Created By : Abhinandan Kumar

Reviewed By : Phani Ponnapalli

Last Updated : May 17, 2023

The greatest and least principal stress, as well as the principal plane, can be discovered and learned with this principal stress calculator. You may rapidly enter the stress components on our website and obtain your results in no time! 

Principal Stress - Definition

The typical stress that a body can experience at any time is known as principal stress.

When more than two stresses are present (compound stress), the resultant stress at a place is a combination of normal and shear stress. The stress at that location is a function of the inclination to the horizontal in both circumstances. In other words, the principal stress is the difference between the minimum and maximum stresses obtained from normal stress at a plane angle where shear stress is zero.

What is the purpose of the principal stress calculation?

  • When a material is naturally brittle, it will fail owing to primary stresses rather than shear stresses.
  • Furthermore, any material deterioration is caused by the maximum primary stress.
  • As a result, components are built to endure the highest possible primary stress.
  • The material's fracture will occur along the primary plane.

How do I use our Principal Stress Calculator?

To use this principal stress calculator with some exemplary values, follow the steps below:

  • Put the σ_x, normal stress in the horizontal direction.
  • Then, put the σ_y, normal stress in the vertical direction.
  • Also, enter the shear stress, τ_xy.

τ_yx is shear stress that has the same amplitude as τ_xy but acts in the opposite direction.

The maximum and minimum principal stresses, as well as the angle of principal stress, will now be calculated using the principal stress equations.

The major stress angle is generally calculated in radians.

You can either convert the value to degrees using our angle converter or simply convert it in this main stress calculator.

Principal Stress Formula

If you want to compute the principal stresses on your own, you can use the following principal stress equations:

Maximum principal stress: 

σmax = ((σx+ σy)/2) + √(((σx - σy)/2)2+ (τxy2))

Minimum principal stress:

σmin = ((σx+ σy)/2) - √(((σx - σy)/2)2+ (τxy2))

Angle of principal stress:

θ = (tan-1((2 * τxy) / (σx - σy)))/2

Maximum shear stress is found at 45 degrees to the major plane.

Maximum angle of shear stress:

θτmax = (tan-1((2 * τxy) / (σx - σy)))/2 + tan-1(1)

Minimum angle of shear stress:

θτmax = (tan-1((2 * τxy) / (σx - σy)))/2 - tan-1(1)

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FAQs on Principal Stress Calculator

1. What are the three principal sources of stress?

The three major stressors are commonly referred to as σ1, σ2, and σ3. σ1 denotes the highest (most tensile) principal stress, σ3 the lowest (most compressive), and σ2 the intermediate principal stress.

2. What is the definition of intermediate principal stress?

The main stress whose magnitude is neither the greatest nor the least (in terms of sign) of the three.

3. What is the difference between principal stress and principal strain?

Primary stress refers to the three stresses normal to shear principal planes, whereas a principal strain refers to the plane in which shear strain is zero.

4. What would be the first principal stress?

The stress value that is normal to the plane in which the shear stress is zero is the 1st principal stress. The greatest tensile stress created in the part as a result of the loading conditions is determined by the 1st principal stress.

5. What is the angle formed by each principal stress?

Maximum shear stress planes are at 45 degrees to the principal planes. The highest shear stress is half the difference between the primary stresses. It's worth noting that the equation of the primary plane, 2θp yields two angles between 0 degrees and 360 degrees.