2025-08-21 14 min

Mohr’s Circle Calculator

Construct Mohr’s circle from plane stress components. Find center, radius, principal stresses, and maximum shear stress.

Calculator

Unit System

Preset Values

Normal Stress sx(MPa) Help

MPa

Normal stress in x-direction

Normal Stress sy(MPa) Help

MPa

Normal stress in y-direction

Shear Stress txy(MPa) Help

MPa

Shear stress on x-y plane

Decimal Precision

Results update live as you type

Results

Normal

Circle Center (Average Stress)

40MPa

Circle Radius (Max Shear)

56.57MPa

Principal Stress s1

96.57MPa

Principal Stress s2

-16.57MPa

Visualizations

Results Overview

40MPaCircle Center

56.6MPaCircle Radius

96.6MPaPrincipal Stress s1

Formula

Mohr’s circle is defined by its center (average stress) and radius (maximum shear stress) derived from the plane stress components.

Center = (sx+sy)/2, R = √[((sx-sy)/2)² + txy²]

Variables

Symbol	Name	Unit	Description
C	Circle Center	MPa	Average normal stress: (sx+sy)/2
R	Circle Radius	MPa	Maximum shear stress: √[((sx-sy)/2)²+txy²]
sx	Normal Stress (x)	MPa	Normal stress in x-direction
sy	Normal Stress (y)	MPa	Normal stress in y-direction
txy	Shear Stress	MPa	Shear stress on x-y plane

Explanation

Mohr’s circle provides a complete graphical representation of the stress state at a point. Every point on the circle represents the stress state (sigma, tau) on a plane oriented at a specific angle.

Derivation

From stress transformation: sigma_x′ = (sx+sy)/2 + (sx-sy)/2 * cos(2θ) + txy * sin(2θ), tau_x′ = -(sx-sy)/2 * sin(2θ) + txy * cos(2θ). Eliminating θ by squaring and adding: (sigma_x′ - C)² + tau_x′² = R², which is the equation of a circle.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Plot points P(sx, txy) and Q(sy, -txy) on the sigma-tau plane.

Result: Reference points identified

Step 2

Find the center: C = (sx+sy)/2 on the sigma axis.

Result: Center located

Step 3

Calculate the radius: R = √[((sx-sy)/2)² + txy²].

Result: Radius computed

Step 4

Draw the circle. Read s1 = C+R and s2 = C-R from the intersections with the sigma axis.

Result: Circle constructed

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Given sx=80 MPa, sy=0, txy=40 MPa, construct Mohr’s circle.

Inputs

sx:80 MPa

sy:0 MPa

txy:40 MPa

Calculation

C = 40, R = √(40² + 40²) = 56.57

Result

s1 = 96.57 MPa, s2 = -16.57 MPa, tau_max = 56.57 MPa

Interpretation

The circle spans from -16.57 to 96.57 on the sigma axis. Maximum shear stress is 56.57 MPa at 45 degrees from the principal planes.

Understanding Your Result

Your Result: Circle Center (Average Stress): 40 MPa

Typical range: For structural steel (yield 250 MPa), Mohr’s circle radii typically range from 0 to 125 MPa. For high-strength alloys, radii can reach 300-500 MPa.

What This Means

Mohr’s circle is a powerful graphical tool that transforms a given stress state into all possible orientations. It shows at a glance the principal stresses, maximum shear stress, and the angles at which they occur.

Why It Matters

Engineers use Mohr’s circle to quickly visualize and verify stress transformation calculations. It is essential for understanding material failure, designing pressure vessels, and analyzing complex loading conditions.

Professional Interpretation

A circle with large radius indicates high shear stress relative to the mean. A circle touching the origin means one principal stress is zero (plane stress). Check if the circle intersects the failure envelope.

