Short Answer:

Mohr’s Circle is a graphical method used to determine the state of stress at a point in a material. It helps find the values and directions of principal stresses, maximum shear stresses, and their corresponding planes.

In simple words, Mohr’s Circle provides a visual representation of how normal and shear stresses vary on different planes passing through a point. It is an easy and accurate tool for analyzing two-dimensional (plane) stress systems and helps engineers understand stress transformation in materials and components.

Detailed Explanation :

Mohr’s Circle

The Mohr’s Circle is a graphical construction developed by the German engineer Christian Otto Mohr. It is a powerful method for representing and analyzing the state of stress at a point in two-dimensional or three-dimensional stress systems. Instead of solving complex equations, Mohr’s Circle allows engineers to determine the magnitudes of principal stresses, maximum shear stress, and their orientations through a simple geometric approach.

The circle gives a clear picture of how stresses act on various inclined planes through a given point in the body. It visually shows how normal and shear stresses change as the plane rotates.

Definition of Mohr’s Circle

Mohr’s Circle can be defined as:

“A graphical representation of the state of stress at a point, used to determine principal stresses, maximum shear stresses, and their directions.”

It is constructed on a coordinate system where:

• The x-axis represents the normal stress (σ).
• The y-axis represents the shear stress (τ).

Each point on the circle corresponds to a particular orientation of the plane passing through the point in the material.

Purpose of Mohr’s Circle

Mohr’s Circle helps in:

1. Determining principal stresses (maximum and minimum normal stresses).
2. Finding maximum shear stress and its corresponding normal stress.
3. Determining the orientation (angle) of principal planes and maximum shear planes.
4. Understanding the relationship between normal and shear stresses on different planes.

It is an effective visualization tool used in engineering stress analysis.

Construction of Mohr’s Circle (for 2D Stress System)

Consider a plane element subjected to normal stresses  and , and shear stress .

Steps to Construct Mohr’s Circle:

1. Draw Coordinate Axes:
   • Draw two perpendicular axes: the horizontal axis for normal stress (σ) and the vertical axis for shear stress (τ).
   • Positive shear stress is taken in the downward direction (as per Mohr’s convention).
2. Plot Stress Points:
   • Point A (σx, τxy) represents the stress condition on the x-face of the element.
   • Point B (σy, -τxy) represents the stress condition on the y-face of the element.
3. Locate the Center of the Circle (C):
   The center of Mohr’s Circle lies at:

1. Determine the Radius (R):
   The radius of the circle is given by:

1. Draw the Circle:
   • Using point C as the center and radius R, draw the circle that passes through both points A and B.
2. Determine Principal Stresses:
   The points where the circle intersects the horizontal (σ) axis give the principal stresses:

 

Here,

1. •  = Major principal stress
   •  = Minor principal stress
2. Determine Maximum Shear Stress:
   The maximum shear stress is equal to the radius of the circle:

The corresponding normal stress at this point is equal to the center of the circle:

1. Determine Orientation (Angle) of Planes:
   The angle between the plane of normal stress and the principal plane in Mohr’s Circle is 2θp, where  is the actual physical angle in the material.

Thus,

Interpretation of Mohr’s Circle

• Every point on the circle represents the state of stress on some plane passing through the point in the material.
• The horizontal distance from the center represents the normal stress.
• The vertical distance from the horizontal axis represents the shear stress.
• The radius of the circle gives the maximum shear stress in the material.
• The extreme points on the σ-axis give the principal stresses.

By rotating the circle, we can determine how stresses change on planes inclined at different angles to the original coordinate axes.

Advantages of Mohr’s Circle

1. Provides a clear graphical view of the stress condition at a point.
2. Simplifies stress transformation without complex mathematical calculations.
3. Helps determine principal stresses and maximum shear stress easily.
4. Indicates the orientation of planes where these stresses act.
5. Useful for both two-dimensional and three-dimensional stress analysis.

Applications of Mohr’s Circle

1. Mechanical Design: Used to check whether machine components can safely handle combined stresses.
2. Structural Engineering: To analyze stresses in beams, trusses, and columns.
3. Material Testing: To study failure in materials under combined stress conditions.
4. Aerospace and Automotive Fields: For analyzing stress distribution in thin shells, wings, and chassis.
5. Finite Element Analysis (FEA): Used for visual verification of computed stress results.

Significance of Mohr’s Circle

• It gives a complete picture of stress variation on all possible planes through a point.
• Helps determine safe stress levels to prevent material failure.
• Offers a quick and accurate method for visualizing and solving stress problems.
• It forms the foundation for understanding strain transformation and failure theories.

Conclusion

The Mohr’s Circle is a graphical tool used to analyze and visualize the state of stress at a point in a material. It provides an easy method to find principal stresses, maximum shear stresses, and their orientations without solving complex equations. The circle simplifies the study of stress transformation and is widely used in mechanical and structural engineering for design and failure analysis. Thus, Mohr’s Circle is an essential concept for ensuring the strength, safety, and reliability of mechanical components and structures.