Merchant’s Circle Diagram of Forces

Merchant’s circle:

For establishing the relationship between measurable and actual forces Merchant’s circle will be used.

Assumptions made in drawing the merchant’s circle:

  1. A shear surface is a plane extending upwards from the cutting edge.
  2. The tool is perfectly sharp and there is no contact along the clearance face.
  3. The cutting edge is a straight line extending perpendicular to the direction of motion and generates a plane surface as the work moves past it.
  4. The chip doesn’t flow to either side, that is chip width is constant.
  5. The depth of the cut remains constant.
  6. The width of the tool is greater than that of the work.
  7. Work moves with uniform velocity relative tooltip.
  8. No Built Up Edge (BUE) is formed.


  • Merchant circle is used to analyze the forces acting in metal cutting.
  • The analysis of three forces system, which balances each other for cutting to occur. Each system is a triangle of forces.

In machining operation there are four actual forces are acting. They are:

F = frictional force acting at the chip-tool interface.

N = force normal to the frictional force

Fs = Shear force acting along the shear plane.

Fsn = force normal to the frictional force.

But all the above four forces are acting in the dynamic environment, hence it is not possible to measure them and these forces are required for analysis of machining operation. Therefore each of the above  actual forces can be resolved into two components of forces and the algebraic sum of forces can be taken as

Let, Fc = algebraic sum of vertical components of forces

FT = algebraic sum of horizontal components of forces

The above two forces can be measurable by using a dynamometer or spring balance but these forces can’t be used in the analysis of machining. Hence to correlate the actual and measurable force the merchant’s circle will be used.

By using a dynamometer, the measurable forces can be measured and by using the merchant’s circle, the actual forces can be calculated. Using this actual force the machining can be analyzed.

merchant circle

From the above circle, it is found that there are three right-angled triangles are present and all will have a common hypotenuse. Using this principle the forces can be related as

The resultant force, R = hypotenuse

merchant circle equation

using the above equations if the force Ft and Fc are known, the friction angle can be determined, the rake angle is already known from the tool designation and the shear angle is known from the chip thickness so that the actual forces Fs, Fsn, F, N can be determined.

Spread the Knowledge

Vinodh Reddy is a Mechanical Engineer and Editor-in-chief of ME Mechanical. He holds a Bachelor of Engineering (Honours) degree in Mechanical Engineering. He also blogs at EduGeneral. He is interested in the farming field.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.