Principal Stress and Mohr's Circle Calculator with von Mises

Enter the plane stress components and get the principal stresses, the maximum shear, the angle of the principal plane and the von Mises equivalent stress.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

At any point in a loaded part, the stress depends on the direction you look. Mohr's circle describes this. The centre of the circle is the average normal stress and the radius is the maximum in-plane shear stress. The principal stresses are the centre plus and minus the radius.

For ductile metals such as steel and aluminium, the von Mises stress combines the whole stress state into one number to compare with the yield strength. For brittle materials such as cast iron, compare the largest principal stress with the tensile strength instead.

Formulas used

  • \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}
  • \sigma_v = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2}

Worked example

With the default numbers in the calculator above:

  1. Centre and radius of Mohr circle: C = 30, R = 58.3 MPa
  2. Principal stresses: 88.3 and -28.3 MPa
  3. Maximum in-plane shear stress: 58.3 MPa
  4. Angle to the principal plane: 15.5°
  5. Von Mises equivalent stress: 105 MPa
  6. Factor of safety against yield (von Mises): 2.37
  7. How far the stresses can grow at FoS 1.5: 1.58 times
  8. Lowest yield strength that reaches FoS 1.5: 158 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is the sign convention?

Tension is positive. Angles are measured counter-clockwise from the x axis.

When do I use von Mises and when the principal stress?

Von Mises for ductile metals. The largest principal stress for brittle materials.

Is this plane stress or plane strain?

Plane stress: the stress through the thickness is taken as zero, as in thin plates and the free surface of a part.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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