For a friction angle of 30 degrees, the bearing capacity factor N_q can be approximated using the formula N_q = e^(π * tan(φ)) * tan(45 + φ/2). For φ = 30 degrees, N_q is approximately 3.5.

Using Terzaghi's equation for cohesive soils, q_u = c*N_c + q, where c = 75 kPa, N_c = 5, and assuming q = 0, q_u = 75 kPa * 5 = 375 kPa.

The primary mode of failure for a foundation subjected to vertical loads is typically bearing capacity failure, which occurs when the soil can no longer support the applied load.

Bearing pressure (q) is calculated as q = Load / Area. The area of the foundation is 2 m * 1 m = 2 m². Therefore, q = 400 kN / 2 m² = 200 kPa.

The bearing capacity is independent of the footing width when the load remains constant; it is a function of the soil properties and the load applied.

The ultimate bearing capacity for a square footing can be calculated as q_u = c*N_c + q. For a depth of 1 m, N_c is approximately 5.0. Thus, q_u = 40 kPa * 5 = 200 kPa.

Factor of safety (FS) is calculated as FS = Allowable Bearing Capacity / Actual Bearing Pressure. Thus, FS = 150 kPa / 100 kPa = 1.5.

The ultimate bearing capacity (q_u) can be calculated using the formula q_u = c*N_c, where c is the cohesion and N_c is the bearing capacity factor. For a depth of 1.5 m, N_c is approximately 5.0. Thus, q_u = 50 kPa * 5 = 250 kPa.