Mohr’s Circle Calculator (2D state of stress)

The Mohr’s circle calculator enables you to determine the primary stresses from a two-dimensional state of stress.

Components of Stress in 2D, MPa


Computed Principal Stresses, their Directions and Maximum Shear Stress

σ1 =
σ2 =
θσ1 =
θσ2 =
τmax =
Modify the angle using the slider below
Angle 0.00

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In the world of structural and mechanical engineering, visual tools often play an important role in translating complex theories into coherent concepts. Stress analysis, a cornerstone of material and structural study, can sometimes boggle the mind with its mathematical intensity. That’s where the Mohr’s Circle comes into play. More than just a visual tool, the Mohr’s Circle offers engineers, both junior and experienced, an intuitive method to analyze and understand stress interactions within materials. Whether you’re an engineer seeking a quick refresher or a curious enthusiast diving into the mechanics, our comprehensive guide on the Mohr’s Circle Calculator is tailored just for you. Dive in, and let’s demystify the world of stress analysis together!

mohr's circle calculator
Stress components at a plane passing through a point in a continuum under plane stress conditions. (wikimedia)

Mohr’s Circle Calculator Definition and Formula

Mohr’s Circle is a graphical representation that showcases the state of stress at a point in a stressed body. Otto Mohr, a German engineer from the late 1800s, came up with this method. It helps us understand different types of stresses, like normal and shear stresses. People in civil, mechanical, and aerospace engineering find Mohr’s Circle really handy.

The principal stresses and maximum shear stresses can be determined from Mohr’s Circle. If σx and σy​ are the normal stresses, and τxy is the shear stress, the center of Mohr’s Circle C is given by:

\text{C} = \frac{{\sigma_x + \sigma_y}}{2}

And the radius R of Mohr’s Circle is determined by:

\text{R} = \sqrt{\left(\frac{{\sigma_x - \sigma_y}}{2}\right)^2 + \tau_{xy}^2}

How to calculate Mohr’s Circle using this calculator

Using the Mohr’s Circle Calculator, you can easily derive the principal stresses, maximum shear stress, and the angle of orientation. Here’s how:

  • Inputting the Known Stresses
    1. Begin by entering the normal stresses σx and σy​.
    2. Next, input the shear stress τxy.
  • Computing the Center and Radius of Mohr’s Circle
    1. After inputting the stresses, the calculator will determine the center C using the formula: \text{C} = \frac{{\sigma_x + \sigma_y}}{2}
    2. Simultaneously, it will calculate the radius R of Mohr’s Circle with the formula:
      \text{R} = \sqrt{\left(\frac{{\sigma_x - \sigma_y}}{2}\right)^2 + \tau_{xy}^2}
  • Deriving the Principal and Shear Stresses
    1. With the center and radius values known, the calculator will then provide the principal stresses σ1 and σ2, as well as the maximum and minimum shear stresses.
    2. Remember that the principal stresses are the normal stresses at the points where the shear stress is zero.
  • Finding the Angle of Rotation
    1. Lastly, the calculator will provide the angle of orientation, θ. This angle indicates the orientation of the plane on which the principal stresses act.
    2. The angle can be derived from the formula:
      \theta = \frac{1}{2} \arctan \left( \frac{{\sigma_x - \sigma_y}}{{2\tau_{xy}}} \right)


Why is Mohr’s Circle important?

Mohr’s Circle is important for engineers and scientists. It helps them see how different pressures work on a material. By looking at Mohr’s Circle, they can understand how stressed something is and where it might break. This helps them make better designs and decisions.

Interesting Fact: While the graphical method of Mohr’s Circle might seem archaic in today’s world of computer-aided design and simulations, it remains a fundamental teaching tool in engineering curricula worldwide. We can better understand the challenging aspects of stress analysis since it is straightforward and simple to look at. That is why it has been cherished and utilized in mechanics for so long.

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