Recommendations

•Always verify analytical calculations with the circle
•Check that the circle passes through the known stress points P and Q
•Use the circle to determine failure plane orientations
•Consider 3D Mohr’s circles for thick components

Educational Content

Introduction

Mohr’s circle, developed by Christian Mohr in 1882, is a fundamental graphical method in mechanics of materials for analyzing stress transformation.

Concept Explanation

The circle maps all possible (sigma, tau) pairs on planes through a point. The center represents average stress, the radius represents maximum shear stress, and the intersections with the sigma axis give principal stresses.

Applications

▸Stress analysis in machine design

▸Pressure vessel design

▸Geotechnical engineering (soil mechanics)

▸Structural engineering

▸Materials testing and characterization

▸Aerospace structural analysis

Benefits

•Visual and intuitive understanding of stress states
•Quick verification of analytical calculations
•Immediate identification of principal stresses
•Helps determine failure plane orientations

Limitations

•Limited to 2D (plane) stress states
•Requires construction for each loading condition
•Does not directly show strain
•Assumes linear elastic behavior

Best Practices

•Use consistent sign conventions
•Verify circle geometry with known stress points
•Check invariants as a calculation verification
•Extend to 3D for thick components

Common Misconceptions

Myth: Mohr’s circle is only for 2D

Truth: While most commonly used for plane stress, 3D Mohr’s circles (three circles using s1, s2, s3) can represent any 3D stress state.

Myth: Mohr’s circle is outdated

Truth: Though developed in 1882, Mohr’s circle remains a standard tool taught in every engineering curriculum and used in practice for stress verification.

Practical Tips

•The center of Mohr’s circle is always on the sigma axis (tau=0)
•An angle of 2*theta on the circle equals theta on the element
•Use invariant checks: s1+s2 = sx+sy and s1*s2 = sx*sy - txy²
•The circle radius equals the maximum in-plane shear stress
•For pure shear, the circle is centered at the origin

Common Mistakes

Confusing the angle on the circle with the physical angle

Angles on Mohr’s circle are doubled (2*theta). A 90-degree rotation on the circle corresponds to 45 degrees on the actual element.

Incorrect sign convention for shear stress

Establish a consistent convention: positive shear causes counterclockwise rotation on the element. Plot positive shear upward on the circle.

Forgetting that the circle must pass through the given stress points

Verify that P(sx, txy) and Q(sy, -txy) both lie on the constructed circle.

Assumptions

This calculator makes the following assumptions to provide estimates:

Plane Stress: Mohr’s circle for 2D applies to plane stress conditions where the out-of-plane stress is zero.

Linear Elastic Material: The stress transformation is valid only within the linear elastic range of the material.

Static Loading: The circle represents a snapshot of the stress state. For dynamic loading, multiple circles may be needed.

Limitations

Be aware of these limitations when using this calculator:

2D Only: A single Mohr’s circle cannot represent a general 3D stress state. Three circles are needed for complete 3D analysis.

No Strain Information: The stress circle does not directly show strains. A separate strain circle must be constructed if needed.

No Material Properties: Mohr’s circle shows the stress state only; it does not indicate whether the material will fail.

Frequently Asked Questions

Answers to common questions about this calculator.

What is Mohr’s circle?

Mohr’s circle is a graphical method for visualizing stress transformation. For a given plane stress state (sx, sy, txy), it plots normal stress on the horizontal axis and shear stress on the vertical axis. The circle reveals principal stresses, maximum shear stress, and stress at any orientation.

How do you construct Mohr’s circle?

Plot point P(sx, txy) and Q(sy, -txy) on the sigma-tau plane. The center C is at ((sx+sy)/2, 0). The radius R = sqrt(((sx-sy)/2)² + txy²). Draw a circle with center C and radius R. The circle intersects the sigma axis at the principal stresses.

What does the center of Mohr’s circle represent?

The center C = (sx+sy)/2 represents the average normal stress. It is also the hydrostatic (volumetric) stress component in 2D. The center lies on the horizontal axis at the mean of the two normal stress components.

