Residual Angle of Twist for Elasto Plastic Case Calculator

Formula Used:

\[ \theta_{res} = \frac{\tau_0}{G \cdot \rho} \cdot \left(1 - \frac{4\rho}{3r_2} \cdot \frac{1 - \frac{1}{4}\left(\frac{\rho}{r_2}\right)^3 - \frac{3r_1}{4\rho}\left(\frac{r_1}{r_2}\right)^3}{1 - \left(\frac{r_1}{r_2}\right)^4}\right) \]

Yield Stress in Shear (τ₀):

Modulus of Rigidity (G):

Radius of Plastic Front (ρ):

Outer Radius of Shaft (r₂):

Inner Radius of Shaft (r₁):

Residual Angle of Twist (θres):

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1. What is Residual Angle of Twist?

The Residual Angle of Twist represents the permanent angular deformation remaining in a shaft after unloading from a partially plastic torsion. It occurs when the applied torque exceeds the elastic limit but doesn't cause complete yielding.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( \theta_{res} \) — Residual Angle of Twist (rad)
\( \tau_0 \) — Yield Stress in Shear (Pa)
\( G \) — Modulus of Rigidity (Pa)
\( \rho \) — Radius of Plastic Front (m)
\( r_2 \) — Outer Radius of Shaft (m)
\( r_1 \) — Inner Radius of Shaft (m)

Explanation: This formula calculates the permanent twist angle remaining after unloading a shaft that has been partially yielded in torsion.

3. Importance of Residual Angle Calculation

Details: Calculating residual twist is crucial for assessing permanent deformation in shafts, predicting service life, and determining if a component needs replacement after overload conditions.

4. Using the Calculator

Tips: Enter all values in consistent SI units. Yield stress and modulus of rigidity in Pascals, all radii in meters. All values must be positive and the plastic front radius should be between inner and outer radii.

5. Frequently Asked Questions (FAQ)

Q1: What causes residual twist in shafts?
A: Residual twist occurs when a shaft is twisted beyond its elastic limit, causing permanent plastic deformation that remains after unloading.

Q2: How is the plastic front radius determined?
A: The plastic front radius is the boundary between the yielded (plastic) and unyielded (elastic) regions of the shaft cross-section.

Q3: Can this formula be used for solid shafts?
A: Yes, for solid shafts set the inner radius (r₁) to zero in the calculation.

Q4: What are typical values for residual twist?
A: Residual twist angles are typically small (fractions of a radian) but depend on material properties and the extent of yielding.

Q5: How does residual twist affect shaft performance?
A: Residual twist can affect alignment, cause vibration, reduce fatigue life, and may require shaft replacement if excessive.