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

✖Yield Stress in Shear is the yield stress of the shaft in shear conditions.ⓘ Yield Stress in Shear [𝝉₀]			+10% -10%
✖Radius Yielded is the remaining stress in a material after the original cause of the stress has been removed, affecting its structural integrity and durability.ⓘ Radius Yielded [r]			+10% -10%
✖Radius of Plastic Front is the distance from the center of the material to the point where plastic deformation occurs due to residual stresses.ⓘ Radius of Plastic Front [ρ]			+10% -10%
✖Outer Radius of Shaft is the distance from the center of the shaft to its outer surface, affecting residual stresses in the material.ⓘ Outer Radius of Shaft [r₂]			+10% -10%
✖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.ⓘ Inner Radius of Shaft [r₁]			+10% -10%

✖Residual Shear Stress in Shaft can be defined as the algebraic sum of applied stress and recovery stress.ⓘ Residual Shear Stress in Shaft when r Lies between r1 and Material Constant [ζ_{shaft_res}]

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

STEP 0: Pre-Calculation Summary

Formula Used

Residual Shear Stress in Shaft = (Yield Stress in Shear*Radius Yielded/Radius of Plastic Front-(((4*Yield Stress in Shear*Radius Yielded)/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))*(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)))
ζ_{shaft_res} = (𝝉₀*r/ρ-(((4*𝝉₀*r)/(3*r₂*(1-(r₁/r₂)^4)))*(1-1/4*(ρ/r₂)^3-(3*r₁)/(4*ρ)*(r₁/r₂)^3)))
This formula uses 6 Variables

Variables Used

Residual Shear Stress in Shaft - (Measured in Pascal) - Residual Shear Stress in Shaft can be defined as the algebraic sum of applied stress and recovery stress.
Yield Stress in Shear - (Measured in Pascal) - Yield Stress in Shear is the yield stress of the shaft in shear conditions.
Radius Yielded - (Measured in Meter) - Radius Yielded is the remaining stress in a material after the original cause of the stress has been removed, affecting its structural integrity and durability.
Radius of Plastic Front - (Measured in Meter) - Radius of Plastic Front is the distance from the center of the material to the point where plastic deformation occurs due to residual stresses.
Outer Radius of Shaft - (Measured in Meter) - Outer Radius of Shaft is the distance from the center of the shaft to its outer surface, affecting residual stresses in the material.
Inner Radius of Shaft - (Measured in Meter) - 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.

STEP 1: Convert Input(s) to Base Unit

Yield Stress in Shear: 145 Megapascal --> 145000000 Pascal (Check conversion here)
Radius Yielded: 60 Millimeter --> 0.06 Meter (Check conversion here)
Radius of Plastic Front: 80 Millimeter --> 0.08 Meter (Check conversion here)
Outer Radius of Shaft: 100 Millimeter --> 0.1 Meter (Check conversion here)
Inner Radius of Shaft: 40 Millimeter --> 0.04 Meter (Check conversion here)

STEP 2: Evaluate Formula

Substituting Input Values in Formula

ζ_{shaft_res} = (𝝉₀*r/ρ-(((4*𝝉₀*r)/(3*r₂*(1-(r₁/r₂)^4)))*(1-1/4*(ρ/r₂)^3-(3*r₁)/(4*ρ)*(r₁/r₂)^3))) --> (145000000*0.06/0.08-(((4*145000000*0.06)/(3*0.1*(1-(0.04/0.1)^4)))*(1-1/4*(0.08/0.1)^3-(3*0.04)/(4*0.08)*(0.04/0.1)^3)))

Evaluating ... ...

ζ_{shaft_res} = 7797619.04761907

STEP 3: Convert Result to Output's Unit

7797619.04761907 Pascal -->7.79761904761907 Megapascal (Check conversion here)

FINAL ANSWER

7.79761904761907 ≈ 7.797619 Megapascal <-- Residual Shear Stress in Shaft

(Calculation completed in 00.004 seconds)

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

LaTeX Go Residual Shear Stress in Shaft = (Yield Stress in Shear*Radius Yielded/Radius of Plastic Front-(((4*Yield Stress in Shear*Radius Yielded)/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))*(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)))

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

LaTeX Go Residual Shear Stress in Shaft = Yield Stress in Shear*(1-(4*Radius Yielded*(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)))/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))

Recovery Elasto Plastic Torque

LaTeX Go Recovery Elasto Plastic Torque = -(pi*Yield Stress in Shear*(Radius of Plastic Front^3/2*(1-(Inner Radius of Shaft/Radius of Plastic Front)^4)+(2/3*Outer Radius of Shaft^3)*(1-(Radius of Plastic Front/Outer Radius of Shaft)^3)))

Residual Shear Stress in Shaft for Fully Plastic Case

LaTeX Go Residual Shear Stress in fully Plastic Yielding = Yield Stress in Shear*(1-(4*Radius Yielded*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^3))/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))

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

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How Residual Stresses are Generated in Shafts?

