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Residual Shear Stress in Shaft when r Lies between r1 and Material Constant Formula

GoResidual Shear Stress in Shaft when r Lies between Material Constant and r2 Formula

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Residual Shear Stress in Shaft can be defined as the algebraic sum of applied stress and recovery stress. Check FAQs

ζ shaft_res = (𝝉 0 \cdot \frac{r}{ρ} - ((\frac{4 \cdot 𝝉 0 \cdot r}{3 \cdot r 2 \cdot (1 - {(\frac{r 1}{r 2})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{ρ}{r 2})}^{3} - \frac{3 \cdot r 1}{4 \cdot ρ} \cdot {(\frac{r 1}{r 2})}^{3})))

ζ_{shaft_res} - Residual Shear Stress in Shaft?𝝉₀ - Yield Stress in Shear?r - Radius Yielded?ρ - Radius of Plastic Front?r₂ - Outer Radius of Shaft?r₁ - Inner Radius of Shaft?

Residual Shear Stress in Shaft when r Lies between r1 and Material Constant Example

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Here is how the Residual Shear Stress in Shaft when r Lies between r1 and Material Constant equation looks like with Values.

Here is how the Residual Shear Stress in Shaft when r Lies between r1 and Material Constant equation looks like with Units.

Here is how the Residual Shear Stress in Shaft when r Lies between r1 and Material Constant equation looks like.

7.7976 = (145 \cdot \frac{60}{80} - ((\frac{4 \cdot 145 \cdot 60}{3 \cdot 100 \cdot (1 - {(\frac{40}{100})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{80}{100})}^{3} - \frac{3 \cdot 40}{4 \cdot 80} \cdot {(\frac{40}{100})}^{3})))

7.7976MPa = (145MPa \cdot \frac{60mm}{80mm} - ((\frac{4 \cdot 145MPa \cdot 60mm}{3 \cdot 100mm \cdot (1 - {(\frac{40mm}{100mm})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{80mm}{100mm})}^{3} - \frac{3 \cdot 40mm}{4 \cdot 80mm} \cdot {(\frac{40mm}{100mm})}^{3})))

7.7976 = (145 \cdot \frac{60}{80} - ((\frac{4 \cdot 145 \cdot 60}{3 \cdot 100 \cdot (1 - {(\frac{40}{100})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{80}{100})}^{3} - \frac{3 \cdot 40}{4 \cdot 80} \cdot {(\frac{40}{100})}^{3})))

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Residual Shear Stress in Shaft when r Lies between r1 and Material Constant Solution

Follow our step by step solution on how to calculate Residual Shear Stress in Shaft when r Lies between r1 and Material Constant?

FIRST Step Consider the formula

ζ shaft_res = (𝝉 0 \cdot \frac{r}{ρ} - ((\frac{4 \cdot 𝝉 0 \cdot r}{3 \cdot r 2 \cdot (1 - {(\frac{r 1}{r 2})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{ρ}{r 2})}^{3} - \frac{3 \cdot r 1}{4 \cdot ρ} \cdot {(\frac{r 1}{r 2})}^{3})))

Next Step Substitute values of Variables

∴

ζ shaft_res = (145MPa \cdot \frac{60mm}{80mm} - ((\frac{4 \cdot 145MPa \cdot 60mm}{3 \cdot 100mm \cdot (1 - {(\frac{40mm}{100mm})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{80mm}{100mm})}^{3} - \frac{3 \cdot 40mm}{4 \cdot 80mm} \cdot {(\frac{40mm}{100mm})}^{3})))

Next Step Convert Units

∴

ζ shaft_res = (1.5E+8Pa \cdot \frac{0.06m}{0.08m} - ((\frac{4 \cdot 1.5E+8Pa \cdot 0.06m}{3 \cdot 0.1m \cdot (1 - {(\frac{0.04m}{0.1m})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{0.08m}{0.1m})}^{3} - \frac{3 \cdot 0.04m}{4 \cdot 0.08m} \cdot {(\frac{0.04m}{0.1m})}^{3})))

Next Step Prepare to Evaluate

∴

ζ shaft_res = (1.5E+8 \cdot \frac{0.06}{0.08} - ((\frac{4 \cdot 1.5E+8 \cdot 0.06}{3 \cdot 0.1 \cdot (1 - {(\frac{0.04}{0.1})}^{4})}) \cdot (1 - \frac{1}{4} \cdot {(\frac{0.08}{0.1})}^{3} - \frac{3 \cdot 0.04}{4 \cdot 0.08} \cdot {(\frac{0.04}{0.1})}^{3})))

Next Step Evaluate

∴

ζ shaft_res = 7797619.04761907Pa

Next Step Convert to Output's Unit

∴

ζ shaft_res = 7.79761904761907MPa

LAST Step Rounding Answer

∴

ζ shaft_res = 7.7976MPa

Residual Shear Stress in Shaft when r Lies between r1 and Material Constant Formula Elements

Variables

Residual Shear Stress in Shaft

Residual Shear Stress in Shaft can be defined as the algebraic sum of applied stress and recovery stress.

Symbol: ζ_{shaft_res}

Measurement: StressUnit: MPa

Note: Value can be positive or negative.

Formulas to find Residual Shear Stress in Shaft

Yield Stress in Shear

Yield Stress in Shear is the yield stress of the shaft in shear conditions.

Symbol: 𝝉₀

Measurement: StressUnit: MPa