What are major and minor principal stresses?

Short Answer:

The major and minor principal stresses are the two extreme normal stresses that act on the principal planes of a material where the shear stress is zero. The major principal stress is the maximum normal stress at that point, while the minor principal stress is the minimum normal stress.

In simple terms, these stresses represent the highest and lowest normal stresses that a material experiences under a given load. They are very important in determining whether a material or structure will fail, yield, or remain safe under combined loading conditions.

Detailed Explanation :

Major and Minor Principal Stresses

When a material or structural component is subjected to complex loading, it experiences stresses in multiple directions — typically normal stresses (tensile or compressive) and shear stresses. At any given point inside the material, there exist certain planes known as principal planes where the shear stresses become zero and only normal stresses act.

The normal stresses acting on these principal planes are called principal stresses. Among them, the largest one is called the major principal stress, and the smallest one is called the minor principal stress. These two stresses describe the complete state of normal stress at a point in a two-dimensional (2D) stress system.

Definition

  • Major Principal Stress (σ₁):
    The major principal stress is the maximum normal stress acting on a plane where the shear stress is zero. It is usually tensile in nature and occurs on one of the principal planes.
  • Minor Principal Stress (σ₂):
    The minor principal stress is the minimum normal stress acting on the plane perpendicular to the one carrying the major principal stress. It is often compressive but can also be tensile, depending on the loading condition.

Thus, the two principal stresses act at right angles to each other on perpendicular principal planes.

Concept of Principal Stresses

Consider a two-dimensional element subjected to:

  • A normal stress  in the x-direction,
  • A normal stress  in the y-direction,
  • A shear stress  on the element.

These stresses can be represented on an infinitesimal element. When the element is rotated at a particular angle, the shear stress on one of the planes becomes zero. On that plane, the normal stress reaches an extreme value (either maximum or minimum).

The normal stress on this plane is the major principal stress (σ₁), and the normal stress on the plane perpendicular to it is the minor principal stress (σ₂).

Both stresses occur on planes where the shear stress is zero, meaning only pure normal stresses exist on principal planes.

Mathematical Expression for Major and Minor Principal Stresses

From the plane stress transformation equations, the general expressions for normal and shear stresses on an inclined plane are:

 

At the principal planes, the shear stress is zero (τθ = 0).
Substituting this condition and solving for σθ, we get the two principal stresses:

Here:

  •  = Major principal stress (maximum)
  •  = Minor principal stress (minimum)

The “+” sign corresponds to the major principal stress (σ₁), while the “−” sign gives the minor principal stress (σ₂).

Angle of Principal Planes

The planes on which these principal stresses act are known as principal planes, and their orientation is given by:

Here,

  •  is the angle of orientation of the principal plane measured from the x-axis.

The two principal planes are always perpendicular to each other, meaning they differ by 90°.

Graphical Representation Using Mohr’s Circle

The Mohr’s Circle method provides a clear visual understanding of major and minor principal stresses.

  • The center (C) of Mohr’s Circle is located at the average normal stress:
  • The radius (R) of the circle represents the maximum shear stress:
  • The points where the circle intersects the x-axis correspond to the major and minor principal stresses:

The right intersection gives the major principal stress (σ₁), and the left intersection gives the minor principal stress (σ₂).

Significance of Major and Minor Principal Stresses

  1. Determination of Failure:
    Principal stresses help determine whether a material will yield, fracture, or remain safe under combined loading.
  2. Design of Structures:
    Knowing the extreme stresses ensures that structures and machine parts are designed to withstand the highest stress levels safely.
  3. Simplifies Stress Analysis:
    By converting complex stress systems into two pure normal stresses, analysis becomes much simpler.
  4. Basis for Failure Theories:
    Most failure criteria, such as Maximum Normal Stress Theory, Von Mises Stress Theory, and Tresca’s Theory, are based on principal stresses.
  5. Useful in Mohr’s Circle Construction:
    The major and minor principal stresses define the size and position of the Mohr’s Circle used for stress visualization.

Example (Conceptual)

Suppose a steel plate is subjected to:

Then:

 

Hence,

Applications

  • Design of beams, shafts, and columns.
  • Pressure vessel and boiler design under combined stresses.
  • Aircraft and bridge component analysis.
  • Stress analysis in mechanical and civil structures.
  • Used in finite element analysis (FEA) to determine safety factors.
Conclusion

The major and minor principal stresses are the maximum and minimum normal stresses acting on perpendicular planes where shear stress is zero. These stresses are extremely important for determining the strength, stability, and safety of materials under combined loading. By identifying the planes of principal stresses, engineers can predict how a material will behave under real-life working conditions and ensure that structural and mechanical components are designed within safe stress limits.