Location of Principal Planes Calculator

✖Shear Stress xy is the Stress acting along xy plane.ⓘ Shear Stress xy [τ_xy]			+10% -10%
✖The Stress along y Direction can be described as axial stress along the given direction.ⓘ Stress along y Direction [σ_y]			+10% -10%
✖The Stress along x Direction can be described as axial stress along the given direction.ⓘ Stress along x Direction [σ_x]			+10% -10%

✖The Theta is the angle subtended by a plane of a body when stress is applied.ⓘ Location of Principal Planes [θ]

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Formula

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Location of Principal Planes

Formula

`"θ" = (((1/2)*atan((2*"τ"_{"xy"})/("σ"_{"y"}-"σ"_{"x"}))))`

Example

`"6.245735°"=(((1/2)*atan((2*"7.2MPa")/("110MPa"-"45MPa"))))`

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Location of Principal Planes Solution

STEP 0: Pre-Calculation Summary

Formula Used

Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction))))
θ = (((1/2)*atan((2*τ_xy)/(σ_y-σ_x))))
This formula uses 2 Functions, 4 Variables

Functions Used

tan - The tangent of an angle is a trigonometric ratio of the length of the side opposite an angle to the length of the side adjacent to an angle in a right triangle., tan(Angle)
atan - Inverse tan is used to calculate the angle by applying the tangent ratio of the angle, which is the opposite side divided by the adjacent side of the right triangle., atan(Number)

Variables Used

Theta - (Measured in Radian) - The Theta is the angle subtended by a plane of a body when stress is applied.
Shear Stress xy - (Measured in Pascal) - Shear Stress xy is the Stress acting along xy plane.
Stress along y Direction - (Measured in Pascal) - The Stress along y Direction can be described as axial stress along the given direction.
Stress along x Direction - (Measured in Pascal) - The Stress along x Direction can be described as axial stress along the given direction.

STEP 1: Convert Input(s) to Base Unit

Shear Stress xy: 7.2 Megapascal --> 7200000 Pascal (Check conversion here)
Stress along y Direction: 110 Megapascal --> 110000000 Pascal (Check conversion here)
Stress along x Direction: 45 Megapascal --> 45000000 Pascal (Check conversion here)

STEP 2: Evaluate Formula

Substituting Input Values in Formula

θ = (((1/2)*atan((2*τ_xy)/(σ_y-σ_x)))) --> (((1/2)*atan((2*7200000)/(110000000-45000000))))

Evaluating ... ...

θ = 0.109008633947581

STEP 3: Convert Result to Output's Unit

0.109008633947581 Radian -->6.24573465568406 Degree (Check conversion here)

FINAL ANSWER

6.24573465568406 ≈ 6.245735 Degree <-- Theta

(Calculation completed in 00.004 seconds)

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Acharya Nagarjuna University College of Engg & Technology (ANU), Guntur

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< 7 Equivalent Bending Moment & Torque Calculators

Location of Principal Planes

Go Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction))))

Diameter of Circular Shaft given Equivalent Bending Stress

Go Diameter of Circular Shaft = ((32*Equivalent Bending Moment)/(pi*(Bending Stress)))^(1/3)

Diameter of Circular Shaft for Equivalent Torque and Maximum Shear Stress

Go Diameter of Circular Shaft = ((16*Equivalent Torque)/(pi*(Maximum Shear Stress)))^(1/3)

Bending Stress of Circular Shaft given Equivalent Bending Moment

Go Bending Stress = (32*Equivalent Bending Moment)/(pi*(Diameter of Circular Shaft^3))

Equivalent Bending Moment of Circular Shaft

Go Equivalent Bending Moment = Bending Stress/(32/(pi*(Diameter of Circular Shaft^3)))

Maximum Shear Stress due to Equivalent Torque

Go Maximum Shear Stress = (16*Equivalent Torque)/(pi*(Diameter of Circular Shaft^3))

Equivalent Torque given Maximum Shear Stress

Go Equivalent Torque = Maximum Shear Stress/(16/(pi*(Diameter of Circular Shaft^3)))

Location of Principal Planes Formula

Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction))))
θ = (((1/2)*atan((2*τ_xy)/(σ_y-σ_x))))

What is Principal stress?

The Principal stresses are maximum and minimum value of normal stresses on a plane (when rotated through an angle) on which there is no shear stress.

What is Principal plane?

The Principal plane is defined as the plane on which the principal stresses act and shear stress is zero.

How to Calculate Location of Principal Planes?

Location of Principal Planes calculator uses Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction)))) to calculate the Theta, The Location of Principal Planes formula is defined as the angle made with the principal planes along which the shear stress is zero. Theta is denoted by θ symbol.

How to calculate Location of Principal Planes using this online calculator? To use this online calculator for Location of Principal Planes, enter Shear Stress xy (τ_xy), Stress along y Direction (σ_y) & Stress along x Direction (σ_x) and hit the calculate button. Here is how the Location of Principal Planes calculation can be explained with given input values -> 357.8542 = (((1/2)*atan((2*7200000)/(110000000-45000000)))).

FAQ

What is Location of Principal Planes?

The Location of Principal Planes formula is defined as the angle made with the principal planes along which the shear stress is zero and is represented as θ = (((1/2)*atan((2*τ_xy)/(σ_y-σ_x)))) or Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction)))). Shear Stress xy is the Stress acting along xy plane, The Stress along y Direction can be described as axial stress along the given direction & The Stress along x Direction can be described as axial stress along the given direction.

How to calculate Location of Principal Planes?

The Location of Principal Planes formula is defined as the angle made with the principal planes along which the shear stress is zero is calculated using Theta = (((1/2)*atan((2*Shear Stress xy)/(Stress along y Direction-Stress along x Direction)))). To calculate Location of Principal Planes, you need Shear Stress xy (τ_xy), Stress along y Direction (σ_y) & Stress along x Direction (σ_x). With our tool, you need to enter the respective value for Shear Stress xy, Stress along y Direction & Stress along x Direction 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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