Residual Shear Stress In Shaft For Fully Plastic Case Calculator

Residual Shear Stress Formula:

\[ \zeta_{f\_res} = \tau_0 \times \left(1 - \frac{4 \times r \times \left(1 - \left(\frac{r_1}{r_2}\right)^3\right)}{3 \times r_2 \times \left(1 - \left(\frac{r_1}{r_2}\right)^4\right)}\right) \]

Yield Stress in Shear (τ₀):

Radius Yielded (r):

Inner Radius of Shaft (r₁):

Outer Radius of Shaft (r₂):

Residual Shear Stress (ζf_res):

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1. What is Residual Shear Stress in Fully Plastic Yielding?

Residual Shear Stress in fully Plastic Yielding can be defined as the algebraic sum of applied stress and recovery stress in a shaft that has undergone plastic deformation.

2. How Does the Calculator Work?

The calculator uses the residual shear stress formula:

\[ \zeta_{f\_res} = \tau_0 \times \left(1 - \frac{4 \times r \times \left(1 - \left(\frac{r_1}{r_2}\right)^3\right)}{3 \times r_2 \times \left(1 - \left(\frac{r_1}{r_2}\right)^4\right)}\right) \]

Where:

\( \zeta_{f\_res} \) — Residual Shear Stress in fully Plastic Yielding (Pa)
\( \tau_0 \) — Yield Stress in Shear (Pa)
\( r \) — Radius Yielded (m)
\( r_1 \) — Inner Radius of Shaft (m)
\( r_2 \) — Outer Radius of Shaft (m)

Explanation: This formula calculates the residual shear stress remaining in a shaft after plastic yielding, accounting for the geometric properties and material yield characteristics.

3. Importance of Residual Shear Stress Calculation

Details: Calculating residual shear stress is crucial for understanding the remaining stress state in structural components after plastic deformation, which affects fatigue life, structural integrity, and safety factors in mechanical design.

4. Using the Calculator

Tips: Enter all values in consistent units (meters for lengths, Pascals for stress). Ensure the inner radius is less than the outer radius, and all values are positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of residual shear stress?
A: Residual shear stress indicates the internal stress remaining in a material after external loads are removed, which can affect subsequent mechanical behavior and fatigue performance.

Q2: How does plastic yielding affect residual stresses?
A: Plastic deformation creates permanent changes in the material structure, leading to residual stresses that remain even after unloading.

Q3: What are typical applications of this calculation?
A: This calculation is used in mechanical engineering for designing shafts, axles, and other rotating components that may experience plastic deformation.

Q4: Are there limitations to this formula?
A: This formula assumes ideal plastic behavior and may not account for strain hardening effects or complex loading conditions.

Q5: How does geometry affect residual stresses?
A: The ratio of inner to outer radius significantly influences the distribution and magnitude of residual stresses in hollow shafts.