2. A stratum of sand 2.5 m thick overlies a stratum of saturated clay 3 m thick. The water table is
1 m below the surface. For the sand, Gs = 2.65, e = 0.50 and for the clay Gs = 2.72, e = 1.1.
Calculate the total and effective vertical stresses at depths of 1 m, 2.5 m and 5.5 m below the
surface assuming that the sand above the water table is completely dry
solution
yes
To calculate the total and effective vertical stresses at different depths, we will use the equation:
𝜎'v = 𝜎v - 𝜎w
where:
𝜎'v = effective vertical stress
𝜎v = total vertical stress
𝜎w = pore water pressure
Let's calculate the stresses at each depth:
Depth: 1 m below the surface
For this depth, we only have the saturated clay layer. The total vertical stress (𝜎v) can be calculated using the equation:
𝜎v = 𝜌c * g * z
where:
𝜌c = unit weight of the clay = Gs * 𝜌w
g = acceleration due to gravity = 9.81 m/s²
z = depth below the surface = 1 m
Given:
Gs of clay = 2.72
e of clay = 1.1
Calculations:
𝜌c = 2.72 * 𝜌w
To find 𝜌w, we need to convert the water content (e) into void ratio (E) using:
E = (1+e)/ (1-e)
=> E = (1+1.1)/ (1-1.1)
=> E = -21
Now, the unit weight of water (𝜌w) can be calculated using:
𝜌w = γw / (1+E)
where:
γw = unit weight of water = 9.81 kN/m³
Calculations:
𝜌w = 9.81 / (1-21)
𝜌c = 2.72 * 𝜌w
Next, we calculate the total vertical stress (𝜎v) using the formula:
𝜎v = 𝜌c * g * z
Calculations:
𝜎v = 𝜌c * g * z
Now, let's calculate the effective vertical stress (𝜎'v):
𝜎'v = 𝜎v - 𝜎w
Calculations:
𝜎'v = 𝜎v - 𝜎w
Depth: 2.5 m below the surface
At this depth, we need to consider both the sand and clay layers.
For the sand layer:
The sand is completely dry, which means there is no pore water pressure (𝜎w = 0).
The total vertical stress (𝜎v) can be calculated using the equation:
𝜎v = 𝜌s * g * z
where:
𝜌s = unit weight of the sand = Gs * 𝜌w
g = acceleration due to gravity = 9.81 m/s²
z = depth below the surface = 2.5 m
Given:
Gs of sand = 2.65
Calculations:
𝜌s = 2.65 * 𝜌w
To calculate 𝜌w, we will use the void ratio (e) of the sand, which is 0.5. Using the equation:
E = (1+e)/ (1-e)
=> E = (1+0.5)/ (1-0.5)
=> E = 3
Now, the unit weight of water (𝜌w) can be calculated using:
𝜌w = γw / (1+E)
where:
γw = unit weight of water = 9.81 kN/m³
Calculations:
𝜌w = 9.81 / (1+3)
𝜌s = 2.65 * 𝜌w
Next, we calculate the total vertical stress (𝜎v) using the formula:
𝜎v = 𝜌s * g * z
Calculations:
𝜎v = 𝜌s * g * z
Now, let's calculate the effective vertical stress (𝜎'v):
𝜎'v = 𝜎v - 𝜎w
Calculations:
𝜎'v = 𝜎v
For the clay layer:
To calculate the total vertical stress (𝜎v), we can use the same formula as before:
𝜎v = 𝜌c * g * z
Calculations:
𝜌c = 2.72 * 𝜌w
𝜎v = 𝜌c * g * z
Now, let's calculate the effective vertical stress (𝜎'v):
𝜎'v = 𝜎v - 𝜎w
Calculations:
𝜎'v = 𝜎v - 𝜎w
Depth: 5.5 m below the surface
At this depth, we only have the clay layer. So, we will calculate the total and effective vertical stresses using the same formulas as before.
For the clay layer:
To calculate the total vertical stress (𝜎v), we can use the formula:
𝜎v = 𝜌c * g * z
Calculations:
𝜌c = 2.72 * 𝜌w
𝜎v = 𝜌c * g * z
Now, let's calculate the effective vertical stress (𝜎'v):
𝜎'v = 𝜎v - 𝜎w
Calculations:
𝜎'v = 𝜎v - 𝜎w
I hope this helps! Let me know if you have more questions.
