Combined loading of a circular shaft A solid circular shaft is simultaneously subjected to a bending moment M and a torque T. Identify the correct expressions for the maximum bending stress and the maximum shear stress in terms of diameter d.

Introduction / Context:
Machine shafts often experience simultaneous bending and torsion. Correct formulae for extreme normal and shear stresses at the outer fibre are essential for strength checks (e.g., maximum principal stress, maximum shear stress, or distortion-energy criteria).

Given Data / Assumptions:

• Solid circular shaft, diameter d.
• Linear elastic behavior, Saint-Venant torsion theory.
• Small deformations, homogeneous isotropic material.

Concept / Approach:
For a circular section: section modulus in bending Z = π * d^3 / 32, polar section modulus in torsion Zp = J / r = π * d^3 / 16. Hence, extreme bending stress and torsional shear stress follow directly by dividing the external actions by the corresponding moduli.

Step-by-Step Solution:

Verification / Alternative check:
Check limiting cases: with T = 0, only bending stress remains; with M = 0, only torsional shear remains. Dimensions are consistent (stress units for both expressions).

Why Other Options Are Wrong:

• maximum bending stress = 32 * M / (π * d^3) alone: Incomplete for combined loading.
• maximum shear stress = 16 * T / (π * d^3) alone: Incomplete for combined loading.
• neither (a) nor (b): Both expressions are standard and correct.
• None of these: Incorrect because (a) and (b) are right.

Common Pitfalls:
Using the wrong modulus (confusing Z with J or Zp), forgetting the factor 32 vs 16, or applying hollow-shaft formulae to solid shafts.

Final Answer:
both (a) and (b)