Difference between 1st 2nd and 3rd principal stress: Problems,Examples,Facts

This article discusses about the difference between 1st 2nd and 3rd principal stress. We will discuss about how to calculate major principal stress and minor principal stress.


When a plane is subjected to stress there are planes where both shear stress and normal stresses act and there are planes where only resultant normal stress acts. That plane is called as principal plane. We will discuss about it further in this article.

What is principal stress?

Principal stresses are those stresses which act on the principal planes. These are the actual stresses acting on the system.

These stresses are normal stresses (that is no shear stress) on a plane (rotated through an angle) This angle can be called as principal angle. It represents the orientation of principal plane with respect to the original axis.

Mathematically the major principal stress can be given as-

And the minor principal stress can be given as-

difference between 1st 2nd and 3rd principal stress
Image: Stresses acting normal to plane
Image credits: Jorge StolfiIsotropic stress noavgCC BY-SA 3.0

What are different types of principal stresses?

Principal stresses can be classified into 1st principal stress, 2nd principal stress and 3rd principal stress.

We shall discuss about these stresses below-

  • First principal stress- This principal stress represents the value of stress that is normal to the plane (note that the shear stress is zero in this plane). It is the maximum stress value (mostly tensile). We shall discuss more about it in further sections.
  • Second principal stress– Similar to first principal stress in characteristics except for the fact that second principal stress is the minimum stress value which is mostly compressive in nature.
  • Third principal stress– Third principal stress is the maximum compressive stress value induced in the material before loading. It has the least value of stress among all three sets of principal stresses.

Difference between 1st and 3rd principal stress

The difference between 1st and 3rd principal stress is shown in the table given below-

First principal stressThird principal stress
 It is normal to the surface of the plane and shows maximum tensile stress.It is normal to the surface of the plane and shows the minimum compressive stress.
 It gives the maximum value of stress in the system. It gives the minimum value of stress in the system.
Table: Difference between 1st and 3rd principal stress

What is 1st principal stress?

This stress is normal to the plane and here the value of shear stress is zero.

This particular principal stress helps us to know the maximum value of tensile stress being induced in the plane. In short, it is the maximum stress value (mostly tensile).

What is 2nd principal stress?

2nd principal stress is the stress normal to the surface of the plane and the value of stress is intermediate to the 1st and 3rd principal stresses that is the value of 2nd principal stress lies in between the values of first and third principal stress.

What is 3rd principal stress?

Third principal stress is the maximum value of compressive stress in the system. It is the least of all the three principal stresses.

Even this stress is also normal to the plane and here also the shear stress is zero. Note that all the principal stresses are normal to the plane and the shear stress is zero in this plane.

Significance of principal stresses

The principal stresses are used to find the yield stress acting on the material under the given loading conditions.

We can calculate Von Mises stress with the help of these principal stresses. Note that Von Mises stress and principal stresses are different, we can find Von Mises stress from principal stress to find whether the material will fail or not. Principal stresses are the actual stresses acting on the material and Von Mises stress is the yield stress used to determine whether the material will yield or not.

 

How to calculate maximum principal stress?

Maximum principal stress is also called as the first principal stress. The formula for maximum principal stress can be given using formula written below- 

How to calculate minimum principal stress?

Minimum principal stress is also called as the third principal stress. The minimum principal stress can be given using the formula given below- 

Maximum principal stress and minimum principal stress example

Let us assume the data for a system of stresses.

Stress in x direction- 20MPa

Stress in Y direction – 20 Mpa

Shear stress- 0 Mpa

Calculate the major principal stress

Substituting values in the formula of major principal stress, we get

Major principal stress or 1st principal stress= 20 MPa

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