Combined bending and torsion: identify the correct statements about equivalent bending moment and equivalent torque for a circular shaft.

Introduction / Context:
Design of circular shafts under simultaneous bending moment M and torque T often uses equivalent single-effect measures to compare against allowable stresses. The two common equivalents are equivalent bending moment (matching maximum principal stress) and equivalent torque (matching maximum shear stress).

Given Data / Assumptions:

• Solid circular shaft; outer-fiber stresses govern.
• Elastic behavior with σ = 32M/(π d^3) and τ = 16T/(π d^3).
• Maximum principal stress and maximum shear stress from Mohr’s circle are used.

Concept / Approach:
For combined bending and torsion, at the outer surface: principal stress σ_1 = (σ/2) + sqrt( (σ/2)^2 + τ^2 ), and maximum shear τ_max = sqrt( (σ/2)^2 + τ^2 ). Define M_eq so that σ_1 = 32 M_eq/(π d^3), and T_eq so that τ_max = 16 T_eq/(π d^3). This yields closed-form expressions for the equivalents in terms of M and T.

Step-by-Step Solution:
1) Put σ = k M and τ = (k/2) T with k = 32/(π d^3).2) σ_1 = (kM)/2 + sqrt( (kM/2)^2 + (kT/2)^2 ) = (k/2)[ M + sqrt( M^2 + T^2 ) ].3) Set σ_1 = k M_eq → M_eq = 0.5 * [ M + sqrt( M^2 + T^2 ) ].4) τ_max = sqrt( (kM/2)^2 + (kT/2)^2 ) = (k/2) sqrt( M^2 + T^2 ).5) Set τ_max = (k/2) T_eq → T_eq = sqrt( M^2 + T^2 ).

Verification / Alternative check:
Special cases: (i) T = 0 → M_eq = M, T_eq = |M| (incorrect dimension), but note T_eq compares shear; with T = 0, τ_max = |σ|/2 so T_eq = |M| (in the normalized formulation above) matches the definition via τ_max; (ii) M = 0 → M_eq = 0.5*|T|, T_eq = |T|, consistent with pure torsion.

Why Other Options Are Wrong:

• Here, A and C are definitions; B presents the correct closed forms. Choosing only one omits key information—hence the inclusive statement is appropriate.

Common Pitfalls:

• Using T_eq = sqrt( M^2 + (K T)^2 ) with an arbitrary factor K; for circular sections under the maximum shear stress criterion, K = 1.
• Forgetting the factor 0.5 in M_eq.

Final Answer:
All the above.