Triaxial Strain and Hooke's Law Calculator

Find the strain in each direction when a material is stressed in two or three directions at once.

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

Each normal stress stretches the material in its own direction and squeezes it across the other two by Poisson's ratio. The strain in a direction is the stress in that direction, minus Poisson's ratio times the other two stresses, all over the modulus.

Enter the three stresses (tension positive), the modulus and Poisson's ratio. Put the third stress at zero for a thin plate in plane stress. The calculator also gives the change in volume, the modulus of rigidity and the bulk modulus.

Formulas used

  • \varepsilon_x = \dfrac{1}{E}\left[\sigma_x - \nu(\sigma_y + \sigma_z)\right]
  • \varepsilon_v = \dfrac{1-2\nu}{E}(\sigma_x+\sigma_y+\sigma_z)
  • G = \dfrac{E}{2(1+\nu)}

Worked example

With the default numbers in the calculator above:

  1. Strain in x: 425 µε
  2. Strain in y: 100 µε
  3. Strain in z: -225 µε
  4. Change in volume (volumetric strain): 300 µε
  5. Modulus of rigidity: 76.923 GPa
  6. Bulk modulus: 166.67 GPa

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

Common questions

What is a microstrain?

A strain of one millionth. A strain of 1000 microstrain is 0.1 percent.

Why is there a strain in z with no stress in z?

The other stresses squeeze or stretch the material in z through Poisson's ratio. A plate in tension gets thinner.

Does it work past yield?

No. It holds only while the material is elastic and the same in every direction.

Related calculators

Further reading: Standard strength of materials texts (generalised Hooke's law).

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