Mohr's Circle Visualizer

Units

Stress Components

σₓ (Normal Stress X)

σᵧ (Normal Stress Y)

τₓᵧ (Shear Stress XY)

Visualization Options

Show Construction Lines

Show Stress Element

Show Step-by-Step

Show Material Assumptions

Angle Transformation

Rotation Angle θ: 0°

Export Options

Enter the normal stresses (σₓ and σᵧ) and shear stress (τₓᵧ) for your 2D stress element.

Note: Positive values indicate tension, negative values indicate compression.

The center (C) of Mohr's Circle is calculated as:

C = (σₓ + σᵧ)/2

The radius (R) is calculated as:

R = √[((σₓ - σᵧ)/2)² + τₓᵧ²]

Tip: The radius represents the maximum shear stress.

The principal stresses (σ₁ and σ₂) are found at the points where the circle intersects the σ-axis:

σ₁ = C + R

σ₂ = C - R

Remember: σ₁ is always the larger principal stress.

The angle to the principal planes is given by:

tan(2θₚ) = 2τₓᵧ / (σₓ - σᵧ)

Note: This gives two angles 90° apart, corresponding to the two principal planes.

To find stresses at any angle θ, use the transformation equations:

σₓ' = (σₓ + σᵧ)/2 + (σₓ - σᵧ)/2 cos(2θ) + τₓᵧ sin(2θ)

σᵧ' = (σₓ + σᵧ)/2 - (σₓ - σᵧ)/2 cos(2θ) - τₓᵧ sin(2θ)

τₓᵧ' = -(σₓ - σᵧ)/2 sin(2θ) + τₓᵧ cos(2θ)

Tip: Use the angle slider to visualize how stresses change with rotation.

Mohr's Circle Fundamentals

Mohr's Circle is a graphical method to represent the state of stress at a point, developed by Christian Otto Mohr in 1882.

Key concepts:

Each point on the circle represents the normal and shear stress components for a specific orientation
The horizontal axis represents normal stress (σ)
The vertical axis represents shear stress (τ)
Principal stresses occur where shear stress is zero
Maximum shear stress equals the radius of the circle

Material Assumptions

Standard assumptions for Mohr's Circle analysis:

Material is homogeneous and isotropic
Linear elastic behavior (Hooke's Law applies)
Small deformations (geometric linearity)
Plane stress or plane strain conditions

References

Mohr's Circle Visualizer

Interactive tool for visualizing 2D stress transformation and principal stresses

Mohr's Circle Plot

Stress Element

σₓ'

σᵧ'

τᵧₓ'

τₓᵧ'

Transformed Stresses

σₓ'

0.00

σᵧ'

0.00

τₓᵧ'

0.00

Results Summary

Parameter	Value	Angle
Center (C)	0.00	-
Radius (R)	0.00	-
σ₁ (Max Principal)	0.00	0°
σ₂ (Min Principal)	0.00	0°
τ_max	0.00	0°

Step-by-Step Construction

Enter stress values to begin calculation