of: a) 1 m b) 3 m c) 5 m