If shaft made with the ductile material then they only follow Rankine Theory because it gives a satisfactory result for the shaft. As all knows guys Rankine theory also known as Maximum principal stress theory.

Failure Theory for Combined Bending and Twisting Moment

When a shaft made from a ductile material is subjected to combined bending and twisting moment, it can experience failure due to shear stress, principal stress, strain energy, and distortion energy. However, the calculations based on different failure theories will give different values of the stress, and it is essential to choose the most conservative (i.e., highest) value to ensure the safety of the shaft.

Different Failure Theories

There are four common failure theories used for mechanical components, including shafts, under combined loading conditions:

1. Maximum Shear Stress Theory: This theory states that failure occurs when the maximum shear stress exceeds the material's shear strength. The maximum shear stress is calculated as (τ_max = (σ_x - σ_y)/2 + √((σ_x - σ_y)/2)^2 + τ_xy^2), where σ_x and σ_y are the normal stresses in the x and y directions, respectively, and τ_xy is the shear stress.

2. Maximum Principal Stress Theory: This theory states that failure occurs when the maximum principal stress exceeds the material's tensile or compressive strength. The maximum principal stress is calculated as (σ_1 = (σ_x + σ_y)/2 + √((σ_x - σ_y)/2)^2 + τ_xy^2)).

3. Maximum Strain Energy Theory: This theory states that failure occurs when the strain energy per unit volume exceeds the material's strain energy per unit volume at yield. The strain energy is calculated as (U = (1/2)σ_xε_x + (1/2)σ_yε_y + (1/2)τ_xyγ_xy), where ε_x and ε_y are the normal strains in the x and y directions, respectively, and γ_xy is the shear strain.

4. Maximum Distortion Energy Theory: This theory states that failure occurs when the distortion energy per unit volume exceeds the material's distortion energy per unit volume at yield. The distortion energy is calculated as (U_d = (1/12)(σ_x + σ_y)^2/μ + (τ_xy)^2/μ), where μ is the shear modulus.

Conservative Value

For a shaft subjected to combined bending and twisting moment, the maximum shear stress theory gives the most conservative value of stress because it assumes that failure occurs due to shear stress alone. In reality, the failure may occur due to a combination of shear and normal stresses, but the maximum shear stress theory provides a safe estimate of the maximum stress that the material can withstand. Therefore, it is commonly used in the design of mechanical components, including shafts.

Maximum shear stress theory (Guest and tresca theory) is used for ductile materials. so correct option is B