Question

A solid circular shaft is subjected to bending moment and torsional moment. What is the nature of maximum principal stress at the outer surface?

• A. Pure bending stress
• B. Pure shear stress
• C. Combination of normal and shear stress
• D. Zero stress

Hint:

For shafts under combined loading, the equivalent bending moment ($M_e$) used in design is:
$M_e = \frac{1}{2} [ M + \sqrt{M^2 + T^2} ]$. This directly relates to the maximum principal stress.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the state of stress and the nature of the maximum principal stress at the outer surface of a shaft subjected to combined bending and torsion.

Step 2: Key Formula or Approach:
- Bending moment ($M$) induces normal stress ($\sigma$) on the outer surface:
\[ \sigma = \frac{32M}{\pi d^3} \]
- Torsional moment ($T$) induces shear stress ($\tau$) on the outer surface:
\[ \tau = \frac{16T}{\pi d^3} \]
- The maximum principal stress ($\sigma_1$) is determined using the combined stress formula:
\[ \sigma_1 = \frac{\sigma}{2} + \sqrt{\left(\frac{\sigma}{2}\right)^2 + \tau^2} \]

Step 3: Detailed Explanation:

• An element at the outer fiber of the shaft experiences a normal stress ($\sigma$) due to bending and a shear stress ($\tau$) due to torsion simultaneously.

• This represents a state of two-dimensional combined loading (biaxial stress state).

• The principal stresses are the maximum and minimum normal stresses acting on planes where the shear stress is zero.

• Because the principal stress calculation formula depends on both the bending-induced normal stress ($\sigma$) and the torsion-induced shear stress ($\tau$), the resulting maximum principal stress is fundamentally a combination of both normal and shear stresses.

Step 4: Final Answer:
The nature of the maximum principal stress is a combination of normal and shear stress.