What does the radius of Mohr’s circle represent?

The radius R = sqrt(((sx-sy)/2)² + txy²) represents the maximum shear stress at the point. It equals (s1-s2)/2. The radius quantifies the deviatoric (distortion) stress component.

How do you find principal stresses from Mohr’s circle?

The circle intersects the horizontal (sigma) axis at two points: s1 = C + R (maximum principal stress) and s2 = C - R (minimum principal stress). At these points, the shear stress is zero by definition.

How do you find maximum shear stress from Mohr’s circle?

Maximum shear stress equals the radius of the circle: tau_max = R = (s1-s2)/2. It occurs at the top and bottom of the circle, oriented 45 degrees from the principal planes.

What is the angle of rotation on Mohr’s circle?

An angle of 2*theta on Mohr’s circle corresponds to an angle of theta on the actual stress element. This doubling effect means the principal planes are at 2*theta_p from the reference, or theta_p = (1/2) * arctan(2*txy/(sx-sy)) in real space.

What is the pole of Mohr’s circle?

The pole (origin of planes) is a unique point on Mohr’s circle such that any line drawn from the pole to a point on the circle is parallel to the plane on which those stresses act. It helps determine the orientation of planes directly from the circle.

How does Mohr’s circle handle pure shear?

For pure shear (sx=sy=0), the circle is centered at the origin with radius = txy. The principal stresses are s1 = +txy and s2 = -txy at 45 degrees. This shows why pure shear creates equal tensile and compressive principal stresses.

What is the significance of Mohr’s circle touching the origin?

When the circle passes through the origin, one principal stress is zero (plane stress). If the circle is entirely on one side of the origin, both principal stresses have the same sign (both tension or both compression).

How is Mohr’s circle used in geotechnical engineering?

In soil mechanics, Mohr’s circle represents the stress state on a soil element. The failure envelope (tangential to the circle) defines the shear strength. The circle is used with the Mohr-Coulomb failure criterion: tau_f = c + sigma * tan(phi).

What is the 3D Mohr’s circle?

For 3D stress states, three Mohr’s circles can be drawn using the three principal stresses (s1, s2, s3). The largest circle (using s1 and s3) gives the absolute maximum shear stress. The other two circles represent intermediate stress states.

What are the limitations of Mohr’s circle?

Mohr’s circle applies to 2D (plane) stress states. It assumes linear elastic behavior. For 3D stress, three circles are needed. It does not directly show strain; a separate Mohr’s circle for strain can be constructed.

How does Mohr’s circle relate to failure theories?

Tresca criterion: yielding occurs when the largest Mohr’s circle touches the failure lines at tau = sigma_y/2. Von Mises: yielding occurs when a related function of the circle dimensions reaches the yield criterion.

Can Mohr’s circle be used for strain?

Yes. A strain Mohr’s circle can be constructed using normal and shear strains. The center is (ex+ey)/2, the radius is sqrt(((ex-ey)/2)² + (gxy/2)²), giving principal strains e1, e2 and maximum shear strain gmax/2.

What is the equation of Mohr’s circle?

The equation is (sigma - C)² + tau² = R², where C = (sx+sy)/2 is the center and R = sqrt(((sx-sy)/2)² + txy²) is the radius. This is the standard circle equation in the sigma-tau coordinate system.

How do you rotate on Mohr’s circle?

To find stresses at angle theta from the reference, rotate 2*theta on the circle from point P(sx, txy). Clockwise rotation on the element corresponds to clockwise on the circle (or counterclockwise depending on sign convention).

What is the sign convention for Mohr’s circle?

Common conventions: tensile stress positive (right), compressive negative (left). Shear stress: positive causes counterclockwise rotation on the element, plotted upward. Different texts may use different conventions; consistency is key.

How is Mohr’s circle used in earthquake engineering?

Mohr’s circle helps analyze stress states in soil and rock during seismic loading. It is used to determine failure planes, liquefaction potential, and the effectiveness of confining pressure in improving soil strength.