When a shaft is twisted, it starts yielding once the shear stress crosses its yield limit. Torque applied may be elasto-plastic or fully plastic. This process is called LOADING. When the shaft so twisted is applied with a torque of same magnitude in the opposite direction, then the recovery of stress takes place. This process is called UNLOADING. The process of UNLOADING is always assumed to be elastic following a linear stress-strain relation. But for a plastically twisted shaft, the recovery doesn’t take place fully. Therefore some amount of stresses are left over or locked. Such stresses are called the residual stresses.

How to Calculate Residual Shear Stress in Shaft when r Lies between r1 and Material Constant?

Residual Shear Stress in Shaft when r Lies between r1 and Material Constant calculator uses Residual Shear Stress in Shaft = (Yield Stress in Shear*Radius Yielded/Radius of Plastic Front-(((4*Yield Stress in Shear*Radius Yielded)/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))*(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))) to calculate the Residual Shear Stress in Shaft, Residual Shear Stress in Shaft when r Lies between r1 and Material Constant formula is defined as a measure of the residual shear stress in a shaft, which is a critical parameter in mechanical engineering, particularly in the design and analysis of shafts subjected to various loads and stresses. Residual Shear Stress in Shaft is denoted by ζ_{shaft_res} symbol.

How to calculate Residual Shear Stress in Shaft when r Lies between r1 and Material Constant using this online calculator? To use this online calculator for Residual Shear Stress in Shaft when r Lies between r1 and Material Constant, enter Yield Stress in Shear (𝝉₀), Radius Yielded (r), Radius of Plastic Front (ρ), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁) and hit the calculate button. Here is how the Residual Shear Stress in Shaft when r Lies between r1 and Material Constant calculation can be explained with given input values -> 7.8E-6 = (145000000*0.06/0.08-(((4*145000000*0.06)/(3*0.1*(1-(0.04/0.1)^4)))*(1-1/4*(0.08/0.1)^3-(3*0.04)/(4*0.08)*(0.04/0.1)^3))).

FAQ

What is Residual Shear Stress in Shaft when r Lies between r1 and Material Constant?

Residual Shear Stress in Shaft when r Lies between r1 and Material Constant formula is defined as a measure of the residual shear stress in a shaft, which is a critical parameter in mechanical engineering, particularly in the design and analysis of shafts subjected to various loads and stresses and is represented as ζ_{shaft_res} = (𝝉₀*r/ρ-(((4*𝝉₀*r)/(3*r₂*(1-(r₁/r₂)^4)))*(1-1/4*(ρ/r₂)^3-(3*r₁)/(4*ρ)*(r₁/r₂)^3))) or Residual Shear Stress in Shaft = (Yield Stress in Shear*Radius Yielded/Radius of Plastic Front-(((4*Yield Stress in Shear*Radius Yielded)/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))*(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))). Yield Stress in Shear is the yield stress of the shaft in shear conditions, Radius Yielded is the remaining stress in a material after the original cause of the stress has been removed, affecting its structural integrity and durability, Radius of Plastic Front is the distance from the center of the material to the point where plastic deformation occurs due to residual stresses, Outer Radius of Shaft is the distance from the center of the shaft to its outer surface, affecting residual stresses in the material & 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.

How to calculate Residual Shear Stress in Shaft when r Lies between r1 and Material Constant?

Residual Shear Stress in Shaft when r Lies between r1 and Material Constant formula is defined as a measure of the residual shear stress in a shaft, which is a critical parameter in mechanical engineering, particularly in the design and analysis of shafts subjected to various loads and stresses is calculated using Residual Shear Stress in Shaft = (Yield Stress in Shear*Radius Yielded/Radius of Plastic Front-(((4*Yield Stress in Shear*Radius Yielded)/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))*(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))). To calculate Residual Shear Stress in Shaft when r Lies between r1 and Material Constant, you need Yield Stress in Shear (𝝉₀), Radius Yielded (r), Radius of Plastic Front (ρ), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁). With our tool, you need to enter the respective value for Yield Stress in Shear, Radius Yielded, Radius of Plastic Front, Outer Radius of Shaft & Inner Radius of Shaft and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

How many ways are there to calculate Residual Shear Stress in Shaft?

In this formula, Residual Shear Stress in Shaft uses Yield Stress in Shear, Radius Yielded, Radius of Plastic Front, Outer Radius of Shaft & Inner Radius of Shaft. We can use 1 other way(s) to calculate the same, which is/are as follows -

Residual Shear Stress in Shaft = Yield Stress in Shear*(1-(4*Radius Yielded*(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)))/(3*Outer Radius of Shaft*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^4)))

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