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

STEP 0: Pre-Calculation Summary
Formula Used
Residual shear stress in Elasto plastic yielding = ((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)))
ζep_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 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*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 ... ...
ζep_res = 7797619.04761907
STEP 3: Convert Result to Output's Unit
7797619.04761907 Pascal -->7.79761904761908 Megapascal (Check conversion here)
FINAL ANSWER
7.79761904761908 Megapascal <-- Residual shear stress in Elasto plastic yielding
(Calculation completed in 00.047 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)))`

Calculator
LaTeX
Go Residual shear stress in Elasto plastic yielding = ((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 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)))`

Calculator
LaTeX
Go 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)))
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)))`

Calculator
LaTeX
Go 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)))
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)))`

Calculator
LaTeX
Go 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)))
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)))`

Calculator
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*Outer radius of shaft^3/3)*(1-(Radius of plastic front/Outer radius of shaft)^3)))
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))))`

Calculator
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))))
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))`

Calculator
LaTeX
Go 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 r1 and material constant) Formula

Residual shear stress in Elasto plastic yielding = ((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)))
ζep_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)))

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 r1 and material constant)?

Residual shear stress in shaft(r lies between r1 and material constant) calculator uses Residual shear stress in Elasto plastic yielding = ((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 Elasto plastic yielding, The Residual shear stress in shaft(r lies between r1 and material constant) 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 r1 and material constant) using this online calculator? To use this online calculator for Residual shear stress in shaft(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(r lies between r1 and material constant) calculation can be explained with given input values -> 7.797619 = ((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(r lies between r1 and material constant)?
The Residual shear stress in shaft(r lies between r1 and material constant) formula is defined as the algebraic sum of applied stress and recovery stress and is represented as ζep_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 Elasto plastic yielding = ((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 condition, Radius Yielded is the yielded portion of shaft under load, Radius of plastic front is the difference between the outer radius of shaft and depth yields plastically, Outer radius of shaft is the external radius of shaft & Inner radius of shaft is the internal radius of shaft.
How to calculate Residual shear stress in shaft(r lies between r1 and material constant)?
The Residual shear stress in shaft(r lies between r1 and material constant) formula is defined as the algebraic sum of applied stress and recovery stress is calculated using Residual shear stress in Elasto plastic yielding = ((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(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 Elasto plastic yielding?
In this formula, Residual shear stress in Elasto plastic yielding 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 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)))
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