Bending stress (σ) is calculated using the formula σ = M/Z, where M is the moment and Z is the section modulus. Here, σ = 50 kNm / 10 cm^3 = 50,000 Nm / 0.00001 m^3 = 5 MPa.

The total load on the cantilever is w*L = 5 kN/m * 4 m = 20 kN. The shear force at the fixed support is equal to the total load, which is 20 kN.

The deflection at the center of a simply supported beam under a uniform load is given by δ = 5wL^4 / (384EI). Substituting w = 4 kN/m, L = 8 m, E = 200 GPa, and I = 0.0001 m^4 gives δ = 0.05 m.

Maximum Bending Moment = wL²/8 = (5 kN/m)(6 m)²/8 = 15 kNm.

The maximum bending moment for a simply supported beam with a central load is given by M = P*L/4. Here, M = 10 kN * 6 m / 4 = 15 kNm.

Using the moment about the right support, R_A * 10 m = 20 kN * 7 m. Solving gives R_A = 14 kN.

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.

The Cone Penetration Test (CPT) is widely used in site investigations to assess the bearing capacity of soil.

The maximum deflection (δ) for a beam fixed at both ends under a central point load (P) is given by δ = PL^3/12EI.

The bending stress (σ) in a beam is related to the moment (M) and the moment of inertia (I) by the formula σ = M/I.

The deflection of a beam is inversely proportional to the moment of inertia (I), meaning as I increases, deflection decreases.

The deflection of a beam is inversely proportional to its moment of inertia (I); as I increases, deflection decreases.

The deflection (δ) at the free end of a cantilever beam with a point load (W) at the end is given by δ = WL^3/(3EI), where E is the modulus of elasticity and I is the moment of inertia.

The reaction force at the fixed support equals the applied load.

The maximum deflection (δ) at the free end of a cantilever beam with a uniform distributed load (w) is given by δ = 5wL^4 / 384EI, where E is the modulus of elasticity and I is the moment of inertia.

The maximum shear force (V) at the fixed support of a cantilever beam with a uniform distributed load (w) is V = w * L, where L is the length of the beam.

The maximum deflection δ for a cantilever beam with a uniformly distributed load w is given by δ = 5wL^4/384EI.

The maximum moment in a fixed beam subjected to a point load at the center is twice that of a simply supported beam, hence the ratio is 2.

The maximum bending moment M for a fixed beam with a central point load P is given by M = PL/2.

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.

The stability of a frame structure is affected by all of the above factors: material properties, geometry of the frame, and load conditions.

The Stiffness Method is commonly used to analyze frame structures by considering the stiffness of each member and the overall deformation of the structure.

The centroid of a triangular load is located at L/3 from the larger end of the triangle.

The deflection (δ) at the center of a simply supported beam subjected to a uniformly distributed load (w) is given by δ = 5WL^3/48EI.

The reaction at the left support (R_A) is given by R_A = P(L-a)/L.

The reaction at the other end of a simply supported beam with a point load P at one end is equal to P.

The reaction force at the support opposite to the load is equal to the point load P.

The reaction force at each support of a simply supported beam with a point load (P) at the center is P/2.

The reaction force at each support for a simply supported beam with a uniform load (w) is R = wL/2.