Nth Polar Moment of Inertia Calculator

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Torsion of Bars

Bending of Beams

Residual Stresses

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Elastic Work Hardening Material

Elastic Perfectly Plastic Materials

Residual Stresses for Non-Linear stress strain Law

✖Material Constant is the constant used when the beam yielded plastically.ⓘ Material Constant [n]			+10% -10%
✖Outer Radius of Shaft is the external radius of shaft.ⓘ Outer Radius of Shaft [r₂]			+10% -10%
✖Inner Radius of Shaft is the internal radius of shaft.ⓘ Inner Radius of Shaft [r₁]			+10% -10%

✖Nth Polar Moment of Inertia can be defined as Integral arising from nonlinear behaviour of material.ⓘ Nth Polar Moment of Inertia [J_n]

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Formula

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Nth Polar Moment of Inertia

Formula

`"J"_{"n"} = ((2*pi)/("n"+3))*("r"_{"2"}^("n"+3)-"r"_{"1"}^("n"+3))`

Example

`"1E^9mm⁴"=((2*pi)/("0.25"+3))*(("100mm")^("0.25"+3)-("40mm")^("0.25"+3))`

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Nth Polar Moment of Inertia Solution

STEP 0: Pre-Calculation Summary

Formula Used

Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3))
J_n = ((2*pi)/(n+3))*(r₂^(n+3)-r₁^(n+3))
This formula uses 1 Constants, 4 Variables

Constants Used

pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288

Variables Used

Nth Polar Moment of Inertia - (Measured in Meter⁴) - Nth Polar Moment of Inertia can be defined as Integral arising from nonlinear behaviour of material.
Material Constant - Material Constant is the constant used when the beam yielded 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

Material Constant: 0.25 --> No Conversion Required
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

J_n = ((2*pi)/(n+3))*(r₂^(n+3)-r₁^(n+3)) --> ((2*pi)/(0.25+3))*(0.1^(0.25+3)-0.04^(0.25+3))

Evaluating ... ...

J_n = 0.00103183369075116

STEP 3: Convert Result to Output's Unit

0.00103183369075116 Meter⁴ -->1031833690.75116 Millimeter⁴ (Check conversion here)

FINAL ANSWER

1031833690.75116 ≈ 1E+9 Millimeter⁴ <-- Nth Polar Moment of Inertia

(Calculation completed in 00.004 seconds)

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< 7 Elastic Work Hardening Material Calculators

Elasto Plastic Yielding Torque in Work Hardening for Hollow Shaft

Go Elasto Plastic Yielding Torque = (2*pi*Yield Shear Stress(non-linear)*Outer Radius of Shaft^3)/3*((3*Radius of Plastic Front^3)/(Outer Radius of Shaft^3*(Material Constant+3))-(3/(Material Constant+3))*(Inner Radius of Shaft/Radius of Plastic Front)^Material Constant*(Inner Radius of Shaft/Outer Radius of Shaft)^3+1-(Radius of Plastic Front/Outer Radius of Shaft)^3)

Elasto Plastic Yielding Torque in Work Hardening for Solid Shaft

Go Elasto Plastic Yielding Torque = (2*pi*Yield Shear Stress(non-linear)*Outer Radius of Shaft^3)/3*(1-(Material Constant/(Material Constant+3))*(Radius of Plastic Front/Outer Radius of Shaft)^3)

Nth Polar Moment of Inertia

Go Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3))

Full Yielding Torque in Work Hardening for Hollow Shaft

Go Full Yielding Torque = (2*pi*Yield Shear Stress(non-linear)*Outer Radius of Shaft^3)/3*(1-(Inner Radius of Shaft/Outer Radius of Shaft)^3)

Incipient Yielding Torque in Work Hardening for Hollow Shaft

Go Incipient Yielding Torque = (Yield Shear Stress(non-linear)*Nth Polar Moment of Inertia)/Outer Radius of Shaft^Material Constant

Incipient Yielding Torque in Work Hardening Solid Shaft

Go Incipient Yielding Torque = (Yield Shear Stress(non-linear)*Nth Polar Moment of Inertia)/Outer Radius of Shaft^Material Constant

Full Yielding Torque in Work Hardening for Solid Shaft

Go Full Yielding Torque = (2*pi*Yield Shear Stress(non-linear)*Outer Radius of Shaft^3)/3

Nth Polar Moment of Inertia Formula

Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3))
J_n = ((2*pi)/(n+3))*(r₂^(n+3)-r₁^(n+3))

What is nth polar moment of inertia?

The nth polar moment of inertia formula is defined as Integral arising from nonlinear behaviour of material.

How to Calculate Nth Polar Moment of Inertia?

Nth Polar Moment of Inertia calculator uses Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)) to calculate the Nth Polar Moment of Inertia, The Nth polar moment of inertia formula is defined as Integral arising from nonlinear behaviour of material. Nth Polar Moment of Inertia is denoted by J_n symbol.

How to calculate Nth Polar Moment of Inertia using this online calculator? To use this online calculator for Nth Polar Moment of Inertia, enter Material Constant (n), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁) and hit the calculate button. Here is how the Nth Polar Moment of Inertia calculation can be explained with given input values -> 1E+21 = ((2*pi)/(0.25+3))*(0.1^(0.25+3)-0.04^(0.25+3)).

FAQ

What is Nth Polar Moment of Inertia?

The Nth polar moment of inertia formula is defined as Integral arising from nonlinear behaviour of material and is represented as J_n = ((2*pi)/(n+3))*(r₂^(n+3)-r₁^(n+3)) or Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)). Material Constant is the constant used when the beam yielded plastically, Outer Radius of Shaft is the external radius of shaft & Inner Radius of Shaft is the internal radius of shaft.

How to calculate Nth Polar Moment of Inertia?

The Nth polar moment of inertia formula is defined as Integral arising from nonlinear behaviour of material is calculated using Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)). To calculate Nth Polar Moment of Inertia, you need Material Constant (n), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁). With our tool, you need to enter the respective value for Material Constant, 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.

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