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🔵 Mohr's Circle Calculator
Calculate principal stress, maximum shear stress, strain transformations, and rotated-plane values with a visual Mohr's Circle.
1. Choose Analysis
2. Enter Values
Positive θ rotates the coordinate axes counterclockwise. Positive shear follows the tensor convention τxy or γxy. Mohr's Circle is plotted with positive shear upward. A physical rotation θ corresponds to a rotation of −2θ around the circle.
3. Principal Results
Maximum Principal Stress — MPa—Principal Stress σ1—Principal Stress σ2—Circle Center—Circle Radius—Maximum In-Plane Shear—σ1 / ε1 Plane Angle—Positive Max Shear Plane—Principal Value Difference—Average Normal Component4. Rotated Plane Results
—Rotated Normal Stress σx'—Rotated Normal Stress σy'—Rotated Shear Stress τx'y'—Physical Rotation θ—Circle Rotation −2θ—Von Mises Stress5. Mohr's Circle Diagram
Original x-plane Rotated x'-plane Principal valuesThe diagram uses the normal component on the horizontal axis and shear component on the vertical axis. In strain mode, the vertical coordinate is γ/2, not the full engineering shear strain.
6. Formulas
Circle Center: C = (X + Y) / 2Circle Radius: R = √[((X − Y) / 2)² + S²]Principal Values: P1 = C + R, P2 = C − RRotated Normal: X' = C + D cos(2θ) + S sin(2θ)Rotated Shear: S' = −D sin(2θ) + S cos(2θ)For stress: X = σx, Y = σy, S = τxy. For strain: X = εx, Y = εy, S = γxy/2. In strain mode, the engineering shear strain after rotation is γx'y' = 2S'.
Plane Stress Von Mises: σv = √(σx² − σxσy + σy² + 3τxy²)7. Important Notes
Plane Stress: The stress mode assumes σz = τxz = τyz = 0. The circle radius is the maximum in-plane shear stress, which is not necessarily the absolute maximum three-dimensional shear stress.
Strain: The strain mode uses infinitesimal strain transformation and engineering shear strain γxy = 2εxy. Its circle radius represents half the maximum engineering shear strain.
Angle Convention: Angles are measured counterclockwise from the original x-axis. Principal directions are 90° apart; maximum in-plane shear directions are 45° from principal directions.
Materials: This tool transforms measured or calculated stresses and strains. It does not independently determine material failure, safety factors, or allowable loading.
Audio Applications: Mohr's Circle can help analyze stresses in speaker brackets, structural mounts, instrument hardware, and other components subjected to combined loading. -
A Mohr's Circle Calculator helps engineers, instrument builders, audio equipment designers, and physics students analyze how stress and strain behave when a material is subjected to forces from different directions.
Mohr's Circle is a graphical method used in mechanics to determine principal stresses, maximum shear stress, principal angles, and stress or strain values on rotated planes.
This calculator provides a visual diagram along with numerical results, making it easier to understand how forces affect materials and structural components.
What Is Mohr's Circle?
Mohr's Circle represents the relationship between normal stress and shear stress at a point in a material.
Normal stress acts perpendicular to a surface, while shear stress acts parallel to it.
By plotting these stresses on a circle, you can determine the maximum and minimum normal stresses and the orientations at which they occur.
The same concept can also be used to analyze two-dimensional strain.
How Does Mohr's Circle Work?
The calculator uses normal stresses in the x and y directions, along with shear stress, to determine the circle's center and radius.
The center represents the average normal stress, while the radius represents the maximum in-plane shear stress.
The leftmost and rightmost points of the circle represent the two principal stresses.
You can also enter a rotation angle to calculate the normal and shear stresses acting on a differently oriented plane.
Mohr's Circle and Musical Instruments
Musical instruments contain structural components that experience tension, compression, bending, and other mechanical forces.
For example, string tension creates loads on bridges, necks, and other supporting hardware.
Mohr's Circle can help analyze combined stresses in idealized structural models of these components.
However, actual instrument behavior may require more detailed analysis because wood and other materials can have direction-dependent mechanical properties.
Mohr's Circle and Speaker Design
Speaker frames, mounting brackets, and structural supports experience mechanical loads during installation and operation.
Understanding stress distribution can help engineers evaluate how these components respond to forces.
Mohr's Circle can be useful when examining the stress state at a point in a speaker mounting structure or other mechanical component.
It does not directly predict sound quality or speaker frequency response.
Principal Stress vs. Shear Stress
Principal stresses are the normal stresses acting on planes where the shear stress is zero.
Maximum in-plane shear stress is the greatest shear stress magnitude that occurs as the plane orientation changes within the analyzed two-dimensional stress state.
These values help engineers understand how different orientations affect the internal loading of a material.
Mohr's Circle for Strain
Mohr's Circle can also be used to analyze strain.
Strain describes how a material deforms relative to its original dimensions.
The strain version of Mohr's Circle helps determine principal strains and the maximum engineering shear strain.
For engineering shear strain, the vertical coordinate of the circle is half the engineering shear strain value.
When Is a Mohr's Circle Calculator Useful?
This calculator is useful for mechanical engineering, structural analysis, material testing, instrument construction research, speaker hardware design, and physics education.
It can help visualize stress transformations, identify principal directions, and compare loading conditions.
Tip: When using Mohr's Circle to analyze instrument or audio equipment components, make sure your stress values come from a suitable mechanical model or measurement. Mohr's Circle transforms an existing stress state; it does not calculate the original loads or determine whether a component is safe by itself.
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