Incipient Yielding Torque For Hollow Shaft Calculator

Formula Used:

\[ T_i = \frac{\pi}{2} \times r_2^3 \times \tau_0 \times \left(1 - \left(\frac{r_1}{r_2}\right)^4\right) \]

Outer Radius of Shaft (r₂):

Yield Stress in Shear (τ₀):

Inner Radius of Shaft (r₁):

Incipient Yielding Torque (Tᵢ):

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1. What is Incipient Yielding Torque?

Incipient Yielding Torque refers to the torque at which a hollow shaft begins to yield. At this stage, the shaft regains its original configuration upon removal of torque, and the stresses are assumed to be fully recovered.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ T_i = \frac{\pi}{2} \times r_2^3 \times \tau_0 \times \left(1 - \left(\frac{r_1}{r_2}\right)^4\right) \]

Where:

\( T_i \) — Incipient Yielding Torque (N·m)
\( r_2 \) — Outer Radius of Shaft (m)
\( \tau_0 \) — Yield Stress in Shear (Pa)
\( r_1 \) — Inner Radius of Shaft (m)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: This formula calculates the torque at which yielding begins in a hollow shaft, considering both the outer and inner radii and the material's yield stress in shear.

3. Importance of Incipient Yielding Torque

Details: Calculating the incipient yielding torque is crucial for designing shafts that can withstand torsional loads without permanent deformation, ensuring structural integrity and safety in mechanical systems.

4. Using the Calculator

Tips: Enter the outer radius and inner radius in meters, and yield stress in shear in Pascals. Ensure the inner radius is less than the outer radius and all values are positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between incipient yielding and full yielding?
A: Incipient yielding is when the material first begins to yield but can recover upon unloading, while full yielding results in permanent deformation.

Q2: Can this formula be used for solid shafts?
A: Yes, for solid shafts, set the inner radius to zero, and the formula simplifies accordingly.

Q3: What units should be used for input values?
A: Use meters for radii and Pascals for yield stress in shear to get torque in Newton-meters.

Q4: How does the inner radius affect the yielding torque?
A: Increasing the inner radius (making the shaft thinner) decreases the yielding torque, as there is less material to resist the torque.

Q5: Is this formula applicable to all materials?
A: This formula is valid for materials that exhibit linear elastic behavior up to the yield point and follow the maximum shear stress theory of failure.