Answer :
To calculate the ultimate bearing capacity of a strip footing 1.6 meters wide resting on soil, we need to consider the given soil characteristics and analyze the conditions provided. We will calculate the ultimate bearing capacity for three distinct conditions regarding the position of the water table.
Given parameters:
- Width of footing ([tex]\(B\)[/tex]) = 1.6 meters
- Unit weight of soil ([tex]\(\gamma\)[/tex]) = 18 kN/m³
- Soil cohesion ([tex]\(C_z\)[/tex]) = 25 kN/m²
- Bearing capacity factor ([tex]\(N_q\)[/tex]) = 1.0
- Unit weight of soil below water table ([tex]\(\gamma_{ses}\)[/tex]) = 20 kN/m³
- Bearing capacity factor ([tex]\(N_c\)[/tex]) = 5.7
- Bearing capacity factor ([tex]\(N_y\)[/tex]) = 0
### 1. Condition i: Water table at ground level
When the water table is at the ground level, the effective unit weight ([tex]\(\gamma'\)[/tex]) of the soil is reduced due to the buoyant effect of water:
[tex]\[ \gamma' = \gamma - \gamma_w \][/tex]
where [tex]\(\gamma_w\)[/tex] (unit weight of water) is typically 9.81 kN/m³.
Therefore:
[tex]\[ \gamma' = 18 - 9.81 \][/tex]
[tex]\[ \gamma' = 8.19 \text{ kN/m}³ \][/tex]
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table at ground level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma' B N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 8.19 \times 1.6 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### 2. Condition ii: Water table at footing bed level
When the water table is at the level of the footing bed, the unit weight of soil remains the same ([tex]\(\gamma = 18 \text{ kN/m}³\)[/tex]).
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table at footing bed level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma B N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 18 \times 1.6 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### 3. Condition iii: Water table is at 2 meters below bed level
In this scenario, the water table is 2 meters below the footing bed level. We need to consider the soil's unit weight above and below the water table.
Given that the depth below bed ([tex]\( d \)[/tex]) is 2 meters and width of footing ([tex]\( B \)[/tex]) is also 1.6 meters, we analyze the soil above and below the water table:
- Above the water table: [tex]\( \gamma = 18 \text{ kN/m}³ \)[/tex]
- Below the water table: [tex]\( \gamma_{ses} = 20 \text{ kN/m}³ \)[/tex]
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table 2 meters below bed level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma (B - d) N_y + 0.5 \gamma_{ses} d N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 18 \times (1.6 - 2) \times 0 + 0.5 \times 20 \times 2 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### Summary of Ultimate Bearing Capacities:
1. Water table at ground level: 142.5 kN/m²
2. Water table at footing bed level: 142.5 kN/m²
3. Water table at 2 meters below bed level: 142.5 kN/m²
Hence, for the provided soil characteristics and conditions, the ultimate bearing capacity of the strip footing in each scenario is 142.5 kN/m².
Given parameters:
- Width of footing ([tex]\(B\)[/tex]) = 1.6 meters
- Unit weight of soil ([tex]\(\gamma\)[/tex]) = 18 kN/m³
- Soil cohesion ([tex]\(C_z\)[/tex]) = 25 kN/m²
- Bearing capacity factor ([tex]\(N_q\)[/tex]) = 1.0
- Unit weight of soil below water table ([tex]\(\gamma_{ses}\)[/tex]) = 20 kN/m³
- Bearing capacity factor ([tex]\(N_c\)[/tex]) = 5.7
- Bearing capacity factor ([tex]\(N_y\)[/tex]) = 0
### 1. Condition i: Water table at ground level
When the water table is at the ground level, the effective unit weight ([tex]\(\gamma'\)[/tex]) of the soil is reduced due to the buoyant effect of water:
[tex]\[ \gamma' = \gamma - \gamma_w \][/tex]
where [tex]\(\gamma_w\)[/tex] (unit weight of water) is typically 9.81 kN/m³.
Therefore:
[tex]\[ \gamma' = 18 - 9.81 \][/tex]
[tex]\[ \gamma' = 8.19 \text{ kN/m}³ \][/tex]
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table at ground level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma' B N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 8.19 \times 1.6 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### 2. Condition ii: Water table at footing bed level
When the water table is at the level of the footing bed, the unit weight of soil remains the same ([tex]\(\gamma = 18 \text{ kN/m}³\)[/tex]).
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table at footing bed level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma B N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 18 \times 1.6 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### 3. Condition iii: Water table is at 2 meters below bed level
In this scenario, the water table is 2 meters below the footing bed level. We need to consider the soil's unit weight above and below the water table.
Given that the depth below bed ([tex]\( d \)[/tex]) is 2 meters and width of footing ([tex]\( B \)[/tex]) is also 1.6 meters, we analyze the soil above and below the water table:
- Above the water table: [tex]\( \gamma = 18 \text{ kN/m}³ \)[/tex]
- Below the water table: [tex]\( \gamma_{ses} = 20 \text{ kN/m}³ \)[/tex]
The ultimate bearing capacity ([tex]\( q_u \)[/tex]) for water table 2 meters below bed level is calculated using the formula:
[tex]\[ q_u = C_z N_c + 0.5 \gamma (B - d) N_y + 0.5 \gamma_{ses} d N_y \][/tex]
Substituting the values:
[tex]\[ q_u = 25 \times 5.7 + 0.5 \times 18 \times (1.6 - 2) \times 0 + 0.5 \times 20 \times 2 \times 0 \][/tex]
Since [tex]\(N_y = 0\)[/tex]:
[tex]\[ q_u = 25 \times 5.7 \][/tex]
[tex]\[ q_u = 142.5 \text{ kN/m}² \][/tex]
### Summary of Ultimate Bearing Capacities:
1. Water table at ground level: 142.5 kN/m²
2. Water table at footing bed level: 142.5 kN/m²
3. Water table at 2 meters below bed level: 142.5 kN/m²
Hence, for the provided soil characteristics and conditions, the ultimate bearing capacity of the strip footing in each scenario is 142.5 kN/m².
