Residual Angle Of Twist In Fully Plastic Case Calculator

Formula Used:

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

Yield Stress in Shear (τ₀):

Modulus of Rigidity (G):

Inner Radius of Shaft (r₁):

Outer Radius of Shaft (r₂):

Residual Angle of Twist (θres):

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1. What is Residual Angle of Twist?

The Residual Angle of Twist is the angle of twist that remains in a shaft after unloading, due to residual stresses developed during plastic deformation. It represents the permanent angular deformation in a shaft subjected to torsional loading beyond its yield point.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the permanent twist angle in a hollow circular shaft that has been loaded beyond its yield point in torsion and then unloaded.

3. Importance of Residual Angle Calculation

Details: Calculating residual angle of twist is crucial for assessing the permanent deformation in shafts after plastic torsion, which is important for structural integrity analysis, failure prediction, and design of mechanical components subjected to torsional loads.

4. Using the Calculator

Tips: Enter yield stress in shear (Pa), modulus of rigidity (Pa), inner radius (m), and outer radius (m). All values must be positive, and outer radius must be greater than inner radius.

5. Frequently Asked Questions (FAQ)

Q1: What causes residual angle of twist?
A: Residual angle of twist occurs when a shaft is twisted beyond its elastic limit into the plastic region and then unloaded, leaving permanent angular deformation.

Q2: How does this differ from elastic twist?
A: Elastic twist is recoverable when load is removed, while residual twist is permanent deformation that remains after unloading.

Q3: What materials exhibit this behavior?
A: Ductile materials like steel, aluminum, and copper show residual twist when loaded beyond their yield point in torsion.

Q4: How is this relevant in engineering applications?
A: Understanding residual twist helps in designing shafts for automotive, aerospace, and machinery applications where torsional overloads might occur.

Q5: Can residual twist be eliminated?
A: Residual twist can be reduced through stress relief heat treatment, but complete elimination may not be possible once plastic deformation has occurred.