What is the relationship between Mohr’s circle and the stress tensor?

Mohr’s circle is the graphical representation of the eigenvalue problem for the 2x2 stress tensor. The circle’s center and radius are derived from the tensor invariants I1 and I2.

How do you use Mohr’s circle for stress at an angle?

Draw a line from point P(sx, txy) at angle 2*theta from the sigma axis. The intersection with the circle gives (sigma_x′, tau_x′). This is the stress state on a plane oriented theta from the reference plane.

What is the diameter of Mohr’s circle in terms of principal stresses?

The diameter equals s1 - s2, the difference between maximum and minimum principal stresses. The diameter also equals 2*tau_max (twice the maximum shear stress).

How does hydrostatic stress affect Mohr’s circle?

Hydrostatic stress shifts the circle horizontally without changing its radius. Adding hydrostatic pressure moves the center to the right (more compression) while keeping the same size, meaning the maximum shear stress is unchanged.

What is the maximum in-plane shear stress from Mohr’s circle?

The maximum in-plane shear stress is the radius R = (s1-s2)/2. It occurs at 45 degrees from the principal planes. For 3D stress, the absolute maximum shear may be out-of-plane.

How do you verify Mohr’s circle calculations?

Use the invariants: the sum of stresses on the circle should equal sx+sy (constant), and the product s1*s2 should equal sx*sy - txy². These serve as check calculations.

What is the average stress on Mohr’s circle?

Average stress = (sx+sy)/2 = (s1+s2)/2 = center of the circle. It represents the mean normal stress and does not cause distortion or yielding according to the von Mises criterion.

Can Mohr’s circle be drawn for dynamic loading?

Yes, but the circle changes with time as stresses vary. For cyclic loading, the largest and smallest circles over a load cycle define the critical stress states for fatigue analysis.

How is Mohr’s circle taught in engineering courses?

Mohr’s circle is typically covered in Mechanics of Materials / Strength of Materials courses. It provides an intuitive graphical understanding of stress transformation that complements the analytical formulas.

What is the numerical range of Mohr’s circle?

The circle spans from s2 (minimum) to s1 (maximum) on the sigma axis, and from -tau_max to +tau_max on the tau axis. All possible stress states for the given loading lie on or inside the circle.

How does Mohr’s circle help with pressure vessel design?

For a thin-walled pressure vessel, Mohr’s circle shows that the hoop stress is the maximum principal stress and the longitudinal stress is the minimum principal stress. The maximum shear stress is (hoop - longitudinal)/2.

What is the Mohr-Coulomb failure criterion?

Mohr-Coulomb states that failure occurs when the Mohr’s circle touches the failure envelope: tau = c + sigma * tan(phi), where c is cohesion and phi is the friction angle. It is widely used for soils, rocks, and concrete.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Mohr’s Circle: A graphical representation of the stress state at a point, showing all possible (sigma, tau) pairs on planes through that point.
Center of Circle: The point C = (sx+sy)/2 on the sigma axis, representing the average normal stress.
Radius of Circle: R = √[((sx-sy)/2)² + txy²], equal to the maximum in-plane shear stress.
Pole: A unique point on the circle from which lines to any point on the circle are parallel to the corresponding plane.
Principal Stresses: The stresses where the circle intersects the sigma axis (tau=0): s1 = C+R, s2 = C-R.
Maximum Shear Stress: Equal to the radius of the circle, occurring at the top and bottom points of the circle.

Authoritative References

Trusted sources and standards referenced by this calculator.

Hibbeler, Mechanics of Materials Beer & Johnston, Mechanics of Materials ASME BPVC Section VIII

Applicable Standards

ASTM E8 - Tension TestingASME BPVC Section VIIIAISC 360 - Steel DesignISO 6892-1 - Metallic Materials Tensile Testing

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Last Updated:

2025-08-21

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.