## Residual shear stress in shaft(r lies between material constant and r2) Solution

STEP 0: Pre-Calculation Summary
Formula Used
ζep_res = 𝝉0*(1-(4*r*(1-((1/4)*(ρ/r2)^3)-(((3*r1)/(4*ρ))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4)))
This formula uses 6 Variables
Variables Used
Residual shear stress in Elasto plastic yielding - (Measured in Pascal) - Residual shear stress in Elasto plastic yielding 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 condition.
Radius Yielded - (Measured in Meter) - Radius Yielded is the yielded portion of shaft under load.
Radius of plastic front - (Measured in Meter) - Radius of plastic front is the difference between the outer radius of shaft and depth yields plastically.
Outer radius of shaft - (Measured in Meter) - Outer radius of shaft is the external radius of shaft.
Inner radius of shaft - (Measured in Meter) - Inner radius of shaft is the internal radius of shaft.
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
ζep_res = 𝝉0*(1-(4*r*(1-((1/4)*(ρ/r2)^3)-(((3*r1)/(4*ρ))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4))) --> 145000000*(1-(4*0.06*(1-((1/4)*(0.08/0.1)^3)-(((3*0.04)/(4*0.08))*(0.04/0.1)^3)))/(3*0.1*(1-(0.04/0.1)^4)))
Evaluating ... ...
ζep_res = 44047619.0476191
STEP 3: Convert Result to Output's Unit
44047619.0476191 Pascal -->44.0476190476191 Megapascal (Check conversion here)
44.0476190476191 Megapascal <-- Residual shear stress in Elasto plastic yielding
(Calculation completed in 00.031 seconds)
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## < 7 Residual Stresses For Idealized Stress Strain Law Calculators

Residual shear stress in shaft(r lies between r1 and material constant)

## Residual shear stress in shaft(r lies between r1 and material constant)

Formula
"ζ"_{"ep_res"} = (("𝝉"_{"0"}*"r"/"ρ")- (((4*"𝝉"_{"0"}*"r")/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4)))*(1-(1/4)*("ρ"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"ρ"))*("r"_{"1"}/"r"_{"2"})^3)))

Example
"7.797619MPa"=(("145MPa"*"60mm"/"80mm")- (((4*"145MPa"*"60mm")/(3*"100mm"*(1-("40mm"/"100mm")^4)))*(1-(1/4)*("80mm"/"100mm")^3-((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)))

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Residual angle of twist for Elasto plastic case

## Residual angle of twist for Elasto plastic case

Formula
"θ"_{"res"} = ("𝝉"_{"0"}/("G"*"ρ"))*(1-((4*"ρ")/(3*"r"_{"2"}))* ((1-(1/4)*("ρ"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"ρ"))*("r"_{"1"}/"r"_{"2"})^3)/(1-("r"_{"1"}/"r"_{"2"})^4)))

Example
"0.001547rad"=("145MPa"/("84e3MPa"*"80mm"))*(1-((4*"80mm")/(3*"100mm"))* ((1-(1/4)*("80mm"/"100mm")^3-((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)/(1-("40mm"/"100mm")^4)))

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Residual shear stress in shaft(r lies between material constant and r2)

## Residual shear stress in shaft(r lies between material constant and r2)

Formula
"ζ"_{"ep_res"} = "𝝉"_{"0"}*(1-(4*"r"*(1-((1/4)*("ρ"/"r"_{"2"})^3)-(((3*"r"_{"1"})/(4*"ρ"))*("r"_{"1"}/"r"_{"2"})^3)))/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4)))

Example
"44.04762MPa"="145MPa"*(1-(4*"60mm"*(1-((1/4)*("80mm"/"100mm")^3)-(((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)))/(3*"100mm"*(1-("40mm"/"100mm")^4)))

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Residual angle of twist in fully plastic case

## Residual angle of twist in fully plastic case

Formula
"θ"_{"res"} = ("𝝉"_{"0"}/("G"*"r"_{"1"}))*(1-((4*"r"_{"1"})/(3*"r"_{"2"}))*((1-("r"_{"1"}/"r"_{"2"})^3)/(1-("r"_{"1"}/"r"_{"2"})^4)))

Example
"0.021046rad"=("145MPa"/("84e3MPa"*"40mm"))*(1-((4*"40mm")/(3*"100mm"))*((1-("40mm"/"100mm")^3)/(1-("40mm"/"100mm")^4)))

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Recovery Elasto plastic Torque

## Recovery Elasto plastic Torque

Formula
"T"_{"rec"} = -(pi*"𝝉"_{"0"}*(("ρ"^3/2)*(1-("r"_{"1"}/"ρ")^4)+(2*"r"_{"2"}^3/3)*(1-("ρ"/"r"_{"2"})^3)))

Example
"-257526821.790267N*mm"=-(pi*"145MPa"*(("80mm"^3/2)*(1-("40mm"/"80mm")^4)+(2*"100mm"^3/3)*(1-("80mm"/"100mm")^3)))

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Residual shear stress in shaft for fully plastic case

## Residual shear stress in shaft for fully plastic case

Formula
"ζ"_{"f_res"} = "𝝉"_{"0"}*(1-((4*"r"*(1-("r"_{"1"}/"r"_{"2"})^3))/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4))))

Example
"33.57143MPa"="145MPa"*(1-((4*"60mm"*(1-("40mm"/"100mm")^3))/(3*"100mm"*(1-("40mm"/"100mm")^4))))

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Recovery Torque in Fully plastic case

## Recovery Torque in Fully plastic case

Formula
"T"_{"f_rec"} = -((2/3)*pi*"r"_{"2"}^3*"𝝉"_{"0"}*(1-("r"_{"1"}/"r"_{"2"})^3))

Example
"-284251303.296805N*mm"=-((2/3)*pi*"100mm"^3*"145MPa"*(1-("40mm"/"100mm")^3))

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Fully plastic recovery torque = -((2/3)*pi*Outer radius of shaft^3*Yield stress in shear*(1-(Inner radius of shaft/Outer radius of shaft)^3))

## Residual shear stress in shaft(r lies between material constant and r2) Formula

ζep_res = 𝝉0*(1-(4*r*(1-((1/4)*(ρ/r2)^3)-(((3*r1)/(4*ρ))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4)))

## 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(r lies between material constant and r2)?

Residual shear stress in shaft(r lies between material constant and r2) calculator uses Residual shear stress in Elasto plastic yielding = 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))) to calculate the Residual shear stress in Elasto plastic yielding, The Residual shear stress in shaft(r lies between material constant and r2) formula is defined as the algebraic sum of applied stress and recovery stress. Residual shear stress in Elasto plastic yielding is denoted by ζep_res symbol.

How to calculate Residual shear stress in shaft(r lies between material constant and r2) using this online calculator? To use this online calculator for Residual shear stress in shaft(r lies between material constant and r2), 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(r lies between material constant and r2) calculation can be explained with given input values -> 44.04762 = 145000000*(1-(4*0.06*(1-((1/4)*(0.08/0.1)^3)-(((3*0.04)/(4*0.08))*(0.04/0.1)^3)))/(3*0.1*(1-(0.04/0.1)^4))).

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