Mohr’s Circle Visualizer

What Is a Mohr’s Circle Visualizer?

A Mohr’s Circle Visualizer is an interactive tool that graphically represents the state of stress at a point in a material. It shows how normal stress and shear stress change when the orientation of the plane changes.

Instead of relying only on formulas, the visualizer:

• Draws Mohr’s Circle on a stress graph
• Shows the circle center and radius
• Highlights principal stresses and maximum shear stress
• Displays rotated stresses for a given angle

This visual approach makes stress transformation intuitive and easier to learn.

Why Mohr’s Circle Matters in Engineering

Mohr’s Circle is widely used in:

• Strength of Materials
• Structural Engineering
• Mechanical Design
• Failure analysis
• Academic learning and teaching

It helps answer key questions such as:

• What are the principal stresses?
• Where does maximum shear stress occur?
• At what angle do stresses become purely normal?

A visualizer makes these answers instantly visible.

Key Inputs in the Mohr’s Circle Visualizer

The calculator you shared uses four main input values. Each input directly affects the shape and position of the circle.

1. Normal Stress in X Direction (σₓ)

This is the normal stress acting on the vertical face of the element.

• Positive value → tensile stress
• Negative value → compressive stress

2. Normal Stress in Y Direction (σᵧ)

This is the normal stress acting on the horizontal face of the element.

Together, σₓ and σᵧ define the average stress and influence the circle’s center.

3. Shear Stress (τₓᵧ)

Shear stress causes distortion in the material.

• It controls the radius of Mohr’s Circle
• Larger shear stress means a larger circle

4. Rotation Angle (θ)

This angle represents the physical rotation of the stress element.

• In Mohr’s Circle, the rotation appears as 2θ
• The visualizer clearly shows this relationship

Core Calculations Behind the Visualizer

The calculator performs standard Mohr’s Circle equations behind the scenes. These calculations are displayed clearly in the results section.

Average Normal Stress

The center of Mohr’s Circle is defined by:

σ_avg = (σₓ + σᵧ) / 2

This value sets the horizontal position of the circle.

Radius of Mohr’s Circle

The radius controls the size of the circle:

R = √[((σₓ − σᵧ)/2)² + τₓᵧ²]

The radius also represents the maximum shear stress.

Principal Stresses

Principal stresses occur where shear stress is zero.

• Principal Stress 1 (σ₁) = σ_avg + R
• Principal Stress 2 (σ₂) = σ_avg − R

These are shown on the horizontal axis of the circle.

Maximum Shear Stress

Maximum shear stress is equal to the radius of the circle:

τ_max = R

The visualizer clearly marks this value for quick understanding.

Principal Angle

The principal plane angle is calculated using:

θ_p = ½ tan⁻¹(2τₓᵧ / (σₓ − σᵧ))

This angle tells you where principal stresses act in the material.

Rotated Stresses (σₓ′ and τₓ′ᵧ′)

When the element is rotated by θ:

• Rotated normal stress:
  σₓ′ = σ_avg + R cos(2θ)
• Rotated shear stress:
  τₓ′ᵧ′ = −R sin(2θ)

The visualizer plots this rotation directly on the circle.

Understanding the Visual Output

The canvas-based visual output is the strongest feature of this tool.

Stress Axes

• Horizontal axis → Normal stress (σ)
• Vertical axis → Shear stress (τ)

A grid background improves scale understanding.

Circle Center and Diameter

• The center lies at (σ_avg, 0)
• The dashed diameter shows σ₁ and σ₂

This makes the concept of principal stresses very clear.

Stress Points

• (σₓ, τₓᵧ) and (σᵧ, −τₓᵧ) are plotted
• A line connects these two points

This line always passes through the center of the circle.

Rotation Line (2θ)

The rotation angle is visualized as a line rotated by 2θ from the center. This helps users understand why Mohr’s Circle uses double the physical angle.

Educational Benefits of the Visualizer

This Mohr’s Circle Visualizer is especially useful for:

• Engineering students learning stress transformation
• Teachers explaining abstract concepts visually
• Quick checks during preliminary design
• Understanding the relationship between equations and graphs

It turns theory into something you can see and interact with.

User-Friendly and Responsive Design

The calculator layout is designed for clarity and usability:

• Clean form inputs with units shown
• Large, readable result values
• Responsive layout for mobile and desktop
• Clear reset and calculate buttons

The dark theme reduces eye strain and improves contrast.

Practical Limitations and Disclaimer

While the visualizer is accurate for learning and early-stage analysis, it should not replace professional tools.

• Intended for education and preliminary design
• Not a substitute for certified engineering software
• Always verify results for critical applications

This disclaimer is clearly included in the interface.