To calculate the total and effective vertical stresses at different depths, we need to consider the unit weight, thickness, and properties of the soil layers.
Total Vertical Stress:
The total vertical stress at a particular depth is a combination of the self-weight of the overlying soil and any externally applied loads.
1. Calculate the total vertical stress at a depth of 1 m below the surface:
The total vertical stress at this depth is equal to the weight of the sand layer above it plus any surcharge or external load applied.
Total Vertical Stress at 1 m = Unit Weight of Sand × Thickness of Sand above 1 m + Surcharge (if any)
As the sand above the water table is completely dry, the unit weight is the regular unit weight of dry sand, which is equal to Gs × γw, where Gs is the specific gravity of solids and γw is the unit weight of water.
Unit Weight of Dry Sand = Gs × γw
Given Gs = 2.65 and γw = 9.81 kN/m^3 (standard value for water), we can substitute these values to find the unit weight of dry sand.
Unit Weight of Dry Sand = 2.65 × 9.81 = 25.97 kN/m^3
Now, we can calculate the total vertical stress at 1 m below the surface.
Total Vertical Stress at 1 m = (Unit Weight of Dry Sand × Thickness of Sand above 1 m) + Surcharge
The thickness of the sand layer above 1 m is 2.5 m, so the calculation becomes:
Total Vertical Stress at 1 m = (25.97 kN/m^3 × 2.5 m) + Surcharge
2. Calculate the total vertical stress at a depth of 2.5 m below the surface:
Similarly, the total vertical stress at this depth will also be influenced by the weight of the sand layer above it.
Total Vertical Stress at 2.5 m = (Unit Weight of Dry Sand × Thickness of Sand above 2.5 m) + Surcharge
The thickness of the sand layer above 2.5 m is 2.5 m, so the calculation becomes:
Total Vertical Stress at 2.5 m = (25.97 kN/m^3 × 2.5 m) + Surcharge
3. Calculate the total vertical stress at a depth of 5.5 m below the surface:
At this depth, both the weight of the sand layer and the weight of the clay layer above it contribute to the total vertical stress.
Total Vertical Stress at 5.5 m = (Unit Weight of Dry Sand × Thickness of Sand above 5.5 m) + (Unit Weight of Clay × Thickness of Clay above 5.5 m) + Surcharge
Given that the thickness of the sand layer above 5.5 m is 2.5 m and the thickness of the clay layer above 5.5 m is 3 m, we can substitute these values to calculate the total vertical stress at 5.5 m below the surface.
Total Vertical Stress at 5.5 m = (25.97 kN/m^3 × 2.5 m) + (Unit Weight of Clay × 3 m) + Surcharge
Effective Vertical Stress:
The effective vertical stress at a particular depth is equal to the difference between the total vertical stress and the pore water pressure.
1. Calculate the effective vertical stress at 1 m below the surface:
As the sand above the water table is completely dry, there is no pore water pressure at this depth.
Effective Vertical Stress at 1 m = Total Vertical Stress at 1 m
2. Calculate the effective vertical stress at 2.5 m below the surface:
Again, there is no pore water pressure at this depth because the sand is dry.
Effective Vertical Stress at 2.5 m = Total Vertical Stress at 2.5 m
3. Calculate the effective vertical stress at 5.5 m below the surface:
At this depth, the pore water pressure in the clay layer needs to be considered.
Effective Vertical Stress at 5.5 m = Total Vertical Stress at 5.5 m - Pore Water Pressure
To determine the pore water pressure, we need to know the void ratio (e) of the saturated clay.
Pore Water Pressure = Unit Weight of Water × Height of Water Column
Given that the water table is 1 m below the surface, the height of the water column in the clay layer is 3 m - 1 m = 2 m.
Unit Weight of Water = 9.81 kN/m^3 (standard value for water)
Pore Water Pressure = 9.81 kN/m^3 × 2 m
Now, substitute the values to calculate the effective vertical stress at 5.5 m below the surface.
Effective Vertical Stress at 5.5 m = Total Vertical Stress at 5.5 m - Pore Water Pressure
I hope this explanation helps you in calculating the total and effective vertical stresses at different depths in the given soil profile.