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

STEP 0: Pre-Calculation Summary
Formula Used
ΞΆep_res = ((π0*r/Ο)- (((4*π0*r)/(3*r2*(1-(r1/r2)^4)))*(1-(1/4)*(Ο/r2)^3-((3*r1)/(4*Ο))*(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
ΞΆep_res = ((π0*r/Ο)- (((4*π0*r)/(3*r2*(1-(r1/r2)^4)))*(1-(1/4)*(Ο/r2)^3-((3*r1)/(4*Ο))*(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 ... ...
ΞΆep_res = 7797619.04761907
STEP 3: Convert Result to Output's Unit
7797619.04761907 Pascal -->7.79761904761908 Megapascal (Check conversion here)
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)
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Residual shear stress in shaft(r lies between r1 and material constant)

Formula
"ΞΆ"_{"ep_res"} = (("π"_{"0"}*"r"/"Ο")- (((4*"π"_{"0"}*"r")/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4)))*(1-(1/4)*("Ο"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"Ο"))*("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
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Residual angle of twist for Elasto plastic case

Formula
"ΞΈ"_{"res"} = ("π"_{"0"}/("G"*"Ο"))*(1-((4*"Ο")/(3*"r"_{"2"}))* ((1-(1/4)*("Ο"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"Ο"))*("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)
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Residual shear stress in shaft(r lies between material constant and r2)

Formula
"ΞΆ"_{"ep_res"} = "π"_{"0"}*(1-(4*"r"*(1-((1/4)*("Ο"/"r"_{"2"})^3)-(((3*"r"_{"1"})/(4*"Ο"))*("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
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Residual angle of twist in fully plastic case

Formula
"ΞΈ"_{"res"} = ("π"_{"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
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Recovery Elasto plastic Torque

Formula
"T"_{"rec"} = -(pi*"π"_{"0"}*(("Ο"^3/2)*(1-("r"_{"1"}/"Ο")^4)+(2*"r"_{"2"}^3/3)*(1-("Ο"/"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
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Residual shear stress in shaft for fully plastic case

Formula
"ΞΆ"_{"f_res"} = "π"_{"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
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Recovery Torque in Fully plastic case

Formula
"T"_{"f_rec"} = -((2/3)*pi*"r"_{"2"}^3*"π"_{"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 r1 and material constant) Formula

ΞΆep_res = ((π0*r/Ο)- (((4*π0*r)/(3*r2*(1-(r1/r2)^4)))*(1-(1/4)*(Ο/r2)^3-((3*r1)/(4*Ο))*(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.
When the shaft so twisted is applied with a torque of same magnitude in the opposite direction, then
The process of UNLOADING is always assumed to be elastic following a linear stress-strain
relation. But for a plastically twisted shaft, the recovery doesnβ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)?

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 (π0), Radius Yielded (r), Radius of plastic front (Ο), 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))).

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