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Residual Angle of Twist in Fully Plastic Case Formula

GoResidual Angle of Twist for Elasto Plastic Case Formula

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Residual Angle of Twist is the angle of twist due to residual stresses. Check FAQs

θ res = \frac{𝝉 0}{G \cdot r 1} \cdot (1 - \frac{4 \cdot r 1}{3 \cdot r 2} \cdot (\frac{1 - {(\frac{r 1}{r 2})}^{3}}{1 - {(\frac{r 1}{r 2})}^{4}}))

θ_res - Residual Angle of Twist?𝝉₀ - Yield Stress in Shear?G - Modulus of Rigidity?r₁ - Inner Radius of Shaft?r₂ - Outer Radius of Shaft?

Residual Angle of Twist in Fully Plastic Case Example

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Here is how the Residual Angle of Twist in Fully Plastic Case equation looks like with Values.

Here is how the Residual Angle of Twist in Fully Plastic Case equation looks like with Units.

Here is how the Residual Angle of Twist in Fully Plastic Case equation looks like.

0.021 = \frac{145}{84000 \cdot 40} \cdot (1 - \frac{4 \cdot 40}{3 \cdot 100} \cdot (\frac{1 - {(\frac{40}{100})}^{3}}{1 - {(\frac{40}{100})}^{4}}))

0.021rad = \frac{145MPa}{84000MPa \cdot 40mm} \cdot (1 - \frac{4 \cdot 40mm}{3 \cdot 100mm} \cdot (\frac{1 - {(\frac{40mm}{100mm})}^{3}}{1 - {(\frac{40mm}{100mm})}^{4}}))

0.021 = \frac{145}{84000 \cdot 40} \cdot (1 - \frac{4 \cdot 40}{3 \cdot 100} \cdot (\frac{1 - {(\frac{40}{100})}^{3}}{1 - {(\frac{40}{100})}^{4}}))

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Residual Angle of Twist in Fully Plastic Case Solution

Follow our step by step solution on how to calculate Residual Angle of Twist in Fully Plastic Case?

FIRST Step Consider the formula

θ res = \frac{𝝉 0}{G \cdot r 1} \cdot (1 - \frac{4 \cdot r 1}{3 \cdot r 2} \cdot (\frac{1 - {(\frac{r 1}{r 2})}^{3}}{1 - {(\frac{r 1}{r 2})}^{4}}))

Next Step Substitute values of Variables

∴

θ res = \frac{145MPa}{84000MPa \cdot 40mm} \cdot (1 - \frac{4 \cdot 40mm}{3 \cdot 100mm} \cdot (\frac{1 - {(\frac{40mm}{100mm})}^{3}}{1 - {(\frac{40mm}{100mm})}^{4}}))

Next Step Convert Units

∴

θ res = \frac{1.5E+8Pa}{8.4E+10Pa \cdot 0.04m} \cdot (1 - \frac{4 \cdot 0.04m}{3 \cdot 0.1m} \cdot (\frac{1 - {(\frac{0.04m}{0.1m})}^{3}}{1 - {(\frac{0.04m}{0.1m})}^{4}}))

Next Step Prepare to Evaluate

∴

θ res = \frac{1.5E+8}{8.4E+10 \cdot 0.04} \cdot (1 - \frac{4 \cdot 0.04}{3 \cdot 0.1} \cdot (\frac{1 - {(\frac{0.04}{0.1})}^{3}}{1 - {(\frac{0.04}{0.1})}^{4}}))

Next Step Evaluate

∴

θ res = 0.0210459183673469rad

LAST Step Rounding Answer

∴

θ res = 0.021rad

Residual Angle of Twist in Fully Plastic Case Formula Elements

Variables

Residual Angle of Twist

Residual Angle of Twist is the angle of twist due to residual stresses.

Symbol: θ_res

Measurement: AngleUnit: rad

Note: Value should be greater than 0.

Formulas to find Residual Angle of Twist

Yield Stress in Shear

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

Symbol: 𝝉₀

Measurement: StressUnit: MPa

Note: Value should be greater than 0.

Formulas to find Yield Stress in Shear

Modulus of Rigidity

Modulus of Rigidity is a measure of the stiffness of a solid material, defining the ratio of stress to strain within the proportional limit of elasticity.

Symbol: G

Measurement: StressUnit: MPa

Note: Value can be positive or negative.

Formulas to find Modulus of Rigidity

Inner Radius of Shaft

Inner Radius of Shaft is the internal radius of a shaft, which is a critical dimension in mechanical engineering, affecting stress concentrations and structural integrity.

Symbol: r₁

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Inner Radius of Shaft

Outer Radius of Shaft

Outer Radius of Shaft is the distance from the center of the shaft to its outer surface, affecting residual stresses in the material.

