Question

A circular shaft has been designed for the twisting moment of $5\text{ kN}\cdot\text{m}$. If the twisting moment is reduced to $4\text{ kN}\cdot\text{m}$, then what will be the maximum value of bending moment that can be applied for the same designed condition?

• A. $1\text{ kN}\cdot\text{m}$
• B. $1.5\text{ kN}\cdot\text{m}$
• C. $2\text{ kN}\cdot\text{m}$
• D. $3\text{ kN}\cdot\text{m}$

Hint:

This problem follows the standard 3-4-5 right-angle triangle relationship! Since $T_e = \sqrt{M^2 + T^2}$, if the total capacity vector length is 5 and one leg is 4, the remaining leg must be 3.

Answer 1

The Correct Option is D

Concept: For a solid circular transmission shaft subjected to a combined loading of a bending moment $M$ and a twisting moment $T$, the design limit is determined using the Equivalent Twisting Moment ($T_e$) based on the Maximum Shear Stress Theory (Tresca's Criterion): \[ T_e = \sqrt{M^2 + T^2} \] The value $T_e$ represents the structural torque capacity for which the shaft's diameter was originally sized.

Step 1: Finding the baseline designed torque capacity ($T_e$).
The problem states that the circular shaft was initially designed to handle a pure twisting moment of $5\text{ kN}\cdot\text{m}$ without any accompanying bending moment ($M = 0$): \[ T_e = \sqrt{0^2 + (5)^2} = 5\text{ kN}\cdot\text{m} \] This establishes that the designed structural capacity limit of the shaft under this criterion is exactly $5\text{ kN}\cdot\text{m}$.

Step 2: Calculating the allowable bending moment ($M$) under the new twisting load.
Now, the twisting moment is reduced to $T = 4\text{ kN}\cdot\text{m}$. We need to find the maximum allowable bending moment $M$ that keeps the shaft within its designed capacity ($T_e = 5\text{ kN}\cdot\text{m}$): \[ 5 = \sqrt{M^2 + (4)^2} \] Squaring both sides of the equation to clear the radical sign: \[ 5^2 = M^2 + 4^2 \quad \Rightarrow \quad 25 = M^2 + 16 \] Isolating the squared bending moment term: \[ M^2 = 25 - 16 = 9 \] Taking the positive square root gives the maximum allowable bending moment: \[ M = \sqrt{9} = 3\text{ kN}\cdot\text{m} \] This perfectly matches Option (D).