Symbol: r₂

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Outer Radius of Shaft

Other Formulas to find Residual Angle of Twist

Go Residual Angle of Twist for Elasto Plastic Case

θ res = \frac{𝝉 0}{G \cdot ρ} \cdot (1 - (\frac{4 \cdot ρ}{3 \cdot r 2}) \cdot (\frac{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}}{1 - {(\frac{r 1}{r 2})}^{4}}))

Other formulas in Residual Stresses For Idealized Stress Strain Law category

Go Recovery Elasto Plastic Torque

T rec = - (π \cdot 𝝉 0 \cdot (\frac{ρ^{3}}{2} \cdot (1 - {(\frac{r 1}{ρ})}^{4}) + (\frac{2}{3} \cdot {r 2}^{3}) \cdot (1 - {(\frac{ρ}{r 2})}^{3})))

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

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})))

Go Residual Shear Stress in Shaft when r Lies between Material Constant and r2

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

Go Residual Shear Stress in Shaft for Fully Plastic Case

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

How to Evaluate Residual Angle of Twist in Fully Plastic Case?

Residual Angle of Twist in Fully Plastic Case evaluator uses Residual Angle of Twist = Yield Stress in Shear/(Modulus of Rigidity*Inner Radius of Shaft)*(1-(4*Inner Radius of Shaft)/(3*Outer Radius of Shaft)*((1-(Inner Radius of Shaft/Outer Radius of Shaft)^3)/(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4))) to evaluate the Residual Angle of Twist, Residual Angle of Twist in Fully Plastic Case formula is defined as a measure of the residual twist angle in a fully plastic shaft, which occurs due to the uneven distribution of residual stresses, affecting the shaft's overall strength and stability. Residual Angle of Twist is denoted by θ_res symbol.

How to evaluate Residual Angle of Twist in Fully Plastic Case using this online evaluator? To use this online evaluator for Residual Angle of Twist in Fully Plastic Case, enter Yield Stress in Shear (𝝉₀), Modulus of Rigidity (G), Inner Radius of Shaft (r₁) & Outer Radius of Shaft (r₂) and hit the calculate button.

FAQs on Residual Angle of Twist in Fully Plastic Case

What is the formula to find Residual Angle of Twist in Fully Plastic Case?

The formula of Residual Angle of Twist in Fully Plastic Case is expressed as Residual Angle of Twist = Yield Stress in Shear/(Modulus of Rigidity*Inner Radius of Shaft)*(1-(4*Inner Radius of Shaft)/(3*Outer Radius of Shaft)*((1-(Inner Radius of Shaft/Outer Radius of Shaft)^3)/(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4))). Here is an example- 0.021046 = 145000000/(84000000000*0.04)*(1-(4*0.04)/(3*0.1)*((1-(0.04/0.1)^3)/(1-(0.04/0.1)^4))).

How to calculate Residual Angle of Twist in Fully Plastic Case?

With Yield Stress in Shear (𝝉₀), Modulus of Rigidity (G), Inner Radius of Shaft (r₁) & Outer Radius of Shaft (r₂) we can find Residual Angle of Twist in Fully Plastic Case using the formula - Residual Angle of Twist = Yield Stress in Shear/(Modulus of Rigidity*Inner Radius of Shaft)*(1-(4*Inner Radius of Shaft)/(3*Outer Radius of Shaft)*((1-(Inner Radius of Shaft/Outer Radius of Shaft)^3)/(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4))).

What are the other ways to Calculate Residual Angle of Twist?

Here are the different ways to Calculate Residual Angle of Twist-

Residual Angle of Twist=Yield Stress in Shear/(Modulus of Rigidity*Radius of Plastic Front)*(1-((4*Radius of Plastic Front)/(3*Outer Radius of Shaft))*((1-1/4*(Radius of Plastic Front/Outer Radius of Shaft)^3-(3*Inner Radius of Shaft)/(4*Radius of Plastic Front)*(Inner Radius of Shaft/Outer Radius of Shaft)^3)/(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))

Can the Residual Angle of Twist in Fully Plastic Case be negative?

No, the Residual Angle of Twist in Fully Plastic Case, measured in Angle cannot be negative.

Which unit is used to measure Residual Angle of Twist in Fully Plastic Case?

Residual Angle of Twist in Fully Plastic Case is usually measured using the Radian[rad] for Angle. Degree[rad], Minute[rad], Second[rad] are the few other units in which Residual Angle of Twist in Fully Plastic Case can be measured.

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Residual Angle of Twist in Fully Plastic Case Formula

​GoResidual Angle of Twist for Elasto Plastic Case Formula

Residual Angle of Twist in Fully Plastic Case Example

Residual Angle of Twist in Fully Plastic Case Solution

Residual Angle of Twist in Fully Plastic Case Formula Elements

Other Formulas to find Residual Angle of Twist

Other formulas in Residual Stresses For Idealized Stress Strain Law category

How to Evaluate Residual Angle of Twist in Fully Plastic Case?

FAQs on Residual Angle of Twist in Fully Plastic Case

Variable Name

GoResidual Angle of Twist for Elasto Plastic Case Formula